Finite Group Databasebeta

← #10777929

Subgroups — #10777929

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3106C211yesmaximal, characteristic, complemented
2#83291C211yesmaximal, characteristic, complemented
3#82529C211yesmaximal, characteristic, complemented
4#82517C211yesmaximal, characteristic, complemented
5#71805C211yesmaximal, characteristic, complemented
6#71154C211yesmaximal, characteristic, complemented
7#71153C211yesmaximal, characteristic, complemented
8#71138C211yesmaximal, characteristic, complemented
9#71137C211yesmaximal, characteristic, complemented
10#71026C211yesmaximal, characteristic, complemented
11#71025C211yesmaximal, characteristic, complemented
12#71010C211yesmaximal, characteristic, complemented
13#71009C211yesmaximal, characteristic, complemented
14#70773C211yesmaximal, characteristic, complemented
15#70745C211yesmaximal, characteristic, complemented
16#19160—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5477C2211yescharacteristic
20C3×#482—21no—
21C3×#477C2211yescharacteristic
22C3×#476C2211yescharacteristic
23C3×#460C2211yescharacteristic
24C3×#459C2211yescharacteristic
25C3×#449—21no—
26C3×C4×SD16—21no—
27C3×C4×D8—21no—
28C3×#431C2211yescharacteristic
29#5238—21no—
30#5088C2211yescharacteristic
31#5078C2211yescharacteristic
32#5073C2211yescharacteristic
33#5070C2211yescharacteristic
34#5038—21no—
35#5038—21no—
36#5027C2211yescharacteristic
37C4×#215—21no—
38#5024C2211yescharacteristic
39#5024C2211yescharacteristic
40#5021C2211yescharacteristic
41#5015C2211yescharacteristic
42C4×#213—21no—
43C4×#213—21no—
44C4×#213—21no—
45#5012C2211yescharacteristic
46#5012C2211yescharacteristic
47#5011—21no—
48#5011—21no—
49#5008C2211yescharacteristic
50#5003—21no—
51#4925—21no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54C4×#208—21no—
55C2×C4×C3⋊C8C2211yescharacteristic
56#4546C2211yescharacteristic
57#4546C2211yescharacteristic
58#4544C2211yescharacteristic
59#4544C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3169—31no—
69#3106—31no—
70#3106D311yescharacteristic, complemented
71#2965—31no—
72#1698—31no—
73#1698—31no—
74#1697—31no—
75#1697—31no—
76#1696—31no—
77#1695—31no—
78#1688—31no—
79#1687—31no—
80#1564—31no—
81#1543—31no—
82#1543—31no—
83C2×C3×#41—21no—
84C2×C3×#41D411yescharacteristic
85C2×#235—21no—
86C2×C3×SD16—21no—
87C2×C3×D8—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×#127C2311yescharacteristic
90C3×#124—21no—
91C3×#121—21no—
92C3×#121—21no—
93C3×C4×Q8D411yescharacteristic
94C3×C4×Q8C2311yescharacteristic
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4D411yescharacteristic
100C3×C4×D4C2311yescharacteristic
101C3×#117—21no—
102C3×#117D411yescharacteristic
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×#215—21no—
106C2×#213—21no—
107C2×#213—21no—
108C2×#213—21no—
109#930—21no—
110#930—21no—
111C2×#208—21no—
112C2×#208—21no—
113C22×C3⋊C8D411yescharacteristic
114C22×C3⋊C8D411yescharacteristic
115#875—21no—
116#874C2311yescharacteristic
117C4×D12—21no—
118C4×D12—21no—
119#869C2311yescharacteristic
120C3×#107—21no—
121C3×#106—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#103—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C4×C8—21no—
130#835—21no—
131#832—21no—
132#832—21no—
133#832—21no—
134#830—21no—
135#830—21no—
136#830—21no—
137#830—21no—
138#809—21no—
139#809—21no—
140#809—21no—
141#809—21no—
142#808—21no—
143#807—21no—
144#807—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152#803—21no—
153#803—21no—
154C4×C3⋊C8—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8D411yescharacteristic
157C4×C3⋊C8D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161#532C2×D311yescharacteristic
