Finite Group Databasebeta

← #72261794

Subgroups — #72261794

Order 2048.

At least 7220 conjugacy classes of subgroups; only the 1140 normal subgroups are shown.

The subgroup lattice has 503 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#146707C212yesmaximal, complemented
2C2×#7672740C211yesmaximal, characteristic
3#72261758C211yesmaximal, characteristic, complemented
4C2×#149443C212yesmaximal, complemented
5#72261738C211yesmaximal, characteristic, complemented
6C2×#146488C212yesmaximal, complemented
7#72238514C212yesmaximal, complemented
8#72236572C212yesmaximal, complemented
9C2×#147577C212yesmaximal, complemented
10C22×#3494C411yescharacteristic, complemented
11C22×#3488C2211yescharacteristic
12C22×#2856C411yescharacteristic, complemented
13C2×#35532C2211yescharacteristic
14C2×#35530C411yescharacteristic, complemented
15C2×#35528C411yescharacteristic, complemented
16C2×#35527C2211yescharacteristic
17C2×#35264C411yescharacteristic, complemented
18C2×#35262C411yescharacteristic, complemented
19C2×#35256C411yescharacteristic, complemented
20C2×#35250C2211yescharacteristic
21C2×#34275C2211yescharacteristic
22C2×#34258C2211yescharacteristic
23C2×C4×#525C411yescharacteristic, complemented
24C22×#2297C2211yescharacteristic
25C22×#2122C2214yes—
26C22×#1795C2214yes—
27C2×#20591C2212yes—
28C2×#20584C2212yes—
29C2×#12585C2212yes—
30C2×#12585C2212yes—
31C2×#12581C2212yes—
32C2×#12581C2212yes—
33C2×#12575C2214yes—
34C2×#10765C2214yes—
35C23×H(5,𝔽2)C2311yescharacteristic
36C24×#41C2311yescharacteristic
37C22×#544C2×C411yescharacteristic
38C22×#537C2×C411yescharacteristic
39C22×#537D412yes—
40C22×#535C2×C411yescharacteristic
41C22×#535C2×C411yescharacteristic
42C22×#535C2×C411yescharacteristic
43C22×#535D412yes—
44C22×#535C2311yescharacteristic
45C23×#127C2×C411yescharacteristic
46C22×#525C2×C411yescharacteristic
47C22×#525D412yes—
48C23×#121C2×C411yescharacteristic
49C23×#121C2×C411yescharacteristic
50C23×#121D412yes—
51C22×#520C2×C411yescharacteristic
52C22×#520C2311yescharacteristic
53C22×#518C2×C411yescharacteristic
54C22×C4×#41C2×C411yescharacteristic
55C23×#117C2×C411yescharacteristic
56C24×C42C2×C411yescharacteristic
57C22×#413C2314yes—
58C22×#410C2314yes—
59#35532D418yes—
60#35527D418yes—
61C2×#2404D414yes—
62C2×#2403D414yes—
63#34275D418yes—
64#34258D418yes—
65C2×#2317C2×C411yescharacteristic
66C2×#2317C2×C411yescharacteristic
67C2×#2311C2×C411yescharacteristic
68C2×#2311C2311yescharacteristic
69C2×#2309C2×C411yescharacteristic
70C2×#2309C2×C411yescharacteristic
71C2×#2297C2×C411yescharacteristic
