Finite Group Databasebeta

← #10765772

Subgroups — #10765772

Order 768.

662 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79903C211yesmaximal, characteristic, complemented
2#79664C211yesmaximal, characteristic
3#79422C211yesmaximal, characteristic, complemented
4C3×C8×SD16C211yesmaximal, characteristic, complemented
5C8×#215C211yesmaximal, characteristic, complemented
6#70900C211yesmaximal, characteristic, complemented
7C8×#213C211yesmaximal, characteristic, complemented
8#68531C211yesmaximal, characteristic, complemented
9#68525C211yesmaximal, characteristic, complemented
10#68242C211yesmaximal, characteristic, complemented
11#68226C211yesmaximal, characteristic, complemented
12#68179C211yesmaximal, characteristic, complemented
13#68163C211yesmaximal, characteristic, complemented
14C8×D24C211yesmaximal, characteristic, complemented
15C8×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
16#17964—31nomaximal, Sylow
17C2×#901—21no—
18#5580C2211yescharacteristic
19#5548C2211yescharacteristic
20C3×C8×Q8C2211yescharacteristic
21C3×C4×SD16C2211yescharacteristic
22C3×C8×D4C2211yescharacteristic
23#5176C411yescharacteristic
24#5174C411yescharacteristic
25C8×#67—21no—
26C4×#215C2211yescharacteristic
27#5024C2211yescharacteristic
28C4×#213C2211yescharacteristic
29#5012C2211yescharacteristic
30#4865C411yescharacteristic
31#4856C411yescharacteristic
32#4840—21no—
33#4838C2211yescharacteristic
34#4838C2211yescharacteristic
35#4837C2211yescharacteristic
36#4836—21no—
37#4835C2211yescharacteristic
38#4812C411yescharacteristic
39#4794C411yescharacteristic
40#4782C411yescharacteristic
41#4764C411yescharacteristic
42#4729—21no—
43#4727—21no—
44C4×D24C2211yescharacteristic
45#4689—21no—
46C4×ΓL(1,𝔽25)C2211yescharacteristic
47C8×D12—21no—
48C8×D12C2211yescharacteristic
49C8×D12C2211yescharacteristic
50#4685C2211yescharacteristic
51C4×D3×C8C2211yescharacteristic
52C3×#334C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×C82C2211yescharacteristic
56#4535C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4483C2211yescharacteristic
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic
61#4479C2211yescharacteristic
62#4459C2211yescharacteristic
63#4459C2211yescharacteristic
64#4458C2211yescharacteristic
65#4455C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3171—31no—
69#2966—31no—
70#2931—31no—
71#1594—31no—
72#1594—31no—
73#1592—31no—
74#1592—31no—
75C8×SD16—31no—
76C8×SD16D311yescharacteristic, complemented
77C8×D8—31no—
78C8×D8—31no—
79#1569—31no—
80#1496—31no—
81#1495—31no—
82C8×M16—31no—
83C2×#235—21no—
84C2×C3×SD16C2×C411yescharacteristic
85C22×C3×C8—21no—
86C3×C4×Q8C2311yescharacteristic
87C3×C4×D4C2311yescharacteristic
88#947C2×C411yescharacteristic
89#946C2×C411yescharacteristic
90C2×#215C2×C411yescharacteristic
91#940C2×C411yescharacteristic
92#938C2×C411yescharacteristic
93C2×#213C2×C411yescharacteristic
94C4×#67—21no—
95C2×#208—21no—
96C2×D24C2×C411yescharacteristic
97#901—44no—
98C2×ΓL(1,𝔽25)—21no—
99C2×ΓL(1,𝔽25)—21no—
100C2×ΓL(1,𝔽25)—21no—
101C2×ΓL(1,𝔽25)C2×C411yescharacteristic
102C2×ΓL(1,𝔽25)C2×C411yescharacteristic
103C2×D3×C8—21no—
104C2×D3×C8—21no—
105C2×D3×C8D411yescharacteristic
106C2×D3×C8D411yescharacteristic
107#898—21no—
108#895C2×C411yescharacteristic