162C4×#41C2×D311yescharacteristic
163#500—31no—
164#493—31no—
165#486—31no—
166#486—31no—
167#482—21no—
168#482—61no—
169#477—31no—
170#477C2×D311yescharacteristic
171#476—31no—
172#476C2×D311yescharacteristic
173#466—61no—
174#466—61no—
175#460—31no—
176#460C2×D311yescharacteristic
177#459—31no—
178#459C2×D311yescharacteristic
179#449—21no—
180#449—61no—
181#446—31no—
182#446—31no—
183C4×SD16—21no—
184C4×SD16—61no—
185C4×D8—21no—
186C4×D8—61no—
187C4×D8—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#435—61no—
192#435—61no—
193#431—31no—
194#431—31no—
195#431—61no—
196#431C2×D311yescharacteristic
197C4×M16—61no—
198C2×C4×C8—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#331—31no—
208#331—31no—
209#331—31no—
210#331—31no—
211C3×#41—41no—
212C3×#41—41no—
213C3×#41—82no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—41no—
221C2×C3×D4C2×D411yescharacteristic
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—41no—
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4#4111yescharacteristic
229C2×#67—41no—
230#235—82no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×C4×D3—41no—
235C2×Dic6—21no—
236C3×SD16—42no—
237C3×D8—42no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—41no—
250C3×C22⋊C4—41no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42C2×D411yescharacteristic
255C3×C42C2×D411yescharacteristic
256C3×C42#4111yescharacteristic
257C3×C42C2411yescharacteristic
258#215—42no—
259#213—42no—
260#213—42no—
261#213—42no—
262(C2×D3)⋊C4—41no—
263(C2×D3)⋊C4—41no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266#208—42no—
267#208—42no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—41no—
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277C2×C3⋊C8C2×D411yescharacteristic
278C2×#41—21no—
279C2×#41—61no—
280C2×#41#6711yescharacteristic
281C2×SD16—21no—
282C2×SD16—61no—
283C2×D8—21no—
284C2×D8—61no—
285C2×D8—61no—
286C2×D8—61no—
287C2×M16—61no—
288C2×M16—61no—
289C22×C8—31no—
290C22×C8—31no—
291C4⋊Q8—31no—
292C4⋊Q8C22×D311yescharacteristic
293#127—31no—
294#127C22×D311yescharacteristic
295#124—21no—
296#124—61no—
297#121—21no—
298#121—21no—
299#121—61no—
300#121—61no—
301C4×Q8#6711yescharacteristic
302C4×Q8C22×D311yescharacteristic
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307C4×D4—61no—
308C4×D4—61no—
309C4×D4#6711yescharacteristic
310C4×D4C22×D311yescharacteristic
311#117—21no—
312#117#6711yescharacteristic
313C2×C42—21no—
314C2×C42C22×D311yescharacteristic
315#107—21no—
316#107—61no—
317#107—61no—
318#107—61no—
319#106—21no—
320#106—61no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—61no—
326C4⋊C8—61no—
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—21no—
330#103—61no—
331#102—21no—
332#102—21no—
333#102—21no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—21no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346#97—61no—
347#97—61no—
348C4×C8—21no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—31no—
353C4×C8—61no—
354C4×C8—61no—
355C23×C3—21no—
356C23×C3—41no—
357C23×C3#12111yescharacteristic
358C22×D3—41no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—81no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—42no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—81no—
370C3×D4—82no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—42no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—42no—
383C2×C3×C4—41no—
384C2×C3×C4—82no—
385C2×C3×C4#12111yescharacteristic
386C2×C3×C4#12111yescharacteristic
387C2×C3×C4#12111yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391#67—82no—
392C2×Dic3—41no—
393D12—41no—
394D12—41no—
395D12—41no—
396D12—41no—
397D12—81no—
398C4×D3—82no—
