Finite Group Databasebeta

← #10786607

Subgroups — #10786607

Order 768.

624 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×SD16C211yesmaximal, characteristic, complemented
2#83706C211yesmaximal, characteristic, complemented
3#83560C211yesmaximal, characteristic
4#79688C211yesmaximal, characteristic
5#71941C211yesmaximal, characteristic, complemented
6#71915C211yesmaximal, characteristic, complemented
7#71344C211yesmaximal, characteristic, complemented
8#71333C211yesmaximal, characteristic, complemented
9#71140C211yesmaximal, characteristic
10#71137C211yesmaximal, characteristic, complemented
11#71125C211yesmaximal, characteristic, complemented
12#71121C211yesmaximal, characteristic, complemented
13#70943C211yesmaximal, characteristic, complemented
14C8×#215C211yesmaximal, characteristic, complemented
15#70899C211yesmaximal, characteristic, complemented
16#19895—31nomaximal, Sylow
17C3×Q82C2211yescharacteristic
18C3×D4×Q8C2211yescharacteristic
19C2×#942—21no—
20#5579C2211yescharacteristic
21#5551C2211yescharacteristic
22C3×#498—21no—
23C3×#498C2211yescharacteristic
24C3×#476—21no—
25C3×#476C2211yescharacteristic
26C3×#475—21no—
27C3×#475C2211yescharacteristic
28C3×C8×Q8C2211yescharacteristic
29C3×C4×SD16—21no—
30C3×C4×SD16C2211yescharacteristic
31#5228—21no—
32#5171—21no—
33#5093C2211yescharacteristic
34#5092C2211yescharacteristic
35#5087—21no—
36#5084C2211yescharacteristic
37#5083C2211yescharacteristic
38#5081—21no—
39#5079C2211yescharacteristic
40#5078C2211yescharacteristic
41#5052—21no—
42#5047—21no—
43#5031C2211yescharacteristic
44#5026—21no—
45C4×#215C2211yescharacteristic
46#5024C2211yescharacteristic
47#5022—21no—
48#5021C2211yescharacteristic
49#5020C2211yescharacteristic
50#5012C2211yescharacteristic
51#5008C2211yescharacteristic
52D3×#106—21no—
53#4805—21no—
54#4777—21no—
55C8×D12C2211yescharacteristic
56#4680C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4535C2211yescharacteristic
60#4535C2211yescharacteristic
61D4⋊(C3⋊C8)C2211yescharacteristic
62#4486C2211yescharacteristic
63#4483C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4457C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3385—31no—
69Q8×SD16D311yescharacteristic, complemented
70#3335—31no—
71#3276—31no—
72#1707—31no—
73#1695—31no—
74#1690—31no—
75#1672—31no—
76#1671—31no—
77#1645—31no—
78#1643—31no—
79#1637—31no—
80#1635—31no—
81#1590—31no—
82C8×SD16—31no—
83C22×C3×Q8—21no—
84C2×#235—21no—
85C2×C3×SD16D411yescharacteristic
86C3×C4⋊Q8—21no—
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—21no—
89C3×C4⋊Q8—41no—
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—41no—
94C3×C4×Q8—21no—
95C3×C4×Q8—21no—
96C3×C4×Q8C2311yescharacteristic
97C3×C4×Q8C2311yescharacteristic
98C3×C4×D4—21no—
99C3×C4×D4C2311yescharacteristic
100#946—21no—
101C2×#216—21no—
102C2×#216—21no—
103C2×#216—21no—
104#942—44no—
105C2×#215—21no—
106C2×#215D411yescharacteristic
107#938—21no—
108#924—21no—
109C2×#208—21no—
110C2×D3×C8—21no—
111#896—41no—
112D3×C4⋊C4—21no—
113D3×C4⋊C4—21no—
114#883—41no—
115C4×D12—21no—
116C4×D12C2311yescharacteristic
117#870—21no—
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C3×#106—21no—
121C3×#106—21no—
122C3×#106—21no—
123C3×#106—21no—
124C3×#106D411yescharacteristic
125C3×C4⋊C8—21no—
126C3×C4⋊C8C2311yescharacteristic
127C3×#103—21no—
128C3×#103D411yescharacteristic
129C3×#102—21no—
130C3×#102D411yescharacteristic
131C3×C4×C8—21no—
132C3×C4×C8C2311yescharacteristic
133#835—21no—
134#835—21no—
135#835—21no—
136#835—41no—
137#835D411yescharacteristic
138#832—21no—
139#830—21no—
140(C2×D3)⋊C8—21no—
141#817—21no—
142#817—21no—
143#810—21no—
144#810—41no—
145#809—21no—
146#809D411yescharacteristic
147#808—21no—
148#808—21no—
149#808—21no—
150#808D411yescharacteristic
