Finite Group Databasebeta

← #10765689

Subgroups — #10765689

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79947C211yesmaximal, characteristic, complemented
2#79652C211yesmaximal, characteristic
3#79523C211yesmaximal, characteristic, complemented
4C3×#1590C211yesmaximal, characteristic, complemented
5#70952C211yesmaximal, characteristic, complemented
6#70896C211yesmaximal, characteristic, complemented
7#70861C211yesmaximal, characteristic, complemented
8#68532C211yesmaximal, characteristic, complemented
9#68519C211yesmaximal, characteristic, complemented
10#68234C211yesmaximal, characteristic, complemented
11#68221C211yesmaximal, characteristic, complemented
12#68174C211yesmaximal, characteristic, complemented
13#68158C211yesmaximal, characteristic, complemented
14#68055C211yesmaximal, characteristic, complemented
15#68048C211yesmaximal, characteristic, complemented
16#17955—31nomaximal, Sylow
17#5580C2211yescharacteristic
18C4×#237C2211yescharacteristic
19C3×#446C2211yescharacteristic
20C3×C4×SD16C2211yescharacteristic
21C3×#436C2211yescharacteristic
22#5173C411yescharacteristic
23#5165C411yescharacteristic
24#5127—21no—
25#5118—21no—
26C4×#216C2211yescharacteristic
27Q8×C3⋊C8C2211yescharacteristic
28C4×#213C2211yescharacteristic
29#5012C2211yescharacteristic
30#4866C411yescharacteristic
31#4857C411yescharacteristic
32#4839—21no—
33#4838—21no—
34#4837C2211yescharacteristic
35#4834C2211yescharacteristic
36D3×C4⋊C8C2211yescharacteristic
37#4831C2211yescharacteristic
38#4815C411yescharacteristic
39#4798C411yescharacteristic
40#4781C411yescharacteristic
41#4763C411yescharacteristic
42#4728—21no—
43#4719—21no—
44#4709—21no—
45#4706—21no—
46C4×#204C2211yescharacteristic
47#4688C2211yescharacteristic
48C4×ΓL(1,𝔽25)C2211yescharacteristic
49#4685C2211yescharacteristic
50#4685C2211yescharacteristic
51#4679C2211yescharacteristic
52C3×#334C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×C82C2211yescharacteristic
56#4535C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4484C2211yescharacteristic
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic
61#4480C2211yescharacteristic
62#4459C2211yescharacteristic
63#4457C2211yescharacteristic
64#4456C2211yescharacteristic
65#4455C2211yescharacteristic
66#4454C2211yescharacteristic
67#4454C2211yescharacteristic
68#3102—31no—
69#2993—31no—
70#2989—31no—
71#1594—31no—
72#1593—31no—
73#1590—31no—
74#1590D311yescharacteristic, complemented
75#1587—31no—
76#1586—31no—
77#1583—31no—
78#1582—31no—
79#1570—31no—
80#1489—31no—
81#1483—31no—
82#1453—31no—
83C2×#237C2×C411yescharacteristic
84C2×C3×SD16C2×C411yescharacteristic
85C2×C3×M16—21no—
86C3×C4×Q8C2311yescharacteristic
87C3×C4×D4C2311yescharacteristic
88#947C2×C411yescharacteristic
89#946C2×C411yescharacteristic
90C2×#216C2×C411yescharacteristic
91D4×Dic3C2×C411yescharacteristic
92C2×#213C2×C411yescharacteristic
93C4×#67—21no—
94C2×C4×Dic3—21no—
95C2×#208—21no—
96C2×#204C2×C411yescharacteristic
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×ΓL(1,𝔽25)—21no—
100C2×D3×C8—21no—
101C2×D3×C8—21no—
102#898—21no—
103#895C2×C411yescharacteristic
104#894—21no—
105#892C2×C411yescharacteristic
106#890C2×C411yescharacteristic
107#887C2×C411yescharacteristic
108#67⋊C4—21no—
109#877—21no—
110C4×D12—21no—
111C4×D12C2311yescharacteristic
112#872—21no—
113C42×D3C2311yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×#106C2×C411yescharacteristic
116C3×C4⋊C8—21no—
117C3×C4⋊C8C2311yescharacteristic
118C3×C4⋊C8C2311yescharacteristic
119C3×#103C2×C411yescharacteristic
120C3×#102C2×C411yescharacteristic
121C3×C22⋊C8—21no—
122C3×#97—21no—
123C3×C4×C8C2311yescharacteristic
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126#835C2×C411yescharacteristic
127#832C2×C411yescharacteristic
128#830—21no—
129#821C2×C411yescharacteristic
130(C2×D3)⋊C8—21no—
131(C2×D3)⋊C8—21no—
132(C2×D3)⋊C8C2×C411yescharacteristic
133(C2×D3)⋊C8C2×C411yescharacteristic
134(C2×D3)⋊C8D411yescharacteristic
135(C2×D3)⋊C8D411yescharacteristic
136#817C2×C411yescharacteristic
