Finite Group Databasebeta

← #10943885

Subgroups — #10943885

Order 768.

4070 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×#5241C214yesmaximal, complemented
3#84082C212yesmaximal, complemented
4#84049C211yesmaximal, characteristic, complemented
5#72140C214yesmaximal, complemented
6#72138C212yesmaximal, complemented
7#72129C211yesmaximal, characteristic, complemented
8#32888—31nomaximal, Sylow
9C2×C3×H(5,𝔽2)—216no—
10C2×C3×H(5,𝔽2)—24no—
11C2×C3×H(5,𝔽2)—22no—
12C2×C3×H(5,𝔽2)C2214yes—
13C22×C3×#41—22no—
14C22×C3×#41C2212yes—
15C23×C3×D4—22no—
16C23×C3×D4—22no—
17C23×C3×D4C2211yescharacteristic
18C2×#953—22no—
19C2×#949C2214yescomplemented
20C2×#940C2212yescomplemented
21C2×#930C2211yescharacteristic, complemented
22#5241—216no—
23#5238—24no—
24C2×#837C2218yescomplemented
25#5232C2214yescomplemented
26#5227—22no—
27#5223—22no—
28#5221C2212yescomplemented
29#5217—22no—
30C2×#833C2214yescomplemented
31C4×#217C2211yescharacteristic, complemented
32#5207C2212yescomplemented
33C22×H(5,𝔽2)D311yescharacteristic, complemented
34C2×#453—34no—
35#3006—32no—
36#2433—31no—
37#1792—34no—
38#1789—32no—
39#1783—31no—
40C25×C3—22no—
41C25×C3D412yes—
42C3×H(5,𝔽2)—232no—
43C3×H(5,𝔽2)—232no—
44C3×H(5,𝔽2)—232no—
45C3×H(5,𝔽2)—216no—
46C3×H(5,𝔽2)—28no—
47C3×H(5,𝔽2)D4116yescomplemented
48C2×C3×#41—232no—
49C2×C3×#41—28no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41D414yescomplemented
55C2×C3×#41C2314yes—
56C22×C3×Q8D412yescomplemented
57C22×C3×D4—24no—
58C22×C3×D4—232no—
59C22×C3×D4—232no—
60C22×C3×D4—28no—
61C22×C3×D4—28no—
62C22×C3×D4—22no—
63C22×C3×D4—28no—
64C22×C3×D4—28no—
65C22×C3×D4—28no—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4D412yes—
69C22×C3×D4D412yescomplemented
70C22×C3×D4C2312yes—
71C23×C3×C4—22no—
72C23×C3×C4—22no—
73C23×C3×C4C2311yescharacteristic
74C22×#67—21no—
75C22×#67—24no—
76C22×D12C2311yescharacteristic, complemented
77#953—48no—
78C2×#217—21no—
79C2×#217—24no—
80#949—216no—
81C2×#215—24no—
82#940—28no—
83C2×#213—24no—
84#930—22no—
85#930—22no—
86C2×(C2×D3)⋊C4—21no—
87C2×C4⋊Dic3C2311yescharacteristic, complemented
88C2×C4×Dic3C2312yescomplemented
89C2×#208C2314yescomplemented
90#837—216no—
91#835—24no—
92#833—28no—
93#832—24no—
94C3⋊#95—21no—
95#830—22no—
96#830—22no—
97C2×H(5,𝔽2)—216no—
98C2×H(5,𝔽2)—24no—
99C2×H(5,𝔽2)—22no—
100C2×H(5,𝔽2)C2×D314yes—
101C22×#41—22no—
102C22×#41C2×D312yes—
103C23×D4—22no—
104C23×D4—22no—
105C23×D4C2×D311yescharacteristic
106C2×AGL(1,ℤ/8)—34no—
107C2×#127—32no—
108C2×#121—31no—
109C2×#120—62no—
110#453—616no—
111#449—64no—
112#448—62no—
113#447—62no—
114C2×#104—38no—
115#417—34no—
116C2×#100—34no—
117#407—32no—
118#392—62no—
119#390—32no—
120C4×C22⋊C4—31no—
121C24×C3—216no—
122C24×C3—28no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—28no—
126C24×C3—22no—
127C24×C3—44no—
128C24×C3—82no—
129C24×C3—84no—
130C24×C3—84no—
131C24×C3C2×D411yescharacteristic
132C24×C3C2×D414yes—
133C23×D3—21no—
134C3×#41—264no—
135C3×#41—216no—
136C3×#41—232no—
137C3×#41—216no—
138C3×#41—232no—
139C3×#41—28no—
140C3×#41—28no—
141C3×#41—28no—
142C3×#41C2×D4116yes—
143C2×C3×Q8—28no—
144C2×C3×Q8—22no—
145C2×C3×Q8C2×D414yes—
146C2×C3×D4—232no—
147C2×C3×D4—216no—
148C2×C3×D4—216no—
149C2×C3×D4—216no—
150C2×C3×D4—264no—
151C2×C3×D4—264no—
152C2×C3×D4—232no—
153C2×C3×D4—232no—
154C2×C3×D4—216no—
155C2×C3×D4—24no—
156C2×C3×D4—232no—
157C2×C3×D4—28no—
158C2×C3×D4—24no—
159C2×C3×D4—28no—
160C2×C3×D4—232no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4C2×D418yes—
166C2×C3×D4C2×D414yes—
167C22×C3×C4—216no—
