Finite Group Databasebeta

← #72256582

Subgroups — #72256582

Order 2048.

At least 8100 conjugacy classes of subgroups; only the 1708 normal subgroups are shown.

The subgroup lattice has 542 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72256581C211yesmaximal, characteristic, complemented
2#72256566C211yesmaximal, characteristic, complemented
3#72256553C211yesmaximal, characteristic, complemented
4#72251235C211yesmaximal, characteristic, complemented
5#72251215C211yesmaximal, characteristic, complemented
6#72251060C212yesmaximal, complemented
7#72238119C212yesmaximal, complemented
8#72236789C211yesmaximal, characteristic, complemented
9#72236681C212yesmaximal, complemented
10#72235823C212yesmaximal, complemented
11#72234806C211yesmaximal, characteristic, complemented
12#72233895C218yesmaximal, complemented
13#72210600C218yesmaximal, complemented
14C2×#62818C2211yescharacteristic
15C23×D42C2211yescharacteristic
16C23×#535C2211yescharacteristic, complemented
17#7642244C2214yes—
18#7630939C2214yes—
19#7628649C2212yescomplemented
20C2×#35250C2211yescharacteristic
21C2×#35222C2212yes—
22#7628282C2212yes—
23C2×#34736C2211yescharacteristic, complemented
24#7626203C2211yescharacteristic
25#6470225C2218yescomplemented
26#6374576C2212yescomplemented
27#6374576C2212yescomplemented
28C2×#21793C2211yescharacteristic
29C2×#21793C2211yescharacteristic
30C2×#20983C2212yescomplemented
31C2×#20722C2212yescomplemented
32C2×#20584C2212yescomplemented
33#635984C2214yescomplemented
34#612424C2214yescomplemented
35#517415C2211yescharacteristic, complemented
36#517415C2212yescomplemented
37#516385C2211yescharacteristic
38#514052C2211yescharacteristic, complemented
39#394058C22116yescomplemented
40#392109C2214yescomplemented
41#382422C2214yescomplemented
42#353407C2212yescomplemented
43#233542C2218yescomplemented
44#233541C2218yescomplemented
45#201664C2218yescomplemented
46#187928C2218yescomplemented
47#181342C2212yescomplemented
48#181342C2212yescomplemented
49#115690C22116yescomplemented
50#115205C2218yescomplemented
51#115204C2218yescomplemented
52#106766C2218yescomplemented
53C24×#41C2311yescharacteristic
54C25×D4D411yescharacteristic, complemented
55C25×D4C2312yes—
56C25×D4C2311yescharacteristic
57C2×#3505D414yescomplemented
58C2×#3496D414yescomplemented
59C2×#3484C2314yes—
60C22×#535D414yescomplemented
61C23×#127D411yescharacteristic, complemented
62C22×#525C2312yes—
63C23×#121D411yescharacteristic, complemented
64C23×#121D412yescomplemented
65C23×#121C2311yescharacteristic
66C23×#120D412yescomplemented
67C23×#120C2314yes—
68C23×#120C2312yes—
69C23×C4×D4D411yescharacteristic, complemented
70C23×C4×D4C2311yescharacteristic
71C23×#117C2311yescharacteristic
72C2×#2847D414yescomplemented
73C2×#2846C2318yes—
74#35084D414yescomplemented
75#35076D414yescomplemented
76#34992C2314yes—
