Finite Group Databasebeta

← #11664505

Subgroups — #11664505

Order 768.

2206 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic
2#85725C211yesmaximal, characteristic, complemented
3C3×#2896C211yesmaximal, characteristic, complemented
4#84072C211yesmaximal, characteristic, complemented
5C2×#5131C211yesmaximal, characteristic, complemented
6#83747C211yesmaximal, characteristic, complemented
7C2×#5001C211yesmaximal, characteristic, complemented
8#78954C214yesmaximal, complemented
9#78921C214yesmaximal, complemented
10#78876C218yesmaximal, complemented
11#78827C218yesmaximal, complemented
12#60619—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17#5830—22no—
18#5830—22no—
19#5830C2214yes—
20#5830C2214yes—
21C2×#947C2212yes—
22C2×#940C2212yes—
23C2×#937—22no—
24C2×#930—22no—
25C2×#926C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C22×#208C2211yescharacteristic
28#5519—22no—
29#5519C2214yescomplemented
30#5517—22no—
31#5517C2214yescomplemented
32#5500C2214yescomplemented
33#5483—22no—
34#5470—22no—
35C3×#419C2212yescomplemented
36C3×#418C2212yescomplemented
37C3×#417C2214yescomplemented
38C3×#416C2214yescomplemented
39#5233C2214yescomplemented
40#5232C2214yescomplemented
41#5225C2212yescomplemented
42#5215C2212yescomplemented
43#5131C2214yes—
44#5131C2214yescomplemented
45#5122C2214yescomplemented
46Dic3×M16C2214yescomplemented
47C2×#821C2214yescomplemented
48#5101C2214yescomplemented
49#5001C2214yes—
50#5001C2214yescomplemented
51C2×#809C2214yescomplemented
52#4800C22116yescomplemented
53#4794C22116yescomplemented
54#4764C22116yescomplemented
55#4760C22116yescomplemented
56#3458—31no—
57C2×#535—31no—
58#3140—34no—
59#3138—34no—
60#3136—38no—
61#3136—38no—
62#2896—31no—
63#2896D311yescharacteristic, complemented
64C2×#418—31no—
65C2×#418—31no—
66#2873—31no—
67C24×D3—21no—
68C2×C3×#41—21no—
69C2×C3×#41—21no—
70C2×C3×#41—24no—
71C2×C3×#41—24no—
72C2×C3×#41C2312yes—
73C2×C3×#41C2312yes—
74C22×C3×Q8C2311yescharacteristic
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×#67—24no—
80C22×D12—24no—
81C22×D12—24no—
82C22×D12—21no—
83C22×D12—21no—
84C22×D12—21no—
85C22×D12C2312yes—
86C22×D12C2312yes—
87C22×D12C2312yes—
88C2×C3×M16—24no—
89C2×C3×M16C2312yes—
90C2×C3×M16C2312yes—
91C22×C3×C8C2312yes—
92C3×#117—21no—
93C3×#117—21no—
94C3×#117C2312yes—
95C3×#117C2312yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—21no—
99C2×#217—21no—
100C2×#217D412yes—
101#947—24no—
102#947C2318yes—
103#940—24no—
104#940C2318yes—
105#937—28no—
106#937—28no—
107#930—28no—
108#930—28no—
109C2×(C2×D3)⋊C4—24no—
110#926—21no—
111#926—21no—
112#926D412yes—
113#926D412yes—
114#926C2312yes—
115#926C2312yes—
116#926C2312yes—
117#926C2312yes—
118C2×C4⋊Dic3D412yes—
119C2×C4⋊Dic3C2312yes—
120#924—28no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2312yes—
123C2×#208—24no—
124C2×#208C2312yes—
125C2×#208C2312yes—
126C22×C3⋊C8C2312yes—
127#890—24no—
128#890C2318yescomplemented
129#888—24no—
130#888C2318yescomplemented
131#878—28no—
132#878—28no—
133#870—24no—
134#869—24no—
135C4×Dic6—28no—
136C3×#103C2318yescomplemented
137C3×#102C2318yescomplemented
138#835C2318yescomplemented
139#832C2318yescomplemented
140#821C23116yes—
141#815C2318yescomplemented
