Finite Group Databasebeta

← #11665165

Subgroups — #11665165

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#86219C212yesmaximal, complemented
2C3×#3184C211yesmaximal, characteristic, complemented
3#83294C212yesmaximal, complemented
4#81891C211yesmaximal, characteristic, complemented
5#81889C211yesmaximal, characteristic, complemented
6#81782C211yesmaximal, characteristic, complemented
7#81777C211yesmaximal, characteristic, complemented
8#81562C212yesmaximal, complemented
9#81559C212yesmaximal, complemented
10#80863C214yesmaximal, complemented
11#80852C214yesmaximal, complemented
12#79864C214yesmaximal, complemented
13#79862C214yesmaximal, complemented
14#78540C211yesmaximal, characteristic, complemented
15C4×D3×M16C211yesmaximal, characteristic, complemented
16#60693—31nomaximal, Sylow
17C3×#532C2212yes—
18#5828—22no—
19Dic3×#41—22no—
20C2×#912C2214yes—
21C2×D3×M16C2212yes—
22#5728C2212yes—
23#5728C2212yes—
24#5716C2212yes—
25#5716C2212yes—
26#5680—22no—
27#5680—22no—
28#5606—22no—
29#5606C2214yes—
30#5600—22no—
31#5600C2214yes—
32#5557—22no—
33#5557C2214yes—
34C4×#237—22no—
35C4×#237C2214yes—
36C2×C42×D3C2211yescharacteristic
37#5470C2212yes—
38C3×#498C2211yescharacteristic, complemented
39C3×#493C2211yescharacteristic, complemented
40C3×#486C2212yescomplemented
41C3×#469C2214yes—
42C3×#442C2214yescomplemented
43C3×C4×M16C2211yescharacteristic
44#5159C2214yes—
45#5155C2214yescomplemented
46#5152C2214yescomplemented
47#5137C2214yes—
48Dic3×M16C2212yes—
49#5093C2212yescomplemented
50#5078C2212yescomplemented
51#5076C2212yescomplemented
52#5070C2212yescomplemented
53#5042C2218yes—
54#5018C2218yescomplemented
55C4×#208C2211yescharacteristic
56#4867C2214yes—
57#4860C2214yes—
58#4858C2214yescomplemented
59#4853C2214yescomplemented
60#4780C2218yes—
61#4772C2218yes—
62#4768C2218yescomplemented
63#4757C2218yescomplemented
64#4716C2214yescomplemented
65#4707C2212yes—
66D3×#97C2212yes—
67#4699C2211yescharacteristic, complemented
68C4×ΓL(1,𝔽25)C2211yescharacteristic
69C4×ΓL(1,𝔽25)C2211yescharacteristic
70C4×D3×C8C2211yescharacteristic
71C4×D3×C8C2211yescharacteristic
72#4681C2212yescomplemented
73#4680C2211yescharacteristic, complemented
74#3444—32no—
75#3185—31no—
76#3185—31no—
77#3184—31no—
78#3184D311yescharacteristic, complemented
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×C3×#41—22no—
90C2×#237—22no—
91C2×#237C2314yes—
92C22×C4×D3D412yes—
93C22×C4×D3C2311yescharacteristic
94C2×C3×D8C2314yes—
95C2×C3×M16C2312yes—
96C3×C4⋊Q8C2312yes—
97C3×#127C2312yes—
98C3×#124—22no—
99C3×#121—22no—
100C3×#121C2314yes—
101C3×C4×D4—22no—
102C3×C4×D4C2314yes—
103C2×C3×C42C2311yescharacteristic
104#946—22no—
105#946—22no—
106Q8×Dic3—22no—
107Q8×Dic3—22no—
108#938—22no—
109#938—22no—
110#938—22no—
111#938C2314yes—
112#937—24no—
113#936—22no—
114#936C2314yes—
115#935—24no—
116D4×Dic3—24no—
117D4×Dic3—24no—
118D4×Dic3—22no—
119D4×Dic3—22no—
120D4×Dic3—22no—
121D4×Dic3C2314yes—
122C2×#214C2318yes—
123C4×#67—24no—
124C4×#67—24no—
