Finite Group Databasebeta

← #10765315

Subgroups — #10765315

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79920C211yesmaximal, characteristic, complemented
2#79653C211yesmaximal, characteristic
3#79431C211yesmaximal, characteristic
4C3×SD16⋊C8C211yesmaximal, characteristic, complemented
5C8×#215C211yesmaximal, characteristic, complemented
6#70898C211yesmaximal, characteristic, complemented
7C8×#214C211yesmaximal, characteristic, complemented
8#68573C211yesmaximal, characteristic, complemented
9#68562C211yesmaximal, characteristic, complemented
10#68239C211yesmaximal, characteristic, complemented
11#68217C211yesmaximal, characteristic, complemented
12#68171C211yesmaximal, characteristic
13#68155C211yesmaximal, characteristic, complemented
14#68076C211yesmaximal, characteristic, complemented
15#68072C211yesmaximal, characteristic, complemented
16#17923—31nomaximal, Sylow
17C4×D3×Q8C2211yescharacteristic
18#5548C2211yescharacteristic
19C3×C8×Q8C2211yescharacteristic
20C3×C4×SD16C2211yescharacteristic
21C3×C8×D4C2211yescharacteristic
22#5170C411yescharacteristic
23#5163C411yescharacteristic
24#5111—21no—
25#5110—21no—
26C4×#215C2211yescharacteristic
27Q8×C3⋊C8C2211yescharacteristic
28C4×#214C2211yescharacteristic
29#5012C2211yescharacteristic
30#4863C411yescharacteristic
31#4854C411yescharacteristic
32#4838C2211yescharacteristic
33#4835C2211yescharacteristic
34#4833—21no—
35#4833C2211yescharacteristic
36#4830—21no—
37#4830C2211yescharacteristic
38#4806C411yescharacteristic
39#4791C411yescharacteristic
40#4777C411yescharacteristic
41#4762C411yescharacteristic
42#4728—21no—
43#4727—21no—
44C4×#204C2211yescharacteristic
45C4×ΓL(1,𝔽25)C2211yescharacteristic
46C8×D12C2211yescharacteristic
47#4685—21no—
48#4685C2211yescharacteristic
49#4685C2211yescharacteristic
50C8×Dic6—21no—
51C8×Dic6C2211yescharacteristic
52C3×#335C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×#322C2211yescharacteristic
56#4535C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4484C2211yescharacteristic
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic
61#4480C2211yescharacteristic
62#4459C2211yescharacteristic
63#4458C2211yescharacteristic
64#4456C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3103—31no—
69#2982—31no—
70#2981—31no—
71#1588—31no—
72#1588—31no—
73#1583—31no—
74#1583—31no—
75SD16⋊C8—31no—
76SD16⋊C8D311yescharacteristic, complemented
77C8×SD16—31no—
78C8×SD16—31no—
79#1572—31no—
80#1491—31no—
81#1484—31no—
82#1453—31no—
83C2×D3×Q8C2×C411yescharacteristic
84C2×#235—21no—
85C2×C3×SD16C2×C411yescharacteristic
86C22×C3×C8—21no—
87C3×C4×Q8C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89Q8×Dic3C2×C411yescharacteristic
90C2×#215C2×C411yescharacteristic
91#940C2×C411yescharacteristic
92#938C2×C411yescharacteristic
93C2×#214C2×C411yescharacteristic
94C4×#67—21no—
95C2×#208—21no—
96C2×#204C2×C411yescharacteristic
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100#895C2×C411yescharacteristic
101D3×C4⋊C4—21no—
102D3×C4⋊C4C2×C411yescharacteristic
103#888C2×C411yescharacteristic
104#887—21no—
105#887C2×C411yescharacteristic
106#885—21no—
107#884—21no—
108C4×D12C2311yescharacteristic
109C42×D3—21no—
110C42×D3C2311yescharacteristic
111C4×Dic6—21no—
112C4×Dic6C2311yescharacteristic
113C3×#106C2×C411yescharacteristic
114C3×C4⋊C8—21no—
115C3×C4⋊C8C2311yescharacteristic
116C3×C4⋊C8C2311yescharacteristic
117C3×#103C2×C411yescharacteristic
118C3×#102C2×C411yescharacteristic
119C3×C22⋊C8—21no—
120C3×C4×C8—21no—
121C3×C4×C8C2311yescharacteristic
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124#835C2×C411yescharacteristic
125#832C2×C411yescharacteristic
126#830—21no—
127#821C2×C411yescharacteristic
128(C2×D3)⋊C8—21no—
129(C2×D3)⋊C8—21no—
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—
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#809C2×C411yescharacteristic
151#808C2×C411yescharacteristic
