Finite Group Databasebeta

← #11664971

Subgroups — #11664971

Order 768.

1998 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3444C211yesmaximal, characteristic, complemented
2#85680C211yesmaximal, characteristic, complemented
3#83345C214yesmaximal, complemented
4#83340C211yesmaximal, characteristic, complemented
5#83335C211yesmaximal, characteristic, complemented
6#83292C214yesmaximal, complemented
7#83286C211yesmaximal, characteristic, complemented
8#83282C211yesmaximal, characteristic, complemented
9#83136C214yesmaximal, complemented
10#83135C214yesmaximal, complemented
11#82923C214yesmaximal, complemented
12C4×#950C214yesmaximal, complemented
13C2×C4×#208C211yesmaximal, characteristic, complemented
14#60676—31nomaximal, Sylow
15C3×#533C2212yes—
16C3×#533C2211yescharacteristic
17C3×#532C2211yescharacteristic
18C3×#532C2212yes—
19C3×#529—22no—
20C3×#529—22no—
21C3×#529C2214yes—
22C3×#525—22no—
23C3×#525C2212yes—
24C3×C4×#41—22no—
25C3×C4×#41C2212yes—
26C22×C3×C42C2211yescharacteristic
27C2×#950C2214yes—
28#5791—22no—
29#5791C2212yes—
30C22×#208C2212yes—
31#5479—22no—
32#5479—22no—
33#5479C2214yes—
34#5477C2211yescharacteristic
35#5477C2212yes—
36C4×#235—22no—
37C4×#235C2212yes—
38#5470C2211yescharacteristic
39#5470C2212yes—
40#5233C2214yes—
41#5089C2214yescomplemented
42#5086C2212yescomplemented
43#5082C2212yescomplemented
44#5081C2214yescomplemented
45#5074C2212yescomplemented
46#5071C2212yescomplemented
47#5069C2212yescomplemented
48#5067C2212yescomplemented
49#5064C2214yescomplemented
50#5060C2212yescomplemented
51#5058C2212yescomplemented
52#5055C2214yescomplemented
53#5052C2218yes—
54#5049C2218yes—
55#5042C2218yes—
56#5039C2218yes—
57C4×#216C2218yescomplemented
58C4×#215C2218yescomplemented
59C4×#214C2218yescomplemented
60C4×#213C2218yescomplemented
61#5002C2214yes—
62#4998C2214yes—
63C4×#208C2214yes—
64C4×#208C2214yes—
65C2×#803C2211yescharacteristic
66C2×#803C2211yescharacteristic
67C2×C4×C3⋊C8C2212yes—
68#3444—31no—
69#3444D311yescharacteristic, complemented
70#3172—34no—
71#3172—34no—
72#3171—34no—
73#3171—34no—
74#3162—31no—
75#3150—31no—
76#3135—31no—
77#3135—31no—
78#3054—34no—
79C4×#135—34no—
80C2×C4×M16—31no—
81C2×C3×#41—22no—
82C2×C3×#41C2312yes—
83C23×C3×C4D412yes—
84C23×C3×C4C2311yescharacteristic
85C2×#235—22no—
86C2×#235C2312yes—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2312yes—
89C3×#127C2311yescharacteristic
90C3×#126—24no—
91C3×#125—22no—
92C3×#125C2312yes—
93C3×#124—22no—
94C3×#124C2312yes—
95C3×#123—24no—
96C3×#123—22no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8C2314yes—
100C3×#121—22no—
101C3×#121—22no—
102C3×#121C2314yes—
103C3×#120—22no—
104C3×C4×Q8—22no—
105C3×C4×Q8—22no—
106C3×C4×Q8C2314yes—
107C3×C4×D4—24no—
108C3×C4×D4—24no—
109C3×C4×D4—24no—
110C3×C4×D4—22no—
111C3×C4×D4—22no—
112C3×C4×D4C2314yes—
113C3×#117—22no—
114C3×#117—24no—
115C3×#117—22no—
116C3×#117C2312yes—
117C2×C3×C42—22no—
118C2×C3×C42D412yes—
119C2×C3×C42D412yes—
120C2×C3×C42D412yes—
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C2×C3×C42C2312yes—
124C2×C3×C42C2312yes—
125#953—22no—
126#950—216no—
127C2×#216C2314yes—
128C2×#215C2314yes—
129C2×#214C2314yes—
130C2×#213C2314yes—
131#930—22no—
132#930—22no—
133#930C2314yes—
134#929—22no—
135#929—24no—
136C4×#67—24no—
137C4×#67—24no—
138C4×#67—24no—
139#926—22no—
140#926C2312yes—
141#924—22no—
142#924—22no—
143#924C2314yes—
144C2×#208—28no—
145C2×#208C2318yes—
146C22×C3⋊C8C2312yes—
147C22×C3⋊C8C2312yes—
148#876—24no—
149#875—22no—
150#875C2312yes—
151#874C2311yescharacteristic
152C4×D12—22no—
153C4×D12—22no—
154C4×D12C2314yes—
155#872—24no—
156#872—22no—
157C42×D3—22no—
158#870—22no—
159#870C2312yes—
160#869C2311yescharacteristic
161#869C2312yes—
162C4×Dic6—22no—
163C4×Dic6—22no—
164C4×Dic6C2314yes—
165#835C2318yes—
166#832C2318yes—
167#810C2318yes—
168#809C2318yes—
169#808C2318yes—
170#807C2318yes—
171#803C2312yes—
172#803C2312yes—
173#803C2312yes—
174#803C2312yes—
175C4×C3⋊C8C2318yes—
176C2×#135—34no—
177C22×M16—32no—
