Finite Group Databasebeta

← #10778193

Subgroups — #10778193

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79302C211yesmaximal, characteristic, complemented
2#79021C211yesmaximal, characteristic
3#78533C211yesmaximal, characteristic, complemented
4#78512C211yesmaximal, characteristic, complemented
5C3×#1565C211yesmaximal, characteristic, complemented
6#70775C211yesmaximal, characteristic, complemented
7#70439C211yesmaximal, characteristic, complemented
8#68590C211yesmaximal, characteristic, complemented
9#68588C211yesmaximal, characteristic, complemented
10#68585C211yesmaximal, characteristic, complemented
11#68582C211yesmaximal, characteristic, complemented
12#68549C211yesmaximal, characteristic, complemented
13#68546C211yesmaximal, characteristic, complemented
14#68544C211yesmaximal, characteristic, complemented
15#68542C211yesmaximal, characteristic, complemented
16#19182—31nomaximal, Sylow
17C2×#904—21no—
18#5477C2211yescharacteristic
19C4×#235C2211yescharacteristic
20#5470C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C3×C4×M16C2211yescharacteristic
24#5134—21no—
25#5134—21no—
26#5128—21no—
27#5128—21no—
28#5113—21no—
29#4926C2211yescharacteristic
30#4851C2211yescharacteristic
31#4849C2211yescharacteristic
32#4849C2211yescharacteristic
33#4848C2211yescharacteristic
34#4846C2211yescharacteristic
35#4845—21no—
36#4844—21no—
37#4843C2211yescharacteristic
38#4841C2211yescharacteristic
39#4841C2211yescharacteristic
40#4733—21no—
41#4723—21no—
42#4710—21no—
43#4710—21no—
44#4707—21no—
45#4705—21no—
46C4×Dic12—21no—
47#4694C2211yescharacteristic
48#4693C2211yescharacteristic
49C4×#204—21no—
50#4688C2211yescharacteristic
51#4688C2211yescharacteristic
52#4682C2211yescharacteristic
53#4681C2211yescharacteristic
54C8×Dic6C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4482C2211yescharacteristic
65#4482C2211yescharacteristic
66#4482C2211yescharacteristic
67#4482C2211yescharacteristic
68#3182—31no—
69#3107—31no—
70#3035—31no—
71#2971—31no—
72#1701—31no—
73#1699—31no—
74#1693—31no—
75#1685—31no—
76#1682—31no—
77#1680—31no—
78#1674—31no—
79#1666—31no—
80#1565D311yescharacteristic, complemented
81#1553—31no—
82#1552—31no—
83C2×#235—21no—
84C2×#235D411yescharacteristic
85C2×C3×M16D411yescharacteristic
86C2×C3×M16D411yescharacteristic
87C22×C3×C8—21no—
88C2×C3×C42C2311yescharacteristic
89#930—21no—
90#930—21no—
91C4×#67—21no—
92C4×#67—21no—
93#926—21no—
94#926D411yescharacteristic
95#924—21no—
96#924—21no—
97C2×Dic12—21no—
98C2×Dic12—21no—
99#904—44no—
100C2×D24—21no—
101C2×#204—21no—
102C2×#204—21no—
103C2×#204—21no—
104C2×#204—21no—
105C2×#204—21no—
106C2×ΓL(1,𝔽25)—21no—
107C2×ΓL(1,𝔽25)—21no—
108#875—21no—
109#874C2311yescharacteristic
110C4×D12—21no—
111C4×D12—21no—
112C4×D12D411yescharacteristic
113C4×D12C2311yescharacteristic
114#872—21no—
115C42×D3—21no—
116#870—21no—
117#869C2311yescharacteristic
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C4×Dic6—21no—
121C4×Dic6—21no—
122C4×Dic6D411yescharacteristic
123C4×Dic6C2311yescharacteristic
124C3×C4⋊C8—21no—
125C3×C4⋊C8C2311yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×C4⋊C8C2311yescharacteristic
128C3×C4⋊C8C2311yescharacteristic
129C3×C22⋊C8—21no—
130C3×#97D411yescharacteristic
131C3×#97D411yescharacteristic
132C3×C4×C8—21no—
133C3×C4×C8C2311yescharacteristic
134C3×C4×C8C2311yescharacteristic
135#830—21no—
136#821—21no—
137#821—21no—
138#821—21no—
139#821—21no—
140#821—21no—
141(C2×D3)⋊C8—21no—
142(C2×D3)⋊C8—21no—
143#818—21no—
144#818—21no—
145#818—21no—
146#817—21no—
147#816—21no—
148#816—21no—
149#816—21no—
150#816—21no—
151#816—21no—
152#816—21no—
153#816—21no—
154Dic3⋊C8—21no—
155Dic3⋊C8—21no—
156C8×Dic3—21no—
157C8×Dic3—21no—
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160#803—21no—
161C2×#135—61no—
162#533—31no—
163#532—31no—
164C4×#41—31no—
165#500—31no—
