Finite Group Databasebeta

← #10778203

Subgroups — #10778203

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#81595C211yesmaximal, characteristic, complemented
2#81093C211yesmaximal, characteristic
3#81010C211yesmaximal, characteristic, complemented
4#79905C211yesmaximal, characteristic, complemented
5C3×#1684C211yesmaximal, characteristic, complemented
6#71235C211yesmaximal, characteristic, complemented
7#71149C211yesmaximal, characteristic, complemented
8#71116C211yesmaximal, characteristic, complemented
9#70940C211yesmaximal, characteristic, complemented
10#68584C211yesmaximal, characteristic, complemented
11#68568C211yesmaximal, characteristic, complemented
12#68501C211yesmaximal, characteristic, complemented
13#68485C211yesmaximal, characteristic, complemented
14#68448C211yesmaximal, characteristic, complemented
15#68427C211yesmaximal, characteristic, complemented
16#19183—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5725C2211yescharacteristic
19C4×#239C2211yescharacteristic
20C3×#487C2211yescharacteristic
21C3×#477C2211yescharacteristic
22C3×#475C2211yescharacteristic
23C3×C8×Q8C2211yescharacteristic
24#5087C2211yescharacteristic
25#5085C2211yescharacteristic
26#5084C2211yescharacteristic
27#5078C2211yescharacteristic
28#5027C2211yescharacteristic
29Q8×C3⋊C8C2211yescharacteristic
30#5021C2211yescharacteristic
31#4885—21no—
32#4872—21no—
33#4871—21no—
34#4860—21no—
35#4843C2211yescharacteristic
36#4842C2211yescharacteristic
37#4838—21no—
38#4838C2211yescharacteristic
39#4835—21no—
40#4835C2211yescharacteristic
41#4833C2211yescharacteristic
42D3×C4⋊C8C2211yescharacteristic
43#4817—21no—
44#4817—21no—
45#4812—21no—
46#4812—21no—
47#4810—21no—
48#4804—21no—
49#4799—21no—
50#4788—21no—
51#4694C2211yescharacteristic
52#4688—21no—
53#4688C2211yescharacteristic
54C4×ΓL(1,𝔽25)—21no—
55C4×ΓL(1,𝔽25)C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4546C2211yescharacteristic
61#4485C2211yescharacteristic
62#4485C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4457C2211yescharacteristic
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#1684D311yescharacteristic, complemented
77#1678—31no—
78#1676—31no—
79#1666—31no—
80#1565—31no—
81#1554—31no—
82#1551—31no—
83C2×#239—21no—
84C2×#239D411yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4×Q8C2311yescharacteristic
88#947—21no—
89#946—21no—
90Q8×Dic3—21no—
91Q8×Dic3D411yescharacteristic
92C2×#215—21no—
93C2×#215—21no—
94C2×#215—21no—
95C2×D24—21no—
96C2×ΓL(1,𝔽25)—21no—
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)D411yescharacteristic
99C2×ΓL(1,𝔽25)D411yescharacteristic
100C2×D3×C8—21no—
101#897—21no—
102#895—21no—
103#895—21no—
104#893—21no—
105#893—21no—
106#893—21no—
107#893D411yescharacteristic
108#892—21no—
109#892—21no—
110#892D411yescharacteristic
111#890—21no—
112#887—21no—
113#874C2311yescharacteristic
114C4×D12—21no—
115C4×D12C2311yescharacteristic
116C42×D3—21no—
117C42×D3C2311yescharacteristic
118#869C2311yescharacteristic
119C3×#107—21no—
120C3×#106—21no—
121C3×C4⋊C8—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×#103—21no—
126C3×#103—21no—
127C3×#103—21no—
128C3×#103—21no—
129C3×C4×C8—21no—
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#821—21no—
133#821—21no—
134#821—21no—
135#821—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140#817—21no—
141#815—21no—
142#815D411yescharacteristic
143#815D411yescharacteristic
144Dic3⋊C8—21no—
145C8×Dic3—21no—
146#809—21no—
147#809—21no—
148#809—21no—
149#808—21no—
150#807—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#487C2×D311yescharacteristic
166#486—31no—
167#482—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#477C2×D311yescharacteristic
172#476—31no—
173#476—31no—
174#475C2×D311yescharacteristic
175#474—31no—
176#468—61no—
177#468—61no—
178#460—31no—
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#335C2×D311yescharacteristic
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333C2×D311yescharacteristic
207#332—31no—
208#332C2×D311yescharacteristic
209#331—31no—
210#331—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214#239—22no—
215#239—41no—
216#239—82no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—41no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—41no—
227C2×C4×D3C2×D411yescharacteristic
228C2×C4×D3C2×D411yescharacteristic
229C2×C4×D3#4111yescharacteristic
230C2×Dic6—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4C2×D411yescharacteristic
