Finite Group Databasebeta

← #11664509

Subgroups — #11664509

Order 768.

2014 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic
2#85725C211yesmaximal, characteristic, complemented
3C2×C3×#419C211yesmaximal, characteristic, complemented
4C2×#5225C211yesmaximal, characteristic, complemented
5#83768C211yesmaximal, characteristic
6#83747C211yesmaximal, characteristic, complemented
7#82213C211yesmaximal, characteristic
8#78954C214yesmaximal, complemented
9#78953C214yesmaximal, complemented
10#78952C218yesmaximal, complemented
11#78785C218yesmaximal, complemented
12#60620—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17#5816—22no—
18#5816—22no—
19#5816C2214yes—
20#5816C2214yes—
21C2×#947C2212yes—
22C2×#946—22no—
23C2×#944C2212yes—
24C2×#929—22no—
25C2×#926C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C22×#208C2211yescharacteristic
28#5519—22no—
29#5519C2214yescomplemented
30#5517—22no—
31#5517C2214yes—
32#5500C2214yescomplemented
33#5483—22no—
34#5470—22no—
35C3×#419C2214yes—
36C3×#419C2214yescomplemented
37C2×C3×#103C2214yescomplemented
38#5225C2214yes—
39#5225C2214yescomplemented
40C2×#835C2214yescomplemented
41#5133C2214yes—
42#5131C2212yes—
43#5122C2214yescomplemented
44#5120C2212yes—
45Dic3×M16C2214yes—
46#5113C2214yes—
47#5101C2214yescomplemented
48#5004C2212yes—
49#5002C2214yes—
50#5001C2212yes—
51#4966C2214yes—
52#4800C22116yescomplemented
53#4797C22116yescomplemented
54#4794C22116yescomplemented
55#4792C22116yescomplemented
56#3458—31no—
57C2×#536—31no—
58#3140—34no—
59#3139—34no—
60#3137—38no—
61#3137—38no—
62#2896—31no—
63#2896—31no—
64C2×#419—31no—
65C2×#419D311yescharacteristic, complemented
66#2873—31no—
67C22×C3×Q8—21no—
68C22×C3×Q8—21no—
69C22×C3×Q8—21no—
70C22×C3×Q8—24no—
71C22×C3×Q8—24no—
72C22×C3×Q8C2312yes—
73C22×C3×Q8C2312yes—
74C22×C3×Q8C2312yes—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C2×#235—21no—
79C2×#235—21no—
80C2×#235—24no—
81C2×#235—24no—
82C2×#235C2312yes—
83C2×#235C2312yes—
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×Dic6C2311yescharacteristic
87C2×C3×M16—24no—
88C2×C3×M16C2312yes—
89C2×C3×M16C2312yes—
90C22×C3×C8C2312yes—
91C3×#117—21no—
92C3×#117—21no—
93C3×#117C2312yes—
94C3×#117C2312yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×#217—21no—
99C2×#217D412yes—
100#947—24no—
101#947C2318yes—
102#946—28no—
103#946—28no—
104#944—24no—
105#944C2318yes—
106#929—28no—
107#929—28no—
108C2×(C2×D3)⋊C4—24no—
109#926—21no—
110#926—21no—
111#926D412yes—
112#926D412yes—
113#926C2312yes—
114#926C2312yes—
115#926C2312yes—
116#926C2312yes—
117C2×C4⋊Dic3D412yes—
118C2×C4⋊Dic3C2312yes—
119#924—28no—
120C2×Dic3⋊C4—24no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2312yes—
123C2×#208—24no—
124C2×#208C2312yes—
125C2×#208C2312yes—
126C22×C3⋊C8C2312yes—
127#890—24no—
128#890C2318yes—
129#888—24no—
130#888C2318yes—
131#878—28no—
132#878—28no—
133#870—24no—
134#869—24no—