72C2×#2297C2×C411yescharacteristic
73C2×#2297C2×C411yescharacteristic
74C2×#2297D411yescharacteristic
75C2×#2297D411yescharacteristic
76C2×#2297D411yescharacteristic
77C2×#2297D411yescharacteristic
78C2×#2297C2311yescharacteristic
79C2×#2297C2311yescharacteristic
80C2×#2296C2×C411yescharacteristic
81C22×#391D411yescharacteristic
82C22×#391D411yescharacteristic
83C22×#390C2×C411yescharacteristic
84C22×#390C2×C411yescharacteristic
85C22×#390D411yescharacteristic
86C22×#390D411yescharacteristic
87C22×#384C2×C411yescharacteristic
88C2×#2280C2×C411yescharacteristic
89C2×C4×#117C2×C411yescharacteristic
90C22×H(5,𝔽2)C2×D414yes—
91C22×H(5,𝔽2)C2×D414yes—
92C23×#41C22×C411yescharacteristic
93C23×#41C22×C411yescharacteristic
94C23×#41C2×D412yes—
95C23×#41C2×D412yes—
96C23×#41C2×D411yescharacteristic
97C23×#41C2×D412yes—
98C23×#41C2×D411yescharacteristic
99C23×#41C2411yescharacteristic
100C24×Q8C2×D411yescharacteristic
101C24×D4C22⋊C412yes—
102C24×D4C22×C411yescharacteristic
103C24×D4C22×C411yescharacteristic
104C24×D4C2×D411yescharacteristic
105C24×D4C2×D412yes—
106C24×D4C2×D412yes—
107C25×C4C22×C411yescharacteristic
108C25×C4C2×D412yes—
109C2×#544C22⋊C414yescomplemented
110C2×#535C22⋊C414yescomplemented
111C2×#535C2×D414yes—
112C2×#535C2×D414yes—
113C22×#127C22⋊C414yescomplemented
114C22×C22⋊Q8C22⋊C411yescharacteristic
115C22×C22⋊Q8C22⋊C411yescharacteristic
116C22×C22⋊Q8C22⋊C412yes—
117C22×#121C22⋊C411yescharacteristic
118C22×#121C22⋊C411yescharacteristic
119C22×#121C22⋊C412yes—
120C22×#121C22⋊C412yes—
121C22×#121C22⋊C412yes—
122C22×#121C22⋊C412yes—
123C22×#121C22⋊C412yes—
124C22×#120C22⋊C414yes—
125C22×#120C2×D414yes—
126C2×#520C22⋊C414yescomplemented
127C2×#520C2×D414yes—
128C2×#520C2×D414yes—
129C2×#518C22⋊C414yescomplemented
130C2×C4×#41C22⋊C414yescomplemented
131C22×#117C22⋊C411yescharacteristic
132C22×#117C22⋊C411yescharacteristic
133C22×#117C22×C411yescharacteristic
134C22×#117C22×C411yescharacteristic
135C22×#117C22×C411yescharacteristic
136C22×#117C22×C411yescharacteristic
137C22×#117C22×C411yescharacteristic
138C22×#117C2×D412yes—
139C23×C4⋊C4C22⋊C411yescharacteristic
140C23×C4⋊C4C22⋊C411yescharacteristic
141C23×C4⋊C4C22×C411yescharacteristic
142C23×C4⋊C4C22×C411yescharacteristic
143C23×C4⋊C4C2×D412yes—
144C23×C22⋊C4C22⋊C412yes—
145C23×C42C22⋊C411yescharacteristic
146C23×C42C22⋊C411yescharacteristic
147C23×C42C22⋊C411yescharacteristic
148C23×C42C22⋊C411yescharacteristic
149C23×C42C22×C411yescharacteristic
150C23×C42C22×C411yescharacteristic
151C2×#413#4118yescomplemented
152C2×#410#4118yescomplemented
