Finite Group Databasebeta

← #10764392

Subgroups — #10764392

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79339C211yesmaximal, characteristic
2#79292C211yesmaximal, characteristic
3#79283C211yesmaximal, characteristic
4C3×#1572C211yesmaximal, characteristic, complemented
5#70797C211yesmaximal, characteristic, complemented
6#70749C211yesmaximal, characteristic, complemented
7#70748C211yesmaximal, characteristic, complemented
8#68575C211yesmaximal, characteristic, complemented
9#68574C211yesmaximal, characteristic, complemented
10#68571C211yesmaximal, characteristic, complemented
11#68571C211yesmaximal, characteristic, complemented
12#68533C211yesmaximal, characteristic, complemented
13#68532C211yesmaximal, characteristic, complemented
14#68530C211yesmaximal, characteristic, complemented
15#68529C211yesmaximal, characteristic, complemented
16#17824—31nomaximal, Sylow
17C4×#235C2211yescharacteristic
18C3×#432C2211yescharacteristic
19C3×#431C2211yescharacteristic
20C3×#431C2211yescharacteristic
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4840C412yes—
25#4839C412yes—
26#4837C2211yescharacteristic
27#4837C2211yescharacteristic
28#4837C2211yescharacteristic
29#4835C2211yescharacteristic
30#4835C2211yescharacteristic
31#4835C2211yescharacteristic
32#4833C2211yescharacteristic
33D3×C4⋊C8C2211yescharacteristic
34#4730—21no—
35#4730—21no—
36#4729—21no—
37#4728—21no—
38#4727C412yes—
39#4720C412yes—
40#4706—21no—
41#4701—21no—
42#4688—21no—
43C4×ΓL(1,𝔽25)C2211yescharacteristic
44#4685C2211yescharacteristic
45#4685C2211yescharacteristic
46C4×D3×C8C2211yescharacteristic
47#4679—21no—
48C3×#335C2211yescharacteristic
49C3×#335C2211yescharacteristic
50C3×#334C2211yescharacteristic
51C3×#334C2211yescharacteristic
52#4480C2211yescharacteristic
53#4480C2211yescharacteristic
54#4480C2211yescharacteristic
55#4480C2211yescharacteristic
56#4479C2211yescharacteristic
57#4479C2211yescharacteristic
58#4479C2211yescharacteristic
59#4479C2211yescharacteristic
60#4457C2211yescharacteristic
61#4457C2211yescharacteristic
62#4457C2211yescharacteristic
63#4457C2211yescharacteristic
64#2989—31no—
65#2989—31no—
66#2922—31no—
67#1574—31no—
68#1574—31no—
69#1573—31no—
70#1573—31no—
71#1572—31no—
72#1572D311yescharacteristic, complemented
73#1570—31no—
74#1570—31no—
75#1569—31no—
76#1569—31no—
77#1568—31no—
78#1566—31no—
79C2×#235C2×C411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C4×#67D411yescharacteristic
82C4×#67D411yescharacteristic
83C4×#67Q811yescharacteristic
84C4×#67Q811yescharacteristic
85#926C2×C411yescharacteristic
86C2×#208—21no—
87C22×C3⋊C8—21no—
88C2×ΓL(1,𝔽25)—21no—
89C2×ΓL(1,𝔽25)—21no—
90C2×ΓL(1,𝔽25)—21no—
91C2×ΓL(1,𝔽25)C2×C412yes—
92C2×D3×C8—21no—
93C2×D3×C8C2×C412yes—
94C4×D12D411yescharacteristic
95C4×D12Q811yescharacteristic
96#872C2×C411yescharacteristic
97#872C2×C411yescharacteristic
98C42×D3C2311yescharacteristic
99C42×D3C2311yescharacteristic