168C22×C3×C4—216no—
169C22×C3×C4—28no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—48no—
179C2×#67—44no—
180C2×#67—41no—
181C2×#67—41no—
182C2×#67—48no—
183C2×#67—84no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C2×D12—22no—
187C2×D12—24no—
188#217—24no—
189#217—48no—
190#215—48no—
191#213—48no—
192(C2×D3)⋊C4—42no—
193C4⋊Dic3—22no—
194C4×Dic3—24no—
195#208—28no—
196C2×C3⋊C8—24no—
197C25—22no—
198C25#6712yes—
199H(5,𝔽2)—232no—
200H(5,𝔽2)—232no—
201H(5,𝔽2)—232no—
202H(5,𝔽2)—216no—
203H(5,𝔽2)—28no—
204H(5,𝔽2)#67116yescomplemented
205C2×#41—232no—
206C2×#41—28no—
207C2×#41—216no—
208C2×#41—28no—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41#6714yescomplemented
212C2×#41C22×D314yes—
213C22×Q8#6712yescomplemented
214C22×D4—24no—
215C22×D4—232no—
216C22×D4—232no—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—28no—
221C22×D4—28no—
222C22×D4—28no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—31no—
226C22×D4—61no—
227C22×D4—64no—
228C22×D4#6712yes—
229C22×D4#6712yescomplemented
230C22×D4C22×D312yes—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C22×D311yescharacteristic
234AGL(1,ℤ/8)—616no—
235C2×SD16—64no—
236C2×D8—64no—
237C2×M16—34no—
238#127—68no—
239#121—62no—
240#121—62no—
241#120—128no—
242C2×C4⋊C4—31no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—64no—
245C2×C22⋊C4—61no—
246C2×C42—32no—
247#104—616no—
248#103—64no—
249#102—64no—
250#100—68no—
251C22⋊C8—62no—
252C22⋊C8—62no—
253#95—61no—
254C23×C3—28no—
255C23×C3—28no—
256C23×C3—28no—
257C23×C3—232no—
258C23×C3—22no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—24no—
263C23×C3—41no—
264C23×C3—44no—
265C23×C3—44no—
266C23×C3—44no—
267C23×C3—44no—
268C23×C3—44no—
269C23×C3—44no—
270C23×C3—82no—
271C23×C3—82no—
272C23×C3—816no—
273C23×C3—832no—
274C23×C3—88no—
275C23×C3—88no—
276C23×C3#12011yescharacteristic
277C23×C3#12011yescharacteristic
278C23×C3#12018yes—
279C23×C3C22×D411yescharacteristic
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—41no—
283C22×D3—42no—
284C3×Q8—28no—
285C3×Q8—24no—
286C3×Q8#12018yescomplemented
287C3×D4—264no—
288C3×D4—216no—
289C3×D4—216no—
290C3×D4—28no—
291C3×D4—216no—
292C3×D4—216no—
293C3×D4—24no—
294C3×D4#12018yescomplemented
295C2×C3×C4—28no—
296C2×C3×C4—232no—
297C2×C3×C4—216no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—216no—
302C2×C3×C4—24no—
303C2×C3×C4#12012yes—
304C2×C3×C4C22×D412yes—
305C2×C3×C4C22×D414yes—
306#67—816no—
307#67—82no—
308#67—82no—
309C2×Dic3—24no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—41no—
314D12—44no—
315C3⋊C8—44no—
316C24—216no—
317C24—28no—
318C24—22no—
319C24—22no—
320C24—28no—
321C24—22no—
322C24—44no—
323C24—61no—
324C24—82no—
325C24—84no—
326C24—84no—
327C24C2×#6711yescharacteristic
328C24C2×#6714yes—
329#41—264no—
330#41—216no—
331#41—232no—
332#41—216no—
333#41—232no—
334#41—28no—
335#41—28no—
336#41—28no—
337#41C2×#67116yes—
338C2×Q8—28no—
339C2×Q8—22no—
340C2×Q8C2×#6714yes—
341C2×D4—232no—
342C2×D4—216no—
343C2×D4—216no—
344C2×D4—216no—
345C2×D4—264no—
346C2×D4—264no—
347C2×D4—232no—
348C2×D4—232no—
349C2×D4—216no—
350C2×D4—24no—
351C2×D4—232no—
352C2×D4—28no—
353C2×D4—24no—
354C2×D4—28no—
355C2×D4—232no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—62no—
361C2×D4—64no—
362C2×D4—128no—
363C2×D4—124no—
364C2×D4—121no—
365C2×D4—121no—
366C2×D4—128no—
367C2×D4—244no—
368C2×D4C2×#6718yes—
369C2×D4C2×#6714yes—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6712yes—
381C22×C4C2×#6712yes—
382C22×C4C23×D311yescharacteristic
383SD16—128no—
384D8—128no—
