Finite Group Databasebeta

← #11665170

Subgroups — #11665170

Order 768.

1998 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86257C212yesmaximal, complemented
2C3×#3185C211yesmaximal, characteristic, complemented
3#83347C212yesmaximal, complemented
4#81891C211yesmaximal, characteristic, complemented
5#81888C211yesmaximal, characteristic, complemented
6#81799C211yesmaximal, characteristic, complemented
7#81794C211yesmaximal, characteristic, complemented
8#81598C212yesmaximal, complemented
9#81595C212yesmaximal, complemented
10#80903C214yesmaximal, complemented
11#80892C214yesmaximal, complemented
12#79828C214yesmaximal, complemented
13#79826C214yesmaximal, complemented
14#78535C211yesmaximal, characteristic, complemented
15C4×D3×M16C211yesmaximal, characteristic, complemented
16#60693—31nomaximal, Sylow
17C3×#533C2212yes—
18C2×#919C2214yes—
19C2×D3×M16C2212yes—
20#5728C2212yes—
21#5728C2212yes—
22#5725C2212yes—
23#5725C2212yes—
24#5691—22no—
25#5691—22no—
26#5627—22no—
27#5627C2214yes—
28#5624—22no—
29#5624C2214yes—
30#5580—22no—
31#5580C2214yes—
32C4×#239—22no—
33C4×#239C2214yes—
34#5524—22no—
35#5522—22no—
36#5477C2212yes—
37C2×C42×D3C2211yescharacteristic
38C3×#498C2211yescharacteristic, complemented
39C3×#494C2211yescharacteristic, complemented
40C3×#487C2212yescomplemented
41C3×#470C2214yes—
42C3×#441C2214yescomplemented
43C3×C4×M16C2211yescharacteristic
44#5191C2214yescomplemented
45#5189C2214yes—
46#5184C2214yescomplemented
47#5135C2214yes—
48Dic3×M16C2212yes—
49#5088C2212yescomplemented
50#5085C2212yescomplemented
51#5079C2212yescomplemented
52#5078C2212yescomplemented
53#5049C2218yes—
54#5026C2218yescomplemented
55C4×#208C2211yescharacteristic
56#4869C2214yescomplemented
57#4865C2214yes—
58#4860C2214yes—
59#4858C2214yescomplemented
60#4817C2218yescomplemented
61#4812C2218yes—
62#4804C2218yes—
63#4800C2218yescomplemented
64#4711C2214yescomplemented
65#4707C2212yes—
66D3×#97C2212yes—
67#4695C2211yescharacteristic, complemented
68#4694C2212yescomplemented
69C4×ΓL(1,𝔽25)C2211yescharacteristic
70C4×ΓL(1,𝔽25)C2211yescharacteristic
71C4×D3×C8C2211yescharacteristic
72C4×D3×C8C2211yescharacteristic
73#4680C2211yescharacteristic, complemented
74#3444—32no—
75#3185—31no—
76#3185D311yescharacteristic, complemented
77#3184—31no—
78#3184—31no—
79#3183—34no—
80#3183—34no—
81#3182—32no—
82#3182—32no—
83#3160—31no—
84#3147—31no—
85#3131—32no—
86#3054—34no—
87#2943—34no—
88C2×C4×M16—31no—
89C2×#239—22no—
90C2×#239C2314yes—
91C2×#235—22no—
92C22×C4×D3D412yes—
93C22×C4×D3C2311yescharacteristic
94C2×C3×Q16C2314yes—
95C2×C3×M16C2312yes—
96C3×C4⋊Q8C2312yes—
97C3×C4⋊Q8C2312yes—
98C3×#125—22no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8C2314yes—
101C3×C4×Q8—22no—
102C3×C4×Q8C2314yes—
103C3×#117—22no—
104C2×C3×C42C2311yescharacteristic
105#947—22no—
106#947C2314yes—
107#946—22no—
108#946C2314yes—
109Q8×Dic3—22no—
110Q8×Dic3C2314yes—
111C2×#215C2318yes—
112#930—22no—
113#930C2314yes—
114C2×C4×Dic3D412yes—
115C2×C4×Dic3C2311yescharacteristic
116C2×#208C2312yes—
117#919—216no—
118D3×M16—28no—