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158#803—21no—
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#137—61no—
162Q82C2×D311yescharacteristic
163#554—31no—
164D4×Q8C2×D311yescharacteristic
165#547—31no—
166#501—61no—
167#500—31no—
168#498—21no—
169#498—31no—
170#498C2×D311yescharacteristic
171#494—31no—
172#487—31no—
173#487—31no—
174#483—61no—
175#481—61no—
176#479—61no—
177#477—31no—
178#476—21no—
179#476—31no—
180#476—31no—
181#476C2×D311yescharacteristic
182#475—21no—
183#475—31no—
184#475C2×D311yescharacteristic
185#474—61no—
186#470—61no—
187#465—61no—
188#462—31no—
189#461—31no—
190#452—61no—
191#451—61no—
192#446—31no—
193C8×Q8C2×D311yescharacteristic
194#440—61no—
195C4×SD16—21no—
196C4×SD16—31no—
197C4×SD16C2×D311yescharacteristic
198#436—31no—
199C8×D4—31no—
200C2×#106—61no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207Q8⋊C8—31no—
208Q8⋊C8—31no—
209D4⋊C8—31no—
210D4⋊C8—31no—
211C82—31no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—41no—
215C2×C3×Q8—41no—
216C2×C3×Q8—42no—
217C2×C3×Q8—81no—
218C2×C3×Q8C2×D411yescharacteristic
219C2×C3×Q8C2×D411yescharacteristic
220C2×C3×D4C2×D411yescharacteristic
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C2×#67—41no—
224#235—42no—
225#235—81no—
226C2×D12C2×D411yescharacteristic
227C2×C4×D3—21no—
228C2×C4×D3—41no—
229C2×C4×D3—41no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C3×SD16—22no—
233C2×C3×C8—21no—
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C4⋊C4C2×D411yescharacteristic
248C3×C22⋊C4—21no—
249C3×C22⋊C4—41no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—41no—
253C3×C42C2411yescharacteristic
254#217—41no—
255#216—42no—
256#216—42no—
257#216—81no—
258#215—42no—
259#215SD1614yescomplemented
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—41no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3C2×D411yescharacteristic
267Dic3⋊C4—41no—
268Dic3⋊C4—41no—
269Dic3⋊C4—41no—
270#208—42no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277D3×C8—42no—
278C2×#41—61no—
279C22×Q8—21no—
280#137—124no—
281C2×Q16—61no—
282C2×Q16—61no—
283C2×Q16—61no—
284C2×SD16—31no—
285C2×SD16—61no—
286C2×SD16#6711yescharacteristic
287C2×M16—61no—
288C22×C8—61no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—31no—
293C4⋊Q8—31no—
294C4⋊Q8—41no—
295C4⋊Q8C22×D311yescharacteristic
296C4⋊Q8C22×D311yescharacteristic
297#125—61no—
298#123—121no—
299C22⋊Q8—21no—
300C22⋊Q8—41no—
301C22⋊Q8—61no—
302C22⋊Q8—61no—
303C22⋊Q8—121no—
304#121—61no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8C22×D311yescharacteristic
308C4×Q8C22×D311yescharacteristic
309C4×D4—21no—
310C4×D4—31no—
311C4×D4—61no—
312C4×D4C22×D311yescharacteristic
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315#107—61no—
316#107—61no—
317#107—61no—
318#106—21no—
319#106—21no—
320#106—21no—
321#106—21no—
322#106—31no—
323#106—61no—
324#106—61no—
325#106—61no—
326#106—61no—
327#106—61no—
328#106#6711yescharacteristic
329C4⋊C8—21no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—31no—
333C4⋊C8—61no—
334C4⋊C8C22×D311yescharacteristic
335#103—21no—
336#103—31no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—121no—
342#103—121no—
343#103#6711yescharacteristic
344#102—21no—
345#102—31no—
346#102—61no—
347#102—61no—
348#102#6711yescharacteristic
349C22⋊C8—61no—
350C22⋊C8—61no—
351#97—61no—
352C4×C8—21no—
353C4×C8—31no—
354C4×C8—31no—
355C4×C8C22×D311yescharacteristic
356C23×C3—21no—
357C22×D3—21no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—82no—
364C3×Q8—81no—
365C3×Q8—161no—
366C3×Q8C2×SD1612yes—
367C3×D4—21no—
368C3×D4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—82no—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382#67—82no—