77#34980C2314yes—
78#34980C2318yes—
79#34956D418yescomplemented
80C2×#2436D412yescomplemented
81C2×#2433D412yescomplemented
82C2×#2405C2312yes—
83C2×#2403C2314yes—
84C2×#2403C2312yes—
85C2×#2613D418yescomplemented
86#34363D418yescomplemented
87#34171C2314yes—
88C2×#2309D414yescomplemented
89C2×#2280C2312yes—
90C2×#2280C2311yescharacteristic
91C2×#2280C2311yescharacteristic
92C2×#2280C2312yes—
93C2×#2280C2312yes—
94#21746C2318yes—
95#21746C2318yes—
96C2×#1964D418yescomplemented
97C2×#1898C2318yes—
98C2×#1819C2314yes—
99C2×#1781C2314yes—
100C2×#1781C2318yes—
101C2×#1781C2314yes—
102C2×#1781C2314yes—
103#14679C2314yes—
104#14679C2318yes—
105#14257D414yescomplemented
106#14136D414yescomplemented
107#13869C2312yes—
108#13867C2312yes—
109#13867C2314yes—
110#10735C2318yes—
111#8101C2318yes—
112#8101C2318yes—
113C27C2×D412yes—
114C27C2412yes—
115C22×H(5,𝔽2)C2×D414yes—
116C23×#41C2×D412yes—
117C24×Q8C2×D411yescharacteristic, complemented
118C24×D4C2×D411yescharacteristic
119C24×D4C2×D414yes—
120C24×D4C2×D411yescharacteristic
121C24×D4C2×D412yes—
122C24×D4C2×D414yes—
123C24×D4C2×D412yes—
124C24×D4C2×D412yes—
125C24×D4C2×D412yes—
126C24×D4C2×D412yes—
127C24×D4C2×D411yescharacteristic
128C24×D4C2×D411yescharacteristic, complemented
129C24×D4C2412yes—
130C24×D4C2411yescharacteristic
131C24×D4C2412yes—
132C25×C4C2×D411yescharacteristic
133C25×C4C2×D412yes—
134C25×C4C2×D412yescomplemented
135C25×C4C2411yescharacteristic
136C2×#534C2×D418yes—
137C22×#123C2×D414yes—
138C22×C22⋊Q8C2×D412yes—
139C22×C22⋊Q8C2×D412yes—
140C22×#121C2×D412yes—
141C22×#121C2×D414yes—
142C22×#121C2×D412yes—
143C22×#121C2×D412yes—
144C22×#121C2×D414yes—
145C22×#121C2×D412yes—
146C22×#121C2×D412yes—
147C22×#121C2×D412yes—
148C22×#121C2×D414yes—
149C22×#121#4114yes—
150C22×#120C2×D4116yes—
151C22×#120C2×D418yescomplemented
152C22×#120C2414yes—
153C2×#518C2×D414yes—
154C22×C4×D4C2×D414yes—
155C22×C4×D4C2×D414yes—
156C22×C4×D4C2×D414yes—
157C22×C4×D4C2×D414yes—
158C22×#117C2×D414yes—
159C22×#117C2×D412yes—
160C22×#117C2×D412yescomplemented
161C22×#117C2×D412yescomplemented
162C23×C4⋊C4C2×D411yescharacteristic
163C23×C4⋊C4C2×D411yescharacteristic
164C23×C22⋊C4C2×D411yescharacteristic
165C23×C22⋊C4C2×D412yes—
166C23×C22⋊C4C2×D412yes—
167C23×C22⋊C4C2×D412yes—
168C23×C22⋊C4C2411yescharacteristic
169C23×C22⋊C4C2412yes—
170C23×C42C2×D411yescharacteristic
171C22×#99C2×D4116yescomplemented
172#2483C2×D418yes—
173#2405C2×D418yes—
174#2405C2×D418yes—
175#2403C2×D418yes—
176#2309C2×D418yes—
177#2308C2×D418yes—
178C2×#386C2×D418yescomplemented
179#2280C2×D414yes—
180#2280C2×D414yes—
181#2280C2×D414yes—
182#2280C2×D414yes—
183C2×#381C2×D414yes—
184C2×#380C2×D412yescomplemented
185C2×#380C2×D412yescomplemented
186C2×#380C2×D414yescomplemented
187C2×#379C2×D4116yes—
188C2×#379C2414yes—
189C2×#379C2412yes—
190C2×#379C2412yes—
