Finite Group Databasebeta

← #10780328

Subgroups — #10780328

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3141C211yesmaximal, characteristic, complemented
2#83312C211yesmaximal, characteristic
3#83214C211yesmaximal, characteristic, complemented
4#78511C211yesmaximal, characteristic, complemented
5#71808C211yesmaximal, characteristic, complemented
6#71773C211yesmaximal, characteristic, complemented
7#71242C211yesmaximal, characteristic, complemented
8#71240C211yesmaximal, characteristic, complemented
9#71234C211yesmaximal, characteristic, complemented
10#71232C211yesmaximal, characteristic, complemented
11#71211C211yesmaximal, characteristic, complemented
12#71209C211yesmaximal, characteristic, complemented
13#71203C211yesmaximal, characteristic, complemented
14#71201C211yesmaximal, characteristic, complemented
15#69851C211yesmaximal, characteristic, complemented
16#19383—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19#5479C2211yescharacteristic
20C3×#489—21no—
21C3×#489—21no—
22C3×#487C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#486C2211yescharacteristic
26C3×#417—21no—
27C3×#417—21no—
28C3×C4×M16C2211yescharacteristic
29#5228—21no—
30#5227—21no—
31#5128—21no—
32#5127—21no—
33#5094C2211yescharacteristic
34#5090C2211yescharacteristic
35#5090C2211yescharacteristic
36#5087C2211yescharacteristic
37#5086—21no—
38#5083C2211yescharacteristic
39#5083C2211yescharacteristic
40#5082—21no—
41#5075C2211yescharacteristic
42#5072C2211yescharacteristic
43#5042—21no—
44#5039—21no—
45#5016—21no—
46#5009—21no—
47#4926C2211yescharacteristic
48#4925C2211yescharacteristic
49#4924C2211yescharacteristic
50#4709—21no—
51#4708—21no—
52#4688C2211yescharacteristic
53C8×D12C2211yescharacteristic
54#4679C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4547C2211yescharacteristic
59#4547C2211yescharacteristic
60#4547C2211yescharacteristic
61#4547C2211yescharacteristic
62#4544C2211yescharacteristic
63#4544C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3251—31no—
69#3218—31no—
70#3141D311yescharacteristic, complemented
71#2969—31no—
72#1702—31no—
73#1701—31no—
74#1700—31no—
75#1699—31no—
76#1682—31no—
77#1681—31no—
78#1680—31no—
79#1679—31no—
80#1569—31no—
81#1553—31no—
82#1546—31no—
83C2×C3×#41—21no—
84C22×C3×Q8—21no—
85C2×C3×M16D411yescharacteristic
86C2×C3×M16D411yescharacteristic
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—21no—
89C3×C4⋊Q8—21no—
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C3×C4⋊Q8C2311yescharacteristic
93C3×#127C2311yescharacteristic
94C3×#124—21no—
95C3×#121—21no—
96C3×#121—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C42C2311yescharacteristic
102C2×#216—21no—
103C2×#216—21no—
104C2×#215—21no—
105C2×#215—21no—
106C2×#214—21no—
107C2×#213—21no—
108#930D411yescharacteristic
109#929—21no—
110C4×#67—21no—
111#924D411yescharacteristic
112C2×#208—21no—
113C2×ΓL(1,𝔽25)—21no—
114C2×D3×C8—21no—
115#876—21no—
116#875D411yescharacteristic
117C4×D12C2311yescharacteristic
118#872—21no—
119#870D411yescharacteristic
120C4×Dic6C2311yescharacteristic
121C3×#103—21no—
122C3×#103—21no—
123C3×#103—21no—
124C3×#103—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×#102—21no—
129C3×#97D411yescharacteristic
130C3×#97D411yescharacteristic
131C3×C4×C8C2311yescharacteristic
132C3×C4×C8C2311yescharacteristic
133#835—21no—
134#835—21no—
135#835—21no—
136#835—21no—
137#832—21no—
138#832—21no—
139#830—21no—
140#830—21no—
141(C2×D3)⋊C8—21no—
142(C2×D3)⋊C8—21no—
143#815—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146C8×Dic3—21no—
