Finite Group Databasebeta

← #11675923

Subgroups — #11675923

Order 768.

1810 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86599C212yesmaximal, complemented
2C3×#3163C211yesmaximal, characteristic, complemented
3#83474C212yesmaximal, complemented
4#81970C214yesmaximal, complemented
5#81964C214yesmaximal, complemented
6#81931C214yesmaximal, complemented
7#81905C214yesmaximal, complemented
8#79242C212yesmaximal, complemented
9#79239C212yesmaximal, complemented
10#79171C211yesmaximal, characteristic, complemented
11#79167C211yesmaximal, characteristic
12#79110C211yesmaximal, characteristic, complemented
13C2×#4869C211yesmaximal, characteristic
14#78561C211yesmaximal, characteristic
15#78543C211yesmaximal, characteristic, complemented
16#61378—31nomaximal, Sylow
17C3×#544C2212yes—
18C22×D24C2211yescharacteristic
19#5730—22no—
20#5730C2214yescomplemented
21#5727—22no—
22#5727C2214yescomplemented
23#5690—22no—
24#5690—22no—
25#5690C2214yescomplemented
26#5690C2214yescomplemented
27#5634—24no—
28D3×C22⋊Q8—24no—
29#5521C2212yes—
30#5521C2212yes—
31#5517C2212yes—
32#5517C2211yescharacteristic
33C2×#893C2212yes—
34#5500C2212yes—
35C3×#501C2214yescomplemented
36C3×#499C2212yescomplemented
37C3×#498C2212yescomplemented
38C3×#428C2212yes—
39C3×#427C2211yescharacteristic
40C2×C3×#106C2211yescharacteristic
41C3×#405C2211yescharacteristic
42#5131C2212yes—
43#5119C2211yescharacteristic
44#5118C2211yescharacteristic
45#5106C2211yescharacteristic
46#5101C2211yescharacteristic
47C2×#815C2211yescharacteristic
48#5087C2214yescomplemented
49#5084C2214yescomplemented
50#5066C2214yescomplemented
51#5066C2214yescomplemented
52#5001C2212yes—
53#4999C2211yescharacteristic
54#4998C2212yes—
55C2×#809C2212yes—
56#4964C2212yes—
57#4891C2214yescomplemented
58#4891C2214yescomplemented
59#4890C2214yescomplemented
60#4890C2214yescomplemented
61#4876C2212yescomplemented
62#4876C2212yescomplemented
63#4872C2214yes—
64#4871C2214yescomplemented
65#4871C2214yescomplemented
66#4870C2214yescomplemented
67#4870C2214yescomplemented
68#4869C2214yes—
69#4869C2214yes—
70#4869C2212yes—
71#4869C2212yes—
72#4858C2212yescomplemented
73#4856C2214yescomplemented
74#4856C2212yescomplemented
75#4856C2212yescomplemented
76#4855C2212yescomplemented
77#4711C2212yes—
78#4711C2212yes—
79#4701C2214yescomplemented
80C4×D24C2214yes—
81#4679C2214yescomplemented
82#3501—32no—
83#3389—34no—
84#3389—34no—
85#3384—34no—
86#3384—34no—
87#3163—31no—
88#3163D311yescharacteristic, complemented
89#3162—31no—
90#3081—32no—
91#3081—32no—
92#3077—32no—
93#2961—31no—
94#2948—31no—
95#2948—31no—
96C2×#442—31no—
97C2×D3×Q8—24no—
98C22×D12C2312yes—
99C22×C4×D3—22no—
100C2×C3×M16C2311yescharacteristic
101C22×C3×C8C2311yescharacteristic
102C3×C4⋊Q8—22no—
103C3×C4⋊Q8C2314yes—
104C3×#125—22no—
105C3×#125C2314yes—
106C3×C22⋊Q8—24no—
107C3×#117C2312yes—
108C3×#117C2311yescharacteristic
109C2×C3×C4⋊C4C2312yes—
110#946—24no—
111#946—24no—
112#946—24no—
113C2×(C2×D3)⋊C4—22no—
114#926C2312yes—
115#926C2311yescharacteristic
116#924—24no—
117C2×C4×Dic3C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×D24Q818yescomplemented
