Finite Group Databasebeta

← #10778200

Subgroups — #10778200

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
1C3×#3182C211yesmaximal, characteristic, complemented
2#83349C211yesmaximal, characteristic, complemented
3#83299C211yesmaximal, characteristic
4#78512C211yesmaximal, characteristic, complemented
5#71832C211yesmaximal, characteristic, complemented
6#71807C211yesmaximal, characteristic, complemented
7#71378C211yesmaximal, characteristic, complemented
8#71374C211yesmaximal, characteristic, complemented
9#71272C211yesmaximal, characteristic, complemented
10#71268C211yesmaximal, characteristic, complemented
11#71241C211yesmaximal, characteristic, complemented
12#71235C211yesmaximal, characteristic, complemented
13#71208C211yesmaximal, characteristic, complemented
14#71202C211yesmaximal, characteristic, complemented
15#69850C211yesmaximal, characteristic, complemented
16#19183—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×#532C2211yescharacteristic
19C4×#235C2211yescharacteristic
20C3×#500C2211yescharacteristic
21C3×#492C2211yescharacteristic
22C3×#487C2211yescharacteristic
23C3×#486C2211yescharacteristic
24C3×#468—21no—
25C3×#468—21no—
26C3×#440—21no—
27C3×#440—21no—
28C3×C4×M16C2211yescharacteristic
29#5128—21no—
30#5128—21no—
31#5094C2211yescharacteristic
32#5091C2211yescharacteristic
33#5091C2211yescharacteristic
34#5087C2211yescharacteristic
35#5084C2211yescharacteristic
36#5084C2211yescharacteristic
37#5075C2211yescharacteristic
38#5072C2211yescharacteristic
39#5066—21no—
40#5063—21no—
41#5037—21no—
42#5035—21no—
43C4×#214—21no—
44C4×#213—21no—
45#5002—21no—
46#4998—21no—
47#4926C2211yescharacteristic
48#4925C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4707—21no—
51#4705—21no—
52#4688C2211yescharacteristic
53#4688C2211yescharacteristic
54C8×Dic6C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4546C2211yescharacteristic
60#4546C2211yescharacteristic
61#4546C2211yescharacteristic
62#4544C2211yescharacteristic
63#4544C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3182D311yescharacteristic, complemented
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#1678—31no—
78#1676—31no—
79#1666—31no—
80#1565—31no—
81#1554—31no—
82#1551—31no—
83C2×C3×#41—21no—
84C2×#235D411yescharacteristic
85C2×C3×SD16—21no—
86C2×C3×SD16—21no—
87C2×C3×M16D411yescharacteristic
88C2×C3×M16D411yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C3×#127C2311yescharacteristic
93C3×#125—21no—
94C3×#124—21no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—21no—
97C3×#121—21no—
98C3×#121—21no—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×#117—21no—
104C2×C3×C42C2311yescharacteristic
105C2×#214—21no—
106C2×#213—21no—
107C4×#67—21no—
108C4×#67—21no—
109#926D411yescharacteristic
110C22×C3⋊C8—21no—
111C2×ΓL(1,𝔽25)—21no—
112C2×ΓL(1,𝔽25)—21no—
113C4×D12D411yescharacteristic
114C4×D12C2311yescharacteristic
115#872—21no—
116C42×D3—21no—
117C4×Dic6D411yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×#107—21no—
120C3×#107—21no—
121C3×#103—21no—
122C3×#103—21no—
123C3×#102—21no—
124C3×#102—21no—
125C3×#97D411yescharacteristic
126C3×#97D411yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129#832—21no—
130#832—21no—
131#830—21no—
132#830—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135Dic3⋊C8—21no—
136Dic3⋊C8—21no—
137C8×Dic3—21no—
138C8×Dic3—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#809—21no—
143#809—21no—
144#809—21no—
145#808—21no—
146#808—21no—
147#808—21no—
148#807—21no—
149#807—21no—
150#807—21no—
151C4⋊(C3⋊C8)—21no—
152C4⋊(C3⋊C8)C2311yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158#803—21no—
159C4×C3⋊C8—21no—
160#533C2×D311yescharacteristic
161#532C2×D311yescharacteristic
162C4×#41—31no—
163#500C2×D311yescharacteristic
164#492C2×D311yescharacteristic
165#487C2×D311yescharacteristic
166#486C2×D311yescharacteristic
167#482—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#477—31no—
172#476—31no—
173#476—31no—
174#475—31no—
175#474—31no—
176#468—21no—
177#468—21no—
178#460—31no—
179#459—31no—
180#449—61no—
181C8×Q8—31no—
182C8×Q8—31no—
183#440—21no—
184#440—21no—
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—61no—
198C4×M16C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C3×#41—82no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—41no—
219C22×C3×C4—41no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4#4111yescharacteristic
223C2×#67—21no—
224#235—22no—
225#235—41no—
226C2×D12C2×D411yescharacteristic
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×Dic6C2×D411yescharacteristic
231C3×SD16—42no—
232C3×SD16—42no—
233C3×M16—22no—
234C3×M16—41no—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—41no—
246C3×C4⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C22⋊C4—41no—
249C3×C22⋊C4—41no—
250C3×C42—41no—
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42#4111yescharacteristic
254C3×C42C2411yescharacteristic
255#217—21no—
256#214—42no—
257#213—42no—
258(C2×D3)⋊C4—21no—
259(C2×D3)⋊C4—21no—
260C4⋊Dic3C2×D411yescharacteristic
261C4⋊Dic3C2×D411yescharacteristic
262Dic3⋊C4—21no—
263Dic3⋊C4—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—42no—
274ΓL(1,𝔽25)—42no—
275ΓL(1,𝔽25)—42no—
276C2×#41—21no—
277C2×#41—31no—
