Finite Group Databasebeta

← #10778192

Subgroups — #10778192

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#81936C211yesmaximal, characteristic, complemented
2#80475C211yesmaximal, characteristic
3#80457C211yesmaximal, characteristic, complemented
4#79905C211yesmaximal, characteristic, complemented
5C3×#1674C211yesmaximal, characteristic, complemented
6#71373C211yesmaximal, characteristic, complemented
7#71055C211yesmaximal, characteristic, complemented
8#71054C211yesmaximal, characteristic, complemented
9#70938C211yesmaximal, characteristic, complemented
10#68582C211yesmaximal, characteristic, complemented
11#68568C211yesmaximal, characteristic, complemented
12#68454C211yesmaximal, characteristic, complemented
13#68453C211yesmaximal, characteristic, complemented
14#68426C211yesmaximal, characteristic, complemented
15#68425C211yesmaximal, characteristic, complemented
16#19182—31nomaximal, Sylow
17C2×#919—21no—
18#5728C2211yescharacteristic
19#5725C2211yescharacteristic
20C4×#239C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#462C2211yescharacteristic
23C3×#462C2211yescharacteristic
24C3×C8×Q8C2211yescharacteristic
25#5187—21no—
26#5185—21no—
27Dic3×Q16—21no—
28#5093C2211yescharacteristic
29#5088C2211yescharacteristic
30#5083C2211yescharacteristic
31#5083C2211yescharacteristic
32#5027C2211yescharacteristic
33Q8×C3⋊C8C2211yescharacteristic
34#5020C2211yescharacteristic
35#4889—21no—
36#4889—21no—
37#4886—21no—
38#4886—21no—
39#4846C2211yescharacteristic
40#4843C2211yescharacteristic
41#4838—21no—
42#4838C2211yescharacteristic
43#4835—21no—
44#4835C2211yescharacteristic
45#4833C2211yescharacteristic
46D3×C4⋊C8C2211yescharacteristic
47#4819—21no—
48#4814—21no—
49#4804—21no—
50#4788—21no—
51#4693C2211yescharacteristic
52#4688—21no—
53#4688C2211yescharacteristic
54C4×ΓL(1,𝔽25)—21no—
55C4×ΓL(1,𝔽25)C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4547C2211yescharacteristic
61#4486C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4457C2211yescharacteristic
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#1674D311yescharacteristic, complemented
79#1666—31no—
80#1565—31no—
81#1553—31no—
82#1552—31no—
83C2×#239—21no—
84C2×#239D411yescharacteristic
85C2×C3×Q16—21no—
86C2×C3×Q16—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4×Q8C2311yescharacteristic
90#947—21no—
91#946—21no—
92Q8×Dic3—21no—
93Q8×Dic3D411yescharacteristic
94C2×#215—21no—
95C2×#215—21no—
96C2×#215—21no—
97#919—44no—
98C2×D24—21no—
99C2×#204—21no—
100C2×#204—21no—
101C2×ΓL(1,𝔽25)—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×ΓL(1,𝔽25)D411yescharacteristic
104C2×ΓL(1,𝔽25)D411yescharacteristic
105C2×D3×C8—21no—
106#897—21no—
107#895—21no—
108#895—21no—
109#893—21no—
110#893—21no—
111#893—21no—
112#893D411yescharacteristic
113#892—21no—
114#892—21no—
115#892D411yescharacteristic
116#890—21no—
117#887—21no—
118#874C2311yescharacteristic
119C4×D12—21no—
120C4×D12C2311yescharacteristic
121C42×D3—21no—
122C42×D3C2311yescharacteristic
123#869C2311yescharacteristic
124C3×#107—21no—
125C3×#107—21no—
126C3×C4⋊C8—21no—
127C3×C4⋊C8C2311yescharacteristic
128C3×C4⋊C8C2311yescharacteristic
129C3×C4⋊C8C2311yescharacteristic
130C3×#103—21no—
131C3×#103—21no—
132C3×C4×C8—21no—
133C3×C4×C8C2311yescharacteristic
134#835—21no—
135#835—21no—
136#835—21no—
137#821—21no—
138#821—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142(C2×D3)⋊C8—21no—
143#818—21no—
144#815—21no—
145#815D411yescharacteristic
146#815D411yescharacteristic
147Dic3⋊C8—21no—
148C8×Dic3—21no—
149#810—21no—
150#810—21no—
151#809—21no—
152#809—21no—
153#809—21no—
154#808—21no—
155C4⋊(C3⋊C8)—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8—21no—