121C2×D24C2312yes—
122C2×D24C2312yes—
123#897—24no—
124#897—24no—
125#897—24no—
126#896—24no—
127#896—24no—
128#896—24no—
129#896—24no—
130#896—24no—
131#896—24no—
132#893—22no—
133#893—22no—
134#893—22no—
135#893—22no—
136#893C2314yes—
137#893C2314yes—
138#893C2314yes—
139#893C2314yes—
140D3×C4⋊C4—24no—
141D3×C4⋊C4—24no—
142D3×C4⋊C4—24no—
143#890—22no—
144#890D412yes—
145#890C2312yes—
146#890C2314yes—
147#890C2312yes—
148#888—22no—
149#888D412yes—
150#888C2312yes—
151#888C2314yes—
152#888C2312yes—
153D3×C22⋊C4—24no—
154D3×C22⋊C4—24no—
155#878—22no—
156#878—22no—
157#878D414yes—
158C4×D12—22no—
159C4×D12—22no—
160C4×D12C2314yes—
161C4×D12C2314yes—
162C4×Dic6—22no—
163C4×Dic6—22no—
164C4×Dic6C2314yes—
165C3×#107C2312yes—
166C3×#107C2314yes—
167C3×#106C2312yes—
168C3×#106C2312yes—
169C3×#106C2311yescharacteristic
170C3×#106C2311yescharacteristic
171C3×#106C2314yes—
172C3×#97C2312yes—
173C3×C4×C8C2312yes—
174#821C2314yes—
175#821C2314yes—
176#818C2312yes—
177#817C2311yescharacteristic
178#817C2311yescharacteristic
179#815C2312yes—
180#815C2312yes—
181#815C2312yes—
182Dic3⋊C8C2314yes—
183Dic3⋊C8C2314yes—
184C8×Dic3C2312yes—
185#809C2314yes—
186#809C2314yes—
187#809C2314yes—
188#809C2314yes—
189#808C2312yes—
190#808C2314yes—
191#808C2312yes—
192#807C2312yes—
193#807C2314yes—
194#807C2312yes—
195C4⋊(C3⋊C8)C2314yes—
196C22×D8—31no—
197#554—34no—
198#554—34no—
199#554—34no—
200#554—62no—
201#554—62no—
202#554—62no—
203D4×Q8—34no—
204D4×Q8—62no—
205#544—32no—
206#544C2×D312yes—
207#543—64no—
208#533—32no—
209#533—31no—
210C2×C22⋊Q8—64no—
211#518—32no—
212#518—32no—
213C2×C4×D4—32no—
214#501—32no—
215#501—32no—
216#501—32no—
217#501—34no—
218#501—32no—
219#501C2×D314yescomplemented
220#499—32no—
221#499C2×D312yescomplemented
222#498—32no—
223#498C2×D312yescomplemented
224#478—34no—
225#478—34no—
226#478—34no—
227#478—34no—
228#478—34no—
229#478—34no—
230#476—34no—
231#476—34no—
232#476—34no—
233#474—34no—
234#474—34no—
235#474—34no—
236#446—34no—
237#446—34no—
238#442—32no—
239#442—34no—
240#442—34no—
241#442—32no—
242#442—32no—
243#442—32no—
244C4×D8—34no—
245C4×D8—34no—
246#428—32no—
247#428C2×D312yes—
248#427—31no—
249#427—32no—
250#427C2×D311yescharacteristic
251C2×#106C2×D311yescharacteristic
252#424—31no—
253#424—31no—
254#423—31no—
255#418—32no—
256#418—32no—
257C2×#102—32no—
258#405—31no—
259#405C2×D311yescharacteristic
260C2×#97—31no—
261C23×D3—22no—
262C2×C3×Q8—24no—
263C22×C3×C4—22no—
264C22×C3×C4C2411yescharacteristic
265C22×Dic3C2×D412yes—
266D3×Q8—88no—
267C2×D12—22no—
268C2×D12—24no—
269C2×D12C2×Q818yescomplemented
270C2×D12C2414yes—
271C2×C4×D3—24no—
272C2×C4×D3—24no—
273C2×C4×D3—24no—
274C2×C4×D3—24no—
275C2×C4×D3—24no—
276C2×C4×D3—24no—
277C2×C4×D3—84no—
278C2×Dic6—22no—
279C2×Dic6—22no—
280C2×Dic6—24no—
281C2×Dic6C2×D414yes—
282C3×M16—22no—
283C2×C3×C8C2×Q814yescomplemented
284C2×C3×C8C2412yes—
285C2×C3×C8C2411yescharacteristic
286C2×C3×C8C2411yescharacteristic
287C3×C4⋊C4—22no—
288C3×C4⋊C4—24no—
289C3×C4⋊C4—24no—
290C3×C4⋊C4—24no—
291C3×C4⋊C4C2×D412yes—