142C8×Dic3C2318yescomplemented
143#809C23116yes—
144C22×#41C2×D311yescharacteristic
145C23×D4—31no—
146C22×M16—31no—
147C22×M16C2×D311yescharacteristic
148#544—34no—
149#544—34no—
150#544—62no—
151#543—62no—
152#535—34no—
153#535—34no—
154#535—62no—
155#535—62no—
156#533—34no—
157#533—62no—
158#533—62no—
159C2×#127—32no—
160C2×#124—32no—
161C2×#121—62no—
162C2×#120—62no—
163C2×#117—31no—
164C2×#117—31no—
165C2×#117C2×D311yescharacteristic
166#490—316no—
167#489—316no—
168#488—316no—
169#486—316no—
170#419—32no—
171#419C2×D312yescomplemented
172#418—32no—
173#418—34no—
174#418—34no—
175#418—34no—
176#418—34no—
177#418C2×D312yescomplemented
178#417—34no—
179#417C2×D314yescomplemented
180#416—34no—
181#416C2×D314yescomplemented
182C2×#102—34no—
183C2×#102—34no—
184#405—34no—
185#405—34no—
186C4×M16—34no—
187C24×C3—21no—
188C24×C3C2×D411yescharacteristic
189C23×D3—22no—
190C23×D3—22no—
191C23×D3—22no—
192C23×D3—44no—
193C23×D3—84no—
194C3×#41—24no—
195C3×#41—24no—
196C3×#41—24no—
197C3×#41—24no—
198C3×#41—44no—
199C3×#41—44no—
200C2×C3×Q8—21no—
201C2×C3×Q8—22no—
202C2×C3×Q8—22no—
203C2×C3×Q8C2412yes—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—21no—
207C2×C3×D4—22no—
208C2×C3×D4—22no—
209C2×C3×D4—44no—
210C2×C3×D4C2412yes—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—44no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4#4114yes—
220C22×C3×C4#4114yes—
221C22×C3×C4C2411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C2×#67—216no—
225C2×#67—88no—
226C22×Dic3—22no—
227C22×Dic3C2×D414yes—
228C2×D12—24no—
229C2×D12—24no—
230C2×D12—24no—
231C2×D12—24no—
232C2×D12—24no—
233C2×D12—24no—
234C2×D12—22no—
235C2×D12—22no—
236C2×D12—22no—
237C2×D12—44no—
238C2×D12—44no—
239C2×D12—44no—
240C2×D12C2414yes—
241C2×Dic6—28no—
242C3×M16—24no—
243C3×M16—24no—
244C2×C3×C8—24no—
245C2×C3×C8C2414yes—
246C3×C4⋊C4—22no—
247C3×C4⋊C4C2414yes—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C42—22no—
251C3×C42—22no—
252#217—22no—
253#217—22no—
254#217C2×D414yes—
255#217C2×D414yes—
256(C2×D3)⋊C4—216no—
257C4⋊Dic3—22no—
258C4⋊Dic3—28no—
259C4⋊Dic3C2×D414yes—
260C4⋊Dic3C2×D414yes—
261C4⋊Dic3C2414yes—
262Dic3⋊C4—28no—
263Dic3⋊C4—28no—
264C4×Dic3—22no—
265C4×Dic3—22no—
266C4×Dic3C2×D414yes—
267C4×Dic3C2414yes—
268#208—24no—
269#208—24no—
270C2×C3⋊C8—24no—
271C2×C3⋊C8C2414yes—
272C25—61no—
273C2×#41—21no—
274C2×#41—21no—
275C2×#41—24no—
276C2×#41—24no—
277C2×#41C22×D312yes—
278C2×#41C22×D312yes—
279C22×Q8C22×D311yescharacteristic
280C22×D4—21no—
281C22×D4—32no—
282C22×D4—32no—
283C22×D4—32no—
284C22×D4—64no—
285C22×D4—64no—
286C22×D4—61no—
287C22×D4—61no—
288C22×D4—61no—
289C22×D4—64no—
290C22×D4C22×D311yescharacteristic
291C23×C4—21no—
292C23×C4C22×D311yescharacteristic
293C2×M16—24no—
294C2×M16—32no—
295C2×M16—32no—
296C2×M16—64no—
297C2×M16C22×D312yes—
298C2×M16C22×D312yes—
299C22×C8—32no—
300C22×C8C22×D312yes—
301C4⋊Q8—38no—
302C4⋊Q8—64no—
303C4⋊Q8—64no—
304#127—38no—
305#127—64no—
306#125—38no—
307#125—64no—
308#125—64no—
309#124—38no—
310#124—64no—
311C22⋊Q8—68no—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314#121—68no—
315#121—68no—
316#120—68no—
317#120—68no—
318C4×Q8—68no—
319#117—21no—
320#117—21no—
321#117—32no—
322#117—32no—
323#117—32no—