166#492—31no—
167#487—31no—
168#486—31no—
169#485—61no—
170#475—31no—
171#474—31no—
172#468—61no—
173#468—61no—
174#464—61no—
175#463—61no—
176#462—31no—
177#462—31no—
178#461—31no—
179#461—31no—
180#460—31no—
181#459—31no—
182#452—61no—
183C8×Q8—31no—
184C8×Q8—31no—
185#440—61no—
186#440—61no—
187C4×Q16—61no—
188C4×SD16—61no—
189#436—31no—
190#436—31no—
191#436—61no—
192#436—61no—
193#432—31no—
194#432—61no—
195#431C2×D311yescharacteristic
196C2×C4⋊C8C2×D311yescharacteristic
197#416—61no—
198C4×M16—61no—
199C4×M16C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#335C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#334C2×D311yescharacteristic
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#331—31no—
211#331—31no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4#4111yescharacteristic
215C2×#67—21no—
216C2×#67—41no—
217#235—22no—
218#235—41no—
219#235—42no—
220#235—42no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×D411yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—41no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6C2×D411yescharacteristic
233C3×M16—22no—
234C3×M16—41no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—42no—
240C2×C3×C8C2×D411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2×D411yescharacteristic
243C2×C3×C8C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42#4111yescharacteristic
247C3×C42C2411yescharacteristic
248#217—21no—
249#217—41no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—41no—
253(C2×D3)⋊C4—41no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D411yescharacteristic
261C4⋊Dic3C2×D411yescharacteristic
262Dic3⋊C4—21no—
263Dic3⋊C4—21no—
264Dic3⋊C4—41no—
265Dic3⋊C4—41no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—41no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272Dic12—42no—
273Dic12—42no—
274D24—42no—
275#204—42no—
276#204—42no—
277#204—42no—
278#204—42no—
279#204—42no—
280ΓL(1,𝔽25)—42no—
281ΓL(1,𝔽25)—42no—
282C2×#41—31no—
283C2×#41—61no—
284#135—124no—
285C2×Q16—61no—
286C2×Q16—61no—
287C2×SD16—61no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—61no—
293C2×M16—61no—
294C2×M16—61no—
295C2×M16D1211yescharacteristic
296C2×M16D1211yescharacteristic
297C22×C8—21no—
298C4⋊Q8—31no—
299C4⋊Q8—31no—
300C4⋊Q8—31no—
301#127—31no—
302#125—61no—
303#124—61no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306#121—61no—
307#121—61no—
308C4×Q8—31no—
309C4×Q8—31no—
310C4×Q8—61no—
311C4×Q8—61no—
312C4×D4—31no—
313C4×D4—31no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317C4×D4—61no—
318#117—31no—
319#117—61no—
320#117—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—61no—
324#107—61no—
325#107—61no—
326#106—61no—
327C4⋊C8—21no—
328C4⋊C8—31no—
329C4⋊C8—31no—
330C4⋊C8—61no—
331C4⋊C8—61no—
332C4⋊C8C22×D311yescharacteristic
333C4⋊C8C22×D311yescharacteristic
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348C22⋊C8—21no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352#97—61no—
353#97D1211yescharacteristic
354#97D1211yescharacteristic
355C4×C8—21no—
356C4×C8—61no—
357C4×C8—61no—
358C4×C8C22×D311yescharacteristic
359C4×C8C22×D311yescharacteristic
360C23×C3#12111yescharacteristic
361C22×D3—21no—
362C22×D3—41no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4#12111yescharacteristic
369C2×C3×C4#12111yescharacteristic
370C2×C3×C4#12111yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374#67—42no—
375#67—82no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382D12—21no—
383D12—21no—
384D12—41no—
385D12—41no—
386D12—41no—
387D12—41no—
388D12—81no—
389C4×D3—42no—
390C4×D3—41no—
391C4×D3—41no—
392C4×D3—81no—
393C4×D3—82no—
394Dic6—21no—
395Dic6—41no—
396Dic6—41no—
397Dic6—41no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6AGL(1,ℤ/8)12yes—
402C3×C8—21no—
403C3×C8—21no—
404C3×C8—21no—
405C3×C8—41no—
406C3×C8—41no—