292C3×C4⋊C4C2414yes—
293C3×C4⋊C4C2412yes—
294C3×C4⋊C4C2412yes—
295C3×C22⋊C4—22no—
296C3×C22⋊C4C2×D412yes—
297C3×C42—22no—
298C3×C42C2412yes—
299#217—22no—
300#217C2×D412yes—
301(C2×D3)⋊C4—24no—
302(C2×D3)⋊C4—24no—
303(C2×D3)⋊C4—24no—
304(C2×D3)⋊C4—24no—
305(C2×D3)⋊C4—24no—
306(C2×D3)⋊C4—24no—
307C4⋊Dic3—22no—
308C4⋊Dic3—22no—
309C4⋊Dic3—24no—
310C4⋊Dic3C2×D414yes—
311C4⋊Dic3C2×D412yes—
312C4⋊Dic3C2412yes—
313C4⋊Dic3C2412yes—
314Dic3⋊C4—22no—
315Dic3⋊C4—22no—
316Dic3⋊C4—22no—
317Dic3⋊C4—22no—
318Dic3⋊C4—24no—
319Dic3⋊C4—24no—
320Dic3⋊C4C2×D414yes—
321Dic3⋊C4C2×D414yes—
322C4×Dic3—22no—
323C4×Dic3C2×D411yescharacteristic
324C4×Dic3C2×D411yescharacteristic
325C4×Dic3C2412yes—
326C4×Dic3C2412yes—
327#208—22no—
328C2×C3⋊C8—22no—
329C2×C3⋊C8C2412yes—
330C2×C3⋊C8C2412yes—
331D24C2×Q818yescomplemented
332D24C2×Q818yescomplemented
333C22×Q8—64no—
334C22×D4—32no—
335C23×C4—62no—
336C2×D8—38no—
337C2×D8—32no—
338C2×D8—32no—
339C2×M16—31no—
340C2×M16C22×D311yescharacteristic
341C22×C8—31no—
342C22×C8C22×D311yescharacteristic
343C4⋊Q8—22no—
344C4⋊Q8—32no—
345C4⋊Q8—32no—
346C4⋊Q8—32no—
347C4⋊Q8—34no—
348C4⋊Q8—62no—
349C4⋊Q8C22×D314yes—
350#125—22no—
351#125—32no—
352#125—32no—
353#125—32no—
354#125—34no—
355#125—62no—
356#125C22×D314yes—
357C22⋊Q8—24no—
358C22⋊Q8—34no—
359C22⋊Q8—62no—
360C22⋊Q8—62no—
361C22⋊Q8—64no—
362C22⋊Q8—64no—
363C22⋊Q8—64no—
364C22⋊Q8—64no—
365C22⋊Q8—64no—
366C22⋊Q8—64no—
367C22⋊Q8—64no—
368C22⋊Q8—64no—
369C22⋊Q8—64no—
370C22⋊Q8—64no—
371C22⋊Q8—64no—
372C22⋊Q8—64no—
373C22⋊Q8—64no—
374C4×Q8—34no—
375C4×Q8—62no—
376C4×Q8—62no—
377C4×D4—34no—
378C4×D4—34no—
379C4×D4—34no—
380C4×D4—34no—
381C4×D4—34no—
382C4×D4—34no—
383C4×D4—62no—
384C4×D4—62no—
385C4×D4—62no—
386C4×D4—62no—
387C4×D4—62no—
388C4×D4—62no—
389#117—31no—
390#117—32no—
391#117C22×D312yes—
392#117C22×D311yescharacteristic
393C2×C4⋊C4—64no—
394C2×C4⋊C4—64no—
395C2×C4⋊C4—64no—
396C2×C4⋊C4C22×D312yes—
397C2×C22⋊C4—64no—
398C2×C22⋊C4—62no—
399C2×C22⋊C4—64no—
400C2×C42—31no—
401#107—32no—
402#107—32no—
403#107—32no—
404#107—34no—
405#107C22×D312yes—
406#107C22×D314yes—
407#106—31no—
408#106—31no—
409#106—32no—
410#106—32no—
411#106—34no—
412#106C22×D312yes—
413#106C22×D312yes—
414#106C22×D311yescharacteristic
415#106C22×D311yescharacteristic
416#106C22×D314yes—
417C4⋊C8—34no—
418C4⋊C8—34no—
419C4⋊C8—34no—
420#102—34no—
421#102—34no—
422#102—34no—
423#102—34no—
424#102—34no—
425#102—34no—
426#97—32no—
427#97—32no—
428#97—32no—
429#97C22×D312yes—
430C4×C8—32no—
431C4×C8C22×D312yes—
432C23×C3C22×D411yescharacteristic
433C22×D3—24no—
434C22×D3—28no—
435C22×D3—82no—
436C22×D3—82no—
437C3×Q8—84no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—24no—
441C2×C3×C4—82no—
442C2×C3×C4C22×D414yes—
443C2×C3×C4C22×D411yescharacteristic
444C2×C3×C4C22×Q812yes—
445C2×C3×C42−1+412yes—
446C2×C3×C4C2511yescharacteristic
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—42no—
450C2×Dic3C22×D412yes—
451C2×Dic3C22×D412yes—
452C2×Dic3C22×D414yes—
453D12—28no—
454D12—24no—
455D12C22×Q818yes—
456C4×D3—48no—
457C4×D3—84no—
458C4×D3—84no—
459C4×D3—84no—