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—61no—
162#533—31no—
163#532—31no—
164C4×#41—31no—
165#500C2×D311yescharacteristic
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#462C2×D311yescharacteristic
177#462C2×D311yescharacteristic
178#461—31no—
179#461—31no—
180#460—31no—
181#459—31no—
182#452—61no—
183C8×Q8—31no—
184C8×Q8C2×D311yescharacteristic
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#431—31no—
196C2×C4⋊C8—31no—
197#416—61no—
198C4×M16—31no—
199C4×M16—61no—
200#335—31no—
201#335C2×D311yescharacteristic
202#334—31no—
203#334—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332C2×D311yescharacteristic
209#332C2×D311yescharacteristic
210#331—31no—
211#331—31no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D411yescharacteristic
215#239—22no—
216#239—41no—
217#239—42no—
218#239—42no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—41no—
224C2×D12C2×D411yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—41no—
229C2×C4×D3C2×D411yescharacteristic
230C2×C4×D3C2×D411yescharacteristic
231C2×C4×D3#4111yescharacteristic
232C2×Dic6—21no—
233C3×Q16—42no—
234C3×Q16—42no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C42—21no—
247C3×C42C2411yescharacteristic
248#215—42no—
249#215—42no—
250#215—42no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—41no—
255(C2×D3)⋊C4—41no—
256(C2×D3)⋊C4—41no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—41no—
261C4⋊Dic3C2×D411yescharacteristic
262Dic3⋊C4—41no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—41no—
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268C4×Dic3#4111yescharacteristic
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274D24—42no—
275#204—42no—
276#204—42no—
277ΓL(1,𝔽25)—22no—
278ΓL(1,𝔽25)—41no—
279ΓL(1,𝔽25)—42no—
280ΓL(1,𝔽25)—42no—
281D3×C8—42no—
282C2×#41—31no—
283C2×#41—61no—
284#135—124no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×SD16—61no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—61no—
293C2×M16—31no—
294C2×M16—31no—
295C2×M16—61no—
296C2×M16—61no—
297C22×C8—61no—
298C4⋊Q8—31no—
299C4⋊Q8C22×D311yescharacteristic
300C4⋊Q8C22×D311yescharacteristic
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—61no—
310C4×Q8—61no—
311C4×Q8C22×D311yescharacteristic
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—31no—
322C2×C42—61no—
323#107—21no—
324#107—21no—
325#107—61no—
326#106—61no—
327C4⋊C8—21no—
328C4⋊C8—31no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—61no—
332C4⋊C8—61no—
333C4⋊C8C22×D311yescharacteristic
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336#103—21no—
337#103—21no—
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—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352#97—31no—
353#97—31no—
354#97—61no—
355C4×C8—21no—
356C4×C8—31no—
357C4×C8—61no—
358C4×C8—61no—
359C4×C8C22×D311yescharacteristic
360C22×D3—21no—
361C22×D3—41no—
362C22×D3#12111yescharacteristic
363C3×Q8—21no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8AGL(1,ℤ/8)12yes—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3#12111yescharacteristic
379C2×Dic3#12111yescharacteristic
380C2×Dic3#12111yescharacteristic
381D12—21no—
382D12—21no—
383D12—42no—
384D12—41no—
385D12—41no—
386D12—41no—
387D12—41no—
388D12—81no—
389D12—82no—
390C4×D3—21no—
391C4×D3—21no—
392C4×D3—21no—
393C4×D3—21no—
394C4×D3—41no—
395C4×D3—42no—
396C4×D3—41no—
397C4×D3—41no—
398C4×D3—81no—
399C4×D3—82no—
400Dic6—41no—
401Dic6—41no—
402C3×C8—21no—
403C3×C8—41no—
404C3×C8—41no—
405C3×C8—41no—
406C3×C8#13512yescomplemented
407C3⋊C8—21no—
408C3⋊C8—21no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412#41—62no—