178#533—31no—
179#533—32no—
180#533C2×D312yes—
181#533C2×D311yescharacteristic
182#532—32no—
183#532—31no—
184#532C2×D311yescharacteristic
185#532C2×D312yes—
186#529—22no—
187#529—22no—
188#529—34no—
189#529—62no—
190#529—62no—
191#529C2×D314yes—
192#525—22no—
193#525—32no—
194#525—62no—
195#525C2×D312yes—
196C4×#41—22no—
197C4×#41—32no—
198C4×#41—62no—
199C4×#41C2×D312yes—
200C22×C42C2×D311yescharacteristic
201#501—34no—
202#501—34no—
203#497—32no—
204#497—32no—
205#496—32no—
206#496—32no—
207#491—32no—
208#491—32no—
209#490—32no—
210#490—32no—
211#489—34no—
212#489—34no—
213#470—38no—
214#469—38no—
215#468—38no—
216#468—38no—
217C4×Q16—38no—
218C4×SD16—38no—
219C4×SD16—38no—
220C4×D8—38no—
221#427—34no—
222#416—34no—
223#416—34no—
224C4×M16—34no—
225C4×M16—34no—
226C2×#97—31no—
227C2×#97—31no—
228C2×C4×C8—32no—
229C24×C3C2×D411yescharacteristic
230C3×#41—28no—
231C3×#41—48no—
232C2×C3×Q8—22no—
233C2×C3×Q8C2412yes—
234C2×C3×D4—22no—
235C2×C3×D4—24no—
236C2×C3×D4—22no—
237C2×C3×D4C2412yes—
238C22×C3×C4—24no—
239C22×C3×C4—22no—
240C22×C3×C4—24no—
241C22×C3×C4—44no—
242C22×C3×C4C2×D412yes—
243C22×C3×C4C2×D412yes—
244C22×C3×C4C2×D412yes—
245C22×C3×C4C2×D412yes—
246C22×C3×C4C2×D411yescharacteristic
247C22×C3×C4C2×D412yes—
248C22×C3×C4C2×D412yes—
249C22×C3×C4C2×D412yes—
250C22×C3×C4C2412yes—
251C22×C3×C4C2411yescharacteristic
252C2×#67—22no—
253C2×#67—24no—
254#235—28no—
255#235—48no—
256C2×D12—22no—
257C2×D12C2412yes—
258C2×C4×D3—22no—
259C2×C4×D3—24no—
260C2×Dic6—22no—
261C2×Dic6C2412yes—
262C3×C4⋊C4—22no—
263C3×C4⋊C4—22no—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—24no—
266C3×C4⋊C4—24no—
267C3×C4⋊C4C2414yes—
268C3×C22⋊C4—24no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—24no—
271C3×C22⋊C4—24no—
272C3×C22⋊C4—24no—
273C3×C42—24no—
274C3×C42—22no—
275C3×C42C2×D412yes—
276C3×C42C2×D412yes—
277C3×C42C2×D412yes—
278C3×C42C2×D411yescharacteristic
279C3×C42C2×D412yes—
280C3×C42C2×D412yes—
281C3×C42C2×D411yescharacteristic
282C3×C42C2412yes—
283C3×C42C2411yescharacteristic
284C3×C42C2411yescharacteristic
285#217—22no—
286#217—24no—
287#216—28no—
288#215—28no—
289#214—28no—
290#213—28no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—24no—
293(C2×D3)⋊C4—24no—
294C4⋊Dic3—22no—
295C4⋊Dic3—22no—
296C4⋊Dic3C2414yes—
297Dic3⋊C4—24no—
298Dic3⋊C4—24no—
299Dic3⋊C4—24no—
300C4×Dic3—22no—
301C4×Dic3—24no—
302#208—216no—
303C2×C3⋊C8—24no—
304C2×C3⋊C8#4118yescomplemented
305C2×C3⋊C8C2414yes—
306C2×C3⋊C8C2414yes—
307C2×#41—22no—
308C2×#41—32no—
309C2×#41—62no—
310C2×#41C22×D312yes—
311C23×C4#6712yes—
312C23×C4C22×D311yescharacteristic
313#135—616no—
314C2×Q16—34no—
315C2×SD16—34no—
316C2×SD16—34no—
317C2×D8—34no—
318C2×M16—38no—
319C2×M16—68no—
320C22×C8—32no—
321C22×C8—32no—
322C4⋊Q8—32no—
323C4⋊Q8—31no—
324C4⋊Q8C22×D311yescharacteristic
325C4⋊Q8C22×D312yes—
326#127—31no—
327#127C22×D311yescharacteristic
328#126—24no—
329#126—64no—
330#125—22no—
331#125—32no—
332#125—62no—
333#125C22×D312yes—
334#124—22no—
335#124—32no—
336#124—62no—
337#124C22×D312yes—
338#123—24no—
339#123—22no—
340#123—62no—
341#123—64no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—34no—
345C22⋊Q8—62no—
346C22⋊Q8—62no—
347C22⋊Q8C22×D314yes—
348#121—22no—
349#121—22no—
350#121—34no—
351#121—62no—
352#121—62no—
353#121C22×D314yes—
354#120—22no—
355#120—62no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—34no—
359C4×Q8—62no—
360C4×Q8—62no—
361C4×Q8C22×D314yes—
362C4×D4—24no—
363C4×D4—24no—
364C4×D4—24no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—34no—
368C4×D4—64no—
369C4×D4—64no—
370C4×D4—64no—
371C4×D4—62no—
372C4×D4—62no—
373C4×D4C22×D314yes—
374#117—22no—
375#117—24no—
376#117—22no—
377#117—32no—
378#117—62no—
379#117—62no—
380#117—64no—
381#117C22×D312yes—
382C2×C42—22no—
383C2×C42—62no—
384C2×C42#6712yes—
385C2×C42#6712yes—
386C2×C42#6712yes—
387C2×C42C22×D312yes—
388C2×C42C22×D312yes—
389C2×C42C22×D311yescharacteristic
390C2×C42C22×D311yescharacteristic
391#107—38no—
392#106—38no—
393#103—38no—