100C4×Dic6D411yescharacteristic
101C4×Dic6Q811yescharacteristic
102C3×C4⋊C8C2×C412yes—
103C3×C4⋊C8C2311yescharacteristic
104C3×C4⋊C8C2311yescharacteristic
105C3×C4⋊C8C2311yescharacteristic
106C3×C4⋊C8C2311yescharacteristic
107C3×C22⋊C8—21no—
108C3×C22⋊C8—21no—
109C3×C22⋊C8C2×C412yes—
110C3×#97—21no—
111C3×C4×C8—21no—
112C3×C4×C8C2311yescharacteristic
113C3×C4×C8C2311yescharacteristic
114#830C2×C412yes—
115(C2×D3)⋊C8—21no—
116(C2×D3)⋊C8—21no—
117(C2×D3)⋊C8—21no—
118(C2×D3)⋊C8—21no—
119(C2×D3)⋊C8C2×C412yes—
120(C2×D3)⋊C8C2×C412yes—
121#815—21no—
122#815—21no—
123#815—21no—
124#815C2×C412yes—
125Dic3⋊C8—21no—
126Dic3⋊C8—21no—
127Dic3⋊C8—21no—
128Dic3⋊C8—21no—
129Dic3⋊C8C2×C412yes—
130Dic3⋊C8C2×C412yes—
131C8×Dic3—21no—
132C8×Dic3C2×C412yes—
133C4⋊(C3⋊C8)—21no—
134C4⋊(C3⋊C8)—21no—
135C4⋊(C3⋊C8)C2×C412yes—
136C4⋊(C3⋊C8)C2311yescharacteristic
137C4⋊(C3⋊C8)C2311yescharacteristic
138C4⋊(C3⋊C8)C2311yescharacteristic
139C4⋊(C3⋊C8)C2311yescharacteristic
140C4×C3⋊C8C2311yescharacteristic
141C4×C3⋊C8C2311yescharacteristic
142C4×#41—31no—
143#446—61no—
144#446—61no—
145#436—61no—
146#436—61no—
147#435—61no—
148#435—61no—
149#435—61no—
150#435—61no—
151#433—32no—
152#433—32no—
153#432—31no—
154#432—31no—
155#432—31no—
156#432C2×D311yescharacteristic
157#431—31no—
158#431—31no—
159#431—31no—
160#431—31no—
161#431C2×D311yescharacteristic
162#431C2×D311yescharacteristic
163#424—32no—
164#423—31no—
165#423—31no—
166#423—31no—
167#423—31no—
168C2×C4⋊C8—31no—
169C2×C4⋊C8—31no—
170#408—32no—
171C4×M16—31no—
172C2×C4×C8—31no—
173#335—31no—
174#335—31no—
175#335—31no—
176#335—31no—
177#335C2×D311yescharacteristic
178#335C2×D311yescharacteristic
179#334—31no—
180#334—31no—
181#334—31no—
182#334—31no—
183#334—31no—
184#334—31no—
185#334—31no—
186#334—31no—
187#334C2×D311yescharacteristic
188#334C2×D311yescharacteristic
189C22×C3×C4C22×C411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×Q811yescharacteristic
192C2×#67C4⋊C411yescharacteristic
193C2×#67C4⋊C411yescharacteristic
194#235—42no—
195C2×D12C4⋊C411yescharacteristic
196C2×C4×D3C22×C411yescharacteristic
197C2×C4×D3C22×C411yescharacteristic
198C2×C4×D3C2×D411yescharacteristic
199C2×C4×D3C2×D411yescharacteristic
200C2×C4×D3C2×Q811yescharacteristic
201C2×C4×D3C2×Q811yescharacteristic
202C2×Dic6C4⋊C411yescharacteristic
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8C22×C412yes—
208C2×C3×C8C22×C412yes—
209C3×C42C22×C411yescharacteristic
210C3×C42C2×D411yescharacteristic
211C3×C42C2×Q811yescharacteristic
212C3×C42C2411yescharacteristic
213#217C4⋊C411yescharacteristic
214#217C4⋊C411yescharacteristic
215(C2×D3)⋊C4C4⋊C411yescharacteristic
216(C2×D3)⋊C4C4⋊C411yescharacteristic
217(C2×D3)⋊C4C4⋊C411yescharacteristic
218(C2×D3)⋊C4C4⋊C411yescharacteristic
219C4⋊Dic3C4⋊C411yescharacteristic
220C4⋊Dic3C4⋊C411yescharacteristic
221Dic3⋊C4C4⋊C411yescharacteristic
222Dic3⋊C4C4⋊C411yescharacteristic