147#810—21no—
148#809—21no—
149#808—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#532C2×D311yescharacteristic
161C2×C4⋊Q8C2×D311yescharacteristic
162#529—31no—
163#497—61no—
164#491—61no—
165#489—21no—
166#489—21no—
167#487C2×D311yescharacteristic
168#487C2×D311yescharacteristic
169#486C2×D311yescharacteristic
170#486C2×D311yescharacteristic
171#478—61no—
172#475—31no—
173#474—31no—
174#469—61no—
175#468—61no—
176#463—61no—
177#462—31no—
178#462—31no—
179#461—31no—
180#461—31no—
181#460—31no—
182#459—31no—
183#451—61no—
184#448—61no—
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432—31no—
193#431—31no—
194#423—31no—
195#417—21no—
196#417—21no—
197#405—61no—
198C4×M16C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C3×#41—42no—
212C3×#41—42no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—41no—
217C2×C3×Q8—81no—
218C2×C3×Q8—81no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—41no—
223C22×C3×C4—41no—
224C22×C3×C4—41no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4#4111yescharacteristic
228C2×#67—21no—
229C2×D12C2×D411yescharacteristic
230C2×C4×D3—21no—
231C2×Dic6C2×D411yescharacteristic
232C3×M16—41no—
233C3×M16—41no—
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—41no—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C42C2×D411yescharacteristic
249C3×C42C2×D411yescharacteristic
250C3×C42#4111yescharacteristic
251C3×C42C2411yescharacteristic
252#217—21no—
253#216—81no—
254#216—81no—
255#215—81no—
256#215—81no—
257#214—81no—
258#213—81no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—21no—
262(C2×D3)⋊C4—21no—
263C4⋊Dic3C2×D411yescharacteristic
264C4⋊Dic3C2×D411yescharacteristic
265Dic3⋊C4—21no—
266Dic3⋊C4—21no—
267Dic3⋊C4—21no—
268Dic3⋊C4—21no—
269C4×Dic3—21no—
270#208—81no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277ΓL(1,𝔽25)—42no—
278D3×C8—42no—
279C2×#41—21no—
280C22×Q8—21no—
281C2×Q16—61no—
282C2×Q16—61no—
283C2×SD16—61no—
284C2×SD16—61no—
285C2×SD16—61no—
286C2×D8—61no—
287C2×M16—61no—
288C2×M16—61no—
289C2×M16#6711yescharacteristic
290C2×M16#6711yescharacteristic
291C22×C8—61no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8C22×D311yescharacteristic
296C4⋊Q8C22×D311yescharacteristic
297C4⋊Q8C22×D311yescharacteristic
298#127C22×D311yescharacteristic
299#126—61no—
300#125—31no—
301#124—21no—
302#124—31no—
303#123—61no—
304C22⋊Q8—31no—
305#121—21no—
306#121—21no—
307#121—31no—
308C4×Q8—31no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—31no—
312C4×D4—61no—
313#117—61no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C42C22×D311yescharacteristic
317#107—61no—
318#106—61no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—61no—
326C4⋊C8—61no—
327C4⋊C8—61no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—21no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—21no—
338#102—21no—
339#102—21no—
340#102—21no—
341#102—61no—
342#102—61no—
343#102—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347C22⋊C8—61no—
348#97—61no—
349#97—61no—
350#97#6711yescharacteristic
351#97#6711yescharacteristic
352C4×C8—61no—
353C4×C8—61no—
354C4×C8C22×D311yescharacteristic
355C4×C8C22×D311yescharacteristic
356C23×C3—41no—
357C23×C3#12111yescharacteristic
358C22×D3—21no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—82no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—81no—
371C3×D4—82no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—82no—
381C2×C3×C4—82no—
382C2×C3×C4#12111yescharacteristic
383C2×C3×C4#12111yescharacteristic