153#2317C22⋊C414yescomplemented
154#2317C22⋊C414yescomplemented
155#2311C2×D414yes—
156#2311C2×D414yes—
157#2309C22⋊C414yescomplemented
158#2309C22⋊C414yescomplemented
159#2297C22⋊C414yescomplemented
160#2297C2×D414yes—
161#2297C2×D414yes—
162#2297C2×D414yes—
163#2297C2×D414yes—
164#2296C22⋊C414yescomplemented
165C2×#391C22⋊C411yescharacteristic
166C2×#391C22⋊C411yescharacteristic
167C2×#391C22⋊C411yescharacteristic
168C2×#391C22⋊C411yescharacteristic
169C2×#391C2×D411yescharacteristic
170C2×#391C2×D411yescharacteristic
171C2×#391C2×D411yescharacteristic
172C2×#391C2×D411yescharacteristic
173C2×#390C22⋊C411yescharacteristic
174C2×#390C22⋊C411yescharacteristic
175C2×#390C22⋊C411yescharacteristic
176C2×#390C22⋊C411yescharacteristic
177C2×#390C22⋊C414yescomplemented
178C2×#390C22⋊C414yescomplemented
179C2×#390C2×D411yescharacteristic
180C2×#390C2×D411yescharacteristic
181C2×#390C2×D411yescharacteristic
182C2×#390C2×D411yescharacteristic
183C2×#386C22⋊C414yes—
184C2×#386C2×D414yes—
185C2×#384C22⋊C414yescomplemented
186C2×#381C22⋊C412yes—
187C2×#379C22⋊C412yes—
188C4×#117C22⋊C414yescomplemented
189C26C2×C22⋊C411yescharacteristic
190C26C2×C22⋊C412yes—
191C26#12012yes—
192C22×#41C2×C22⋊C412yes—
193C22×#41C2×C22⋊C412yes—
194C22×#41#12018yes—
195C22×#41C22×D412yes—
196C22×#41C22×D412yes—
197C22×#41C22×D412yes—
198C22×#41C22×D412yes—
199C22×#41C2×#4114yes—
200C22×#41C2×#4114yes—
201C22×#41H(5,𝔽2)12yes—
202C22×#41H(5,𝔽2)12yes—
203C22×#412−1+412yes—
204C22×#412−1+412yes—
205C23×Q8C2×C22⋊C411yescharacteristic
206C23×Q8C2×C22⋊C411yescharacteristic
207C23×Q8#12012yes—
208C23×Q8C22×D411yescharacteristic
209C23×D4C2×C22⋊C412yes—
210C23×D4C2×C22⋊C411yescharacteristic
211C23×D4C2×C22⋊C412yes—
212C23×D4C2×C22⋊C412yes—
213C23×D4C2×C22⋊C411yescharacteristic
214C23×D4C2×C22⋊C412yes—
215C23×D4C2×C22⋊C412yes—
216C23×D4C2×C22⋊C412yes—
217C23×D4#12012yes—
218C23×D4#12018yes—
219C23×D4#12012yes—
220C23×D4#12012yes—
221C23×D4C22×D411yescharacteristic
222C23×D4C22×D412yes—
223C24×C4C2×C22⋊C412yes—
224C24×C4C2×C22⋊C411yescharacteristic
225C24×C4C2×C22⋊C411yescharacteristic
226C24×C4C2×C22⋊C411yescharacteristic
227C24×C4C2×C22⋊C411yescharacteristic
228C24×C4C2×C22⋊C412yes—
229C24×C4C2×C22⋊C411yescharacteristic
230C24×C4#12012yes—
231C24×C4#12012yes—
232C24×C4#12012yes—
233C24×C4C23×C411yescharacteristic
234C24×C4C22×D412yes—
235C2×C22⋊Q8#9914yes—
236C2×C22⋊Q8#9914yes—
237C2×#121#9914yes—
238C2×#121#9914yes—
239C2×#120#120116yes—
240C2×#117C2×C22⋊C412yes—
241C2×#117C2×C22⋊C412yes—
242C2×#117C2×C22⋊C412yes—
243C2×#117C2×C22⋊C412yes—
