Finite Group Databasebeta

← #10778211

Subgroups — #10778211

Order 768.

614 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81559C211yesmaximal, characteristic, complemented
2#81107C211yesmaximal, characteristic, complemented
3#80361C211yesmaximal, characteristic
4#79905C211yesmaximal, characteristic, complemented
5C3×#1678C211yesmaximal, characteristic, complemented
6#71202C211yesmaximal, characteristic, complemented
7#71149C211yesmaximal, characteristic, complemented
8#71020C211yesmaximal, characteristic, complemented
9#70947C211yesmaximal, characteristic, complemented
10#68545C211yesmaximal, characteristic, complemented
11#68528C211yesmaximal, characteristic, complemented
12#68501C211yesmaximal, characteristic, complemented
13#68486C211yesmaximal, characteristic, complemented
14#68398C211yesmaximal, characteristic, complemented
15#68385C211yesmaximal, characteristic, complemented
16#19183—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5716C2211yescharacteristic
19C4×#239C2211yescharacteristic
20C3×#486C2211yescharacteristic
21C3×#477C2211yescharacteristic
22C3×#460C2211yescharacteristic
23C3×C8×Q8C2211yescharacteristic
24#5172—21no—
25Dic3×SD16—21no—
26#5084C2211yescharacteristic
27#5078C2211yescharacteristic
28#5076C2211yescharacteristic
29#5072C2211yescharacteristic
30Q8×C3⋊C8C2211yescharacteristic
31#5021C2211yescharacteristic
32#5020C2211yescharacteristic
33#4891—21no—
34#4880—21no—
35#4846C2211yescharacteristic
36#4842C2211yescharacteristic
37#4838—21no—
38#4838C2211yescharacteristic
39#4835—21no—
40#4835C2211yescharacteristic
41#4833C2211yescharacteristic
42D3×C4⋊C8C2211yescharacteristic
43#4804—21no—
44#4796—21no—
45#4780—21no—
46#4780—21no—
47#4779—21no—
48#4757—21no—
49#4757—21no—
50#4756—21no—
51#4688—21no—
52#4688C2211yescharacteristic
53C4×ΓL(1,𝔽25)—21no—
54C4×ΓL(1,𝔽25)C2211yescharacteristic
55#4681C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4546C2211yescharacteristic
60#4544C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3182—31no—
69#3106—31no—
70#3076—31no—
71#2971—31no—
72#1703—31no—
73#1697—31no—
74#1695—31no—
75#1685—31no—
76#1684—31no—
77#1678D311yescharacteristic, complemented
78#1676—31no—
79#1666—31no—
80#1565—31no—
81#1554—31no—
82#1551—31no—
83C2×#239D411yescharacteristic
84C2×#237—21no—
85C2×C3×SD16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×#127C2311yescharacteristic
88C3×C4×Q8C2311yescharacteristic
89#946—21no—
90Q8×Dic3—21no—
91Q8×Dic3D411yescharacteristic
92#938—21no—
93#938—21no—
94#936—21no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97C2×#214—21no—
98C2×#214—21no—
99C2×#213—21no—
100C2×ΓL(1,𝔽25)—21no—
101C2×ΓL(1,𝔽25)—21no—
102C2×ΓL(1,𝔽25)D411yescharacteristic
103C2×ΓL(1,𝔽25)D411yescharacteristic
104C2×D3×C8—21no—
105#897—21no—
106#893—21no—
107#893D411yescharacteristic
108#892—21no—
109#892—21no—
110#892D411yescharacteristic
111#890—21no—
112#887—21no—
113C4×D12—21no—
114C4×D12C2311yescharacteristic
115C42×D3—21no—
116C42×D3C2311yescharacteristic
117#869C2311yescharacteristic
118#869C2311yescharacteristic
119C3×#107—21no—
120C3×C4⋊C8—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#103—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×C4×C8—21no—
129C3×C4×C8C2311yescharacteristic
130#835—21no—
131#835—21no—
132#832—21no—
133#821—21no—
134#821—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139#818—21no—
140#817—21no—
141#816—21no—
142#816—21no—
143#815—21no—
144#815D411yescharacteristic
145#815D411yescharacteristic
146Dic3⋊C8—21no—
147C8×Dic3—21no—
148#809—21no—
149#808—21no—
150#808—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8—21no—
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#532—31no—
162C4×#41—31no—
163#500—31no—
164#492—31no—
165#487—31no—
166#486C2×D311yescharacteristic
167#482—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#477C2×D311yescharacteristic
172#476—31no—
173#476—31no—
174#475—31no—
175#474—31no—
176#468—61no—
177#468—61no—
178#460C2×D311yescharacteristic
179#459—31no—
180#449—61no—
181C8×Q8—31no—
182C8×Q8C2×D311yescharacteristic
183#440—61no—
184#440—61no—
185C4×SD16—61no—
186C4×D8—61no—
187#436—31no—
188#436—31no—
189#436—61no—
190#436—61no—
191#432—31no—
192#432—61no—
193#431—31no—
194#427—61no—
195C2×C4⋊C8—31no—
196#416—61no—
197C4×M16—31no—
198C4×M16—61no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#333—31no—
206#333C2×D311yescharacteristic
207#332—31no—
208#332—31no—
209#331—31no—
210#331C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D411yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×#67—41no—
217C22×Dic3—41no—
218#239—22no—
219#239—41no—
220#237—82no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3C2×D411yescharacteristic
227C2×C4×D3C2×D411yescharacteristic
228C2×C4×D3#4111yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—21no—
231C3×SD16—42no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4C2×D411yescharacteristic
241C3×C42—21no—
242C3×C42C2411yescharacteristic
243#217—41no—
244#217—41no—
245#214—42no—
246#214—42no—
247#213—42no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—41no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—41no—
258C4⋊Dic3C2×D411yescharacteristic
259Dic3⋊C4—41no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—41no—
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3#4111yescharacteristic
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271ΓL(1,𝔽25)—22no—
272ΓL(1,𝔽25)—41no—
273ΓL(1,𝔽25)—42no—
274ΓL(1,𝔽25)—42no—
275D3×C8—42no—
