Finite Group Databasebeta

← #10765293

Subgroups — #10765293

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79922C211yesmaximal, characteristic, complemented
2#79655C211yesmaximal, characteristic
3#79405C211yesmaximal, characteristic
4C3×SD16⋊C8C211yesmaximal, characteristic, complemented
5C8×#216C211yesmaximal, characteristic, complemented
6#70895C211yesmaximal, characteristic, complemented
7C8×#214C211yesmaximal, characteristic, complemented
8#68574C211yesmaximal, characteristic, complemented
9#68560C211yesmaximal, characteristic, complemented
10#68232C211yesmaximal, characteristic, complemented
11#68222C211yesmaximal, characteristic, complemented
12#68173C211yesmaximal, characteristic
13#68156C211yesmaximal, characteristic, complemented
14#68084C211yesmaximal, characteristic, complemented
15#68074C211yesmaximal, characteristic, complemented
16#17921—31nomaximal, Sylow
17#5578C2211yescharacteristic
18C4×#237C2211yescharacteristic
19C3×C8×Q8C2211yescharacteristic
20C3×C4×SD16C2211yescharacteristic
21C3×C8×D4C2211yescharacteristic
22#5173C411yescharacteristic
23#5162C411yescharacteristic
24C8×#67—21no—
25C2×Dic3⋊C8—21no—
26C4×#216C2211yescharacteristic
27#5024C2211yescharacteristic
28C4×#214C2211yescharacteristic
29D4×C3⋊C8C2211yescharacteristic
30#4868C411yescharacteristic
31#4854C411yescharacteristic
32#4840—21no—
33#4835C2211yescharacteristic
34D3×C4⋊C8C2211yescharacteristic
35#4831C2211yescharacteristic
36#4830C2211yescharacteristic
37#4828—21no—
38#4814C411yescharacteristic
39#4795C411yescharacteristic
40#4783C411yescharacteristic
41#4759C411yescharacteristic
42#67⋊C8—21no—
43#4718—21no—
44C4×Dic12C2211yescharacteristic
45#4690—21no—
46#4685C2211yescharacteristic
47#4685C2211yescharacteristic
48#4685C2211yescharacteristic
49#4679—21no—
50C8×Dic6C2211yescharacteristic
51C8×Dic6C2211yescharacteristic
52C3×#335C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×#322C2211yescharacteristic
56#4535C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4484C2211yescharacteristic
59#4484C2211yescharacteristic
60#4480C2211yescharacteristic
61#4479C2211yescharacteristic
62#4457C2211yescharacteristic
63#4456C2211yescharacteristic
64#4456C2211yescharacteristic
65#4455C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3107—31no—
69#2988—31no—
70#2985—31no—
71#1587—31no—
72#1585—31no—
73#1584—31no—
74#1582—31no—
75#1579—31no—
76SD16⋊C8D311yescharacteristic, complemented
77C8×Q16—31no—
78C8×SD16—31no—
79#1573—31no—
80#1488—31no—
81#1481—31no—
82#1454—31no—
83C2×#237C2×C411yescharacteristic
84C2×C3×SD16C2×C411yescharacteristic
85C22×C3×C8—21no—
86C3×C4×Q8C2311yescharacteristic
87C3×C4×D4C2311yescharacteristic
88#946C2×C411yescharacteristic
89#944C2×C411yescharacteristic
90C2×#216C2×C411yescharacteristic
91D4×Dic3C2×C411yescharacteristic
92C2×#214C2×C411yescharacteristic
93C4×#67—21no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8—21no—
96C2×Dic12C2×C411yescharacteristic
97C2×D3×C8—21no—
98#897C2×C411yescharacteristic
99#896—21no—
100#892C2×C411yescharacteristic