385M16—68no—
386C2×C8—64no—
387C4⋊C4—62no—
388C22⋊C4—64no—
389C22⋊C4—128no—
390C22⋊C4—122no—
391C42—64no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—28no—
398C22×C3—28no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—44no—
402C22×C3—48no—
403C22×C3—416no—
404C22×C3—416no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—84no—
410C22×C3—88no—
411C22×C3—832no—
412C22×C3#45312yescomplemented
413C22×C3C2×#12012yes—
414C22×C3C2×#12011yescharacteristic
415C2×D3—84no—
416C2×D3—81no—
417C2×D3—81no—
418C2×D3—81no—
419C3×C4—28no—
420C3×C4—22no—
421C3×C4C2×#12014yes—
422Dic3—44no—
423Dic3—81no—
424C23—28no—
425C23—28no—
426C23—28no—
427C23—232no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—41no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—82no—
441C23—82no—
442C23—816no—
443C23—832no—
444C23—88no—
445C23—88no—
446C23—122no—
447C23—122no—
448C23—122no—
449C23—121no—
450C23#95311yescharacteristic
451C23#95311yescharacteristic
452C23#95318yes—
453C23C22×#6711yescharacteristic
454Q8—28no—
455Q8—24no—
456Q8#95318yescomplemented
457D4—264no—
458D4—216no—
459D4—216no—
460D4—28no—
461D4—216no—
462D4—216no—
463D4—24no—
464D4—124no—
465D4—2416no—
466D4—242no—
467D4—242no—
468D4#95318yescomplemented
469C2×C4—28no—
470C2×C4—232no—
471C2×C4—216no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—216no—
476C2×C4—24no—
477C2×C4—64no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4#95312yes—
483C2×C4C22×#6714yes—
484C2×C4C22×#6712yes—
485C8—124no—
486C2×C3—22no—
487C2×C3—22no—
488C2×C3—41no—
489C2×C3—48no—
490C2×C3—48no—
491C2×C3#284611yescharacteristic
492C2×C3C2×#45311yescharacteristic
493C2×C3#301911yescharacteristic
494D3—161no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—28no—
503C22—28no—
504C22—44no—
505C22—48no—
506C22—416no—
507C22—416no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—84no—
513C22—88no—
514C22—832no—
515C22—244no—
516C22—241no—
517C22—241no—
518C22—241no—
519C22#524112yescomplemented
520C22C2×#95312yes—
521C22C2×#95311yescharacteristic
522C4—28no—
523C4—22no—
524C4—124no—
525C4—241no—
526C4C2×#95314yes—
527C3#3288811yescharacteristic, complemented, Sylow
528C2—22no—
529C2—22no—
530C2—41no—
531C2—48no—
532C2—48no—
533C2—481no—
534C2C2×#524111yescharacteristic
535C2#8410411yescharacteristic
536C2#8412811yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×H(5,𝔽2)C2
socleC22×C3C2×#120
hypercenterC22×H(5,𝔽2)D3
hypocenterC3#32888
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×C3×H(5,𝔽2)C2semidirect
2. C2×#5241C2semidirect
3. #84082C2semidirect
4. #84049C2semidirect
5. #72140C2semidirect
6. #72138C2semidirect
7. #72129C2semidirect
19. C2×#949C22semidirect
20. C2×#940C22semidirect
21. C2×#930C22semidirect
24. C2×#837C22semidirect
25. #5232C22semidirect
28. #5221C22semidirect
30. C2×#833C22semidirect
31. C4×#217C22semidirect
32. #5207C22semidirect
33. C22×H(5,𝔽2)D3semidirect
47. C3×H(5,𝔽2)D4semidirect
54. C2×C3×#41D4semidirect
56. C22×C3×Q8D4semidirect
69. C22×C3×D4D4semidirect
76. C22×D12C23semidirect
87. C2×C4⋊Dic3C23semidirect
88. C2×C4×Dic3C23semidirect
89. C2×#208C23semidirect
204. H(5,𝔽2)#67semidirect
211. C2×#41#67semidirect
213. C22×Q8#67semidirect
229. C22×D4#67semidirect
286. C3×Q8#120semidirect
294. C3×D4#120semidirect
412. C22×C3#453semidirect
456. Q8#953semidirect
468. D4#953semidirect
519. C22#5241semidirect
527. C3#32888semidirect

Maximal subgroups

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

subgroupindex
C22×C3×H(5,𝔽2)2
C2×#52412
#840822
#840492
#721402
#721382
#721292
#328883