125#926—22no—
126#926—24no—
127#924—22no—
128#924C2314yes—
129C2×C4×Dic3—24no—
130C2×C4×Dic3D412yes—
131C2×C4×Dic3C2311yescharacteristic
132C2×#208C2312yes—
133#912—216no—
134D3×M16—28no—
135C2×Dic12C2314yes—
136C2×ΓL(1,𝔽25)C2312yes—
137C2×ΓL(1,𝔽25)C2312yes—
138C2×D3×C8C2312yes—
139C2×D3×C8C2312yes—
140#897—22no—
141#897C2314yes—
142#892—22no—
143#892C2314yes—
144#890—22no—
145#890C2314yes—
146#887—22no—
147#887C2314yes—
148#883—24no—
149#883—24no—
150#67⋊C4—24no—
151#67⋊C4—24no—
152#879—24no—
153#879—24no—
154#877—24no—
155#877—24no—
156C42×D3D412yes—
157C42×D3D412yes—
158C42×D3C2311yescharacteristic
159C42×D3C2311yescharacteristic
160C42×D3C2311yescharacteristic
161C42×D3C2311yescharacteristic
162#870—22no—
163#869C2312yes—
164#869C2312yes—
165C4×Dic6—22no—
166C4×Dic6C2314yes—
167C3×#106C2314yes—
168C3×#102C2318yes—
169C3×#97C2312yes—
170C3×C4×C8C2311yescharacteristic
171C3×C4×C8C2311yescharacteristic
172#832C2318yes—
173#817C2314yes—
174#816C2318yes—
175#815C2312yes—
176#815C2312yes—
177C8×Dic3C2312yes—
178C8×Dic3C2312yes—
179#810C2318yes—
180#807C2318yes—
181#803C2312yes—
182C4×C3⋊C8C2311yescharacteristic
183C4×C3⋊C8C2311yescharacteristic
184C2×#135—34no—
185C22×M16—32no—
186#533—32no—
187#533—32no—
188#533—32no—
189#532—32no—
190#532—32no—
191#532C2×D312yes—
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#494—31no—
211#493C2×D311yescharacteristic, complemented
212#492—32no—
213#488—38no—
214#487—32no—
215#487—32no—
216#486—32no—
217#486C2×D312yescomplemented
218#470—34no—
219#470—34no—
220#469—34no—
221#469C2×D314yes—
222#468—38no—
223#468—38no—
224#442—34no—
225#442C2×D314yescomplemented
226#441—34no—
227#441—34no—
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
242C3×#41—48no—
243C2×C3×Q8—22no—
244C2×C3×D4—24no—
245C2×C3×D4—22no—
246C2×C3×D4—22no—
247C2×C3×D4C2414yes—
248C22×C3×C4—24no—
249C22×C3×C4C2×D412yes—
250C22×C3×C4C2411yescharacteristic
251C2×#67—24no—
252C2×#67—24no—
253C22×Dic3—24no—
254C22×Dic3—24no—
255C22×Dic3C2×D411yescharacteristic
256C22×Dic3C2×D412yes—
257#237—216no—
258#237—48no—
259C2×C4×D3—24no—
260C2×C4×D3—44no—
261C2×C4×D3C2×D411yescharacteristic
262C2×C4×D3C2×D411yescharacteristic
263C2×C4×D3C2×D412yes—
264C2×C4×D3C2×D412yes—
265C2×C4×D3C2×D412yes—
266C2×C4×D3C2×D412yes—
267C2×C4×D3C2411yescharacteristic
268C2×C4×D3C2411yescharacteristic
269C2×Dic6—22no—
270C2×Dic6C2414yes—
271C3×D8—28no—
272C3×M16—24no—
273C2×C3×C8C2412yes—
274C2×C3×C8C2412yes—
275C3×C4⋊C4—22no—
276C3×C4⋊C4C2414yes—
277C3×C22⋊C4—24no—
278C3×C22⋊C4—24no—
279C3×C42C2×D412yes—
280C3×C42C2411yescharacteristic
281C3×C42C2411yescharacteristic
282#217—24no—
283#217—24no—
284#217—24no—
285#217—24no—
286#217—24no—
287#214—216no—
288(C2×D3)⋊C4—24no—
289(C2×D3)⋊C4—24no—
290C4⋊Dic3—22no—
291C4⋊Dic3—22no—
292C4⋊Dic3—22no—
293C4⋊Dic3—24no—
294C4⋊Dic3—24no—
295C4⋊Dic3C2414yes—
296Dic3⋊C4—24no—
297Dic3⋊C4—24no—
298Dic3⋊C4—24no—
299Dic3⋊C4—24no—
300C4×Dic3—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—
313Dic12—28no—