101#889—21no—
102#888C2×C411yescharacteristic
103#887C2×C411yescharacteristic
104#67⋊C4—21no—
105#877—21no—
106#872—21no—
107C42×D3C2311yescharacteristic
108C4×Dic6—21no—
109C4×Dic6C2311yescharacteristic
110C4×Dic6C2311yescharacteristic
111C3×#106C2×C411yescharacteristic
112C3×C4⋊C8—21no—
113C3×C4⋊C8C2311yescharacteristic
114C3×C4⋊C8C2311yescharacteristic
115C3×#103C2×C411yescharacteristic
116C3×#102C2×C411yescharacteristic
117C3×C22⋊C8—21no—
118C3×C4×C8—21no—
119C3×C4×C8C2311yescharacteristic
120C3×C4×C8C2311yescharacteristic
121C3×C4×C8C2311yescharacteristic
122#835C2×C411yescharacteristic
123#832C2×C411yescharacteristic
124#830—21no—
125(C2×D3)⋊C8—21no—
126(C2×D3)⋊C8—21no—
127(C2×D3)⋊C8C2×C411yescharacteristic
128(C2×D3)⋊C8C2×C411yescharacteristic
129(C2×D3)⋊C8D411yescharacteristic
130(C2×D3)⋊C8D411yescharacteristic
131#818C2×C411yescharacteristic
132#816C2×C411yescharacteristic
133#816C2×C411yescharacteristic
134#815—21no—
135#815—21no—
136Dic3⋊C8—21no—
137Dic3⋊C8—21no—
138Dic3⋊C8—21no—
139Dic3⋊C8—21no—
140Dic3⋊C8—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8C2×C411yescharacteristic
143Dic3⋊C8C2×C411yescharacteristic
144Dic3⋊C8D411yescharacteristic
145Dic3⋊C8D411yescharacteristic
146C8×Dic3—21no—
147C8×Dic3—21no—
148C8×Dic3—21no—
149#810C2×C411yescharacteristic
150#810C2×C411yescharacteristic
151#808C2×C411yescharacteristic
152#807C2×C411yescharacteristic
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156#803—21no—
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161C4×#41—31no—
162#485—31no—
163#485—31no—
164#475—31no—
165#474—31no—
166#462—31no—
167#461—31no—
168#452—31no—
169#452—31no—
170#446—31no—
171#446—31no—
172#446—61no—
173#446—61no—
174C8×Q8—31no—
175C8×Q8—31no—
176C8×Q8—31no—
177C8×Q8C2×D311yescharacteristic
178C4×Q16—31no—
179C4×Q16—31no—
180C4×SD16—31no—
181C4×SD16C2×D311yescharacteristic
182C8×D4—31no—
183C8×D4—61no—
184C8×D4—61no—
185C8×D4C2×D311yescharacteristic
186#433—61no—
187#433—61no—
188#432—31no—
189#431—31no—
190#431—31no—
191#431—31no—
192#431—61no—
193C2×C4⋊C8—31no—
194C2×C4⋊C8—61no—
195#335—31no—
196#335C2×D311yescharacteristic
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8—31no—
203Q8⋊C8—31no—
204Q8⋊C8C2×D311yescharacteristic
205D4⋊C8—31no—
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322C2×D311yescharacteristic
210C82—31no—
211C2×C3×Q8C22×C411yescharacteristic
212C2×C3×D4C22×C411yescharacteristic
213C22×C3×C4—21no—
214C2×#67—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217#237—22no—
218#237—22no—
219C2×C4×D3C22×C411yescharacteristic
220C2×C4×D3C2×D411yescharacteristic
221C2×C4×D3#4111yescharacteristic
222C2×Dic6—21no—
223C2×Dic6C22×C411yescharacteristic
224C2×Dic6C22×C411yescharacteristic
225C3×SD16—22no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—81no—
230C2×C3×C8C22×C411yescharacteristic
231C2×C3×C8C22×C411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C3×C4⋊C4—21no—
235C3×C4⋊C4C22×C411yescharacteristic
236C3×C4⋊C4C22×C411yescharacteristic
237C3×C22⋊C4—21no—
238C3×C42—21no—