109#895C2×C411yescharacteristic
110#894—21no—
111#890C2×C411yescharacteristic
112#888C2×C411yescharacteristic
113#885—21no—
114#884—21no—
115C4×D12—21no—
116C4×D12C2311yescharacteristic
117C4×D12C2311yescharacteristic
118#872—21no—
119C42×D3C2311yescharacteristic
120C3×#106C2×C411yescharacteristic
121C3×C4⋊C8—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#103C2×C411yescharacteristic
125C3×#102C2×C411yescharacteristic
126C3×C22⋊C8—21no—
127C3×C4×C8—21no—
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130C3×C4×C8C2311yescharacteristic
131#835C2×C411yescharacteristic
132#832C2×C411yescharacteristic
133#830—21no—
134#821C2×C411yescharacteristic
135#821C2×C411yescharacteristic
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142#818C2×C411yescharacteristic
143#815—21no—
144#815C2×C411yescharacteristic
145#815C2×C411yescharacteristic
146Dic3⋊C8—21no—
147Dic3⋊C8—21no—
148C8×Dic3D411yescharacteristic
149C8×Dic3D411yescharacteristic
150#809C2×C411yescharacteristic
151#809C2×C411yescharacteristic
152#808C2×C411yescharacteristic
153#807C2×C411yescharacteristic
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157#803—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#131—61no—
162#532—31no—
163#529—31no—
164#501—31no—
165#496—31no—
166#490—31no—
167#489—31no—
168#469—31no—
169#469—31no—
170#468—31no—
171#468—31no—
172#446—31no—
173C8×Q8C2×D311yescharacteristic
174C4×SD16—31no—
175C4×SD16C2×D311yescharacteristic
176C4×D8—31no—
177C4×D8—31no—
178#436—31no—
179#436—31no—
180#436—31no—
181#436—61no—
182#435—61no—
183C8×D4—31no—
184C8×D4—31no—
185C8×D4—61no—
186C8×D4—61no—
187C8×D4C2×D311yescharacteristic
188#433—61no—
189#432—31no—
190#431—31no—
191#423—31no—
192#408—61no—
193#405—61no—
194C4×M16—31no—
195C2×C4×C8—31no—
196#335—31no—
197#335—31no—
198#334—31no—
199#334C2×D311yescharacteristic
200Q8⋊C8—31no—
201Q8⋊C8C2×D311yescharacteristic
202D4⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8—31no—
207D4⋊C8C2×D311yescharacteristic
208#322—31no—
209#322—31no—
210C82—31no—
211C82C2×D311yescharacteristic
212C2×C3×Q8C22×C411yescharacteristic
213C2×C3×D4C22×C411yescharacteristic
214C22×C3×C4—21no—
215C2×#67—21no—
216C2×#67—21no—
217C2×#67—21no—
218#235—42no—
219#235—42no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C22×C411yescharacteristic
223C2×D12C22×C411yescharacteristic
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3C22×C411yescharacteristic
227C2×C4×D3C2×D411yescharacteristic
228C2×C4×D3#4111yescharacteristic
229C2×Dic6—21no—
230C3×SD16—22no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—42no—
235C2×C3×C8C22×C411yescharacteristic
236C2×C3×C8C22×C411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C4⋊C4—21no—
240C3×C4⋊C4C22×C411yescharacteristic
241C3×C4⋊C4C22×C411yescharacteristic
242C3×C22⋊C4—21no—
243C3×C42—21no—
244C3×C42C2411yescharacteristic
245#217—21no—
246#215—22no—
247#213—22no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3C22×C411yescharacteristic
257C4⋊Dic3C22×C411yescharacteristic
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261C4×Dic3C22×C411yescharacteristic
262C4×Dic3C2×D411yescharacteristic
263C4×Dic3#4111yescharacteristic
264#208—42no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8C22×C411yescharacteristic