137#816C2×C411yescharacteristic
138#815—21no—
139#815—21no—
140#815—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8C2×C411yescharacteristic
146Dic3⋊C8C2×C411yescharacteristic
147Dic3⋊C8D411yescharacteristic
148Dic3⋊C8D411yescharacteristic
149#810C2×C411yescharacteristic
150#809C2×C411yescharacteristic
151#807C2×C411yescharacteristic
152#807C2×C411yescharacteristic
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4×C3⋊C8—21no—
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#529—31no—
161C4×#41—31no—
162#485—31no—
163#482—31no—
164#479—31no—
165#478—31no—
166#464—31no—
167#463—31no—
168#452—31no—
169#449—31no—
170#446—31no—
171#446—31no—
172#446C2×D311yescharacteristic
173C8×Q8—31no—
174C4×Q16—31no—
175C4×SD16—31no—
176C4×SD16C2×D311yescharacteristic
177C4×D8—31no—
178#436—31no—
179#436—31no—
180#436—61no—
181#436C2×D311yescharacteristic
182#435—61no—
183#435—61no—
184#435—61no—
185C8×D4—31no—
186#433—61no—
187#432—61no—
188#431—31no—
189#431—31no—
190#424—61no—
191#423—31no—
192#423—61no—
193C2×C4⋊C8—31no—
194C4×M16—31no—
195#334—31no—
196#334—31no—
197#334—31no—
198#334C2×D311yescharacteristic
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8C2×D311yescharacteristic
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322—31no—
210C82C2×D311yescharacteristic
211C2×C3×Q8C22×C411yescharacteristic
212C2×C3×D4C22×C411yescharacteristic
213C22×C3×C4—21no—
214C2×#67—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217#237—41no—
218#237—41no—
219C2×D12—21no—
220C2×D12C22×C411yescharacteristic
221C2×C4×D3—21no—
222C2×C4×D3C22×C411yescharacteristic
223C2×C4×D3C2×D411yescharacteristic
224C2×C4×D3#4111yescharacteristic
225C2×Dic6C22×C411yescharacteristic
226C3×SD16—41no—
227C3×M16—81no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8C22×C411yescharacteristic
232C2×C3×C8C22×C411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C3×C4⋊C4—21no—
236C3×C4⋊C4C22×C411yescharacteristic
237C3×C4⋊C4C22×C411yescharacteristic
238C3×C22⋊C4—21no—
239C3×C42—21no—
240C3×C42C2411yescharacteristic
241#217—21no—
242#216—41no—
243#213—41no—
244(C2×D3)⋊C4—21no—
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—
250C4⋊Dic3—21no—
251C4⋊Dic3C22×C411yescharacteristic
252C4⋊Dic3C22×C411yescharacteristic
253Dic3⋊C4—21no—
254Dic3⋊C4—21no—
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3C22×C411yescharacteristic
261C4×Dic3C2×D411yescharacteristic
262C4×Dic3#4111yescharacteristic
263#208—81no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C22×C411yescharacteristic
268C2×C3⋊C8C22×C411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271#204—41no—
272ΓL(1,𝔽25)—42no—
273ΓL(1,𝔽25)—81no—
274ΓL(1,𝔽25)—81no—
275D3×C8—81no—
276D3×C8—81no—
277C2×#41—31no—
278C2×Q16—31no—
279C2×SD16—31no—
280C2×SD16C4×D311yescharacteristic
281C2×D8—31no—
282C2×M16—21no—
283C2×M16—61no—
284C2×M16—61no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—61no—
288C22×C8—61no—
289#126—61no—
290#125—31no—
291#124—31no—
292#123—61no—
293C22⋊Q8—31no—
294#121—31no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8C22×D311yescharacteristic
298C4×D4—31no—
299C4×D4—31no—
300C4×D4—61no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4C22×D311yescharacteristic
304#117—31no—
305#117—61no—
306#117—61no—
307C2×C42—31no—
308C2×C42—61no—
309#107—31no—
310#107—31no—
311#106—31no—
312#106C4×D311yescharacteristic
313C4⋊C8—21no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—31no—
328#103—31no—
329#103—31no—
330#103C4×D311yescharacteristic
331#102—31no—
332#102—31no—
333#102—31no—
334#102C4×D311yescharacteristic
335C22⋊C8—21no—
336C22⋊C8—31no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—31no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—21no—
344#97—61no—
345#97—61no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C23×C3—21no—
355C22×D3—21no—
356C22×D3C4×D411yescharacteristic
357C3×Q8—21no—
358C3×Q8—21no—
359C3×D4—21no—
360C3×D4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4C23×C411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369#67—42no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—42no—