278C2×SD16—21no—
279C2×SD16—21no—
280C2×SD16—61no—
281C2×D8—61no—
282C2×M16—61no—
283C2×M16—61no—
284C2×M16#6711yescharacteristic
285C2×M16#6711yescharacteristic
286C22×C8—61no—
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
289C4⋊Q8C22×D311yescharacteristic
290#127C22×D311yescharacteristic
291#125—21no—
292#124—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295#121—21no—
296#121—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—31no—
300C4×Q8—31no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—31no—
304C4×D4—31no—
305C4×D4—61no—
306C4×D4—61no—
307#117—21no—
308#117—31no—
309#117—61no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312#107—21no—
313#107—21no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8—61no—
327C4⋊C8—61no—
328C4⋊C8—61no—
329#103—21no—
330#103—21no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—21no—
335#102—21no—
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—61no—
346#97#6711yescharacteristic
347#97#6711yescharacteristic
348C4×C8—61no—
349C4×C8—61no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C23×C3—41no—
354C23×C3#12111yescharacteristic
355C22×D3—21no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—81no—
359C3×Q8—81no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—81no—
365C3×D4—82no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—82no—
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4#12111yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381#67—42no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385D12—21no—
386D12—21no—
387C4×D3—42no—
388C4×D3—41no—
389C4×D3—41no—
390C4×D3—81no—
391Dic6—21no—
392Dic6#13712yes—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—21no—
396C3×C8#13512yes—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403#41—62no—
404#41—82no—
405#41—121no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—31no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—31no—
414C2×D4—41no—
415C2×D4—61no—
416C22×C4—41no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4D3×D411yescharacteristic
421C22×C4#23711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423SD16—42no—
424SD16—42no—
425SD16—122no—
426D8—122no—
427M16—22no—
428M16—41no—
429M16—122no—
430M16—122no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—122no—
438C2×C8C2×#6711yescharacteristic
439C2×C8C2×#6711yescharacteristic
440C2×C8C2×#6711yescharacteristic
441C2×C8C2×#6711yescharacteristic
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—31no—
449C4⋊C4—31no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C42—41no—
461C42—61no—
462C42—61no—
463C42—61no—
464C42D3×D411yescharacteristic
465C42#23711yescharacteristic
466C42C2×#6711yescharacteristic
467C42C23×D311yescharacteristic
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3C2×#12111yescharacteristic
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4C2×#13511yescharacteristic
486C3×C4C2×#13711yescharacteristic
487C3×C4#57511yescharacteristic
488C3×C4#57511yescharacteristic
489Dic3—22no—
490Dic3—41no—
491Dic3—41no—
492C23—41no—
493C23—61no—
494C23#93811yescharacteristic
495Q8—32no—
496Q8—41no—
497Q8—41no—
498Q8—61no—
499Q8—81no—
500Q8—81no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—61no—
506D4—61no—
507D4—81no—
508D4—82no—
509D4—122no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—82no—
522C2×C4—122no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—241no—
526C2×C4#93811yescharacteristic
527C2×C4#93811yescharacteristic
528C2×C4#93811yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23711yescharacteristic
531C2×C4C22×#6711yescharacteristic
532C8—21no—
533C8—21no—
534C8—21no—
535C8—121no—
536C8—121no—
537C8—121no—
538C8—121no—
539C8—121no—
540C8—121no—
541C8#95012yes—
542C2×C3—41no—
543C2×C3—161no—
544C2×C3#303511yescharacteristic
545C2×C3#305611yescharacteristic
546C2×C3#306811yescharacteristic
547D3—81no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22C2×#93811yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—62no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—121no—
567C4—121no—
568C4D3×#13711yescharacteristic
569C4#577711yescharacteristic
570C4C2×#95011yescharacteristic
571C4#582211yescharacteristic
572C3#1918311yescharacteristic, complemented, Sylow
573C2—41no—
574C2—161no—
575C2—241no—
576C2#8329611yescharacteristic
577C2#8396311yescharacteristic
578C2#8398511yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3182C2
socleC22×C3C2×#121
prime radical#83299C2
hypercenter#3182D3
hypocenterC3#19183
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3182

Complements

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

NH ≅ G/Ntype
1. C3×#3182C2semidirect
2. #83349C2semidirect
4. #78512C2semidirect
5. #71832C2semidirect
6. #71807C2semidirect
7. #71378C2semidirect
8. #71374C2semidirect
9. #71272C2semidirect
10. #71268C2semidirect
11. #71241C2semidirect
12. #71235C2semidirect
13. #71208C2semidirect
14. #71202C2semidirect
15. #69850C2semidirect
68. #3182D3semidirect
572. C3#19183semidirect

Maximal subgroups

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

subgroupindex
C3×#31822
#833492
#832992
#785122
#718322
#718072
#713782
#713742
#712722
#712682
#712412
#712352
#712082
#712022
#698502
#191833