Finite Group Databasebeta

← #11664501

Subgroups — #11664501

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×#5805C211yesmaximal, characteristic
2#85725C211yesmaximal, characteristic, complemented
3C2×C3×#418C211yesmaximal, characteristic, complemented
4C2×#5215C211yesmaximal, characteristic, complemented
5#83768C211yesmaximal, characteristic, complemented
6#83747C211yesmaximal, characteristic, complemented
7#82213C211yesmaximal, characteristic, complemented
8#78922C214yesmaximal, complemented
9#78921C214yesmaximal, complemented
10#78920C218yesmaximal, complemented
11#78737C218yesmaximal, complemented
12#60619—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#953—22no—
18#5805—22no—
19#5805—22no—
20#5805C2214yes—
21#5805C2214yes—
22C2×#940C2212yes—
23C2×#938—22no—
24C2×#936C2212yes—
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×#418C2214yes—
36C3×#418C2214yescomplemented
37C2×C3×#102C2214yescomplemented
38#5215C2214yes—
39#5215C2214yescomplemented
40C2×#832C2214yescomplemented
41#5133C2214yescomplemented
42#5131C2212yescomplemented
43#5122C2214yescomplemented
44#5120C2212yescomplemented
45Dic3×M16C2214yescomplemented
46#5113C2214yescomplemented
47#5101C2214yescomplemented
48#5004C2212yescomplemented
49#5002C2214yescomplemented
50#5001C2212yescomplemented
51#4966C2214yescomplemented
52#4768C22116yescomplemented
53#4765C22116yescomplemented
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#2896—31no—
64C2×#418—31no—
65C2×#418D311yescharacteristic, complemented
66#2873—31no—
67C25×C3—21no—
68C22×C3×D4—21no—
69C22×C3×D4—21no—
70C22×C3×D4—21no—
71C22×C3×D4—24no—
72C22×C3×D4—24no—
73C22×C3×D4C2312yes—
74C22×C3×D4C2312yes—
75C22×C3×D4C2312yes—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C22×#67—24no—
79C2×#235—21no—
80C2×#235—21no—
81C2×#235—24no—
82C2×#235—24no—
83C2×#235C2312yes—
84C2×#235C2312yes—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×Dic6C2311yescharacteristic
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—
99#953—28no—
100#953—28no—
101C2×#217—21no—
102C2×#217—24no—
103C2×#217D412yes—
104#940—24no—
105#940C2318yes—
106#938—28no—
107#938—28no—
108#936—24no—
109#936C2318yes—
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×#102C23116yes—
137#832C23116yes—
138#821C2318yescomplemented
139#816C2318yescomplemented
140#815C2318yescomplemented
141C8×Dic3C2318yescomplemented
142#810C2318yescomplemented
143#809C2318yescomplemented
144C22×#41—31no—
145C23×D4C2×D311yescharacteristic
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#419—32no—
172#418—34no—
173#418—34no—
174#418—32no—
175#418—32no—
176#418C2×D314yes—
177#418C2×D314yescomplemented
178#417—34no—
179#417—34no—
180#416—34no—
181#416—34no—
182C2×#102—34no—
183C2×#102C2×D314yescomplemented
184#405—34no—
185#405—34no—
186C4×M16—34no—
187C24×C3—22no—
188C24×C3—22no—
189C24×C3—22no—
190C24×C3—44no—
191C24×C3—84no—
192C24×C3C2×D411yescharacteristic
193C23×D3—21no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—24no—
198C2×C3×D4—24no—
199C2×C3×D4—24no—
200C2×C3×D4—24no—
201C2×C3×D4—24no—