119C2×D24C2314yes—
120C2×ΓL(1,𝔽25)C2312yes—
121C2×ΓL(1,𝔽25)C2312yes—
122C2×D3×C8C2312yes—
123C2×D3×C8C2312yes—
124#898—24no—
125#898—24no—
126#897—22no—
127#897—22no—
128#895—22no—
129#895—22no—
130#895—22no—
131#895C2314yes—
132#894—24no—
133#894—24no—
134#893—24no—
135#893—24no—
136#893—22no—
137#893—22no—
138#893—22no—
139#893C2314yes—
140#892—22no—
141#892—24no—
142#892—24no—
143#892C2314yes—
144#890—22no—
145#890C2314yes—
146#887—22no—
147#887—22no—
148#886—24no—
149#882—24no—
150#67⋊C4—24no—
151#67⋊C4—24no—
152#875—22no—
153#874C2312yes—
154C4×D12—24no—
155C4×D12—24no—
156C4×D12—22no—
157C4×D12C2314yes—
158#872—24no—
159C42×D3—24no—
160C42×D3D412yes—
161C42×D3D412yes—
162C42×D3C2311yescharacteristic
163C42×D3C2311yescharacteristic
164C42×D3C2311yescharacteristic
165C42×D3C2311yescharacteristic
166#869C2312yes—
167C3×#106C2314yes—
168C3×#103C2318yes—
169C3×#97C2312yes—
170C3×C4×C8C2311yescharacteristic
171C3×C4×C8C2311yescharacteristic
172#835C2318yes—
173#821C2318yes—
174#817C2314yes—
175#815C2312yes—
176#815C2312yes—
177C8×Dic3C2312yes—
178C8×Dic3C2312yes—
179#809C2318yes—
180#807C2318yes—
181#803C2312yes—
182C4×C3⋊C8C2311yescharacteristic
183C4×C3⋊C8C2311yescharacteristic
184C2×#135—34no—
185C22×M16—32no—
186#533—32no—
187#533—32no—
188#533C2×D312yes—
189#532—32no—
190#532—32no—
191#532—32no—
192#529—34no—
193#529—34no—
194#529—62no—
195#529—62no—
196#529—62no—
197#529—62no—
198#525—34no—
199#525—62no—
200#525—62no—
201C4×#41—34no—
202C4×#41—62no—
203C4×#41—62no—
204C22×C42—31no—
205#500—32no—
206#499—34no—
207#498—31no—
208#498C2×D311yescharacteristic, complemented
209#495—34no—
210#494C2×D311yescharacteristic, complemented
211#493—31no—
212#492—32no—
213#488—38no—
214#487—32no—
215#487C2×D312yescomplemented
216#486—32no—
217#486—32no—
218#470—34no—
219#470C2×D314yes—
220#469—34no—
221#469—34no—
222#468—38no—
223#468—38no—
224#442—34no—
225#442—34no—
226#441—34no—
227#441C2×D314yescomplemented
228#440—38no—
229#440—38no—
230#427—34no—
231#416—38no—
232C4×M16—32no—
233C4×M16—31no—
234C4×M16—32no—
235C4×M16—31no—
236C4×M16—31no—
237C4×M16C2×D311yescharacteristic
238C2×#97—32no—
239C2×C4×C8—31no—
240C2×C4×C8—31no—
241C23×D3C2×D411yescharacteristic
242C2×C3×Q8—22no—
243C2×C3×Q8C2414yes—
244C22×C3×C4C2×D412yes—
245C22×C3×C4C2411yescharacteristic
246C2×#67—24no—
247C22×Dic3C2×D411yescharacteristic
248C22×Dic3C2×D412yes—
249#239—216no—
250#239—48no—
251#235—48no—
252C2×D12—24no—
253C2×D12—24no—
254C2×D12—22no—
255C2×D12—22no—
256C2×D12C2414yes—
257C2×C4×D3—24no—
258C2×C4×D3—24no—
259C2×C4×D3—24no—
260C2×C4×D3—24no—
261C2×C4×D3—44no—
262C2×C4×D3C2×D411yescharacteristic
263C2×C4×D3C2×D411yescharacteristic
264C2×C4×D3C2×D412yes—
265C2×C4×D3C2×D412yes—
266C2×C4×D3C2×D412yes—
267C2×C4×D3C2×D412yes—
268C2×C4×D3C2411yescharacteristic
269C2×C4×D3C2411yescharacteristic
270C2×Dic6—22no—
271C3×Q16—28no—
272C3×M16—24no—
273C2×C3×C8C2412yes—
274C2×C3×C8C2412yes—
275C3×C4⋊C4—22no—
276C3×C4⋊C4—22no—
277C3×C4⋊C4—22no—
278C3×C4⋊C4—24no—
279C3×C4⋊C4—24no—
280C3×C4⋊C4C2414yes—
281C3×C22⋊C4—24no—
282C3×C42—24no—
283C3×C42C2×D412yes—