239C3×C42C2411yescharacteristic
240#217—21no—
241#216—22no—
242#214—22no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3C22×C411yescharacteristic
248C4⋊Dic3C22×C411yescharacteristic
249Dic3⋊C4—21no—
250Dic3⋊C4—21no—
251Dic3⋊C4—21no—
252Dic3⋊C4—21no—
253Dic3⋊C4—21no—
254Dic3⋊C4—21no—
255Dic3⋊C4—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3C22×C411yescharacteristic
261C4×Dic3C2×D411yescharacteristic
262C4×Dic3#4111yescharacteristic
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—81no—
267C2×C3⋊C8C22×C411yescharacteristic
268C2×C3⋊C8C22×C411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271Dic12—22no—
272D3×C8—81no—
273C2×#41—31no—
274C2×Q16—31no—
275C2×Q16—31no—
276C2×SD16—31no—
277C2×SD16C4×D311yescharacteristic
278C22×C8—21no—
279C22×C8—61no—
280C22×C8—61no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283#125—61no—
284C22⋊Q8—31no—
285C22⋊Q8—31no—
286C22⋊Q8—61no—
287C4×Q8—31no—
288C4×Q8—31no—
289C4×Q8—31no—
290C4×Q8—61no—
291C4×Q8C22×D311yescharacteristic
292C4×D4—31no—
293C4×D4—61no—
294C4×D4—61no—
295C4×D4C22×D311yescharacteristic
296#117—31no—
297#117—61no—
298#117—61no—
299C2×C42—31no—
300C2×C42—61no—
301#107—31no—
302#107—31no—
303#106—31no—
304#106C4×D311yescharacteristic
305C4⋊C8—21no—
306C4⋊C8—31no—
307C4⋊C8—31no—
308C4⋊C8—31no—
309C4⋊C8—31no—
310C4⋊C8—31no—
311C4⋊C8—31no—
312C4⋊C8—61no—
313C4⋊C8—61no—
314C4⋊C8—61no—
315C4⋊C8—61no—
316C4⋊C8—61no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8C22×D311yescharacteristic
320C4⋊C8C22×D311yescharacteristic
321#103—31no—
322#103—31no—
323#103—31no—
324#103—31no—
325#103—31no—
326#103C4×D311yescharacteristic
327#102—31no—
328#102C4×D311yescharacteristic
329C22⋊C8—21no—
330C22⋊C8—31no—
331C22⋊C8—31no—
332C22⋊C8—31no—
333C22⋊C8—31no—
334C22⋊C8—61no—
335C22⋊C8—61no—
336C22⋊C8—61no—
337#97—61no—
338#97—61no—
339#97—61no—
340C4×C8—21no—
341C4×C8—31no—
342C4×C8—31no—
343C4×C8—31no—
344C4×C8—61no—
345C4×C8—61no—
346C4×C8—61no—
347C4×C8C22×D311yescharacteristic
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C23×C3—21no—
351C22×D3C4×D411yescharacteristic
352C3×Q8—21no—
353C3×Q8—21no—
354C3×D4—21no—
355C3×D4#13112yes—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4C23×C411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364#67—24no—
365C2×Dic3—24no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—81no—
371C2×Dic3C4×D411yescharacteristic
372C2×Dic3C4×D411yescharacteristic
373C2×Dic3C4×D411yescharacteristic
374C4×D3—21no—
375C4×D3—21no—
376C4×D3—21no—
377C4×D3—21no—
378C4×D3—41no—
379Dic6—21no—
380Dic6—21no—
381Dic6—21no—
382Dic6—81no—
383Dic6#13112yes—
384C3×C8—21no—
385C3×C8—21no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—62no—
399#41—62no—
400C2×Q8—31no—
401C2×Q8—31no—
402C2×Q8—61no—
403C2×Q8C2×C4×D311yescharacteristic
404C2×D4—61no—
405C2×D4C2×C4×D311yescharacteristic
406C22×C4—21no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—61no—
411C22×C4—61no—
412Q16—62no—
413Q16—62no—
414SD16—22no—