223Dic3⋊C4C4⋊C411yescharacteristic
224Dic3⋊C4C4⋊C411yescharacteristic
225C4×Dic3C22×C411yescharacteristic
226C4×Dic3C22×C411yescharacteristic
227C4×Dic3C2×D411yescharacteristic
228C4×Dic3C2×D411yescharacteristic
229C4×Dic3C2×Q811yescharacteristic
230C4×Dic3C2×Q811yescharacteristic
231#208—81no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—81no—
237C2×C3⋊C8C22×C412yes—
238C2×C3⋊C8C22×C412yes—
239ΓL(1,𝔽25)—42no—
240ΓL(1,𝔽25)—81no—
241ΓL(1,𝔽25)—81no—
242ΓL(1,𝔽25)—81no—
243D3×C8—42no—
244D3×C8—81no—
245C2×#41—31no—
246C2×M16—32no—
247C2×M16—61no—
248C2×M16—61no—
249C2×M16—61no—
250C2×M16—61no—
251C22×C8—32no—
252C22×C8—61no—
253C22×C8—61no—
254C4×Q8—31no—
255C4×Q8—31no—
256C4×D4—31no—
257C4×D4—31no—
258C4×D4—31no—
259C4×D4—31no—
260C4×D4—31no—
261C4×D4—31no—
262#117—31no—
263#117—31no—
264#117—31no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42C22×D311yescharacteristic
268C4⋊C8—32no—
269C4⋊C8—32no—
270C4⋊C8—32no—
271C4⋊C8—31no—
272C4⋊C8—31no—
273C4⋊C8—31no—
274C4⋊C8—31no—
275C4⋊C8—61no—
276C4⋊C8—61no—
277C4⋊C8—61no—
278C4⋊C8—61no—
279C4⋊C8—61no—
280C4⋊C8—61no—
281C4⋊C8C4×D312yes—
282C4⋊C8C22×D311yescharacteristic
283C4⋊C8C22×D311yescharacteristic
284C4⋊C8C22×D311yescharacteristic
285C4⋊C8C22×D311yescharacteristic
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—32no—
289C22⋊C8—32no—
290C22⋊C8—32no—
291C22⋊C8—61no—
292C22⋊C8—61no—
293C22⋊C8—61no—
294C22⋊C8—61no—
295C22⋊C8C4×D312yes—
296#97—21no—
297#97—32no—
298#97—61no—
299#97—61no—
300#97—61no—
301C4×C8—21no—
302C4×C8—32no—
303C4×C8—31no—
304C4×C8—31no—
305C4×C8—61no—
306C4×C8C22×D311yescharacteristic
307C4×C8C22×D311yescharacteristic
308C23×C3C2×C4⋊C411yescharacteristic
309C22×D3C2×C4⋊C411yescharacteristic
310C22×D3C2×C4⋊C411yescharacteristic
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4C2×C4⋊C411yescharacteristic
316C2×C3×C4C2×C4⋊C411yescharacteristic
317C2×C3×C4C2×C4⋊C411yescharacteristic
318C2×C3×C4C23×C411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×Q811yescharacteristic
321#67—41no—
322#67—41no—
323C2×Dic3C2×C4⋊C411yescharacteristic
324C2×Dic3C2×C4⋊C411yescharacteristic
325C2×Dic3C2×C4⋊C411yescharacteristic
326C2×Dic3C2×C4⋊C411yescharacteristic
327C2×Dic3C2×C4⋊C411yescharacteristic
328C2×Dic3C2×C4⋊C411yescharacteristic
329D12—41no—
330C4×D3—22no—
331C4×D3—22no—
332C4×D3—21no—
333C4×D3—21no—
334C4×D3—21no—
335C4×D3—21no—
336C4×D3—41no—
337C4×D3—41no—
338Dic6—41no—
339C3×C8—22no—
340C3×C8—22no—
341C3×C8—41no—
342C3×C8—41no—
343C3×C8—41no—
344C3×C8—41no—
345C3⋊C8—22no—
346C3⋊C8—22no—
347C3⋊C8—41no—
348C3⋊C8—41no—
349C3⋊C8—41no—
350C3⋊C8—41no—
351#41—122no—
352C2×Q8—31no—
353C2×D4—31no—
354C2×D4—31no—
355C2×D4—31no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4C2×C4×D311yescharacteristic
363C22×C4D3×D411yescharacteristic
364C22×C4D3×Q811yescharacteristic
365M16—122no—
366M16—241no—
367M16—241no—
368M16—241no—
369M16—241no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—32no—