191C2×#379C2414yes—
192C2×#379C2414yes—
193C2×C4×C22⋊C4C2×D412yescomplemented
194C2×C4×C22⋊C4C2×D412yescomplemented
195C2×C4×C22⋊C4C2×D414yescomplemented
196#1438C2×D4116yescomplemented
197C26C22×D412yes—
198C26C22×D418yes—
199C26C22×D412yes—
200C26C22×D411yescharacteristic
201C26C22×D412yes—
202C26C22×D412yes—
203C26C2511yescharacteristic
204C23×Q8#12012yescomplemented
205C23×D4#12012yes—
206C23×D4#12012yes—
207C23×D4#12012yes—
208C23×D4#12018yes—
209C23×D4#12012yes—
210C23×D4#12014yes—
211C23×D4#12012yes—
212C23×D4#12012yes—
213C23×D4#12012yescomplemented
214C23×D4#12114yes—
215C23×D4C22×D414yes—
216C23×D4C22×D414yes—
217C23×D4C22×D414yes—
218C23×D4C22×D414yes—
219C23×D4C22×D414yes—
220C23×D4C22×D414yes—
221C23×D4C22×D414yes—
222C23×D4C22×D414yes—
223C23×D4C22×D414yes—
224C23×D4C2×#4114yes—
225C24×C4#12014yes—
226C24×C4#12014yescomplemented
227C24×C4C22×D411yescharacteristic
228C24×C4C22×D412yes—
229C24×C4C22×D412yes—
230C24×C4C22×D411yescharacteristic
231C24×C4C22×D411yescharacteristic
232C24×C4C22×D411yescharacteristic
233C24×C4C22×D412yes—
234C24×C4C22×D412yes—
235C24×C4C22×D411yescharacteristic
236C24×C4C22×D412yes—
237C24×C4C22×D412yes—
238C24×C4C22×D412yes—
239C24×C4C22×D412yes—
240C24×C4C22×D411yescharacteristic
241C24×C4C22×D411yescharacteristic
242C24×C4C22×D411yescharacteristic
243C24×C4C22×D411yescharacteristic
244C24×C4C22×D412yes—
245C24×C4C2×#4112yes—
246C24×C4H(5,𝔽2)12yes—
247C24×C4H(5,𝔽2)12yes—
248C24×C4H(5,𝔽2)12yes—
249C24×C4H(5,𝔽2)12yes—
250C2×#121#12018yes—
251C2×#121#12018yes—
252C2×#121#12118yes—
253C2×#121#12118yes—
254C2×#121#12118yes—
255C2×#121#12118yes—
256C2×#121#12718yes—
257C2×#121#12718yes—
258C2×#117#12018yescomplemented
259C2×#117#12018yescomplemented
260C22×C4⋊C4#12012yes—
261C22×C4⋊C4#12012yes—
262C22×C4⋊C4#12014yes—
263C22×C4⋊C4#12012yes—
264C22×C4⋊C4#12012yes—
265C22×C4⋊C4#12012yescomplemented
266C22×C4⋊C4#12012yescomplemented
267C22×C4⋊C4#12014yescomplemented
268C22×C4⋊C4#12014yescomplemented
269C22×C4⋊C4#12114yes—
270C22×C22⋊C4#12012yes—
271C22×C22⋊C4#12012yes—
272C22×C22⋊C4#12018yes—
273C22×C22⋊C4#12014yes—
274C22×C22⋊C4#12014yes—
275C22×C22⋊C4#12014yescomplemented
276C22×C22⋊C4#12014yescomplemented
277C22×C22⋊C4#12118yes—
278C22×C22⋊C4C22×D414yes—
279C22×C22⋊C4C22×D414yes—
280C22×C22⋊C4C22×D414yes—
281C22×C22⋊C4C22×D414yes—
282C22×C22⋊C4C22×D418yes—
283C22×C22⋊C4C22×D414yes—
284C22×C22⋊C4C22×D414yes—
285C22×C22⋊C4C22×D414yes—
286C22×C22⋊C4C22×D418yes—
287C22×C42#12014yes—
288C22×C42#12014yescomplemented
289#381#12018yescomplemented
290#381#12018yescomplemented
291#379C22×D418yes—
292#379C22×D418yes—
293C25C2×#12014yes—
294C25C2×#12014yes—
295C25C2×#12012yes—
296C25C2×#12014yes—
297C25C2×#12014yes—
298C25C2×#12014yes—
299C25C2×#12114yes—
300C25C23×D411yescharacteristic
301C25C23×D411yescharacteristic
302C25C23×D411yescharacteristic