415SD16—62no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—31no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—31no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—81no—
427C2×C8—241no—
428C2×C8—241no—
429C2×C8C2×C4×D311yescharacteristic
430C2×C8C2×C4×D311yescharacteristic
431C2×C8D3×D411yescharacteristic
432C2×C8D3×D411yescharacteristic
433C4⋊C4—21no—
434C4⋊C4—31no—
435C4⋊C4—31no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4C2×C4×D311yescharacteristic
445C4⋊C4C2×C4×D311yescharacteristic
446C22⋊C4—21no—
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C42—21no—
452C42—31no—
453C42—31no—
454C42—31no—
455C42—61no—
456C42—61no—
457C42—61no—
458C42—61no—
459C42C23×D311yescharacteristic
460C22×C3—22no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3C2×C4×D411yescharacteristic
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—21no—
467C3×C4—21no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4C2×#13111yescharacteristic
472C3×C4C2×#13111yescharacteristic
473C3×C4#57711yescharacteristic
474C3×C4#57811yescharacteristic
475Dic3—22no—
476Dic3—21no—
477Dic3—21no—
478Dic3—21no—
479Dic3—41no—
480Dic3—41no—
481Dic3—41no—
482C23—21no—
483C23—31no—
484Q8—21no—
485Q8—21no—
486Q8—32no—
487Q8—61no—
488Q8—61no—
489Q8—61no—
490Q8—241no—
491D4—21no—
492D4—64no—
493D4#90112yes—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—64no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—121no—
512C2×C4—241no—
513C2×C4C22×C4×D311yescharacteristic
514C2×C4C2×D3×D411yescharacteristic
515C2×C4D3×#4111yescharacteristic
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—61no—
524C8—61no—
525C8—61no—
526C8—61no—
527C8—121no—
528C8—121no—
529C8—121no—
530C2×C3—42no—
531C2×C3#292911yescharacteristic
532C2×C3#294711yescharacteristic
533C2×C3#295711yescharacteristic
534D3—41no—
535C22—22no—
536C22—41no—
537C22—41no—
538C22—61no—
539C22—61no—
540C22—61no—
541C22C4×D3×D411yescharacteristic
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—61no—
547C4—61no—
548C4—62no—
549C4—61no—
550C4—121no—
551C4—121no—
552C4—121no—
553C4C2×#90111yescharacteristic
554C4D3×#13111yescharacteristic
555C4#577511yescharacteristic
556C4#577911yescharacteristic
557C3#1792111yescharacteristic, complemented, Sylow
558C2—42no—
559C2—121no—
560C2#7942111yescharacteristic
561C2#7977611yescharacteristic
562C2#7989911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×SD16⋊C8C2
socleC22×C3C2×C4×D4
prime radicalC2×#237C2×C4
cosocleC3×C42C24
hypercenterSD16⋊C8D3
hypocenterC3#17921
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
1. #79922C2semidirect
4. C3×SD16⋊C8C2semidirect
5. C8×#216C2semidirect
6. #70895C2semidirect
7. C8×#214C2semidirect
8. #68574C2semidirect
9. #68560C2semidirect
10. #68232C2semidirect
11. #68222C2semidirect
13. #68156C2semidirect
14. #68084C2semidirect
15. #68074C2semidirect
76. SD16⋊C8D3semidirect
557. C3#17921semidirect

Maximal subgroups

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

subgroupindex
#799222
#796552
#794052
C3×SD16⋊C82
C8×#2162
#708952
C8×#2142
#685742
#685602
#682322
#682222
#681732
#681562
#680842
#680742
#179213