375C2×C8—32no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—61no—
380C2×C8—122no—
381C2×C8—241no—
382C2×C8—241no—
383C2×C8C2×C4×D312yes—
384C2×C8C2×C4×D312yes—
385C4⋊C4—31no—
386C4⋊C4—31no—
387C4⋊C4—31no—
388C4⋊C4—31no—
389C4⋊C4—31no—
390C4⋊C4—31no—
391C22⋊C4—31no—
392C22⋊C4—31no—
393C22⋊C4—31no—
394C22⋊C4—31no—
395C22⋊C4—31no—
396C22⋊C4—31no—
397C42—31no—
398C42—31no—
399C42—31no—
400C42—31no—
401C42—31no—
402C42—31no—
403C42C2×C4×D311yescharacteristic
404C42D3×D411yescharacteristic
405C42D3×Q811yescharacteristic
406C42C23×D311yescharacteristic
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3C22×C4⋊C411yescharacteristic
411C2×D3—21no—
412C2×D3—21no—
413C2×D3—21no—
414C2×D3—21no—
415C2×D3—21no—
416C2×D3—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4#56812yes—
422C3×C4#57711yescharacteristic
423C3×C4#57711yescharacteristic
424Dic3—21no—
425Dic3—21no—
426Dic3—21no—
427Dic3—21no—
428Dic3—21no—
429Dic3—21no—
430C23—31no—
431C23—31no—
432C23D3×C4⋊C411yescharacteristic
433Q8—121no—
434D4—121no—
435D4—121no—
436D4—121no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—41no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4D3×C4⋊C411yescharacteristic
456C2×C4D3×C4⋊C411yescharacteristic
457C2×C4D3×C4⋊C411yescharacteristic
458C2×C4C22×C4×D311yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C2×C4C2×D3×Q811yescharacteristic
461C8—22no—
462C8—22no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—62no—
468C8—62no—
469C8—121no—
470C8—121no—
471C8—121no—
472C8—121no—
473C2×C3—41no—
474C2×C3#290511yescharacteristic
475C2×C3#291211yescharacteristic
476C2×C3#291611yescharacteristic
477D3—41no—
478D3—41no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—61no—
483C22—61no—
484C22—61no—
485C22—61no—
486C22—61no—
487C22—61no—
488C22C2×D3×C4⋊C411yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—61no—
497C4—61no—
498C4—61no—
499C4#575012yes—
500C4#577011yescharacteristic
501C4#577011yescharacteristic
502C3#1782411yescharacteristic, complemented, Sylow
503C2—41no—
504C2—121no—
505C2—121no—
506C2#7908011yescharacteristic
507C2#7917811yescharacteristic
508C2#7926011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1572C2
socleC22×C3C22×C4⋊C4
prime radicalC2×#235C2×C4
cosocleC3×C42C24
hypercenter#1572D3
hypocenterC3#17824
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
4. C3×#1572C2semidirect
5. #70797C2semidirect
6. #70749C2semidirect
7. #70748C2semidirect
8. #68575C2semidirect
9. #68574C2semidirect
10. #68571C2semidirect
11. #68571C2semidirect
12. #68533C2semidirect
13. #68532C2semidirect
14. #68530C2semidirect
15. #68529C2semidirect
72. #1572D3semidirect
502. C3#17824semidirect

Maximal subgroups

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

subgroupindex
#793392
#792922
#792832
C3×#15722
#707972
#707492
#707482
#685752
#685742
#685712
#685332
#685322
#685302
#685292
#178243