303C25C23×D412yes—
304C25C23×D411yescharacteristic
305C25C23×D412yes—
306C25C23×D411yescharacteristic
307C25C23×D411yescharacteristic
308C25C22×#4111yescharacteristic
309C25C2×H(5,𝔽2)12yes—
310C25C2×H(5,𝔽2)12yes—
311C22×D4#39218yes—
312C22×D4#39218yes—
313C22×D4B(4,𝔽2)18yescomplemented
314C22×D4C2×#12018yes—
315C22×D4C2×#12118yes—
316C22×D4C2×#12118yes—
317C22×D4C2×#12718yes—
318C23×C4C2×#12012yes—
319C23×C4C2×#12012yes—
320C23×C4C2×#12012yes—
321C23×C4C2×#12012yes—
322C23×C4C2×#12011yescharacteristic
323C23×C4C2×#12012yes—
324C23×C4C2×#12012yes—
325C23×C4C2×#12012yes—
326C23×C4C2×#12011yescharacteristic
327C23×C4C2×#12011yescharacteristic
328C23×C4C2×#12014yes—
329C23×C4C2×#12014yes—
330C23×C4C2×#12012yes—
331C23×C4C2×#12012yes—
332C23×C4C2×#12012yes—
333C23×C4C2×#12014yes—
334C23×C4C2×#12014yes—
335C23×C4C2×#12014yes—
336C23×C4C2×#12012yes—
337C23×C4C2×#12011yescharacteristic
338C23×C4C2×#12011yescharacteristic
339C23×C4C2×#12012yes—
340C23×C4C2×#12012yes—
341C23×C4C2×#12012yes—
342C23×C4C2×#12012yes—
343C23×C4C2×#12011yescharacteristic
344C23×C4C2×#12011yescharacteristic
345C23×C4C2×#12012yes—
346C23×C4C2×#12012yes—
347C23×C4C2×#12011yescharacteristic
348C23×C4C2×#12112yes—
349C23×C4C2×#12114yes—
350C23×C4C2×#12112yes—
351C23×C4C2×#12112yes—
352C23×C4C2×#12112yes—
353C23×C4C2×#12112yes—
354C23×C4C2×#12112yes—
355C23×C4C2×#12712yes—
356C23×C4C2×#12712yes—
357C23×C4#53412yes—
358C23×C4#53411yescharacteristic
359C23×C4#53411yescharacteristic
360C23×C4#53412yes—
361C23×C4#53411yescharacteristic
362C23×C4#53411yescharacteristic
363C23×C4#53412yes—
364C23×C4#53412yes—
365C23×C4#53412yes—
366C23×C4#53412yes—
367C23×C4#53512yes—
368C23×C4#53512yes—
369C23×C4#53512yes—
370C23×C4#53512yes—
371C23×C4D4212yes—
372C23×C4D4212yes—
373C23×C4D4212yes—
374C23×C4D4212yes—
375C23×C4D4212yes—
376C23×C4D4212yes—
377C23×C4D4212yes—
378C23×C4D4212yes—
379C23×C4D4212yes—
380C23×C4D4212yes—
381C23×C4D4212yes—
382C23×C4D4212yes—
383C23×C4D4212yes—
384C23×C4D4212yes—
385C23×C4D4212yes—
386C23×C4D4212yes—
387C23×C4D4212yes—
388C23×C4D4212yes—
389C23×C4D4212yes—
390C23×C4D4212yes—
391C23×C4D4212yes—
392C23×C4D4212yes—
393C23×C4D4212yes—
394C23×C4D4212yes—
395C23×C4#54612yes—
396C23×C4#54612yes—
397C23×C4#54612yes—
398C23×C4#54612yes—
399C23×C4#54612yes—
400C23×C4#54612yes—
401C23×C4#54612yes—
402C23×C4#54612yes—
403C2×C4⋊C4#39218yes—
404C2×C4⋊C4#39218yes—
405C2×C22⋊C4#392116yes—
406C2×C22⋊C4#392116yes—
407C2×C22⋊C4C2×#120116yes—
408C24C2×#39218yes—
409C24C2×B(4,𝔽2)18yes—
410C24C22×#12011yescharacteristic
411C24C22×#12011yescharacteristic
412C24C22×#12012yes—
413C24C22×#12011yescharacteristic
414C24C22×#12012yes—
415C24C22×#12011yescharacteristic
416C24C22×#12011yescharacteristic
417C24C22×#12111yescharacteristic
418C24C22×#12111yescharacteristic
419C24C22×#12111yescharacteristic