284C3×C42C2411yescharacteristic
285C3×C42C2411yescharacteristic
286#215—216no—
287(C2×D3)⋊C4—24no—
288(C2×D3)⋊C4—24no—
289(C2×D3)⋊C4—24no—
290(C2×D3)⋊C4—24no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—24no—
293(C2×D3)⋊C4—24no—
294(C2×D3)⋊C4—24no—
295C4⋊Dic3—22no—
296C4⋊Dic3—24no—
297C4⋊Dic3—24no—
298C4⋊Dic3C2414yes—
299Dic3⋊C4—24no—
300Dic3⋊C4—24no—
301C4×Dic3—24no—
302C4×Dic3—24no—
303C4×Dic3C2×D412yes—
304C4×Dic3C2×D412yes—
305C4×Dic3C2×D412yes—
306C4×Dic3C2×D412yes—
307C4×Dic3C2×D412yes—
308C4×Dic3C2411yescharacteristic
309C4×Dic3C2411yescharacteristic
310#208—24no—
311C2×C3⋊C8C2412yes—
312C2×C3⋊C8C2412yes—
313D24—28no—
314ΓL(1,𝔽25)—24no—
315ΓL(1,𝔽25)—24no—
316D3×C8—24no—
317D3×C8#4118yescomplemented
318C2×#41—34no—
319C2×#41—62no—
320C2×#41—62no—
321C23×C4—31no—
322C23×C4—32no—
323#135—616no—
324C2×Q16C22×D314yes—
325C2×SD16—38no—
326C2×D8—34no—
327C2×M16—32no—
328C2×M16—32no—
329C2×M16—32no—
330C2×M16—68no—
331C2×M16C22×D312yes—
332C22×C8—32no—
333C22×C8—32no—
334C4⋊Q8—32no—
335C4⋊Q8C22×D312yes—
336C4⋊Q8C22×D312yes—
337#127—32no—
338#126—64no—
339#126—64no—
340#125—22no—
341#125—34no—
342#125—62no—
343#124—34no—
344#124—62no—
345#124—62no—
346#123—64no—
347#123—64no—
348#123—64no—
349C22⋊Q8—22no—
350C22⋊Q8—34no—
351C22⋊Q8—62no—
352C22⋊Q8—62no—
353C22⋊Q8—62no—
354C22⋊Q8C22×D314yes—
355#121—34no—
356#121—34no—
357#121—62no—
358#121—62no—
359#121—62no—
360#121—62no—
361#120—64no—
362C4×Q8—22no—
363C4×Q8—34no—
364C4×Q8—62no—
365C4×Q8—62no—
366C4×Q8—62no—
367C4×Q8C22×D314yes—
368C4×D4—34no—
369C4×D4—34no—
370C4×D4—64no—
371C4×D4—64no—
372C4×D4—62no—
373C4×D4—62no—
374C4×D4—62no—
375C4×D4—64no—
376C4×D4—64no—
377C4×D4—64no—
378C4×D4—64no—
379C4×D4—62no—
380#117—22no—
381#117—34no—
382#117—64no—
383#117—64no—
384#117—64no—
385#117—62no—
386C2×C42—32no—
387C2×C42—31no—
388C2×C42—32no—
389C2×C42—32no—
390C2×C42—31no—
391C2×C42—31no—
392C2×C42—31no—
393C2×C42—31no—
394C2×C42—64no—
395C2×C42C22×D311yescharacteristic
396#107—38no—
397#106—34no—
398#106C22×D314yes—
399#103—38no—
400#103C22×D318yes—
401#102—38no—
402#102—38no—
403#97—32no—
404#97—32no—
405#97—32no—
406#97C22×D312yes—
407C4×C8—32no—
408C4×C8—32no—
409C4×C8—31no—
410C4×C8—31no—
411C4×C8C22×D311yescharacteristic
412C4×C8C22×D311yescharacteristic
413C23×C3C22×D411yescharacteristic
414C22×D3—22no—
415C22×D3—24no—
416C22×D3—42no—
417C22×D3C22×D411yescharacteristic
418C22×D3C22×D411yescharacteristic
419C3×Q8—28no—
420C3×Q8—44no—
421C2×C3×C4—24no—
422C2×C3×C4—22no—
423C2×C3×C4—24no—
424C2×C3×C4—24no—
425C2×C3×C4C22×D411yescharacteristic
426C2×C3×C4C22×D412yes—
427C2×C3×C4C22×D412yes—
428C2×C3×C4C2511yescharacteristic
429#67—48no—
430C2×Dic3—22no—
431C2×Dic3—24no—
432C2×Dic3—42no—
433C2×Dic3C22×D412yes—
434C2×Dic3C22×D411yescharacteristic
435C2×Dic3C22×D412yes—
436C2×Dic3C22×D411yescharacteristic
437D12—28no—
438D12—48no—
439D12—48no—
440D12—44no—
441D12—44no—
442C4×D3—22no—
443C4×D3—24no—
444C4×D3—24no—
445C4×D3—22no—
446C4×D3—22no—
447C4×D3—42no—
448C4×D3—42no—
449C4×D3—48no—
450C4×D3—48no—
451C4×D3—48no—
452C4×D3C2×#4114yes—