374C2×Dic3—81no—
375C2×Dic3C4×D411yescharacteristic
376C2×Dic3C4×D411yescharacteristic
377C2×Dic3C4×D411yescharacteristic
378D12—21no—
379D12—21no—
380D12—81no—
381C4×D3—21no—
382C4×D3—21no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—41no—
388Dic6—21no—
389Dic6—21no—
390C3×C8—22no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—41no—
396C3×C8—41no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—21no—
400C3⋊C8—21no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404#41—121no—
405#41—121no—
406C2×Q8—31no—
407C2×Q8C2×C4×D311yescharacteristic
408C2×D4—31no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4C2×C4×D311yescharacteristic
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419Q16—121no—
420SD16—41no—
421SD16—121no—
422D8—121no—
423M16—81no—
424M16—122no—
425M16—241no—
426M16—241no—
427M16—241no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—31no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—241no—
439C2×C8—241no—
440C2×C8C2×C4×D311yescharacteristic
441C2×C8C2×C4×D311yescharacteristic
442C2×C8D3×D411yescharacteristic
443C2×C8D3×D411yescharacteristic
444C4⋊C4—21no—
445C4⋊C4—31no—
446C4⋊C4—31no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4C2×C4×D311yescharacteristic
453C4⋊C4C2×C4×D311yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C22⋊C4—61no—
461C22⋊C4—61no—
462C42—21no—
463C42—31no—
464C42—31no—
465C42—31no—
466C42—61no—
467C42—61no—
468C42—61no—
469C42C23×D311yescharacteristic
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3C2×C4×D411yescharacteristic
474C2×D3—21no—
475C2×D3—21no—
476C2×D3—21no—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—41no—
480C3×C4—21no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4#56811yescharacteristic
485C3×C4#56811yescharacteristic
486C3×C4#57711yescharacteristic
487C3×C4#57711yescharacteristic
488Dic3—21no—
489Dic3—21no—
490Dic3—21no—
491Dic3—41no—
492Dic3—41no—
493Dic3—41no—
494C23—21no—
495C23—31no—
496C23—61no—
497Q8—21no—
498Q8—21no—
499Q8—61no—
500Q8—61no—
501D4—21no—
502D4—21no—
503D4—61no—
504D4—61no—
505D4—122no—
506D4—241no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—31no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—241no—
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C2×C4D3×#4111yescharacteristic
530C8—22no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—21no—
535C8—41no—
536C8—41no—
537C8—61no—
538C8—61no—
539C8—61no—
540C8—61no—
541C8—121no—
542C8—121no—
543C8—121no—
544C2×C3—81no—
545C2×C3#293011yescharacteristic
546C2×C3#295211yescharacteristic
547C2×C3#295611yescharacteristic
548D3—41no—
549D3—81no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—61no—
554C22—61no—
555C22—61no—
556C22—121no—
557C22—121no—
558C22—121no—
559C22C4×D3×D411yescharacteristic
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—61no—
565C4—61no—
566C4—61no—
567C4—121no—
568C4—121no—
569C4—121no—
570C4#574511yescharacteristic
571C4#575011yescharacteristic
572C4#576211yescharacteristic
573C4#577011yescharacteristic
574C3#1795511yescharacteristic, complemented, Sylow
575C2—81no—
576C2—121no—
577C2—241no—
578C2#7953711yescharacteristic
579C2#7965011yescharacteristic
580C2#7988911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1590C2
socleC22×C3C2×C4×D4
prime radical#4781C4
cosocleC3×C42C24
hypercenter#1590D3
hypocenterC3#17955
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
1. #79947C2semidirect
3. #79523C2semidirect
4. C3×#1590C2semidirect
5. #70952C2semidirect
6. #70896C2semidirect
7. #70861C2semidirect
8. #68532C2semidirect
9. #68519C2semidirect
10. #68234C2semidirect
11. #68221C2semidirect
12. #68174C2semidirect
13. #68158C2semidirect
14. #68055C2semidirect
15. #68048C2semidirect
74. #1590D3semidirect
574. C3#17955semidirect

Maximal subgroups

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

subgroupindex
#799472
#796522
#795232
C3×#15902
#709522
#708962
#708612
#685322
#685192
#682342
#682212
#681742
#681582
#680552
#680482
#179553