244C2×#117C2×C22⋊C412yes—
245C2×#117C2×C22⋊C412yes—
246C2×#117C2×C22⋊C412yes—
247C2×#117C2×C22⋊C412yes—
248C2×#117C2×C22⋊C412yes—
249C2×#117C2×C22⋊C412yes—
250C2×#117#12018yes—
251C22×C4⋊C4C2×C22⋊C412yes—
252C22×C4⋊C4C2×C22⋊C411yescharacteristic
253C22×C4⋊C4C2×C22⋊C411yescharacteristic
254C22×C4⋊C4C2×C22⋊C412yes—
255C22×C4⋊C4C2×C22⋊C412yes—
256C22×C4⋊C4C2×C22⋊C412yes—
257C22×C4⋊C4C2×C22⋊C412yes—
258C22×C4⋊C4C2×C22⋊C411yescharacteristic
259C22×C4⋊C4C2×C22⋊C411yescharacteristic
260C22×C4⋊C4C2×C22⋊C411yescharacteristic
261C22×C4⋊C4C2×C22⋊C411yescharacteristic
262C22×C4⋊C4#12018yes—
263C22×C22⋊C4C2×C22⋊C412yes—
264C22×C22⋊C4C2×C22⋊C412yes—
265C22×C42C2×C22⋊C411yescharacteristic
266C22×C42C2×C22⋊C412yes—
267C22×C42C2×C22⋊C412yes—
268C22×C42C2×C22⋊C412yes—
269C22×C42C2×C22⋊C412yes—
270C22×C42C2×C22⋊C411yescharacteristic
271C22×C42C2×C22⋊C411yescharacteristic
272C22×C42C2×C22⋊C411yescharacteristic
273#391#12014yes—
274#391#12014yes—
275#391#12014yes—
276#391#12014yes—
277#390#12014yes—
278#390#12014yes—
279#390#12014yes—
280#390#12014yes—
281#386#120116yes—
282C25#37911yescharacteristic
283C25#37912yes—
284C25#37911yescharacteristic
285C25#37914yes—
286C25C22×C22⋊C411yescharacteristic
287C25C2×#12012yes—
288C2×#41#52514yes—
289C2×#41#52514yes—
290C2×#41#53214yes—
291C2×#41#53214yes—
292C2×#41#53512yes—
293C2×#41#53512yes—
294C2×#41#53612yes—
295C2×#41#53612yes—
296C2×#41#53712yes—
297C2×#41#53712yes—
298C2×#41#53712yes—
299C2×#41#53712yes—
300C2×#41#54614yes—
301C2×#41#54614yes—
302C2×#41#54714yes—
303C2×#41#54714yes—
304C2×#41#54814yes—
305C2×#41#54814yes—
306C2×#41#55014yes—
307C2×#41#55014yes—
308C22×Q8#37912yes—
309C22×Q8C2×#9912yes—
310C22×Q8C2×#9912yes—
311C22×Q8C2×#12012yes—
312C22×Q8C2×#12012yes—
313C22×Q8#52512yes—
314C22×Q8#53212yes—
315C22×Q8#53511yescharacteristic
316C22×Q8#53611yescharacteristic
317C22×Q8#53712yes—
318C22×D4#37914yes—
319C22×D4#37912yes—
320C22×D4C2×#9912yes—
321C22×D4C2×#9912yes—
322C22×D4C2×#12014yes—
323C22×D4C2×#12014yes—
324C22×D4C2×#12012yes—
325C22×D4C2×#12012yes—
326C22×D4#52512yes—
327C22×D4#53212yes—
328C22×D4#53511yescharacteristic
329C22×D4#53611yescharacteristic
330C22×D4#53712yes—
331C22×D4#54614yes—
332C22×D4#54614yes—
333C22×D4#54714yes—
334C22×D4#54714yes—
335C23×C4#37912yes—
336C23×C4#37911yescharacteristic
337C23×C4#37911yescharacteristic
338C23×C4#37914yes—
339C23×C4C2×#9912yes—
340C23×C4C2×#9912yes—
341C23×C4C2×#9912yes—
342C23×C4C2×#9912yes—
343C23×C4C22×C22⋊C411yescharacteristic
344C23×C4C22×C22⋊C411yescharacteristic
345C23×C4C22×C22⋊C411yescharacteristic
