Finite Group Databasebeta

← #10780326

Subgroups — #10780326

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81601C211yesmaximal, characteristic, complemented
2#81115C211yesmaximal, characteristic, complemented
3#81035C211yesmaximal, characteristic
4#79904C211yesmaximal, characteristic, complemented
5C3×#1684C211yesmaximal, characteristic, complemented
6#71235C211yesmaximal, characteristic, complemented
7#71154C211yesmaximal, characteristic, complemented
8#71123C211yesmaximal, characteristic, complemented
9#70940C211yesmaximal, characteristic, complemented
10#68584C211yesmaximal, characteristic, complemented
11#68571C211yesmaximal, characteristic, complemented
12#68498C211yesmaximal, characteristic, complemented
13#68483C211yesmaximal, characteristic, complemented
14#68447C211yesmaximal, characteristic, complemented
15#68426C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17#5725C2211yescharacteristic
18D3×C4⋊Q8C2211yescharacteristic
19#5580C2211yescharacteristic
20C3×#487C2211yescharacteristic
21C3×#477C2211yescharacteristic
22C3×#475C2211yescharacteristic
23C3×C8×Q8C2211yescharacteristic
24#5088C2211yescharacteristic
25#5087C2211yescharacteristic
26#5084C2211yescharacteristic
27#5079C2211yescharacteristic
28#5027C2211yescharacteristic
29#5024C2211yescharacteristic
30#5021C2211yescharacteristic
31#4882—21no—
32#4876—21no—
33#4871—21no—
34#4865—21no—
35#4843C2211yescharacteristic
36#4842C2211yescharacteristic
37#4840—21no—
38#4837C2211yescharacteristic
39#4836—21no—
40#4835C2211yescharacteristic
41#4834C2211yescharacteristic
42#4833C2211yescharacteristic
43#4819—21no—
44#4812—21no—
45#4805—21no—
46#4803—21no—
47#4803—21no—
48#4794—21no—
49#4794—21no—
50#4794—21no—
51#4694C2211yescharacteristic
52#4689—21no—
53#4688—21no—
54#4688C2211yescharacteristic
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#3251—31no—
69#3238—31no—
70#3141—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1698—31no—
75#1697—31no—
76#1684D311yescharacteristic, complemented
77#1683—31no—
78#1678—31no—
79#1677—31no—
80#1569—31no—
81#1553—31no—
82#1548—31no—
83C2×#239—21no—
84C2×D3×Q8—21no—
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4×Q8C2311yescharacteristic
88#947—21no—
89#947D411yescharacteristic
90#946D411yescharacteristic
91#944—21no—
92C2×#215—21no—
93C2×D24—21no—
94C2×ΓL(1,𝔽25)—21no—
95C2×ΓL(1,𝔽25)—21no—
96C2×ΓL(1,𝔽25)D411yescharacteristic
97C2×ΓL(1,𝔽25)D411yescharacteristic
98C2×D3×C8—21no—
99#898—21no—
100#895—21no—
101#895—21no—
102#895D411yescharacteristic
103#894—21no—
104#893—21no—
105#893—21no—
106D3×C4⋊C4—21no—
107D3×C4⋊C4—21no—
108#890D411yescharacteristic
109#888—21no—
110#888—21no—
111#874C2311yescharacteristic
112C4×D12—21no—
113C4×D12C2311yescharacteristic
114#872—21no—
115C42×D3C2311yescharacteristic
116#869C2311yescharacteristic
117C3×#107—21no—
118C3×#106—21no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×#103—21no—
124C3×#103—21no—
125C3×#103—21no—
126C3×#103—21no—
127C3×C4×C8—21no—
128C3×C4×C8C2311yescharacteristic
129#835—21no—
130#835—21no—
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#809—21no—
150#809—21no—
151#808—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158#803—21no—
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#489—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487C2×D311yescharacteristic
169#486—31no—
170#486—31no—
171#483—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477C2×D311yescharacteristic
176#476—31no—
177#476—31no—
178#475C2×D311yescharacteristic
179#474—31no—
180#469—61no—
181#468—61no—
182#463—61no—
183#460—31no—
184#459—31no—
185#446—31no—
186C8×Q8C2×D311yescharacteristic
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—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
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—42no—
215#239—42no—
216D3×Q8—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—41no—
225C2×C4×D3—41no—
226C2×C4×D3C2×D411yescharacteristic
227C2×C4×D3C2×D411yescharacteristic
228C2×C4×D3#4111yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—41no—
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—81no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
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—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260C4×Dic3C2×D411yescharacteristic
261C4×Dic3C2×D411yescharacteristic
262C4×Dic3#4111yescharacteristic
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C2×D411yescharacteristic
267C2×C3⋊C8C2×D411yescharacteristic
268D24—81no—
269ΓL(1,𝔽25)—41no—
270ΓL(1,𝔽25)—41no—
271ΓL(1,𝔽25)—42no—
272ΓL(1,𝔽25)—81no—
273D3×C8—42no—
274C2×#41—61no—
275C22×Q8—61no—
276C2×SD16—61no—
277C2×D8—61no—
278C2×M16—31no—
279C2×M16—31no—
280C2×M16—61no—
281C2×M16—61no—
282C22×C8—61no—
283C4⋊Q8—31no—
284C4⋊Q8—61no—
285C4⋊Q8—61no—