383C2×Dic3—21no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386D12—21no—
387D12C2×SD1612yes—
388C4×D3—41no—
389C4×D3—41no—
390C4×D3—82no—
391C4×D3—161no—
392Dic6—41no—
393Dic6—41no—
394Dic6—81no—
395C3×C8—21no—
396C3×C8—41no—
397C3×C8#13712yescomplemented
398C3⋊C8—21no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8C2×SD1612yes—
404#41—122no—
405#41—241no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—41no—
409C2×Q8—41no—
410C2×Q8—42no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—81no—
414C2×Q8D3×D411yescharacteristic
415C2×Q8C2×#6711yescharacteristic
416C2×D4—31no—
417C2×D4—121no—
418C2×D4C2×#6711yescharacteristic
419C22×C4—21no—
420C22×C4—41no—
421C22×C4—61no—
422C22×C4—121no—
423C22×C4—121no—
424Q16—122no—
425Q16—122no—
426Q16—241no—
427SD16—22no—
428SD16—34no—
429SD16—122no—
430M16—122no—
431C2×C8—21no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—122no—
439C2×C8C2×#6711yescharacteristic
440C2×C8C2×#6711yescharacteristic
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—31no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—121no—
456C4⋊C4—121no—
457C4⋊C4—121no—
458C4⋊C4D3×D411yescharacteristic
459C4⋊C4C2×#6711yescharacteristic
460C4⋊C4C2×#6711yescharacteristic
461C22⋊C4—21no—
462C22⋊C4—41no—
463C22⋊C4—61no—
464C22⋊C4—121no—
465C22⋊C4—121no—
466C42—21no—
467C42—21no—
468C42—41no—
469C42C23×D311yescharacteristic
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3D4211yescharacteristic
474C2×D3—22no—
475C2×D3—41no—
476C2×D3—41no—
477C3×C4—22no—
478C3×C4—21no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4C22×SD1611yescharacteristic
486C3×C4C2×#13711yescharacteristic
487C3×C4#57711yescharacteristic
488C3×C4#57811yescharacteristic
489Dic3—41no—
490Dic3—81no—
491Dic3—81no—
492C23—21no—
493C23—61no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—82no—
500Q8—81no—
501Q8—121no—
502Q8—121no—
503Q8—161no—
504Q8—241no—
505Q8D3×SD1612yes—
506D4—21no—
507D4—21no—
508D4—32no—
509D4—61no—
510D4—242no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—61no—
521C2×C4—82no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—242no—
527C2×C4—481no—
528C2×C4C2×D3×D411yescharacteristic
529C2×C4#100411yescharacteristic
530C2×C4C22×#6711yescharacteristic
531C8—21no—
532C8—32no—
533C8—41no—
534C8—61no—
535C8—121no—
536C8—121no—
537C8—121no—
538C8—121no—
539C8#94212yescomplemented
540C2×C3—81no—
541C2×C3#327611yescharacteristic
542C2×C3D4×SD1611yescharacteristic
543C2×C3#329011yescharacteristic
544D3—42no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—62no—
549C22—121no—
550C22—121no—
551C22D4×#6711yescharacteristic
552C4—21no—
553C4—22no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—121no—
561C4—241no—
562C4—241no—
563C4C2×D3×SD1611yescharacteristic
564C4#576211yescharacteristic
565C4C2×#94211yescharacteristic
566C4#583811yescharacteristic
567C3#1989511yescharacteristic, complemented, Sylow
568C2—81no—
569C2—122no—
570C2#8356011yescharacteristic
571C2SD16×#6711yescharacteristic
572C2#8401211yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×Q8×SD16C2
socleC22×C3D42
prime radical#5551C22
hypercenterQ8×SD16D3
hypocenterC3#19895
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SD16C2semidirect
2. #83706C2semidirect
5. #71941C2semidirect
6. #71915C2semidirect
7. #71344C2semidirect
8. #71333C2semidirect
10. #71137C2semidirect
11. #71125C2semidirect
12. #71121C2semidirect
13. #70943C2semidirect
14. C8×#215C2semidirect
15. #70899C2semidirect
69. Q8×SD16D3semidirect
259. #215SD16semidirect
397. C3×C8#137semidirect
539. C8#942semidirect
567. C3#19895semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SD162
#837062
#835602
#796882
#719412
#719152
#713442
#713332
#711402
#711372
#711252
#711212
#709432
C8×#2152
#708992
#198953