413#41—121no—
414#41—122no—
415#41—122no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—61no—
419C2×Q8D3×D411yescharacteristic
420C2×D4—31no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—121no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—121no—
433Q16—42no—
434Q16—42no—
435SD16—122no—
436SD16—122no—
437SD16—122no—
438SD16—122no—
439SD16—122no—
440D8—122no—
441M16—62no—
442M16—121no—
443M16—122no—
444M16—122no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—31no—
449C2×C8—31no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—122no—
454C2×C8D3×D411yescharacteristic
455C2×C8D3×D411yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—31no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—121no—
466C4⋊C4—121no—
467C4⋊C4D3×D411yescharacteristic
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—121no—
472C22⋊C4—121no—
473C22⋊C4—121no—
474C42—21no—
475C42—31no—
476C42—31no—
477C42—31no—
478C42—61no—
479C42—61no—
480C42—121no—
481C42C23×D311yescharacteristic
482C22×C3C2×#12111yescharacteristic
483C2×D3—21no—
484C2×D3—21no—
485C2×D3—21no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—22no—
493C3×C4—21no—
494C3×C4—41no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4C2×#13511yescharacteristic
498C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C3×C4#57511yescharacteristic
500C3×C4#57511yescharacteristic
501Dic3—21no—
502Dic3—21no—
503Dic3—21no—
504Dic3—41no—
505Dic3—81no—
506C23—31no—
507C23—61no—
508C23—121no—
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—121no—
515Q8—121no—
516Q8#91112yes—
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—31no—
529C2×C4—31no—
530C2×C4—31no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—121no—
539C2×C4—122no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—241no—
544C2×C4—242no—
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C8—21no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—61no—
553C8—61no—
554C8—121no—
555C8—121no—
556C8—121no—
557C8#91912yescomplemented
558C2×C3#303511yescharacteristic
559C2×C3#305511yescharacteristic
560C2×C3#306911yescharacteristic
561D3—41no—
562D3—81no—
563D3—161no—
564C22—61no—
565C22—61no—
566C22—61no—
567C22—121no—
568C22—121no—
569C22—121no—
570C22—241no—
571C22—241no—
572C22—241no—
573C22D3×#12111yescharacteristic
574C4—21no—
575C4—22no—
576C4—41no—
577C4—61no—
578C4—61no—
579C4—61no—
580C4—81no—
581C4—81no—
582C4—121no—
583C4—241no—
584C4C2×#91111yescharacteristic
585C4C2×#91911yescharacteristic
586C4#576811yescharacteristic
587C4#577711yescharacteristic
588C3#1918211yescharacteristic, complemented, Sylow
589C2—121no—
590C2—241no—
591C2—481no—
592C2#8047511yescharacteristic
593C2#8071511yescharacteristic
594C2#8094411yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1674C2
socleC22×C3C2×#121
prime radical#80475C2
hypercenter#1674D3
hypocenterC3#19182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1674

Complements

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

NH ≅ G/Ntype
1. #81936C2semidirect
3. #80457C2semidirect
4. #79905C2semidirect
5. C3×#1674C2semidirect
6. #71373C2semidirect
7. #71055C2semidirect
8. #71054C2semidirect
9. #70938C2semidirect
10. #68582C2semidirect
11. #68568C2semidirect
12. #68454C2semidirect
13. #68453C2semidirect
14. #68426C2semidirect
15. #68425C2semidirect
78. #1674D3semidirect
406. C3×C8#135semidirect
557. C8#919semidirect
588. C3#19182semidirect

Maximal subgroups

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

subgroupindex
#819362
#804752
#804572
#799052
C3×#16742
#713732
#710552
#710542
#709382
#685822
#685682
#684542
#684532
#684262
#684252
#191823