269C2×C3⋊C8C22×C411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272D24—22no—
273ΓL(1,𝔽25)—22no—
274ΓL(1,𝔽25)—22no—
275ΓL(1,𝔽25)—42no—
276ΓL(1,𝔽25)—42no—
277ΓL(1,𝔽25)—42no—
278D3×C8—22no—
279D3×C8—41no—
280D3×C8—42no—
281D3×C8—42no—
282C2×#41—61no—
283C2×SD16—31no—
284C2×SD16C4×D311yescharacteristic
285C2×D8—31no—
286C2×D8—31no—
287#131—124no—
288C2×M16—31no—
289C2×M16—31no—
290C2×M16—61no—
291C2×M16—61no—
292C2×M16—61no—
293C2×M16—61no—
294C22×C8—21no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8—61no—
298C22×C8—61no—
299C4⋊Q8—31no—
300#127—31no—
301#126—61no—
302#125—31no—
303#124—31no—
304#124—61no—
305#123—61no—
306C22⋊Q8—31no—
307#121—31no—
308#121—31no—
309#121—31no—
310#121—61no—
311C4×Q8C22×D311yescharacteristic
312C4×D4—31no—
313C4×D4—31no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4C22×D311yescharacteristic
317#117—61no—
318C2×C42—31no—
319#107—31no—
320#107—31no—
321#106—31no—
322#106C4×D311yescharacteristic
323C4⋊C8—21no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8—61no—
327C4⋊C8—61no—
328C4⋊C8—61no—
329C4⋊C8C22×D311yescharacteristic
330C4⋊C8C22×D311yescharacteristic
331#103—31no—
332#103C4×D311yescharacteristic
333#102—31no—
334#102—31no—
335#102—31no—
336#102—31no—
337#102—31no—
338#102C4×D311yescharacteristic
339C22⋊C8—21no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347#97—31no—
348#97—31no—
349#97—61no—
350#97—61no—
351C4×C8—21no—
352C4×C8—31no—
353C4×C8—31no—
354C4×C8—31no—
355C4×C8—31no—
356C4×C8—31no—
357C4×C8C22×D311yescharacteristic
358C4×C8C22×D311yescharacteristic
359C4×C8C22×D311yescharacteristic
360C23×C3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3C4×D411yescharacteristic
364C3×Q8—21no—
365C3×Q8—21no—
366C3×D4—21no—
367C3×D4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4C23×C411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376#67—42no—
377#67—42no—
378#67—42no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3C4×D411yescharacteristic
382C2×Dic3C4×D411yescharacteristic
383C2×Dic3C4×D411yescharacteristic
384D12—21no—
385D12—21no—
386D12—21no—
387D12—21no—
388D12—42no—
389D12—42no—
390C4×D3—21no—
391C4×D3—21no—
392C4×D3—21no—
393C4×D3—21no—
394C4×D3—41no—
395C4×D3—41no—
396C4×D3—41no—
397C4×D3—41no—
398C4×D3—41no—
399Dic6—42no—
400C3×C8—21no—
401C3×C8—21no—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3×C8#13112yescomplemented
406C3×C8AGL(1,ℤ/8)12yes—
407C3⋊C8—21no—
408C3⋊C8—21no—
409C3⋊C8—21no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C3⋊C8#13112yescomplemented
414#41—122no—
415#41—122no—
416C2×Q8—61no—
417C2×Q8C2×C4×D311yescharacteristic
418C2×D4—31no—
419C2×D4—31no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4C2×C4×D311yescharacteristic
426C22×C4—21no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—61no—
431C22×C4—61no—
432SD16—22no—
433SD16—62no—
434D8—62no—
435D8—62no—
436M16—62no—
437M16—62no—
438M16—122no—
439M16—122no—
440M16—122no—
441M16—122no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—31no—
446C2×C8—31no—
447C2×C8—31no—
448C2×C8—31no—
449C2×C8—42no—
450C2×C8—62no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—121no—
455C2×C8—122no—
456C2×C8—122no—
457C2×C8C2×C4×D311yescharacteristic
458C2×C8C2×C4×D311yescharacteristic
