Finite Group Databasebeta

← #10777920

Subgroups — #10777920

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79302C211yesmaximal, characteristic, complemented
2#79301C211yesmaximal, characteristic
3#78468C211yesmaximal, characteristic, complemented
4#78447C211yesmaximal, characteristic, complemented
5C3×#1564C211yesmaximal, characteristic, complemented
6#70777C211yesmaximal, characteristic, complemented
7#70773C211yesmaximal, characteristic, complemented
8#68593C211yesmaximal, characteristic, complemented
9#68592C211yesmaximal, characteristic, complemented
10#68583C211yesmaximal, characteristic, complemented
11#68582C211yesmaximal, characteristic, complemented
12#68555C211yesmaximal, characteristic, complemented
13#68554C211yesmaximal, characteristic, complemented
14#68545C211yesmaximal, characteristic, complemented
15#68544C211yesmaximal, characteristic, complemented
16#19159—31nomaximal, Sylow
17#5477C2211yescharacteristic
18C4×#235C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23#5114—21no—
24#5114—21no—
25#5111—21no—
26#5111—21no—
27#4925C2211yescharacteristic
28#4851C2211yescharacteristic
29#4850C2211yescharacteristic
30#4849C2211yescharacteristic
31#4848C2211yescharacteristic
32#4846C2211yescharacteristic
33#4843C2211yescharacteristic
34#4842C2211yescharacteristic
35#4841C2211yescharacteristic
36#4737—21no—
37#4734—21no—
38#4733—21no—
39#4723—21no—
40C4×Dic12—21no—
41#4694C2211yescharacteristic
42#4693C2211yescharacteristic
43C4×D24—21no—
44C4×#204—21no—
45C4×#204—21no—
46C4×#204—21no—
47C4×#204—21no—
48#4688C2211yescharacteristic
49#4688C2211yescharacteristic
50C4×ΓL(1,𝔽25)—21no—
51#4685—21no—
52#4681C2211yescharacteristic
53#4680C2211yescharacteristic
54#4679C2211yescharacteristic
55#4679C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4482C2211yescharacteristic
65#4482C2211yescharacteristic
66#4481C2211yescharacteristic
67#4481C2211yescharacteristic
68#3167—31no—
69#3107—31no—
70#3106—31no—
71#2965—31no—
72#1704—31no—
73#1703—31no—
74#1702—31no—
75#1701—31no—
76#1692—31no—
77#1691—31no—
78#1686—31no—
79#1685—31no—
80#1564D311yescharacteristic, complemented
81#1544—31no—
82#1543—31no—
83C2×#235—21no—
84C2×#235D411yescharacteristic
85C22×C3×C8D411yescharacteristic
86C22×C3×C8D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88#930—21no—
89#930—21no—
90C4×#67—21no—
91C4×#67—21no—
92#926—21no—
93#926D411yescharacteristic
94#924—21no—
95#924—21no—
96C2×Dic12—21no—
97C2×D24—21no—
98C2×#204—21no—
99C2×#204—21no—
100C2×#204—21no—
101C2×#204—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×ΓL(1,𝔽25)—21no—
104#875—21no—
105#874C2311yescharacteristic
106C4×D12—21no—
107C4×D12—21no—
108C4×D12D411yescharacteristic
109C4×D12C2311yescharacteristic
110#872—21no—
111C42×D3—21no—
112#870—21no—
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6—21no—
117C4×Dic6—21no—
118C4×Dic6D411yescharacteristic
119C4×Dic6C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C22⋊C8—21no—
125C3×C22⋊C8—21no—
126C3×C4×C8—21no—
127C3×C4×C8—21no—
128C3×C4×C8D411yescharacteristic
129C3×C4×C8D411yescharacteristic
130C3×C4×C8C2311yescharacteristic
131C3×C4×C8C2311yescharacteristic
132#830—21no—
133#821—21no—
134#821—21no—
135#821—21no—
136#821—21no—
137#821—21no—
138#821—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141#818—21no—
142#818—21no—
143#817—21no—
144#817—21no—
145#817—21no—
146#817—21no—
147#816—21no—
148#816—21no—
149#816—21no—
150#816—21no—
151#816—21no—
152#816—21no—
153#815—21no—
154#815—21no—
155Dic3⋊C8—21no—
156Dic3⋊C8—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8—21no—
160#533—31no—
161#532—31no—
162C4×#41—31no—
163#498—31no—
164#492—31no—
165#487—31no—
166#486—31no—
167#485—61no—
168#482—61no—
169#477—31no—
170#476—31no—
171#475—31no—
172#474—31no—
173#465—61no—
174#465—61no—
175#462—31no—
176#461—31no—
177#460—31no—
178#459—31no—
179#452—61no—
180#449—61no—
181#446—31no—
182#446—31no—
183C4×Q16—61no—
184C4×SD16—61no—
185C4×SD16—61no—
186C4×SD16—61no—
187C4×SD16—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#435—61no—
192#435—61no—
193#431—31no—
194#431—61no—
195#431C2×D311yescharacteristic
196#431C2×D311yescharacteristic
197C4×M16—61no—
198C2×C4×C8C2×D311yescharacteristic
199#335C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#334C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4111yescharacteristic
214C2×#67—21no—
215C2×#67—41no—
216#235—41no—
217#235—41no—
218#235—82no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—41no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6C2×D411yescharacteristic
231C2×C3×C8—22no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—41no—
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42#4111yescharacteristic
244C3×C42C2411yescharacteristic
245#217—21no—
246#217—41no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—41no—
250(C2×D3)⋊C4—41no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3C2×D411yescharacteristic
258C4⋊Dic3C2×D411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261Dic3⋊C4—41no—
262Dic3⋊C4—41no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—41no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269Dic12—42no—
270D24—42no—
271#204—42no—
272#204—42no—
273#204—42no—
274#204—42no—
275ΓL(1,𝔽25)—42no—
276ΓL(1,𝔽25)—42no—