202C2×C3×D4—24no—
203C2×C3×D4—44no—
204C2×C3×D4—44no—
205C2×C3×D4—44no—
206C2×C3×D4C2414yes—
207C22×C3×C4—22no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4#4114yes—
212C22×C3×C4#4114yes—
213C22×C3×C4C2411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C2×#67—22no—
217C2×#67—22no—
218C2×#67—216no—
219C2×#67—44no—
220C2×#67—88no—
221C22×Dic3—21no—
222C22×Dic3—22no—
223C22×Dic3C2×D414yes—
224#235—24no—
225#235—24no—
226#235—24no—
227#235—24no—
228#235—44no—
229#235—44no—
230C2×D12—22no—
231C2×D12—22no—
232C2×D12—21no—
233C2×D12C2412yes—
234C2×C4×D3—22no—
235C2×C4×D3—22no—
236C2×C4×D3—44no—
237C2×Dic6—22no—
238C2×Dic6—22no—
239C2×Dic6—21no—
240C2×Dic6—28no—
241C2×Dic6C2412yes—
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#217—216no—
255#217C2×D414yes—
256#217C2×D414yes—
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—21no—
273C2×#41—32no—
274C2×#41—32no—
275C2×#41—61no—
276C2×#41—61no—
277C2×#41—64no—
278C2×#41—64no—
279C22×Q8—31no—
280C22×D4—24no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4—21no—
284C22×D4—24no—
285C22×D4—31no—
286C22×D4—64no—
287C22×D4—61no—
288C22×D4C22×D312yes—
289C22×D4C22×D312yes—
290C22×D4C22×D312yes—
291C23×C4—61no—
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#103—38no—
343#102—316no—
344#102—38no—
345#102—38no—
346#102C22×D3116yes—
347#97—38no—
348C4×C8—38no—
349C23×C3—24no—
350C23×C3—42no—
351C23×C3—48no—
352C23×C3—44no—
353C23×C3—44no—
354C23×C3—44no—
355C23×C3—84no—
356C23×C3—84no—
357C23×C3—816no—
358C23×C3#12414yes—
359C23×C3C22×D411yescharacteristic
360C23×C3C22×D411yescharacteristic
361C23×C3C22×D411yescharacteristic
362C22×D3—22no—
363C22×D3—44no—
364C22×D3—42no—
365C3×D4—24no—
366C3×D4—24no—
367C3×D4—44no—
368C3×D4—44no—
369C3×D4—44no—
370C2×C3×C4—24no—
371C2×C3×C4—82no—
372C2×C3×C4#12414yes—
373C2×C3×C4#12414yes—
374C2×C3×C4#12414yes—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4112yes—
379C2×C3×C4C2×#4112yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C4C2×#4112yes—
383C2×C3×C4C2×#4112yes—
384C2×C3×C4C2511yescharacteristic
385#67—44no—
386#67—44no—
387#67—816no—
388C2×Dic3—24no—
389C2×Dic3—22no—
390C2×Dic3—44no—
391C2×Dic3—42no—
392C2×Dic3—82no—
393C2×Dic3C22×D418yes—
394D12—22no—
395D12—22no—
396D12—42no—
397C4×D3—44no—
398C4×D3—44no—
399Dic6—22no—
400Dic6—22no—
401Dic6—42no—
402Dic6—88no—
403C3×C8—24no—
404C3⋊C8—24no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—44no—
409C24—61no—
410C24—84no—
411C24D3×D411yescharacteristic
412#41—64no—
413#41—64no—
414#41—64no—
415#41—64no—
416#41—124no—
417#41—124no—
418C2×Q8—32no—
419C2×Q8—68no—
420C2×Q8—61no—
421C2×Q8—62no—
422C2×Q8—62no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—32no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—616no—
437C2×D4—62no—
438C2×D4—62no—
439C2×D4—62no—
440C2×D4—61no—
441C2×D4—62no—
442C2×D4—124no—
443C2×D4—248no—
444C2×D4C23×D314yes—
445C22×C4—22no—
446C22×C4—34no—
447C22×C4—61no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—124no—
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—48no—
495C22×C3—44no—