324#117—32no—
325#117—32no—
326#117—32no—
327#117—61no—
328#117—61no—
329#117C22×D312yes—
330#117C22×D312yes—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4C22×D312yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—64no—
337C2×C22⋊C4—61no—
338C2×C42—21no—
339C2×C42—32no—
340C2×C42—61no—
341#103—38no—
342#103C22×D318yescomplemented
343#102—38no—
344#102—316no—
345#102—316no—
346#102C22×D318yescomplemented
347#97—38no—
348C4×C8—38no—
349C23×C3—22no—
350C23×C3—42no—
351C23×C3—44no—
352C23×C3—42no—
353C23×C3#12414yes—
354C23×C3C22×D411yescharacteristic
355C23×C3C22×D411yescharacteristic
356C23×C3C22×D411yescharacteristic
357C22×D3—24no—
358C22×D3—48no—
359C22×D3—44no—
360C22×D3—44no—
361C22×D3—44no—
362C22×D3—84no—
363C22×D3—84no—
364C22×D3—816no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—42no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—44no—
371C3×D4—44no—
372C3×D4—42no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—42no—
376C2×C3×C4—44no—
377C2×C3×C4—44no—
378C2×C3×C4—82no—
379C2×C3×C4#12414yes—
380C2×C3×C4#12414yes—
381C2×C3×C4#12414yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4112yes—
386C2×C3×C4C2×#4112yes—
387C2×C3×C4C2×#4112yes—
388C2×C3×C4C2×#4112yes—
389C2×C3×C4C2×#4112yes—
390C2×C3×C4C2×#4112yes—
391C2×C3×C4C2511yescharacteristic
392#67—816no—
393C2×Dic3—24no—
394C2×Dic3—44no—
395C2×Dic3—82no—
396C2×Dic3C22×D418yes—
397D12—24no—
398D12—24no—
399D12—44no—
400D12—44no—
401D12—44no—
402Dic6—88no—
403C3×C8—24no—
404C3⋊C8—24no—
405C24—21no—
406C24—62no—
407C24—62no—
408C24—62no—
409C24—124no—
410C24—244no—
411C24D3×D411yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—44no—
417#41—44no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—68no—
422C2×Q8C23×D312yes—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—34no—
429C2×D4—44no—
430C2×D4—616no—
431C2×D4—64no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—64no—
440C2×D4—124no—
441C2×D4—124no—
442C2×D4—124no—
443C2×D4—248no—
444C2×D4C23×D312yes—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—34no—
450C22×C4—44no—
451C22×C4—62no—
452C22×C4#23514yes—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4D3×D411yescharacteristic
456C22×C4#23714yes—
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D311yescharacteristic
459C22×C4C23×D311yescharacteristic
460M16—24no—
461M16—24no—
462M16—64no—
463M16—64no—
464C2×C8—24no—
465C2×C8—34no—
466C2×C8—64no—
467C2×C8C23×D314yes—
468C4⋊C4—22no—
469C4⋊C4—34no—
470C4⋊C4—34no—
471C4⋊C4—34no—
472C4⋊C4—68no—
473C4⋊C4—68no—
474C4⋊C4—68no—
475C4⋊C4—62no—
476C4⋊C4C23×D314yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—34no—
480C22⋊C4—34no—
481C22⋊C4—616no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C42—22no—
485C42—22no—
486C42—34no—
487C42—34no—
488C42—62no—
489C42—62no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—44no—
494C22×C3—44no—
495C22×C3—42no—
496C22×C3—84no—
497C22×C3C2×#12412yes—
498C22×C3C2×#12412yes—
499C22×C3C2×#12412yes—
500C22×C3C23×D411yescharacteristic
501C2×D3—48no—
502C2×D3—44no—
503C2×D3—88no—
504C2×D3—88no—
505C2×D3—88no—
506C3×C4—44no—
507C3×C4—42no—
508C3×C4C2×#12412yes—
509C3×C4C2×#12412yes—
510C3×C4C2×#12412yes—
511C3×C4C22×#4111yescharacteristic