399Dic6—81no—
400C3×C8—41no—
401C3×C8—41no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C3⋊C8—21no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8#13512yes—
410#41—41no—
411#41—41no—
412#41—82no—
413#41—242no—
414C2×Q8—21no—
415C2×Q8—61no—
416C2×Q8C2×#6711yescharacteristic
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—41no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—121no—
426C2×D4C2×#6711yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—121no—
432C22×C4D3×D411yescharacteristic
433C22×C4#23711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435SD16—42no—
436SD16—122no—
437D8—42no—
438D8—122no—
439D8—122no—
440D8—122no—
441M16—122no—
442M16—122no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—31no—
446C2×C8—31no—
447C2×C8—31no—
448C2×C8—31no—
449C2×C8—62no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—121no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4C2×#6711yescharacteristic
462C4⋊C4C2×#6711yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—121no—
469C22⋊C4—121no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42D3×D411yescharacteristic
474C42#23711yescharacteristic
475C42C2×#6711yescharacteristic
476C42C23×D311yescharacteristic
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3C2×#12111yescharacteristic
487C2×D3—81no—
488C2×D3—81no—
489C2×D3—81no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—81no—
497C3×C4#47212yes—
498C3×C4C2×#13511yescharacteristic
499C3×C4#57511yescharacteristic
500C3×C4#57511yescharacteristic
501C3×C4#57511yescharacteristic
502Dic3—81no—
503C23—21no—
504C23—41no—
505C23—121no—
506C23#93911yescharacteristic
507Q8—21no—
508Q8—21no—
509Q8—81no—
510Q8—241no—
511D4—21no—
512D4—21no—
513D4—42no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—81no—
519D4—82no—
520D4—121no—
521D4—121no—
522D4—121no—
523D4—121no—
524D4—241no—
525D4—242no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—121no—
541C2×C4—242no—
542C2×C4#93911yescharacteristic
543C2×C4#93911yescharacteristic
544C2×C4#93911yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×#23711yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C8—32no—
549C8—41no—
550C8—41no—
551C8—61no—
552C8—61no—
553C8—61no—
554C8—121no—
555C8—121no—
556C8—121no—
557C8—121no—
558C2×C3—41no—
559C2×C3—81no—
560C2×C3—161no—
561C2×C3#303511yescharacteristic
562C2×C3#305411yescharacteristic
563C2×C3C2×#47211yescharacteristic
564D3—161no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22C2×#93911yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—241no—
586C4#519912yes—
587C4D3×#13511yescharacteristic
588C4#576811yescharacteristic
589C4#582211yescharacteristic
590C4#582211yescharacteristic
591C3#1916011yescharacteristic, complemented, Sylow
592C2—41no—
593C2—81no—
594C2—161no—
595C2—481no—
596C2#8292311yescharacteristic
597C2#8390011yescharacteristic
598C2C2×#519911yescharacteristic
subgroupG/H
centerC2×C4#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3106C2
socleC22×C3C2×#121
hypercenter#3106D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. C3×#3106C2semidirect
2. #83291C2semidirect
3. #82529C2semidirect
4. #82517C2semidirect
5. #71805C2semidirect
6. #71154C2semidirect
7. #71153C2semidirect
8. #71138C2semidirect
9. #71137C2semidirect
10. #71026C2semidirect
11. #71025C2semidirect
12. #71010C2semidirect
13. #71009C2semidirect
14. #70773C2semidirect
15. #70745C2semidirect
70. #3106D3semidirect
591. C3#19160semidirect

Maximal subgroups

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

subgroupindex
C3×#31062
#832912
#825292
#825172
#718052
#711542
#711532
#711382
#711372
#710262
#710252
#710102
#710092
#707732
#707452
#191603