460C4×D3—84no—
461C4×D3—84no—
462Dic6—44no—
463Dic6—44no—
464Dic6—82no—
465Dic6—84no—
466C3×C8—22no—
467C3×C8C22×Q812yes—
468C3×C8C22×Q812yes—
469C3⋊C8—22no—
470C3⋊C8—22no—
471C24—62no—
472C2×Q8—24no—
473C2×Q8—34no—
474C2×Q8—62no—
475C2×Q8—62no—
476C2×Q8—64no—
477C2×Q8—248no—
478C2×D4—38no—
479C2×D4—34no—
480C2×D4—62no—
481C2×D4—64no—
482C22×C4—22no—
483C22×C4—32no—
484C22×C4—64no—
485C22×C4—64no—
486C22×C4—64no—
487C22×C4—64no—
488C22×C4—64no—
489C22×C4—64no—
490C22×C4—244no—
491C22×C4C23×D311yescharacteristic
492D8—38no—
493D8—38no—
494M16—22no—
495M16—62no—
496C2×C8—32no—
497C2×C8—32no—
498C2×C8—62no—
499C2×C8D3×Q814yescomplemented
500C2×C8C23×D312yes—
501C2×C8C23×D311yescharacteristic
502C2×C8C23×D311yescharacteristic
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—34no—
508C4⋊C4—34no—
509C4⋊C4—34no—
510C4⋊C4—32no—
511C4⋊C4—32no—
512C4⋊C4—32no—
513C4⋊C4—62no—
514C4⋊C4—62no—
515C4⋊C4—62no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—62no—
520C4⋊C4—62no—
521C4⋊C4—62no—
522C4⋊C4D3×D412yes—
523C4⋊C4C23×D314yes—
524C4⋊C4C23×D312yes—
525C4⋊C4C23×D312yes—
526C22⋊C4—22no—
527C22⋊C4—32no—
528C22⋊C4—64no—
529C22⋊C4—64no—
530C22⋊C4—64no—
531C22⋊C4—64no—
532C22⋊C4—62no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C22⋊C4D3×D412yes—
536C42—22no—
537C42—31no—
538C42—31no—
539C42—32no—
540C42—32no—
541C42—62no—
542C42C23×D312yes—
543C22×C3—22no—
544C22×C3D4×Q812yes—
545C22×C3C23×D411yescharacteristic
546C2×D3—28no—
547C2×D3—44no—
548C2×D3—44no—
549C2×D3—88no—
550C3×C4—24no—
551C3×C4—42no—
552C3×C4—44no—
553C3×C4—42no—
554C3×C4D4×Q812yes—
555C3×C4C23×Q811yescharacteristic
556C3×C4C2×2−1+411yescharacteristic
557Dic3—22no—
558Dic3—24no—
559Dic3—22no—
560Dic3—42no—
561Dic3—42no—
562C23—64no—
563C23—68no—
564C23—242no—
565C23—242no—
566C23C2×D3×D411yescharacteristic
567Q8—84no—
568Q8—124no—
569Q8—124no—
570Q8—244no—
571Q8—242no—
572D4—38no—
573D4—64no—
574D4—68no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—34no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—82no—
584C2×C4—128no—
585C2×C4—122no—
586C2×C4—244no—
587C2×C4—244no—
588C2×C4—244no—
589C2×C4—244no—
590C2×C4—244no—
591C2×C4C2×D3×D414yes—
592C2×C4C2×D3×D411yescharacteristic
593C2×C4C2×D3×Q812yes—
594C2×C4#100712yes—
595C2×C4C24×D311yescharacteristic
596C8—22no—
597C8—62no—
598C8—62no—
599C8C2×D3×Q812yes—
600C8C2×D3×Q812yes—
601C2×C3—22no—
602C2×C3C2×D4×Q811yescharacteristic
603C2×C3#358512yes—
604D3—48no—
605C22—22no—
606C22—68no—
607C22—124no—
608C22—124no—
609C22—248no—
610C22#572712yes—
611C22C22×D3×D411yescharacteristic
612C4—24no—
613C4—44no—
614C4—42no—
615C4—42no—
616C4—62no—
617C4—62no—
618C4—64no—
619C4—122no—
620C4—122no—
621C4#572712yes—
622C4C22×D3×Q811yescharacteristic
623C4C2×#100711yescharacteristic
624C3#6137811yescharacteristic, complemented, Sylow
625C2—22no—
626C2—128no—
627C2C2×#572711yescharacteristic
628C2#8720712yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3163C2
socleC22×C3C23×D4
prime radicalC22×D24C22
hypercenter#3163D3
hypocenterC3#61378
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #3163