420C24C22×#12711yescharacteristic
421C24C2×#53411yescharacteristic
422C24C2×#53412yes—
423C24C2×#53411yescharacteristic
424C24C2×#53512yes—
425C24C2×D4212yes—
426C24C2×D4212yes—
427C24C2×D4212yes—
428C24C2×D4212yes—
429C24C2×D4212yes—
430C24C2×D4212yes—
431C24C2×#54612yes—
432C24C2×#54612yes—
433C22×C4C2×#39218yes—
434C22×C4C2×#39212yes—
435C22×C4C2×#39212yes—
436C22×C4C2×#39212yes—
437C22×C4C2×#39212yes—
438C22×C4#240312yes—
439C22×C4#240312yes—
440C22×C4#240312yes—
441C22×C4#240312yes—
442C22×C4#240311yescharacteristic
443C22×C4#240312yes—
444C22×C4#240312yes—
445C22×C4#240311yescharacteristic
446C22×C4#240312yes—
447C22×C4#240312yes—
448C22×C4#240311yescharacteristic
449C22×C4#240311yescharacteristic
450C22×C4#240311yescharacteristic
451C22×C4#240311yescharacteristic
452C22×C4#240312yes—
453C22×C4#240312yes—
454C22×C4#240312yes—
455C22×C4#240312yes—
456C22×C4#240311yescharacteristic
457C22×C4#240311yescharacteristic
458C22×C4#240812yes—
459C22×C4#240812yes—
460C22×C4#240812yes—
461C22×C4#240812yes—
462C22×C4#240812yes—
463C22×C4#240812yes—
464C22×C4#240812yes—
465C22×C4#240812yes—
466C22×C4#241012yes—
467C22×C4#241012yes—
468C22×C4#241012yes—
469C22×C4#241012yes—
470C22×C4#241012yes—
471C22×C4#241012yes—
472C22×C4#241012yes—
473C22×C4#241012yes—
474C22×C4#241012yes—
475C22×C4#241012yes—
476C22×C4#241012yes—
477C22×C4#241012yes—
478C22×C4#241012yes—
479C22×C4#241012yes—
480C22×C4#241012yes—
481C22×C4#241012yes—
482C22×C4#243312yes—
483C22×C4#243312yes—
484C22×C4#243312yes—
485C22×C4#243312yes—
486C22×C4#243312yes—
487C22×C4#243312yes—
488C22×C4#243312yes—
489C22×C4#243312yes—
490C22×C4#261312yes—
491C22×C4#261312yes—
492C22×C4#261312yes—
493C22×C4#261312yes—
494C22×C4#284612yes—
495C22×C4#284612yes—
496C22×C4#284612yes—
497C22×C4#284612yes—
498C22×C4C2×B(4,𝔽2)14yes—
499C22×C4#302614yes—
500C23C22×#39211yescharacteristic
501C23C2×#261311yescharacteristic
502C23C2×#240312yes—
503C23C2×#240311yescharacteristic
504C23C2×#240311yescharacteristic
505C23C2×#241011yescharacteristic
506C23C2×#241012yes—
507C23C2×#241011yescharacteristic
508C23C2×#240812yes—
509C23C2×#243312yes—
510C23C2×#284611yescharacteristic
511C23C22×B(4,𝔽2)12yes—
512C23C2×#302612yes—
513C23#6039914yes—
514C2×C4#1471512yes—
515C2×C4#1471512yes—
516C2×C4#1471512yes—
517C2×C4#1471512yes—
518C2×C4#1471512yes—
519C2×C4#1471512yes—
520C2×C4#1471512yes—
521C2×C4#1471512yes—
522C2×C4#2390012yes—
523C2×C4#2390012yes—
524C2×C4#2390312yes—
525C2×C4#2390312yes—
526C2×C4#2440418yes—
527C2×C4#3289818yes—
528C22C2×#1471511yescharacteristic
529C22#643179512yes—
530C22C2×#2390011yescharacteristic
531C22C2×#2390311yescharacteristic
532C22#643763012yes—
533C22#643763314yes—
534C22C2×#2440412yes—
535C22C2×#3289812yes—
536C4#39406614yes—
537C4#39406614yes—
538C2C2×#39406611yescharacteristic
539C2#7223803411yescharacteristic