459C2×C8D3×D411yescharacteristic
460C2×C8D3×D411yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—31no—
463C4⋊C4—31no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4C2×C4×D311yescharacteristic
469C4⋊C4C2×C4×D311yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—61no—
477C22⋊C4—61no—
478C22⋊C4—61no—
479C42—21no—
480C42—31no—
481C42—31no—
482C42—31no—
483C42C23×D311yescharacteristic
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3C2×C4×D411yescharacteristic
488C2×D3—21no—
489C2×D3—21no—
490C2×D3—21no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—41no—
496C2×D3—41no—
497C3×C4—21no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4C2×#13111yescharacteristic
502C3×C4C2×#13111yescharacteristic
503C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
504C3×C4#57511yescharacteristic
505Dic3—21no—
506Dic3—21no—
507Dic3—21no—
508Dic3—41no—
509Dic3—41no—
510C23—21no—
511C23—31no—
512C23—61no—
513C23—61no—
514Q8—21no—
515Q8—21no—
516Q8—122no—
517D4—21no—
518D4—21no—
519D4—61no—
520D4—61no—
521D4—61no—
522D4—61no—
523D4—122no—
524D4—122no—
525D4—122no—
526D4—122no—
527D4—122no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—31no—
532C2×C4—31no—
533C2×C4—31no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4C22×C4×D311yescharacteristic
548C2×C4C2×D3×D411yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C8—21no—
551C8—21no—
552C8—32no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—61no—
557C8—61no—
558C8—61no—
559C8—121no—
560C8—121no—
561C8—121no—
562C8#90112yescomplemented
563C8#91412yes—
564C2×C3—81no—
565C2×C3#293111yescharacteristic
566C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
567C2×C3#295511yescharacteristic
568D3—41no—
569D3—81no—
570D3—81no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—61no—
575C22—61no—
576C22—61no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22C4×D3×D411yescharacteristic
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—61no—
589C4—61no—
590C4—61no—
591C4—121no—
592C4—121no—
593C4C2×#90111yescharacteristic
594C4D3×#13111yescharacteristic
595C4C2×#91411yescharacteristic
596C4#576811yescharacteristic
597C3#1796411yescharacteristic, complemented, Sylow
598C2—81no—
599C2—121no—
600C2—241no—
601C2—241no—
602C2#7942211yescharacteristic
603C2C4×#91411yescharacteristic
604C2#7988811yescharacteristic
subgroupG/H
centerC2×C8D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×C8×SD16C2
socleC22×C3C2×C4×D4
prime radical#79664C2
cosocleC3×C42C24
hypercenterC8×SD16D3
hypocenterC3#17964
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
1. #79903C2semidirect
3. #79422C2semidirect
4. C3×C8×SD16C2semidirect
5. C8×#215C2semidirect
6. #70900C2semidirect
7. C8×#213C2semidirect
8. #68531C2semidirect
9. #68525C2semidirect
10. #68242C2semidirect
11. #68226C2semidirect
12. #68179C2semidirect
13. #68163C2semidirect
14. C8×D24C2semidirect
15. C8×ΓL(1,𝔽25)C2semidirect
76. C8×SD16D3semidirect
405. C3×C8#131semidirect
413. C3⋊C8#131semidirect
562. C8#901semidirect
597. C3#17964semidirect

Maximal subgroups

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

subgroupindex
#799032
#796642
#794222
C3×C8×SD162
C8×#2152
#709002
C8×#2132
#685312
#685252
#682422
#682262
#681792
#681632
C8×D242
C8×ΓL(1,𝔽25)2
#179643