394#103—38no—
395#102—38no—
396#102—38no—
397#97—32no—
398#97—32no—
399#97—32no—
400#97—32no—
401C4×C8—38no—
402C23×C3—22no—
403C23×C3—22no—
404C23×C3—42no—
405C23×C3C22×D412yes—
406C23×C3C22×D411yescharacteristic
407C22×D3—22no—
408C3×Q8—24no—
409C3×Q8—44no—
410C3×D4—24no—
411C3×D4—44no—
412C3×D4—48no—
413C3×D4—44no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—24no—
417C2×C3×C4—24no—
418C2×C3×C4—24no—
419C2×C3×C4—24no—
420C2×C3×C4—22no—
421C2×C3×C4—24no—
422C2×C3×C4—42no—
423C2×C3×C4—42no—
424C2×C3×C4—42no—
425C2×C3×C4—44no—
426C2×C3×C4—48no—
427C2×C3×C4C22×D412yes—
428C2×C3×C4C22×D412yes—
429C2×C3×C4C22×D412yes—
430C2×C3×C4C22×D412yes—
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4C22×D412yes—
433C2×C3×C4C2×#4114yes—
434C2×C3×C4C2511yescharacteristic
435#67—48no—
436#67—44no—
437C2×Dic3—22no—
438C2×Dic3—24no—
439D12—24no—
440D12—44no—
441C4×D3—44no—
442C4×D3—48no—
443Dic6—24no—
444Dic6—44no—
445C3⋊C8—24no—
446C3⋊C8C2×#4118yes—
447C24C2×#6711yescharacteristic
448#41—28no—
449#41—48no—
450#41—68no—
451#41—128no—
452C2×Q8—22no—
453C2×Q8—32no—
454C2×Q8—62no—
455C2×Q8C23×D312yes—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—32no—
460C2×D4—64no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4C23×D312yes—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—44no—
468C22×C4—64no—
469C22×C4—62no—
470C22×C4C2×#6712yes—
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6712yes—
473C22×C4C2×#6712yes—
474C22×C4C2×#6712yes—
475C22×C4C2×#6712yes—
476C22×C4C2×#6712yes—
477C22×C4C2×#6712yes—
478C22×C4C23×D312yes—
479C22×C4C23×D311yescharacteristic
480Q16—68no—
481SD16—68no—
482SD16—68no—
483D8—68no—
484M16—616no—
485C2×C8—38no—
486C2×C8—34no—
487C2×C8—34no—
488C2×C8—64no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—34no—
495C4⋊C4—64no—
496C4⋊C4—62no—
497C4⋊C4—62no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4C23×D314yes—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—62no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—64no—
510C22⋊C4—64no—
511C42—24no—
512C42—22no—
513C42—62no—
514C42—64no—
515C42C2×#6711yescharacteristic
516C42C2×#6712yes—
517C42C2×#6712yes—
518C42C2×#6712yes—
519C42C2×#6712yes—
520C42C2×#6712yes—
521C42C2×#6711yescharacteristic
522C42C23×D312yes—
523C42C23×D311yescharacteristic
524C42C23×D311yescharacteristic
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—44no—
530C22×C3—42no—
531C22×C3—44no—
532C22×C3#53212yes—
533C22×C3C23×D411yescharacteristic
534C2×D3—42no—
535C2×D3—44no—
536C3×C4—22no—
537C3×C4—22no—
538C3×C4—22no—
539C3×C4—22no—
540C3×C4—42no—
541C3×C4—44no—
542C3×C4#53212yes—
543C3×C4#57514yes—
544C3×C4C22×#4112yes—
545Dic3—42no—
546Dic3—44no—
547C23—22no—
548C23—22no—
549C23—42no—
550C23—62no—
551C23C22×#6712yes—
552C23C22×#6711yescharacteristic
553Q8—24no—
554Q8—44no—
555Q8—64no—
556Q8—124no—
557D4—24no—
558D4—44no—
559D4—48no—
560D4—44no—
561D4—64no—
562D4—124no—
563D4—128no—
564D4—124no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—128no—
581C2×C4—124no—
582C2×C4D3×#4114yes—
583C2×C4C22×#6712yes—
584C2×C4C22×#6712yes—
585C2×C4C22×#6712yes—
586C2×C4C22×#6712yes—
587C2×C4C22×#6712yes—
588C2×C4C22×#6711yescharacteristic
589C2×C4C24×D311yescharacteristic
590C8—38no—
591C8—64no—
592C2×C3—22no—
593C2×C3—42no—
594C2×C3—82no—
595C2×C3C2×#53211yescharacteristic
596C2×C3C2×#57512yes—
597D3—82no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—124no—
606C22—122no—
607C22#583212yes—
608C22C23×#6711yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—44no—
614C4—42no—
615C4—124no—
616C4—122no—
617C4#582214yes—
618C4#583212yes—
619C4C2×D3×#4112yes—
620C3#6067611yescharacteristic, complemented, Sylow
621C2—22no—
622C2—42no—
623C2—82no—
624C2—242no—
625C2C2×#582212yes—
626C2C2×#583211yescharacteristic
subgroupG/H
centerC42C2×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3444C2
socleC22×C3C23×D4
hypercenter#3444D3
hypocenterC3#60676
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C42 ◁ C22×C42 ◁ #3444