346C23×C4C22×C22⋊C411yescharacteristic
347C23×C4C22×C22⋊C411yescharacteristic
348C23×C4C22×C22⋊C411yescharacteristic
349C23×C4C4×#4112yes—
350C23×C4C4×#4112yes—
351C23×C4#52012yes—
352C23×C4#52012yes—
353C23×C4C2×#12012yes—
354C23×C4C2×#12014yes—
355C23×C4C2×#12014yes—
356C23×C4#54614yes—
357C23×C4#54614yes—
358C23×C4#54714yes—
359C23×C4#54714yes—
360C2×C4⋊C4#37914yes—
361C2×C4⋊C4#37912yes—
362C2×C4⋊C4#37912yes—
363C2×C4⋊C4#37912yes—
364C2×C4⋊C4#37912yes—
365C2×C4⋊C4#37912yes—
366C2×C4⋊C4#37912yes—
367C2×C4⋊C4C2×#9912yes—
368C2×C4⋊C4C2×#9912yes—
369C2×C4⋊C4C2×#9912yes—
370C2×C4⋊C4C2×#9912yes—
371C2×C22⋊C4#37914yes—
372C2×C22⋊C4#37914yes—
373C2×C42#37912yes—
374C2×C42#37912yes—
375C2×C42#37912yes—
376C2×C42#37912yes—
377C24#178112yes—
378C24#178112yes—
379C24C2×#37911yescharacteristic
380C24C2×#37911yescharacteristic
381C24C2×#37911yescharacteristic
382C24#233912yes—
383C24#233912yes—
384C24#240412yes—
385C24#240612yes—
386C24#243014yes—
387C24#284614yes—
388C2×Q8#178114yes—
389C2×Q8#238814yes—
390C2×Q8#240714yes—
391C2×Q8#243112yes—
392C2×Q8#243212yes—
393C2×Q8#261512yes—
394C2×Q8#261612yes—
395C2×D4#178114yes—
396C2×D4#238814yes—
397C2×D4#240414yes—
398C2×D4#240614yes—
399C2×D4#240714yes—
400C2×D4#240714yes—
401C2×D4#240714yes—
402C2×D4#243014yes—
403C2×D4#243014yes—
404C2×D4#243114yes—
405C2×D4#243112yes—
406C2×D4#243214yes—
407C2×D4#243212yes—
408C2×D4#261512yes—
409C2×D4#261612yes—
410C22×C4#178112yes—
411C22×C4#178112yes—
412C22×C4C2×#37911yescharacteristic
413C22×C4C2×#37911yescharacteristic
414C22×C4C2×#37911yescharacteristic
415C22×C4C2×#37911yescharacteristic
416C22×C4#229711yescharacteristic
417C22×C4#229711yescharacteristic
418C22×C4#229712yes—
419C22×C4#229712yes—
420C22×C4#231111yescharacteristic
421C22×C4#231111yescharacteristic
422C22×C4#231112yes—
423C22×C4#231112yes—
424C22×C4#233912yes—
425C22×C4#233912yes—
426C22×C4#233912yes—
427C22×C4#233912yes—
428C22×C4#234112yes—
429C22×C4#234112yes—
430C22×C4#240412yes—
431C22×C4#240414yes—
432C22×C4#240612yes—
433C22×C4#240614yes—
434C22×C4#240714yes—
435C22×C4#240714yes—
436C22×C4#243014yes—
437C22×C4#243014yes—
438C22×C4#243014yes—
439C22×C4#243114yes—
440C22×C4#243214yes—
441C22×C4#284614yes—
442C22×C4C22×#9911yescharacteristic
443C22×C4C22×#9911yescharacteristic
444C22×C4C22×#9911yescharacteristic
445C22×C4C22×#9911yescharacteristic
446C22×C4#288312yes—
447C22×C4#288312yes—
448C4⋊C4#178114yes—
449C4⋊C4#178114yes—
450C23#1271814yes—
451C23#1393712yes—
452C23#1395611yescharacteristic
453C23#1395611yescharacteristic
454C23#1395612yes—
455C23#1467911yescharacteristic