286C4⋊Q8—61no—
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
289#127—31no—
290#126—61no—
291#125—31no—
292#124—31no—
293#124—61no—
294#123—61no—
295C22⋊Q8—31no—
296#121—31no—
297#121—61no—
298#121—61no—
299C4×Q8C22×D311yescharacteristic
300C4×D4—31no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4—61no—
304#117—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C42—31no—
308#107—21no—
309#107—61no—
310#107—61no—
311#106—21no—
312#106—61no—
313#106—61no—
314C4⋊C8—21no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8C22×D311yescharacteristic
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—21no—
324#103—21no—
325#103—21no—
326#103—21no—
327#103—61no—
328#103—61no—
329#103—61no—
330#102—61no—
331#102—61no—
332#102—61no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—31no—
344#97—31no—
345#97—61no—
346#97—61no—
347C4×C8—21no—
348C4×C8—31no—
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C22×D3—21no—
352C22×D3—41no—
353C22×D3#12111yescharacteristic
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—81no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—41no—
368C2×Dic3#12111yescharacteristic
369C2×Dic3#12111yescharacteristic
370C2×Dic3#12111yescharacteristic
371D12—21no—
372D12—21no—
373D12—41no—
374D12—41no—
375D12—41no—
376D12—41no—
377D12—81no—
378D12—81no—
379D12—82no—
380C4×D3—21no—
381C4×D3—21no—
382C4×D3—21no—
383C4×D3—21no—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—82no—
388C4×D3—82no—
389Dic6—82no—
390Dic6—81no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—122no—
402#41—122no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—61no—
406C2×Q8—121no—
407C2×Q8—242no—
408C2×Q8D3×D411yescharacteristic
409C2×D4—31no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—121no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—61no—
419C22×C4—121no—
420C22×C4—121no—
421SD16—241no—
422D8—241no—
423M16—121no—
424M16—121no—
425M16—122no—
426M16—241no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—122no—
436C2×C8D3×D411yescharacteristic
437C2×C8D3×D411yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—31no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—121no—
450C4⋊C4—121no—
451C4⋊C4D3×D411yescharacteristic
452C22⋊C4—61no—
453C22⋊C4—61no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C42—21no—
460C42—31no—
461C42—31no—
462C42—31no—
463C42C23×D311yescharacteristic
464C22×C3C2×#12111yescharacteristic
465C2×D3—21no—
466C2×D3—21no—
467C2×D3—21no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—81no—
472C2×D3—81no—
473C2×D3—81no—
474C3×C4—21no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4#57511yescharacteristic
480C3×C4#57511yescharacteristic
481C3×C4#57711yescharacteristic
482C3×C4#57811yescharacteristic
483Dic3—21no—
484Dic3—21no—
485Dic3—21no—
486Dic3—41no—
487Dic3—81no—
488C23—31no—
489C23—61no—
490C23—121no—
491Q8—21no—
492Q8—21no—
493Q8—41no—
494Q8—41no—
495Q8—81no—
496Q8—241no—
497Q8—242no—
498D4—61no—
499D4—61no—
500D4—121no—
501D4—121no—
502D4—121no—
503D4—121no—
504D4—241no—
505D4—241no—
506D4—242no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—31no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—242no—
524C2×C4—242no—
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4D3×#4111yescharacteristic
528C8—21no—
529C8—21no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—61no—
534C8—61no—
535C8—121no—
536C8—121no—
537C8—121no—
538C2×C3#304011yescharacteristic
539C2×C3#306311yescharacteristic
540C2×C3#306811yescharacteristic
541D3—41no—
542D3—81no—
543D3—161no—
544C22—61no—
545C22—61no—
546C22—61no—
547C22—121no—
548C22—121no—
549C22—121no—
550C22—241no—
551C22—241no—
552C22—241no—
553C22D3×#12111yescharacteristic
554C4—21no—
555C4—41no—
556C4—41no—
557C4—61no—
558C4—61no—
559C4—61no—
560C4—81no—
561C4—81no—
562C4—121no—
563C4—241no—
564C4#576611yescharacteristic
565C4#576811yescharacteristic
566C4#577011yescharacteristic
567C4#577711yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—121no—
570C2—241no—
571C2—481no—
572C2#8059911yescharacteristic
573C2#8076211yescharacteristic
574C2#8093011yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1684C2
socleC22×C3C2×#121
prime radical#81035C2
hypercenter#1684D3
hypocenterC3#19382
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. #81601C2semidirect
2. #81115C2semidirect
4. #79904C2semidirect
5. C3×#1684C2semidirect
6. #71235C2semidirect
7. #71154C2semidirect
8. #71123C2semidirect
9. #70940C2semidirect
10. #68584C2semidirect
11. #68571C2semidirect
12. #68498C2semidirect
13. #68483C2semidirect
14. #68447C2semidirect
15. #68426C2semidirect
76. #1684D3semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
#816012
#811152
#810352
#799042
C3×#16842
#712352
#711542
#711232
#709402
#685842
#685712
#684982
#684832
#684472
#684262
#193823