453Dic6—44no—
454C3×C8—22no—
455C3×C8C2×#4114yes—
456C3⋊C8—22no—
457C3⋊C8C2×#4114yes—
458C24—31no—
459#41—616no—
460#41—128no—
461#41—128no—
462C2×Q8—22no—
463C2×Q8—62no—
464C2×Q8C23×D314yes—
465C2×D4—34no—
466C2×D4—64no—
467C2×D4—64no—
468C2×D4—64no—
469C2×D4—62no—
470C2×D4—62no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—31no—
476C22×C4—32no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—32no—
480C22×C4—32no—
481C22×C4—64no—
482C22×C4—64no—
483C22×C4—64no—
484C22×C4—64no—
485C22×C4—124no—
486C22×C4D3×D412yes—
487C22×C4C23×D311yescharacteristic
488Q16—28no—
489SD16—616no—
490D8—68no—
491M16—24no—
492M16—64no—
493M16—64no—
494M16—64no—
495C2×C8—32no—
496C2×C8—38no—
497C2×C8—32no—
498C2×C8—64no—
499C2×C8C23×D312yes—
500C2×C8C23×D312yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—34no—
507C4⋊C4—64no—
508C4⋊C4—64no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—62no—
512C4⋊C4C23×D314yes—
513C22⋊C4—24no—
514C22⋊C4—64no—
515C22⋊C4—64no—
516C22⋊C4—64no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—64no—
522C42—24no—
523C42—32no—
524C42—32no—
525C42—32no—
526C42—32no—
527C42—32no—
528C42—31no—
529C42—31no—
530C42—64no—
531C42—64no—
532C42D3×D412yes—
533C42C23×D311yescharacteristic
534C42C23×D311yescharacteristic
535C22×C3—21no—
536C22×C3—22no—
537C22×C3C23×D411yescharacteristic
538C2×D3—22no—
539C2×D3—21no—
540C2×D3—22no—
541C2×D3—44no—
542C2×D3—44no—
543C2×D3—44no—
544C2×D3—44no—
545C2×D3#53212yes—
546C3×C4—21no—
547C3×C4—22no—
548C3×C4—44no—
549C3×C4—44no—
550C3×C4#57514yes—
551C3×C4C22×#4112yes—
552Dic3—21no—
553Dic3—22no—
554Dic3—22no—
555Dic3—44no—
556Dic3#53212yes—
557C23—31no—
558C23—31no—
559C23—62no—
560C23—64no—
561C23—122no—
562C23C2×D3×D411yescharacteristic
563Q8—28no—
564Q8—44no—
565Q8—124no—
566D4—68no—
567D4—128no—
568D4—128no—
569D4—128no—
570D4—124no—
571D4—124no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—31no—
577C2×C4—34no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—31no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—64no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—128no—
591C2×C4—128no—
592C2×C4—128no—
593C2×C4—122no—
594C2×C4C2×D3×D412yes—
595C2×C4C2×D3×D411yescharacteristic
596C2×C4C2×D3×D412yes—
597C2×C4C24×D311yescharacteristic
598C8—22no—
599C8—34no—
600C8—62no—
601C8D3×#4114yes—
602C2×C3—41no—
603C2×C3C2×#53211yescharacteristic
604C2×C3C2×#57512yes—
605D3—22no—
606D3—41no—
607D3—84no—
608C22—22no—
609C22—21no—
610C22—32no—
611C22—62no—
612C22—62no—
613C22—61no—
614C22—124no—
615C22—124no—
616C22—124no—
617C22—124no—
618C22C22×D3×D411yescharacteristic
619C4—22no—
620C4—21no—
621C4—32no—
622C4—44no—
623C4—44no—
624C4—62no—
625C4—62no—
626C4—61no—
627C4—124no—
628C4#577714yes—
629C4C2×D3×#4112yes—
630C3#6069311yescharacteristic, complemented, Sylow
631C2—41no—
632C2—62no—
633C2—121no—
634C2—244no—
635C2D3×#53211yescharacteristic
636C2C2×#577712yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3185C2
socleC22×C3C23×D4
hypercenter#3185D3
hypocenterC3#60693
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3185