135C4×Dic6—28no—
136C3×#103C23116yes—
137#835C23116yes—
138#821C2318yes—
139#816C2318yes—
140#815C2318yes—
141C8×Dic3C2318yes—
142#810C2318yes—
143#809C2318yes—
144C22×#41—31no—
145C23×Q8C2×D311yescharacteristic
146C22×M16—31no—
147C22×M16C2×D311yescharacteristic
148#544—34no—
149#544—34no—
150#544—62no—
151#543—62no—
152#536—34no—
153#536—34no—
154#536—62no—
155#536—62no—
156#533—34no—
157#533—62no—
158#533—62no—
159C2×C4⋊Q8—32no—
160C2×#124—32no—
161C2×#123—62no—
162C2×C22⋊Q8—62no—
163C2×#117—31no—
164C2×#117—31no—
165C2×#117C2×D311yescharacteristic
166#491—316no—
167#489—316no—
168#488—316no—
169#487—316no—
170#419—34no—
171#419—34no—
172#419—32no—
173#419—32no—
174#419C2×D314yes—
175#419C2×D314yescomplemented
176#418—32no—
177#418—32no—
178#417—34no—
179#417—34no—
180#416—34no—
181#416—34no—
182C2×#103—34no—
183C2×#103C2×D314yescomplemented
184#405—34no—
185#405—34no—
186C4×M16—34no—
187C24×C3C2×D411yescharacteristic
188C23×D3—21no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—24no—
193C2×C3×Q8—24no—
194C2×C3×Q8—24no—
195C2×C3×Q8—24no—
196C2×C3×Q8—24no—
197C2×C3×Q8—24no—
198C2×C3×Q8—44no—
199C2×C3×Q8—44no—
200C2×C3×Q8—44no—
201C2×C3×Q8C2414yes—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—44no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4#4114yes—
211C22×C3×C4#4114yes—
212C22×C3×C4C2411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C2×#67—22no—
216C2×#67—22no—
217C2×#67—44no—
218C22×Dic3—21no—
219C22×Dic3—22no—
220C22×Dic3C2×D414yes—
221#235—24no—
222#235—24no—
223#235—24no—
224#235—24no—
225#235—44no—
226#235—44no—
227C2×D12—22no—
228C2×D12—22no—
229C2×D12—21no—
230C2×D12C2412yes—
231C2×C4×D3—22no—
232C2×C4×D3—22no—
233C2×C4×D3—44no—
234C2×Dic6—22no—
235C2×Dic6—22no—
236C2×Dic6—21no—
237C2×Dic6—28no—
238C2×Dic6C2412yes—
239C3×M16—24no—
240C3×M16—24no—
241C2×C3×C8—24no—
242C2×C3×C8C2414yes—
243C3×C4⋊C4—22no—
244C3×C4⋊C4C2414yes—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C42—22no—
248C3×C42—22no—
249#217—22no—
250#217—22no—
251#217C2×D414yes—
252#217C2×D414yes—
253(C2×D3)⋊C4—216no—
254C4⋊Dic3—22no—
255C4⋊Dic3—28no—
256C4⋊Dic3C2×D414yes—
257C4⋊Dic3C2×D414yes—
258C4⋊Dic3C2414yes—
259Dic3⋊C4—28no—
260Dic3⋊C4—28no—
261Dic3⋊C4—216no—
262C4×Dic3—22no—
263C4×Dic3—22no—
264C4×Dic3C2×D414yes—
265C4×Dic3C2414yes—
266#208—24no—
267#208—24no—
268C2×C3⋊C8—24no—
269C2×C3⋊C8C2414yes—
270C2×#41—32no—
271C2×#41—32no—
272C2×#41—61no—
273C2×#41—61no—
274C2×#41—64no—
275C2×#41—64no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—24no—
280C22×Q8—24no—
281C22×Q8—31no—
282C22×Q8C22×D312yes—
283C22×Q8C22×D312yes—
284C22×Q8C22×D312yes—
285C22×D4—31no—
286C22×D4—61no—
287C23×C4—21no—
288C23×C4—61no—
289C23×C4C22×D311yescharacteristic
290C2×M16—24no—
291C2×M16—32no—
292C2×M16—32no—
293C2×M16—64no—
294C2×M16C22×D312yes—
295C2×M16C22×D312yes—
296C22×C8—32no—
297C22×C8C22×D312yes—
298C4⋊Q8—38no—
299C4⋊Q8—38no—
300C4⋊Q8—64no—