384C2×C3×C4#12111yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388#67—81no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392D12—21no—
393D12—21no—
394C4×D3—41no—
395C4×D3—41no—
396Dic6—21no—
397Dic6—21no—
398C3×C8—21no—
399C3×C8—21no—
400C3×C8—21no—
401C3×C8—21no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408#41—42no—
409#41—42no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—31no—
414C2×Q8—41no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—31no—
421C2×D4—41no—
422C2×D4—61no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—61no—
426C22×C4D3×D411yescharacteristic
427C22×C4#23711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429Q16—241no—
430Q16—241no—
431SD16—241no—
432SD16—241no—
433SD16—241no—
434D8—241no—
435M16—41no—
436M16—41no—
437M16—122no—
438M16—241no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—122no—
446C2×C8C2×#6711yescharacteristic
447C2×C8C2×#6711yescharacteristic
448C2×C8C2×#6711yescharacteristic
449C2×C8C2×#6711yescharacteristic
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—31no—
457C4⋊C4—31no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C42—61no—
472C42D3×D411yescharacteristic
473C42#23711yescharacteristic
474C42C2×#6711yescharacteristic
475C42C23×D311yescharacteristic
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3C2×#12111yescharacteristic
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4#57511yescharacteristic
494C3×C4#57511yescharacteristic
495C3×C4#57611yescharacteristic
496C3×C4#57711yescharacteristic
497Dic3—41no—
498Dic3—41no—
499Dic3—41no—
500C23—41no—
501C23—61no—
502C23#93811yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—61no—
510Q8—61no—
511Q8—82no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—61no—
517D4—61no—
518D4—81no—
519D4—82no—
520D4—241no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4#93811yescharacteristic
537C2×C4#93811yescharacteristic
538C2×C4#93811yescharacteristic
539C2×C4C2×D3×D411yescharacteristic
540C2×C4C2×#23711yescharacteristic
541C2×C4C22×#6711yescharacteristic
542C8—21no—
543C8—21no—
544C8—21no—
545C8—21no—
546C8—121no—
547C8—121no—
548C8—121no—
549C8—121no—
550C8—121no—
551C8—121no—
552C2×C3—41no—
553C2×C3—161no—
554C2×C3#304011yescharacteristic
555C2×C3#306211yescharacteristic
556C2×C3#306911yescharacteristic
557D3—81no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—121no—
565C22—121no—
566C22—121no—
567C22C2×#93811yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—121no—
576C4—121no—
577C4—121no—
578C4#577411yescharacteristic
579C4#577511yescharacteristic
580C4#582211yescharacteristic
581C4#582211yescharacteristic
582C3#1938311yescharacteristic, complemented, Sylow
583C2—41no—
584C2—161no—
585C2—241no—
586C2#8294511yescharacteristic
587C2#8395111yescharacteristic
588C2#8399811yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3141C2
socleC22×C3C2×#121
prime radical#83312C2
hypercenter#3141D3
hypocenterC3#19383
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
1. C3×#3141C2semidirect
3. #83214C2semidirect
4. #78511C2semidirect
5. #71808C2semidirect
6. #71773C2semidirect
7. #71242C2semidirect
8. #71240C2semidirect
9. #71234C2semidirect
10. #71232C2semidirect
11. #71211C2semidirect
12. #71209C2semidirect
13. #71203C2semidirect
14. #71201C2semidirect
15. #69851C2semidirect
70. #3141D3semidirect
582. C3#19383semidirect

Maximal subgroups

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

subgroupindex
C3×#31412
#833122
#832142
#785112
#718082
#717732
#712422
#712402
#712342
#712322
#712112
#712092
#712032
#712012
#698512
#193833