496C22×C3—88no—
497C22×C3—88no—
498C22×C3—88no—
499C22×C3C2×#12412yes—
500C22×C3C2×#12412yes—
501C22×C3C2×#12412yes—
502C22×C3C23×D411yescharacteristic
503C2×D3—44no—
504C2×D3—42no—
505C2×D3—84no—
506C3×C4—44no—
507C3×C4C2×#12412yes—
508C3×C4C2×#12412yes—
509C3×C4C2×#12412yes—
510C3×C4C22×#4111yescharacteristic
511C3×C4C22×#4111yescharacteristic
512Dic3—28no—
513Dic3—44no—
514Dic3—42no—
515C23—24no—
516C23—42no—
517C23—48no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—62no—
522C23—84no—
523C23—84no—
524C23—816no—
525C23—124no—
526C23—122no—
527C23#88514yes—
528C23C2×D3×D411yescharacteristic
529C23C2×D3×D411yescharacteristic
530C23C2×D3×D411yescharacteristic
531Q8—62no—
532Q8—62no—
533Q8—122no—
534Q8—248no—
535D4—24no—
536D4—24no—
537D4—44no—
538D4—44no—
539D4—44no—
540D4—62no—
541D4—62no—
542D4—124no—
543D4—124no—
544D4—122no—
545D4—2416no—
546C2×C4—24no—
547C2×C4—38no—
548C2×C4—64no—
549C2×C4—62no—
550C2×C4—82no—
551C2×C4—124no—
552C2×C4—122no—
553C2×C4—124no—
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—84no—
575C2×C3C22×#12411yescharacteristic
576C2×C3#358512yes—
577D3—82no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—44no—
582C22—48no—
583C22—44no—
584C22—88no—
585C22—88no—
586C22—88no—
587C22—124no—
588C22—122no—
589C22—244no—
590C22C2×#88512yes—
591C22C2×#88512yes—
592C22C2×#88512yes—
593C22C22×D3×D411yescharacteristic
594C4—44no—
595C4—68no—
596C4—124no—
597C4—122no—
598C4C2×#88512yes—
599C4C2×#88512yes—
600C4C2×#88512yes—
601C4C22×#23511yescharacteristic
602C4C22×#23711yescharacteristic
603C3#6061911yescharacteristic, complemented, Sylow
604C2—22no—
605C2—22no—
606C2—22no—
607C2—84no—
608C2—242no—
609C2C22×#88511yescharacteristic
610C2#8719212yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#418C2
socleC22×C3C23×D4
prime radicalC2×#5805C2
hypercenterC2×#418D3
hypocenterC3#60619
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. #85725C2semidirect
3. C2×C3×#418C2semidirect
4. C2×#5215C2semidirect
5. #83768C2semidirect
6. #83747C2semidirect
7. #82213C2semidirect
8. #78922C2semidirect
9. #78921C2semidirect
10. #78920C2semidirect
11. #78737C2semidirect
29. #5519C22semidirect
31. #5517C22semidirect
32. #5500C22semidirect
36. C3×#418C22semidirect
37. C2×C3×#102C22semidirect
39. #5215C22semidirect
40. C2×#832C22semidirect
41. #5133C22semidirect
42. #5131C22semidirect
43. #5122C22semidirect
44. #5120C22semidirect
45. Dic3×M16C22semidirect
46. #5113C22semidirect
47. #5101C22semidirect
48. #5004C22semidirect
49. #5002C22semidirect
50. #5001C22semidirect
51. #4966C22semidirect
52. #4768C22semidirect
53. #4765C22semidirect
54. #4764C22semidirect
55. #4760C22semidirect
65. C2×#418D3semidirect
128. #890C23semidirect
130. #888C23semidirect
138. #821C23semidirect
139. #816C23semidirect
140. #815C23semidirect
141. C8×Dic3C23semidirect
142. #810C23semidirect
143. #809C23semidirect
177. #418C2×D3semidirect
183. C2×#102C2×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×#58052
#857252
C2×C3×#4182
C2×#52152
#837682
#837472
#822132
#789222
#789212
#789202
#787372
#606193