512C3×C4C22×#4111yescharacteristic
513Dic3—28no—
514Dic3—44no—
515C23—22no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—64no—
520C23—128no—
521C23—124no—
522C23—124no—
523C23—124no—
524C23—244no—
525C23—244no—
526C23—2416no—
527C23#88514yes—
528C23C2×D3×D411yescharacteristic
529C23C2×D3×D411yescharacteristic
530C23C2×D3×D411yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—42no—
534Q8—248no—
535D4—22no—
536D4—22no—
537D4—44no—
538D4—44no—
539D4—42no—
540D4—64no—
541D4—64no—
542D4—124no—
543D4—124no—
544D4—124no—
545D4—2416no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—38no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—64no—
553C2×C4—82no—
554C2×C4—124no—
555C2×C4—242no—
556C2×C4#88514yes—
557C2×C4#88514yes—
558C2×C4#88514yes—
559C2×C4C2×#23512yes—
560C2×C4C2×#23512yes—
561C2×C4C2×#23512yes—
562C2×C4C2×D3×D411yescharacteristic
563C2×C4C2×D3×D411yescharacteristic
564C2×C4C2×D3×D411yescharacteristic
565C2×C4C2×#23712yes—
566C2×C4C2×#23712yes—
567C2×C4C2×#23712yes—
568C2×C4C24×D311yescharacteristic
569C8—24no—
570C8—64no—
571C2×C3—22no—
572C2×C3—22no—
573C2×C3—22no—
574C2×C3—82no—
575C2×C3C22×#12411yescharacteristic
576C2×C3#358512yes—
577D3—84no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—84no—
585C22—128no—
586C22—124no—
587C22—248no—
588C22—248no—
589C22—248no—
590C22C2×#88512yes—
591C22C2×#88512yes—
592C22C2×#88512yes—
593C22C22×D3×D411yescharacteristic
594C4—44no—
595C4—42no—
596C4—68no—
597C4—124no—
598C4C2×#88512yes—
599C4C2×#88512yes—
600C4C2×#88512yes—
601C4C22×#23511yescharacteristic
602C4C22×#23711yescharacteristic
603C3#6061911yescharacteristic, complemented, Sylow
604C2—22no—
605C2—22no—
606C2—22no—
607C2—82no—
608C2—244no—
609C2C22×#88511yescharacteristic
610C2#8720012yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2896C2
socleC22×C3C23×D4
prime radicalC2×#5830C2
hypercenter#2896D3
hypocenterC3#60619
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. #85725C2semidirect
3. C3×#2896C2semidirect
4. #84072C2semidirect
5. C2×#5131C2semidirect
6. #83747C2semidirect
7. C2×#5001C2semidirect
8. #78954C2semidirect
9. #78921C2semidirect
10. #78876C2semidirect
11. #78827C2semidirect
29. #5519C22semidirect
31. #5517C22semidirect
32. #5500C22semidirect
35. C3×#419C22semidirect
36. C3×#418C22semidirect
37. C3×#417C22semidirect
38. C3×#416C22semidirect
39. #5233C22semidirect
40. #5232C22semidirect
41. #5225C22semidirect
42. #5215C22semidirect
44. #5131C22semidirect
45. #5122C22semidirect
46. Dic3×M16C22semidirect
47. C2×#821C22semidirect
48. #5101C22semidirect
50. #5001C22semidirect
51. C2×#809C22semidirect
52. #4800C22semidirect
53. #4794C22semidirect
54. #4764C22semidirect
55. #4760C22semidirect
63. #2896D3semidirect
128. #890C23semidirect
130. #888C23semidirect
136. C3×#103C23semidirect
137. C3×#102C23semidirect
138. #835C23semidirect
139. #832C23semidirect
141. #815C23semidirect
142. C8×Dic3C23semidirect
171. #419C2×D3semidirect
177. #418C2×D3semidirect
179. #417C2×D3semidirect
181. #416C2×D3semidirect
342. #103C22×D3semidirect
346. #102C22×D3semidirect
603. C3#60619semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
#857252
C3×#28962
#840722
C2×#51312
#837472
C2×#50012
#789542
#789212
#788762
#788272
#606193