314ΓL(1,𝔽25)—24no—
315ΓL(1,𝔽25)—24no—
316D3×C8—24no—
317D3×C8#4118yescomplemented
318C2×#41—22no—
319C2×#41—34no—
320C2×#41—62no—
321C23×C4—31no—
322C23×C4—32no—
323#135—616no—
324C2×Q16—34no—
325C2×SD16—38no—
326C2×D8C22×D314yes—
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⋊Q8—32no—
336C4⋊Q8C22×D312yes—
337#127C22×D312yes—
338#126—64no—
339#126—64no—
340#125—34no—
341#125—62no—
342#125—62no—
343#124—22no—
344#124—34no—
345#124—62no—
346#123—64no—
347#123—64no—
348#123—64no—
349C22⋊Q8—34no—
350C22⋊Q8—34no—
351C22⋊Q8—62no—
352C22⋊Q8—62no—
353C22⋊Q8—62no—
354C22⋊Q8—62no—
355#121—22no—
356#121—34no—
357#121—62no—
358#121—62no—
359#121—62no—
360#121C22×D314yes—
361#120—64no—
362C4×Q8—34no—
363C4×Q8—34no—
364C4×Q8—62no—
365C4×Q8—62no—
366C4×Q8—62no—
367C4×Q8—62no—
368C4×D4—22no—
369C4×D4—34no—
370C4×D4—62no—
371C4×D4—62no—
372C4×D4—62no—
373C4×D4—64no—
374C4×D4—64no—
375C4×D4—64no—
376C4×D4—64no—
377C4×D4—64no—
378C4×D4—64no—
379C4×D4C22×D314yes—
380#117—34no—
381#117—64no—
382#117—64no—
383#117—64no—
384#117—62no—
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#103—38no—
401#102—38no—
402#102C22×D318yes—
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×C3—24no—
414C23×C3C22×D411yescharacteristic
415C22×D3—22no—
416C22×D3—42no—
417C22×D3C22×D411yescharacteristic
418C22×D3C22×D411yescharacteristic
419C3×Q8—44no—
420C3×D4—28no—
421C3×D4—48no—
422C3×D4—44no—
423C3×D4—44no—
424C2×C3×C4—24no—
425C2×C3×C4—22no—
426C2×C3×C4—24no—
427C2×C3×C4—48no—
428C2×C3×C4C22×D412yes—
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C22×D412yes—
431C2×C3×C4C2511yescharacteristic
432#67—48no—
433#67—48no—
434C2×Dic3—22no—
435C2×Dic3—24no—
436C2×Dic3—24no—
437C2×Dic3—42no—
438C2×Dic3—48no—
439C2×Dic3—48no—
440C2×Dic3C22×D412yes—
441C2×Dic3C22×D411yescharacteristic
442C2×Dic3C22×D412yes—
443C2×Dic3C22×D411yescharacteristic
444C4×D3—22no—
445C4×D3—24no—
446C4×D3—24no—
447C4×D3—22no—
448C4×D3—22no—
449C4×D3—42no—
450C4×D3—42no—
451C4×D3C2×#4114yes—
452Dic6—28no—
453Dic6—44no—
454C3×C8—22no—
455C3×C8C2×#4114yes—
456C3⋊C8—22no—
457C3⋊C8C2×#4114yes—
458C24—31no—
459#41—48no—
460#41—616no—
461#41—128no—
462C2×Q8—22no—
463C2×Q8—34no—
464C2×Q8—62no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—64no—
469C2×D4—64no—
470C2×D4C23×D314yes—
471C22×C4—24no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—32no—
475C22×C4—32no—
476C22×C4—31no—
477C22×C4—32no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—32no—
481C22×C4—32no—
482C22×C4—64no—
483C22×C4—64no—
484C22×C4—64no—
485C22×C4—124no—
486C22×C4D3×D412yes—
487C22×C4C23×D311yescharacteristic
488Q16—68no—
489SD16—616no—
490D8—28no—
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—34no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—64no—
512C4⋊C4C23×D314yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—64no—
516C22⋊C4—64no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—64no—
522C42—32no—
523C42—32no—
524C42—32no—
525C42—32no—
526C42—32no—