152#808C2×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#532—31no—
161C2×C4×Q8—31no—
162#481—31no—
163#481—31no—
164#475—31no—
165#475—31no—
166#460—31no—
167#460—31no—
168#448—31no—
169#448—31no—
170C8×Q8—31no—
171C8×Q8—31no—
172C8×Q8—31no—
173C8×Q8—61no—
174C8×Q8—61no—
175C8×Q8C2×D311yescharacteristic
176C4×SD16—31no—
177C4×SD16—31no—
178C4×SD16—31no—
179C4×SD16C2×D311yescharacteristic
180#436—31no—
181#436—31no—
182#435—61no—
183#435—61no—
184C8×D4—31no—
185C8×D4C2×D311yescharacteristic
186#432—31no—
187#431—31no—
188#431—31no—
189#431—31no—
190#431—61no—
191#431—61no—
192#408—61no—
193#408—61no—
194C4×M16—31no—
195#335—31no—
196#335C2×D311yescharacteristic
197#334—31no—
198#334—31no—
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#322C2×D311yescharacteristic
210C82—31no—
211C2×C3×Q8C22×C411yescharacteristic
212C2×C3×D4C22×C411yescharacteristic
213C22×C3×C4—21no—
214C2×#67—21no—
215C2×#67—21no—
216C2×#67—21no—
217D3×Q8—22no—
218D3×Q8—22no—
219#235—81no—
220#235—81no—
221C2×D12—21no—
222C2×D12C22×C411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3C22×C411yescharacteristic
227C2×C4×D3C2×D411yescharacteristic
228C2×C4×D3#4111yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6C22×C411yescharacteristic
232C3×SD16—22no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—81no—
237C2×C3×C8C22×C411yescharacteristic
238C2×C3×C8C22×C411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C3×C4⋊C4—21no—
242C3×C4⋊C4C22×C411yescharacteristic
243C3×C4⋊C4C22×C411yescharacteristic
244C3×C22⋊C4—21no—
245C3×C42—21no—
246C3×C42C2411yescharacteristic
247#217—21no—
248#215—22no—
249#214—22no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3C22×C411yescharacteristic
254C4⋊Dic3C22×C411yescharacteristic
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3C22×C411yescharacteristic
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3#4111yescharacteristic
265#208—81no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C22×C411yescharacteristic
270C2×C3⋊C8C22×C411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273#204—22no—
274ΓL(1,𝔽25)—42no—
275ΓL(1,𝔽25)—81no—
276D3×C8—81no—
277C2×#41—61no—
278C22×Q8—31no—
279C2×SD16—31no—
280C2×SD16—31no—
281C2×SD16—31no—
282C2×SD16C4×D311yescharacteristic
283C2×M16—61no—
284C2×M16—61no—
285C2×M16—61no—
286C22×C8—21no—
287C22×C8—61no—
288C4⋊Q8—31no—
289#127—31no—
290#124—61no—
291#121—31no—
292#121—31no—
293#121—61no—
294C4×Q8—31no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×Q8C22×D311yescharacteristic
300C4×D4—31no—
301C4×D4—61no—
302C4×D4C22×D311yescharacteristic
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C42—31no—
306C2×C42—61no—
307#106—31no—
308#106—31no—
309#106—31no—
310#106C4×D311yescharacteristic
311C4⋊C8—21no—
312C4⋊C8—31no—
313C4⋊C8—31no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8—61no—
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—31no—
324#103—31no—
325#103—31no—
326#103C4×D311yescharacteristic
327#102—31no—
328#102—31no—
329#102—31no—
330#102C4×D311yescharacteristic
331C22⋊C8—21no—
332C22⋊C8—31no—
333C22⋊C8—31no—
334C22⋊C8—31no—
335C22⋊C8—31no—
336C22⋊C8—61no—
337C22⋊C8—61no—
338C22⋊C8—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341#97—61no—
342#97—61no—
343C4×C8—21no—
344C4×C8—31no—
345C4×C8—31no—
346C4×C8—31no—
347C4×C8—61no—
348C4×C8—61no—
349C4×C8—61no—
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#13112yes—
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—81no—
370#67—81no—
371#67—81no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3C4×D411yescharacteristic
376C2×Dic3C4×D411yescharacteristic
377C2×Dic3C4×D411yescharacteristic
378D12—21no—
379D12—21no—
380D12—81no—
381C4×D3—24no—
382C4×D3—21no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—21no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—81no—
390Dic6—24no—
391Dic6—21no—
392Dic6—81no—
393Dic6#13112yes—