301C4⋊Q8—64no—
302C4⋊Q8—64no—
303#125—38no—
304#125—64no—
305#125—64no—
306#124—38no—
307#124—64no—
308#123—68no—
309#123—68no—
310C22⋊Q8—68no—
311C22⋊Q8—68no—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314C22⋊Q8—68no—
315C4×Q8—68no—
316#117—21no—
317#117—21no—
318#117—32no—
319#117—32no—
320#117—32no—
321#117—32no—
322#117—32no—
323#117—32no—
324#117—61no—
325#117—61no—
326#117C22×D312yes—
327#117C22×D312yes—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—64no—
331C2×C4⋊C4C22×D312yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—64no—
335C2×C22⋊C4—61no—
336C2×C42—21no—
337C2×C42—32no—
338C2×C42—61no—
339#103—316no—
340#103—38no—
341#103—38no—
342#103C22×D3116yes—
343#102—38no—
344#102—38no—
345#97—38no—
346C4×C8—38no—
347C23×C3—42no—
348C23×C3#12414yes—
349C23×C3C22×D411yescharacteristic
350C23×C3C22×D411yescharacteristic
351C23×C3C22×D411yescharacteristic
352C22×D3—22no—
353C22×D3—44no—
354C22×D3—42no—
355C3×Q8—24no—
356C3×Q8—24no—
357C3×Q8—44no—
358C3×Q8—44no—
359C3×Q8—44no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—44no—
363C2×C3×C4—44no—
364C2×C3×C4—44no—
365C2×C3×C4—82no—
366C2×C3×C4#12414yes—
367C2×C3×C4#12414yes—
368C2×C3×C4#12414yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4112yes—
373C2×C3×C4C2×#4112yes—
374C2×C3×C4C2×#4112yes—
375C2×C3×C4C2×#4112yes—
376C2×C3×C4C2×#4112yes—
377C2×C3×C4C2×#4112yes—
378C2×C3×C4C2511yescharacteristic
379#67—44no—
380#67—44no—
381C2×Dic3—24no—
382C2×Dic3—22no—
383C2×Dic3—44no—
384C2×Dic3—42no—
385C2×Dic3—82no—
386C2×Dic3C22×D418yes—
387D12—22no—
388D12—22no—
389D12—42no—
390C4×D3—44no—
391C4×D3—44no—
392Dic6—22no—
393Dic6—22no—
394Dic6—42no—
395Dic6—88no—
396C3×C8—24no—
397C3⋊C8—24no—
398C24—61no—
399C24D3×D411yescharacteristic
400#41—64no—
401#41—64no—
402#41—64no—
403#41—64no—
404#41—124no—
405#41—124no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—32no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—68no—
420C2×Q8—61no—
421C2×Q8—62no—
422C2×Q8—62no—
423C2×Q8C23×D314yes—
424C2×D4—32no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—61no—
429C2×D4—62no—
430C2×D4—124no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—34no—
436C22×C4—44no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—124no—
442C22×C4#23514yes—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4D3×D411yescharacteristic
446C22×C4#23714yes—
447C22×C4C23×D311yescharacteristic
448C22×C4C23×D311yescharacteristic
449C22×C4C23×D311yescharacteristic
450M16—24no—
451M16—24no—
452M16—64no—
453M16—64no—
454C2×C8—24no—
455C2×C8—34no—
456C2×C8—64no—
457C2×C8C23×D314yes—
458C4⋊C4—22no—
459C4⋊C4—34no—
460C4⋊C4—34no—
461C4⋊C4—34no—
462C4⋊C4—616no—
463C4⋊C4—68no—
464C4⋊C4—68no—
465C4⋊C4—68no—
466C4⋊C4—62no—
467C4⋊C4C23×D314yes—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—34no—
471C22⋊C4—34no—
472C22⋊C4—616no—
473C22⋊C4—62no—
474C22⋊C4—62no—
475C42—22no—
476C42—22no—