407C3×C8—41no—
408C3×C8—41no—
409C3×C8#13512yescomplemented
410C3⋊C8—41no—
411C3⋊C8—41no—
412#41—62no—
413#41—121no—
414#41—122no—
415#41—122no—
416C2×Q8—31no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×D4—31no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—121no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—121no—
430C22×C4C2×D1211yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4#23911yescharacteristic
433Q16—122no—
434Q16—122no—
435SD16—122no—
436SD16—122no—
437SD16—122no—
438SD16—122no—
439SD16—122no—
440D8—122no—
441M16—22no—
442M16—41no—
443M16—122no—
444M16—122no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—42no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8C2×D1211yescharacteristic
453C2×C8C2×D1211yescharacteristic
454C2×C8C2×D1211yescharacteristic
455C2×C8C2×D1211yescharacteristic
456C4⋊C4—31no—
457C4⋊C4—31no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—121no—
467C4⋊C4—121no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—121no—
472C22⋊C4—121no—
473C22⋊C4—121no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—121no—
478C42C2×D1211yescharacteristic
479C42D3×D411yescharacteristic
480C42#23911yescharacteristic
481C42C23×D311yescharacteristic
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3C2×#12111yescharacteristic
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4C2×#13511yescharacteristic
497C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
498C3×C4#57511yescharacteristic
499C3×C4#57511yescharacteristic
500Dic3—22no—
501Dic3—41no—
502Dic3—41no—
503Dic3—81no—
504Dic3—81no—
505Dic3—81no—
506C23—61no—
507C23—121no—
508C23#89511yescharacteristic
509Q8—32no—
510Q8—61no—
511Q8—121no—
512Q8—121no—
513Q8—121no—
514Q8—121no—
515Q8—121no—
516Q8—121no—
517D4—61no—
518D4—61no—
519D4—122no—
520D4—121no—
521D4—121no—
522D4—121no—
523D4—121no—
524D4—241no—
525D4—242no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—122no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—241no—
541C2×C4—242no—
542C2×C4#89511yescharacteristic
543C2×C4#89511yescharacteristic
544C2×C4#89511yescharacteristic
545C2×C4C22×D1211yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×#23911yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—121no—
556C8—121no—
557C8#90412yescomplemented
558C2×C3—41no—
559C2×C3#303511yescharacteristic
560C2×C3#305511yescharacteristic
561C2×C3#306911yescharacteristic
562D3—81no—
563D3—161no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—121no—
568C22—121no—
569C22—121no—
570C22—241no—
571C22—241no—
572C22—241no—
573C22C2×#89511yescharacteristic
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—62no—
579C4—121no—
580C4—121no—
581C4—241no—
582C4—241no—
583C4—241no—
584C4C2×#90411yescharacteristic
585C4#574811yescharacteristic
586C4D3×AGL(1,ℤ/8)11yescharacteristic
587C4#577711yescharacteristic
588C3#1918211yescharacteristic, complemented, Sylow
589C2—41no—
590C2—241no—
591C2—481no—
592C2#7902111yescharacteristic
593C2#7921111yescharacteristic
594C2#7926211yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1565C2
socleC22×C3C2×#121
prime radical#79021C2
hypercenter#1565D3
hypocenterC3#19182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
1. #79302C2semidirect
3. #78533C2semidirect
4. #78512C2semidirect
5. C3×#1565C2semidirect
6. #70775C2semidirect
7. #70439C2semidirect
8. #68590C2semidirect
9. #68588C2semidirect
10. #68585C2semidirect
11. #68582C2semidirect
12. #68549C2semidirect
13. #68546C2semidirect
14. #68544C2semidirect
15. #68542C2semidirect
80. #1565D3semidirect
409. C3×C8#135semidirect
557. C8#904semidirect
588. C3#19182semidirect

Maximal subgroups

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

subgroupindex
#793022
#790212
#785332
#785122
C3×#15652
#707752
#704392
#685902
#685882
#685852
#685822
#685492
#685462
#685442
#685422
#191823