540C2#7225116011yescharacteristic
subgroupG/H
center, socleC22C2×#14715
commutator, Frattini, cosocleC26C25
LCSG ▷ C26 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25×C4 ◁ G
LECSG ▷ C26 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #72256581C2semidirect
2. #72256566C2semidirect
3. #72256553C2semidirect
4. #72251235C2semidirect
5. #72251215C2semidirect
6. #72251060C2semidirect
7. #72238119C2semidirect
8. #72236789C2semidirect
9. #72236681C2semidirect
10. #72235823C2semidirect
11. #72234806C2semidirect
12. #72233895C2semidirect
13. #72210600C2semidirect
16. C23×#535C22semidirect
19. #7628649C22semidirect
23. C2×#34736C22semidirect
25. #6470225C22semidirect
26. #6374576C22semidirect
27. #6374576C22semidirect
30. C2×#20983C22semidirect
31. C2×#20722C22semidirect
32. C2×#20584C22semidirect
33. #635984C22semidirect
34. #612424C22semidirect
35. #517415C22semidirect
36. #517415C22semidirect
38. #514052C22semidirect
39. #394058C22semidirect
40. #392109C22semidirect
41. #382422C22semidirect
42. #353407C22semidirect
43. #233542C22semidirect
44. #233541C22semidirect
45. #201664C22semidirect
46. #187928C22semidirect
47. #181342C22semidirect
48. #181342C22semidirect
49. #115690C22semidirect
50. #115205C22semidirect
51. #115204C22semidirect
52. #106766C22semidirect
54. C25×D4D4semidirect
57. C2×#3505D4semidirect
58. C2×#3496D4semidirect
60. C22×#535D4semidirect
61. C23×#127D4semidirect
63. C23×#121D4semidirect
64. C23×#121D4semidirect
66. C23×#120D4semidirect
69. C23×C4×D4D4semidirect
72. C2×#2847D4semidirect
74. #35084D4semidirect
75. #35076D4semidirect
79. #34956D4semidirect
80. C2×#2436D4semidirect
81. C2×#2433D4semidirect
85. C2×#2613D4semidirect
86. #34363D4semidirect
88. C2×#2309D4semidirect
96. C2×#1964D4semidirect
105. #14257D4semidirect
106. #14136D4semidirect
117. C24×Q8C2×D4semidirect
128. C24×D4C2×D4semidirect
134. C25×C4C2×D4semidirect
151. C22×#120C2×D4semidirect
160. C22×#117C2×D4semidirect
161. C22×#117C2×D4semidirect
171. C22×#99C2×D4semidirect
178. C2×#386C2×D4semidirect
184. C2×#380C2×D4semidirect
185. C2×#380C2×D4semidirect
186. C2×#380C2×D4semidirect
193. C2×C4×C22⋊C4C2×D4semidirect
194. C2×C4×C22⋊C4C2×D4semidirect
195. C2×C4×C22⋊C4C2×D4semidirect
196. #1438C2×D4semidirect
204. C23×Q8#120semidirect
213. C23×D4#120semidirect
226. C24×C4#120semidirect
258. C2×#117#120semidirect
259. C2×#117#120semidirect
265. C22×C4⋊C4#120semidirect
266. C22×C4⋊C4#120semidirect
267. C22×C4⋊C4#120semidirect
268. C22×C4⋊C4#120semidirect
275. C22×C22⋊C4#120semidirect
276. C22×C22⋊C4#120semidirect
288. C22×C42#120semidirect
289. #381#120semidirect
290. #381#120semidirect
313. C22×D4B(4,𝔽2)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#722565812
#722565662
#722565532
#722512352
#722512152
#722510602
#722381192
#722367892
#722366812
#722358232
#722348062
#722338952
#722106002