394C3×C8—21no—
395C3×C8—21no—
396C3×C8—21no—
397C3×C8—21no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3⋊C8—22no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408#41—241no—
409#41—241no—
410C2×Q8—31no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—62no—
414C2×Q8—62no—
415C2×Q8C2×C4×D311yescharacteristic
416C2×D4—31no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4C2×C4×D311yescharacteristic
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429SD16—22no—
430SD16—62no—
431SD16—62no—
432SD16—62no—
433M16—122no—
434M16—241no—
435M16—241no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—31no—
440C2×C8—31no—
441C2×C8—31no—
442C2×C8—31no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—81no—
447C2×C8—241no—
448C2×C8C2×C4×D311yescharacteristic
449C2×C8C2×C4×D311yescharacteristic
450C2×C8D3×D411yescharacteristic
451C2×C8D3×D411yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—31no—
454C4⋊C4—31no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4C2×C4×D311yescharacteristic
461C4⋊C4C2×C4×D311yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—61no—
464C22⋊C4—61no—
465C22⋊C4—61no—
466C42—21no—
467C42—31no—
468C42—31no—
469C42—31no—
470C42—61no—
471C42—61no—
472C42—61no—
473C42C23×D311yescharacteristic
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×C4×D411yescharacteristic
478C2×D3—21no—
479C2×D3—21no—
480C2×D3—21no—
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—41no—
484C3×C4—22no—
485C3×C4—21no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4C2×#13111yescharacteristic
489C3×C4C2×#13111yescharacteristic
490C3×C4#57711yescharacteristic
491C3×C4#57711yescharacteristic
492Dic3—21no—
493Dic3—22no—
494Dic3—21no—
495Dic3—21no—
496Dic3—41no—
497Dic3—41no—
498C23—21no—
499C23—31no—
500C23—61no—
501Q8—21no—
502Q8—32no—
503Q8—64no—
504Q8—61no—
505Q8—241no—
506Q8#90112yes—
507D4—21no—
508D4—21no—
509D4—61no—
510D4—61no—
511D4—241no—
512D4—241no—
513D4—241no—
514D4—241no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—31no—
519C2×C4—31no—
520C2×C4—31no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—64no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—241no—
535C2×C4C22×C4×D311yescharacteristic
536C2×C4C2×D3×D411yescharacteristic
537C2×C4D3×#4111yescharacteristic
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—62no—
546C8—61no—
547C8—61no—
548C8—61no—
549C8—61no—
550C8—121no—
551C8—121no—
552C2×C3—81no—
553C2×C3#292911yescharacteristic
554C2×C3#294911yescharacteristic
555C2×C3#295611yescharacteristic
556D3—41no—
557D3—81no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—61no—
562C22—61no—
563C22—61no—
564C22—121no—
565C22—121no—
566C22—121no—
567C22C4×D3×D411yescharacteristic
568C4—21no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—61no—
573C4—61no—
574C4—62no—
575C4—61no—
576C4—121no—
577C4—121no—
578C4C2×#90111yescharacteristic
579C4D3×#13111yescharacteristic
580C4#577511yescharacteristic
581C4#577811yescharacteristic
582C3#1792311yescharacteristic, complemented, Sylow
583C2—81no—
584C2—121no—
585C2—241no—
586C2#7942111yescharacteristic
587C2#7977711yescharacteristic
588C2#7989811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×SD16⋊C8C2
socleC22×C3C2×C4×D4
prime radical#5548C22
cosocleC3×C42C24
hypercenterSD16⋊C8D3
hypocenterC3#17923
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. #79920C2semidirect
4. C3×SD16⋊C8C2semidirect
5. C8×#215C2semidirect
6. #70898C2semidirect
7. C8×#214C2semidirect
8. #68573C2semidirect
9. #68562C2semidirect
10. #68239C2semidirect
11. #68217C2semidirect
13. #68155C2semidirect
14. #68076C2semidirect
15. #68072C2semidirect
76. SD16⋊C8D3semidirect
582. C3#17923semidirect

Maximal subgroups

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

subgroupindex
#799202
#796532
#794312
C3×SD16⋊C82
C8×#2152
#708982
C8×#2142
#685732
#685622
#682392
#682172
#681712
#681552
#680762
#680722
#179233