527C42—31no—
528C42—31no—
529C42—64no—
530C42—64no—
531C42—64no—
532C42D3×D412yes—
533C42C23×D311yescharacteristic
534C42C23×D311yescharacteristic
535C22×C3—21no—
536C22×C3—22no—
537C22×C3—44no—
538C22×C3—44no—
539C22×C3—44no—
540C22×C3C23×D411yescharacteristic
541C2×D3—22no—
542C2×D3—21no—
543C2×D3—22no—
544C2×D3—44no—
545C2×D3#53212yes—
546C3×C4—21no—
547C3×C4—22no—
548C3×C4—44no—
549C3×C4#57514yes—
550C3×C4C22×#4112yes—
551Dic3—21no—
552Dic3—22no—
553Dic3—22no—
554Dic3—44no—
555Dic3—44no—
556Dic3#53212yes—
557C23—24no—
558C23—31no—
559C23—31no—
560C23—62no—
561C23—122no—
562C23C2×D3×D411yescharacteristic
563Q8—44no—
564Q8—68no—
565Q8—124no—
566D4—28no—
567D4—48no—
568D4—44no—
569D4—44no—
570D4—128no—
571D4—128no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—31no—
576C2×C4—34no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—31no—
580C2×C4—48no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—64no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—128no—
592C2×C4—122no—
593C2×C4—128no—
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×C3—84no—
604C2×C3C2×#53211yescharacteristic
605C2×C3C2×#57512yes—
606D3—22no—
607D3—41no—
608C22—22no—
609C22—21no—
610C22—32no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—62no—
615C22—62no—
616C22—61no—
617C22—124no—
618C22C22×D3×D411yescharacteristic
619C4—22no—
620C4—21no—
621C4—32no—
622C4—44no—
623C4—62no—
624C4—62no—
625C4—61no—
626C4—124no—
627C4—124no—
628C4#577314yes—
629C4C2×D3×#4112yes—
630C3#6069311yescharacteristic, complemented, Sylow
631C2—41no—
632C2—62no—
633C2—84no—
634C2—121no—
635C2D3×#53211yescharacteristic
636C2C2×#577312yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3184C2
socleC22×C3C23×D4
hypercenter#3184D3
hypocenterC3#60693
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3184

Complements

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

NH ≅ G/Ntype
1. #86219C2semidirect
2. C3×#3184C2semidirect
3. #83294C2semidirect
4. #81891C2semidirect
5. #81889C2semidirect
6. #81782C2semidirect
7. #81777C2semidirect
8. #81562C2semidirect
9. #81559C2semidirect
10. #80863C2semidirect
11. #80852C2semidirect
12. #79864C2semidirect
13. #79862C2semidirect
14. #78540C2semidirect
15. C4×D3×M16C2semidirect
38. C3×#498C22semidirect
39. C3×#493C22semidirect
40. C3×#486C22semidirect
42. C3×#442C22semidirect
45. #5155C22semidirect
46. #5152C22semidirect
49. #5093C22semidirect
50. #5078C22semidirect
51. #5076C22semidirect
52. #5070C22semidirect
54. #5018C22semidirect
58. #4858C22semidirect
59. #4853C22semidirect
62. #4768C22semidirect
63. #4757C22semidirect
64. #4716C22semidirect
67. #4699C22semidirect
72. #4681C22semidirect
73. #4680C22semidirect
78. #3184D3semidirect
208. #498C2×D3semidirect
211. #493C2×D3semidirect
217. #486C2×D3semidirect
225. #442C2×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
#862192
C3×#31842
#832942
#818912
#818892
#817822
#817772
#815622
#815592
#808632
#808522
#798642
#798622
#785402
C4×D3×M162
#606933