277C2×#41—31no—
278C2×#41—61no—
279C2×Q16—61no—
280C2×SD16—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×SD16—61no—
284C2×D8—61no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8D1211yescharacteristic
288C22×C8D1211yescharacteristic
289C4⋊Q8—31no—
290C4⋊Q8—31no—
291C4⋊Q8—31no—
292#127—31no—
293#125—61no—
294#124—61no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297#121—61no—
298#121—61no—
299C4×Q8—31no—
300C4×Q8—31no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×D4—31no—
304C4×D4—31no—
305C4×D4—61no—
306C4×D4—61no—
307C4×D4—61no—
308C4×D4—61no—
309#117—31no—
310#117—61no—
311#117—61no—
312C2×C42—61no—
313C2×C42C22×D311yescharacteristic
314#107—61no—
315#107—61no—
316#106—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—61no—
346#97—61no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—61no—
350C4×C8D1211yescharacteristic
351C4×C8D1211yescharacteristic
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C23×C3#12111yescharacteristic
355C22×D3—21no—
356C22×D3—41no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4#12111yescharacteristic
363C2×C3×C4#12111yescharacteristic
364C2×C3×C4#12111yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368#67—42no—
369#67—82no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376D12—21no—
377D12—21no—
378D12—41no—
379D12—41no—
380D12—41no—
381D12—41no—
382D12—81no—
383C4×D3—42no—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—42no—
387C4×D3—82no—
388Dic6—21no—
389Dic6—21no—
390Dic6—41no—
391Dic6—41no—
392Dic6—41no—
393Dic6—41no—
394Dic6—81no—
395C3×C8—21no—
396C3×C8—21no—
397C3×C8—21no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C3×C8#13512yes—
403C3⋊C8—41no—
404C3⋊C8—41no—
405#41—121no—
406#41—121no—
407#41—242no—
408C2×Q8—31no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×D4—31no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—121no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—121no—
422C22×C4C2×D1211yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4#23911yescharacteristic
425Q16—122no—
426SD16—122no—
427SD16—122no—
428SD16—122no—
429SD16—122no—
430D8—122no—
431M16—122no—
432M16—122no—
433C2×C8—22no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—41no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8C2×D1211yescharacteristic
442C2×C8C2×D1211yescharacteristic
443C2×C8C2×D1211yescharacteristic
444C2×C8C2×D1211yescharacteristic
445C4⋊C4—31no—
446C4⋊C4—31no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—121no—
456C4⋊C4—121no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C22⋊C4—121no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C42—61no—
464C42—61no—
465C42—61no—
466C42—121no—
467C42C2×D1211yescharacteristic
468C42D3×D411yescharacteristic
469C42#23911yescharacteristic
470C42C23×D311yescharacteristic
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3C2×#12111yescharacteristic
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4#47112yes—
485C3×C4C2×#13511yescharacteristic
486C3×C4#57511yescharacteristic
487C3×C4#57511yescharacteristic
488C3×C4#57511yescharacteristic
489Dic3—41no—
490Dic3—41no—
491Dic3—41no—
492Dic3—81no—
493Dic3—81no—
494Dic3—81no—
495C23—61no—
496C23—121no—
497C23#89511yescharacteristic
498Q8—61no—
499Q8—61no—
500Q8—121no—
501Q8—121no—
502Q8—121no—
503Q8—121no—
504Q8—241no—
505D4—61no—
506D4—61no—
507D4—122no—
508D4—121no—
509D4—121no—
510D4—121no—
511D4—121no—
512D4—241no—
513D4—242no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—242no—
530C2×C4#89511yescharacteristic
531C2×C4#89511yescharacteristic
532C2×C4#89511yescharacteristic
533C2×C4C22×D1211yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×#23911yescharacteristic
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—121no—
544C8—121no—
545C8#90412yes—
546C2×C3—41no—
547C2×C3#303511yescharacteristic
548C2×C3#305411yescharacteristic
549C2×C3C2×#47111yescharacteristic
550D3—81no—
551D3—161no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22—241no—
559C22—241no—
560C22—241no—
561C22C2×#89511yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—121no—
566C4—121no—
567C4—121no—
568C4—241no—
569C4—241no—
570C4—241no—
571C4#489812yes—
572C4C2×#90411yescharacteristic
573C4#574811yescharacteristic
574C4#576811yescharacteristic
575C4#576811yescharacteristic
576C3#1915911yescharacteristic, complemented, Sylow
577C2—41no—
578C2—241no—
579C2—481no—
580C2#7902111yescharacteristic
581C2#7918111yescharacteristic
582C2C2×#489811yescharacteristic
subgroupG/H
centerC2×C4#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1564C2
socleC22×C3C2×#121
prime radical#79301C2
hypercenter#1564D3
hypocenterC3#19159
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #79302C2semidirect
3. #78468C2semidirect
4. #78447C2semidirect
5. C3×#1564C2semidirect
6. #70777C2semidirect
7. #70773C2semidirect
8. #68593C2semidirect
9. #68592C2semidirect
10. #68583C2semidirect
11. #68582C2semidirect
12. #68555C2semidirect
13. #68554C2semidirect
14. #68545C2semidirect
15. #68544C2semidirect
80. #1564D3semidirect
576. C3#19159semidirect

Maximal subgroups

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

subgroupindex
#793022
#793012
#784682
#784472
C3×#15642
#707772
#707732
#685932
#685922
#685832
#685822
#685552
#685542
#685452
#685442
#191593