Complements

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

NH ≅ G/Ntype
1. #86257C2semidirect
2. C3×#3185C2semidirect
3. #83347C2semidirect
4. #81891C2semidirect
5. #81888C2semidirect
6. #81799C2semidirect
7. #81794C2semidirect
8. #81598C2semidirect
9. #81595C2semidirect
10. #80903C2semidirect
11. #80892C2semidirect
12. #79828C2semidirect
13. #79826C2semidirect
14. #78535C2semidirect
15. C4×D3×M16C2semidirect
38. C3×#498C22semidirect
39. C3×#494C22semidirect
40. C3×#487C22semidirect
42. C3×#441C22semidirect
44. #5191C22semidirect
46. #5184C22semidirect
49. #5088C22semidirect
50. #5085C22semidirect
51. #5079C22semidirect
52. #5078C22semidirect
54. #5026C22semidirect
56. #4869C22semidirect
59. #4858C22semidirect
60. #4817C22semidirect
63. #4800C22semidirect
64. #4711C22semidirect
67. #4695C22semidirect
68. #4694C22semidirect
73. #4680C22semidirect
76. #3185D3semidirect
208. #498C2×D3semidirect
210. #494C2×D3semidirect
215. #487C2×D3semidirect
227. #441C2×D3semidirect
317. D3×C8#41semidirect
630. C3#60693semidirect

Maximal subgroups

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

subgroupindex
#862572
C3×#31852
#833472
#818912
#818882
#817992
#817942
#815982
#815952
#809032
#808922
#798282
#798262
#785352
C4×D3×M162
#606933