477C42—34no—
478C42—34no—
479C42—62no—
480C42—62no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—44no—
485C22×C3C2×#12412yes—
486C22×C3C2×#12412yes—
487C22×C3C2×#12412yes—
488C22×C3C23×D411yescharacteristic
489C2×D3—44no—
490C2×D3—42no—
491C2×D3—84no—
492C3×C4—44no—
493C3×C4—44no—
494C3×C4C2×#12412yes—
495C3×C4C2×#12412yes—
496C3×C4C2×#12412yes—
497C3×C4C22×#4111yescharacteristic
498C3×C4C22×#4111yescharacteristic
499Dic3—28no—
500Dic3—44no—
501Dic3—42no—
502C23—42no—
503C23—62no—
504C23—124no—
505C23—122no—
506C23#88514yes—
507C23C2×D3×D411yescharacteristic
508C23C2×D3×D411yescharacteristic
509C23C2×D3×D411yescharacteristic
510Q8—24no—
511Q8—24no—
512Q8—44no—
513Q8—44no—
514Q8—44no—
515Q8—62no—
516Q8—62no—
517Q8—122no—
518Q8—248no—
519D4—62no—
520D4—62no—
521D4—124no—
522D4—124no—
523D4—122no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—38no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—64no—
531C2×C4—62no—
532C2×C4—82no—
533C2×C4—124no—
534C2×C4—122no—
535C2×C4—124no—
536C2×C4—124no—
537C2×C4—242no—
538C2×C4#88514yes—
539C2×C4#88514yes—
540C2×C4#88514yes—
541C2×C4C2×#23512yes—
542C2×C4C2×#23512yes—
543C2×C4C2×#23512yes—
544C2×C4C2×D3×D411yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×#23712yes—
548C2×C4C2×#23712yes—
549C2×C4C2×#23712yes—
550C2×C4C24×D311yescharacteristic
551C8—24no—
552C8—64no—
553C2×C3—22no—
554C2×C3—22no—
555C2×C3—22no—
556C2×C3C22×#12411yescharacteristic
557C2×C3#358612yes—
558D3—82no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—44no—
563C22—124no—
564C22—122no—
565C22—244no—
566C22C2×#88512yes—
567C22C2×#88512yes—
568C22C2×#88512yes—
569C22C22×D3×D411yescharacteristic
570C4—44no—
571C4—44no—
572C4—68no—
573C4—124no—
574C4—122no—
575C4C2×#88512yes—
576C4C2×#88512yes—
577C4C2×#88512yes—
578C4C22×#23511yescharacteristic
579C4C22×#23711yescharacteristic
580C3#6062011yescharacteristic, complemented, Sylow
581C2—22no—
582C2—22no—
583C2—22no—
584C2—242no—
585C2C22×#88511yescharacteristic
586C2#8719712yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#419C2
socleC22×C3C23×D4
prime radicalC22×#235C22
hypercenterC2×#419D3
hypocenterC3#60620
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. #85725C2semidirect
3. C2×C3×#419C2semidirect
4. C2×#5225C2semidirect
6. #83747C2semidirect
8. #78954C2semidirect
9. #78953C2semidirect
10. #78952C2semidirect
11. #78785C2semidirect
29. #5519C22semidirect
32. #5500C22semidirect
36. C3×#419C22semidirect
37. C2×C3×#103C22semidirect
39. #5225C22semidirect
40. C2×#835C22semidirect
43. #5122C22semidirect
47. #5101C22semidirect
52. #4800C22semidirect
53. #4797C22semidirect
54. #4794C22semidirect
55. #4792C22semidirect
65. C2×#419D3semidirect
175. #419C2×D3semidirect
183. C2×#103C2×D3semidirect
580. C3#60620semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
#857252
C2×C3×#4192
C2×#52252
#837682
#837472
#822132
#789542
#789532
#789522
#787852
#606203