Complements

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

NH ≅ G/Ntype
1. C3×#3444C2semidirect
2. #85680C2semidirect
3. #83345C2semidirect
4. #83340C2semidirect
5. #83335C2semidirect
6. #83292C2semidirect
7. #83286C2semidirect
8. #83282C2semidirect
9. #83136C2semidirect
10. #83135C2semidirect
11. #82923C2semidirect
12. C4×#950C2semidirect
13. C2×C4×#208C2semidirect
41. #5089C22semidirect
42. #5086C22semidirect
43. #5082C22semidirect
44. #5081C22semidirect
45. #5074C22semidirect
46. #5071C22semidirect
47. #5069C22semidirect
48. #5067C22semidirect
49. #5064C22semidirect
50. #5060C22semidirect
51. #5058C22semidirect
52. #5055C22semidirect
57. C4×#216C22semidirect
58. C4×#215C22semidirect
59. C4×#214C22semidirect
60. C4×#213C22semidirect
69. #3444D3semidirect
304. C2×C3⋊C8#41semidirect
620. C3#60676semidirect

Maximal subgroups

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

subgroupindex
C3×#34442
#856802
#833452
#833402
#833352
#832922
#832862
#832822
#831362
#831352
#829232
C4×#9502
C2×C4×#2082
#606763