276C2×#41—31no—
277C2×#41—61no—
278C2×SD16—21no—
279C2×SD16—61no—
280C2×SD16—61no—
281C2×D8—61no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—61no—
285C2×M16—61no—
286C22×C8—61no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8C22×D311yescharacteristic
290#127C22×D311yescharacteristic
291#125—61no—
292#124—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295#121—61no—
296#121—61no—
297C4×Q8—31no—
298C4×Q8—61no—
299C4×Q8—61no—
300C4×Q8C22×D311yescharacteristic
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4—61no—
307#117—31no—
308#117—61no—
309#117—61no—
310C2×C42—31no—
311C2×C42—61no—
312#107—21no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—21no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—21no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—21no—
335#102—21no—
336#102—21no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—31no—
346#97—31no—
347#97—61no—
348C4×C8—21no—
349C4×C8—31no—
350C4×C8—61no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C23×C3—41no—
354C22×D3—21no—
355C22×D3#12111yescharacteristic
356C3×Q8—21no—
357C3×Q8—81no—
358C3×Q8#13712yes—
359C3×D4—41no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—81no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370#67—82no—
371C2×Dic3—21no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—82no—
375C2×Dic3#12111yescharacteristic
376C2×Dic3#12111yescharacteristic
377C2×Dic3#12111yescharacteristic
378D12—21no—
379D12—21no—
380D12—42no—
381C4×D3—21no—
382C4×D3—21no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—41no—
386C4×D3—42no—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—81no—
390Dic6—41no—
391Dic6—41no—
392Dic6—81no—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C3⋊C8—21no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8#13512yes—
403#41—62no—
404#41—121no—
405#41—242no—
406C2×Q8—21no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8D3×D411yescharacteristic
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—31no—
414C2×D4—61no—
415C2×D4—121no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—121no—
423SD16—42no—
424SD16—122no—
425SD16—122no—
426D8—122no—
427M16—62no—
428M16—121no—
429M16—122no—
430M16—122no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—31no—
435C2×C8—31no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—122no—
440C2×C8D3×D411yescharacteristic
441C2×C8D3×D411yescharacteristic
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—31no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—121no—
452C4⋊C4—121no—
453C4⋊C4D3×D411yescharacteristic
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C22⋊C4—121no—
460C42—21no—
461C42—31no—
462C42—31no—
463C42—31no—
464C42—61no—
465C42—61no—
466C42—121no—
467C42C23×D311yescharacteristic
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3C2×#12111yescharacteristic
472C2×D3—21no—
473C2×D3—21no—
474C2×D3—21no—
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C3×C4—22no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4C2×#13511yescharacteristic
483C3×C4C2×#13711yescharacteristic
484C3×C4#57511yescharacteristic
485C3×C4#57511yescharacteristic
486Dic3—21no—
487Dic3—21no—
488Dic3—21no—
489Dic3—41no—
490Dic3—81no—
491Dic3—81no—
492C23—31no—
493C23—41no—
494C23—61no—
495Q8—21no—
496Q8—81no—
497Q8—121no—
498Q8—121no—
499Q8—241no—
500Q8#91512yes—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—61no—
506D4—61no—
507D4—81no—
508D4—122no—
509D4—242no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—31no—
513C2×C4—31no—
514C2×C4—31no—
515C2×C4—41no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—121no—
522C2×C4—122no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—241no—
528C2×C4—242no—
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×D3×D411yescharacteristic
531C2×C4D3×#4111yescharacteristic
532C8—21no—
533C8—21no—
534C8—32no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—61no—
539C8—121no—
540C8—121no—
541C8—121no—
542C2×C3—161no—
543C2×C3#303511yescharacteristic
544C2×C3#305611yescharacteristic
545C2×C3#306811yescharacteristic
546D3—41no—
547D3—81no—
548C22—61no—
549C22—61no—
550C22—61no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22D3×#12111yescharacteristic
558C4—22no—
559C4—21no—
560C4—41no—
561C4—61no—
562C4—61no—
563C4—61no—
564C4—81no—
565C4—121no—
566C4—241no—
567C4—241no—
568C4C2×#91511yescharacteristic
569C4D3×#13511yescharacteristic
570C4#576811yescharacteristic
571C4#577711yescharacteristic
572C3#1918311yescharacteristic, complemented, Sylow
573C2—121no—
574C2—161no—
575C2—241no—
576C2#8059711yescharacteristic
577C2#8065811yescharacteristic
578C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1678C2
socleC22×C3C2×#121
prime radical#80361C2
hypercenter#1678D3
hypocenterC3#19183
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1678

Complements

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

NH ≅ G/Ntype
1. #81559C2semidirect
2. #81107C2semidirect
4. #79905C2semidirect
5. C3×#1678C2semidirect
6. #71202C2semidirect
7. #71149C2semidirect
8. #71020C2semidirect
9. #70947C2semidirect
10. #68545C2semidirect
11. #68528C2semidirect
12. #68501C2semidirect
13. #68486C2semidirect
14. #68398C2semidirect
15. #68385C2semidirect
77. #1678D3semidirect
572. C3#19183semidirect

Maximal subgroups

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

subgroupindex
#815592
#811072
#803612
#799052
C3×#16782
#712022
#711492
#710202
#709472
#685452
#685282
#685012
#684862
#683982
#683852
#191833