456C23#1488514yes—
457C23#1488914yes—
458C23C2×#178111yescharacteristic
459C23C2×#178111yescharacteristic
460C23#2058511yescharacteristic
461C23#2058511yescharacteristic
462C2×C4#1385111yescharacteristic
463C2×C4#1393711yescharacteristic
464C2×C4#1393711yescharacteristic
465C2×C4#1393712yes—
466C2×C4#1395612yes—
467C2×C4#1433711yescharacteristic
468C2×C4#1488514yes—
469C2×C4#1488914yes—
470C2×C4C2×#178111yescharacteristic
471C2×C4C2×#178111yescharacteristic
472C2×C4C2×#178111yescharacteristic
473C2×C4C2×#178111yescharacteristic
474C2×C4#2058512yes—
475C2×C4#2058612yes—
476C2×C4#2247612yes—
477C2×C4#2247612yes—
478C2×C4#2247612yes—
479C2×C4#2247612yes—
480C2×C4#2247612yes—
481C2×C4#2247612yes—
482C2×C4#2247612yes—
483C2×C4#2247612yes—
484C22#15412311yescharacteristic
485C22#36387911yescharacteristic
486C22#36387911yescharacteristic
487C22#36387911yescharacteristic
488C22#36387911yescharacteristic
489C22#39210911yescharacteristic
490C22#39211111yescharacteristic
491C22C2×#1271812yes—
492C22#49341912yes—
493C4#36318912yes—
494C4#36318912yes—
495C4#36387912yes—
496C4#36387912yes—
497C2#7226042612yes—
498C2#7226120712yes—
499C2#7226121011yescharacteristic
500C2#7226140511yescharacteristic
501C2#7226170511yescharacteristic
subgroupG/H
center, socleC23C2×#1781
commutatorC24×C4C23×C4
Frattini, cosocleC23×#41C24
prime radicalC2×#7672740C2
LCSG ▷ C24×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#3488 ◁ G
LECSG ▷ C23×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#146707C2semidirect
3. #72261758C2semidirect
4. C2×#149443C2semidirect
5. #72261738C2semidirect
6. C2×#146488C2semidirect
7. #72238514C2semidirect
8. #72236572C2semidirect
9. C2×#147577C2semidirect
10. C22×#3494C4semidirect
12. C22×#2856C4semidirect
14. C2×#35530C4semidirect
15. C2×#35528C4semidirect
17. C2×#35264C4semidirect
18. C2×#35262C4semidirect
19. C2×#35256C4semidirect
23. C2×C4×#525C4semidirect
109. C2×#544C22⋊C4semidirect
110. C2×#535C22⋊C4semidirect
113. C22×#127C22⋊C4semidirect
126. C2×#520C22⋊C4semidirect
129. C2×#518C22⋊C4semidirect
130. C2×C4×#41C22⋊C4semidirect
151. C2×#413#41semidirect
152. C2×#410#41semidirect
153. #2317C22⋊C4semidirect
154. #2317C22⋊C4semidirect
157. #2309C22⋊C4semidirect
158. #2309C22⋊C4semidirect
159. #2297C22⋊C4semidirect
164. #2296C22⋊C4semidirect
177. C2×#390C22⋊C4semidirect
178. C2×#390C22⋊C4semidirect
185. C2×#384C22⋊C4semidirect
188. C4×#117C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#1467072
C2×#76727402
#722617582
C2×#1494432
#722617382
C2×#1464882
#722385142
#722365722
C2×#1475772