Complements

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

NH ≅ G/Ntype
1. #86599C2semidirect
2. C3×#3163C2semidirect
3. #83474C2semidirect
4. #81970C2semidirect
5. #81964C2semidirect
6. #81931C2semidirect
7. #81905C2semidirect
8. #79242C2semidirect
9. #79239C2semidirect
10. #79171C2semidirect
12. #79110C2semidirect
15. #78543C2semidirect
20. #5730C22semidirect
22. #5727C22semidirect
25. #5690C22semidirect
26. #5690C22semidirect
35. C3×#501C22semidirect
36. C3×#499C22semidirect
37. C3×#498C22semidirect
48. #5087C22semidirect
49. #5084C22semidirect
50. #5066C22semidirect
51. #5066C22semidirect
57. #4891C22semidirect
58. #4891C22semidirect
59. #4890C22semidirect
60. #4890C22semidirect
61. #4876C22semidirect
62. #4876C22semidirect
64. #4871C22semidirect
65. #4871C22semidirect
66. #4870C22semidirect
67. #4870C22semidirect
72. #4858C22semidirect
73. #4856C22semidirect
74. #4856C22semidirect
75. #4856C22semidirect
76. #4855C22semidirect
79. #4701C22semidirect
81. #4679C22semidirect
88. #3163D3semidirect
120. C2×D24Q8semidirect
219. #501C2×D3semidirect
221. #499C2×D3semidirect
223. #498C2×D3semidirect
269. C2×D12C2×Q8semidirect
283. C2×C3×C8C2×Q8semidirect
331. D24C2×Q8semidirect
332. D24C2×Q8semidirect
499. C2×C8D3×Q8semidirect
624. C3#61378semidirect

Maximal subgroups

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

subgroupindex
#865992
C3×#31632
#834742
#819702
#819642
#819312
#819052
#792422
#792392
#791712
#791672
#791102
C2×#48692
#785612
#785432
#613783