242C3×C42—21no—
243C3×C42C2411yescharacteristic
244#215—42no—
245#215—42no—
246#215—42no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—41no—
251(C2×D3)⋊C4—41no—
252(C2×D3)⋊C4—41no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—41no—
257C4⋊Dic3C2×D411yescharacteristic
258Dic3⋊C4—41no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—41no—
262C4×Dic3C2×D411yescharacteristic
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3#4111yescharacteristic
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270D24—42no—
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—61no—
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⋊Q8C22×D311yescharacteristic
289C4⋊Q8C22×D311yescharacteristic
290#127—31no—
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—21no—
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—21no—
331#103—21no—
332#103—21no—
333#103—61no—
334#102—61no—
335#102—61no—
336#102—61no—
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
353C22×D3—21no—
354C22×D3—41no—
355C22×D3#12111yescharacteristic
356C3×Q8—21no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—81no—
360C3×Q8#13712yes—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3#12111yescharacteristic
371C2×Dic3#12111yescharacteristic
372C2×Dic3#12111yescharacteristic
373D12—21no—
374D12—21no—
375D12—42no—
376D12—41no—
377D12—41no—
378D12—41no—
379D12—41no—
380D12—81no—
381D12—82no—
382C4×D3—21no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—21no—
386C4×D3—41no—
387C4×D3—42no—
388C4×D3—41no—
389C4×D3—41no—
390C4×D3—81no—
391C4×D3—82no—
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—21no—
408C2×Q8—61no—
409C2×Q8D3×D411yescharacteristic
410C2×D4—31no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
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—122no—
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—21no—
446C4⋊C4—21no—
447C4⋊C4—31no—
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×C3C2×#12111yescharacteristic
469C2×D3—21no—
470C2×D3—21no—
471C2×D3—21no—
472C2×D3—41no—
473C2×D3—41no—
474C2×D3—41no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—22no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4C2×#13511yescharacteristic
484C3×C4C2×#13711yescharacteristic
485C3×C4#57511yescharacteristic
486C3×C4#57511yescharacteristic
487Dic3—21no—
488Dic3—21no—
489Dic3—21no—
490Dic3—41no—
491Dic3—81no—
492C23—31no—
493C23—61no—
494C23—121no—
495Q8—21no—
496Q8—41no—
497Q8—41no—
498Q8—81no—
499Q8—241no—
500Q8#91812yes—
501D4—61no—
502D4—61no—
503D4—122no—
504D4—121no—
505D4—121no—
506D4—121no—
507D4—121no—
508D4—241no—
509D4—242no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—31no—
513C2×C4—31no—
514C2×C4—31no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—121no—
523C2×C4—122no—
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#303511yescharacteristic
543C2×C3#305611yescharacteristic
544C2×C3#306811yescharacteristic
545D3—41no—
546D3—81no—
547D3—161no—
548C22—61no—
549C22—61no—
550C22—61no—
551C22—121no—
552C22—121no—
553C22—121no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22D3×#12111yescharacteristic
558C4—21no—
559C4—22no—
560C4—41no—
561C4—61no—
562C4—61no—
563C4—61no—
564C4—81no—
565C4—81no—
566C4—121no—
567C4—241no—
568C4C2×#91811yescharacteristic
569C4D3×#13511yescharacteristic
570C4#576811yescharacteristic
571C4#577711yescharacteristic
572C3#1918311yescharacteristic, complemented, Sylow
573C2—121no—
574C2—241no—
575C2—481no—
576C2#8059711yescharacteristic
577C2#8075911yescharacteristic
578C2#8093011yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1684C2
socleC22×C3C2×#121
prime radical#81093C2
hypercenter#1684D3
hypocenterC3#19183
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1684

Complements

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

NH ≅ G/Ntype
1. #81595C2semidirect
3. #81010C2semidirect
4. #79905C2semidirect
5. C3×#1684C2semidirect
6. #71235C2semidirect
7. #71149C2semidirect
8. #71116C2semidirect
9. #70940C2semidirect
10. #68584C2semidirect
11. #68568C2semidirect
12. #68501C2semidirect
13. #68485C2semidirect
14. #68448C2semidirect
15. #68427C2semidirect
76. #1684D3semidirect
572. C3#19183semidirect

Maximal subgroups

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

subgroupindex
#815952
#810932
#810102
#799052
C3×#16842
#712352
#711492
#711162
#709402
#685842
#685682
#685012
#684852
#684482
#684272
#191833