Finite Group Databasebeta

← #10768804

Subgroups — #10768804

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86052C211yesmaximal, characteristic
2#82914C211yesmaximal, characteristic, complemented
3C2×#4736C211yesmaximal, characteristic, complemented
4#73519C211yesmaximal, characteristic, complemented
5C3×#1501C211yesmaximal, characteristic, complemented
6#69921C211yesmaximal, characteristic, complemented
7#69444C211yesmaximal, characteristic, complemented
8#68321C214yesmaximal, complemented
9#68288C214yesmaximal, complemented
10#18264—31nomaximal, Sylow
11C3×#525C2211yescharacteristic
12#5830—21no—
13C2×#930C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic, complemented
16#5666—22no—
17#5644—21no—
18#5644C2212yes—
19#5605—21no—
20#5605C2212yes—
21#5590—22no—
22#5565—21no—
23#5565—21no—
24D4×D12—21no—
25D4×D12—21no—
26#5535—21no—
27C2×C3×C22⋊C8C2211yescharacteristic, complemented
28C3×C4×C22⋊C4C2211yescharacteristic, complemented
29#5232—21no—
30#5207C2211yescharacteristic, complemented
31C2×#821—21no—
32C2×#818—21no—
33#5044—21no—
34#5043C2212yescomplemented
35#5034—21no—
36#5033C2212yescomplemented
37#4995—21no—
38#4958—22no—
39#4939—21no—
40#4936—21no—
41#4736—22no—
42#4736C2214yescomplemented
43#4674C2214yescomplemented
44C3×#328C2212yescomplemented
45C3×#327C2212yescomplemented
46#4468C2214yescomplemented
47#4451C2214yescomplemented
48#3537—31no—
49#3092—31no—
50C2×#480—31no—
51#2504—31no—
52#1612—34no—
53#1607—34no—
54#1507—31no—
55#1501D311yescharacteristic, complemented
56#1471—31no—
57C2×C3×#41D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×#67—21no—
60C22×#67D412yes—
61C23×Dic3—21no—
62C2×D3×D4—22no—
63C2×D3×D4—22no—
64C22×D12—21no—
65C22×D12D411yescharacteristic
66C22×D12D411yescharacteristic
67C22×C3×C8—21no—
68C3×#123—21no—
69C3×C22⋊Q8D411yescharacteristic
70C3×C22⋊Q8C2311yescharacteristic
71C3×#121D411yescharacteristic
72C3×#121C2311yescharacteristic
73C3×#120—21no—
74C3×C4×D4—21no—
75C3×C4×D4—21no—
76C3×#117D411yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79#947—22no—
80#940—22no—
81#939—22no—
82#939—22no—
83#937—22no—
84#937—22no—
85#930—21no—
86#930—21no—
87#930—21no—
88#930—21no—
89#930—21no—
90#930—21no—
91#930—21no—
92#930—21no—
93#930D412yes—
94#930C2312yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—24no—
97C2×(C2×D3)⋊C4C2312yes—
98#926—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2312yes—
107C2×#208—21no—
108#898—22no—
109#898—22no—
110#895—22no—
111#895—22no—
112#894—22no—
113#894—22no—
114#884—22no—
115#884—22no—
116#883—22no—
117#883—22no—
118#882—22no—
119#882—22no—
120#882—22no—
121#882—22no—
122#67⋊C4—22no—
123#67⋊C4—22no—
124#874—21no—
125C4×D12—21no—
126C4×D12—21no—
127#870—21no—
128C3×C22⋊C8—21no—
129C3×C22⋊C8C2312yes—
130C3×#95C2312yes—
131#835—22no—
132#832—22no—
133C3⋊#95—24no—
134#830—21no—
135#830C2312yes—
136#821—24no—
137#818—24no—
138#808—22no—
139#807—22no—
140#560—62no—
141#553—61no—
142#553—61no—
143D42—61no—
144D42—61no—
145D42—62no—
146#540—61no—
147#538—32no—
148#538—61no—
149#535—61no—
150#534—32no—
151#534—61no—
152#525C2×D311yescharacteristic
153C2×#121—31no—
154C2×#121—31no—
155C22×C4⋊C4—31no—
156#484—61no—
157#483—32no—
158#481—61no—
159#480—32no—
160#480—34no—
161#480—62no—
162#428—61no—
163C2×#107—61no—
164#417—61no—
165C2×#102—61no—
166#407—31no—
167C2×C22⋊C8C2×D311yescharacteristic, complemented
168#397—34no—
169#390—61no—
170#386—62no—
171#382—61no—
172C4×C22⋊C4C2×D311yescharacteristic, complemented
173#328C2×D312yescomplemented
174#327—34no—
175#327C2×D312yescomplemented
176#323—34no—
177C24×C3C2×D411yescharacteristic
178C23×D3—21no—
179C23×D3—22no—
180C23×D3C2×D412yes—
181C3×#41—42no—
182C2×C3×Q8C2×D411yescharacteristic
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4C2×D411yescharacteristic
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—42no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C2×#67—22no—
197C2×#67—22no—
198C2×#67—22no—
199C2×#67—22no—
200C2×#67—22no—
201C2×#67—22no—
202C2×#67—44no—
203C2×#67—82no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—41no—
207C22×Dic3—41no—
208C22×Dic3—82no—
209C22×Dic3C2×D412yes—
210D3×D4—84no—
211D3×D4—84no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12—21no—
218C2×D12—42no—
219C2×D12—42no—
220C2×D12—82no—
221C2×D12C2×D412yes—
222C2×C4×D3—22no—
223C2×C4×D3—22no—
224C2×C3×C8—22no—
225C2×C3×C8—42no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4C2×D411yescharacteristic
229C3×C4⋊C4C2×D411yescharacteristic
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—42no—
236C3×C42—21no—
237C3×C42—42no—
238#217—22no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—84no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—22no—
252C4⋊Dic3—22no—
253C4⋊Dic3—41no—
254C4⋊Dic3—41no—
255C4⋊Dic3C2×D412yes—
256Dic3⋊C4—22no—
257Dic3⋊C4—22no—
258C4×Dic3—22no—
259#208—42no—
260C2×C3⋊C8—22no—
261C2×#41D1211yescharacteristic
262C22×D4—32no—
263C22×D4—31no—
264C22×D4—31no—
265C22×D4—61no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—61no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271C2×M16—61no—
272C22×C8—21no—
273#127—61no—
274#127—62no—
275#126—62no—
276#126—62no—
277#125—61no—
278#124—62no—
279#123—21no—
280#123—62no—
281#123—62no—
282#123—62no—
283#123—62no—
284C22⋊Q8D1211yescharacteristic
285C22⋊Q8C22×D311yescharacteristic
286#121—32no—
287#121—32no—
288#121—62no—
289#121—62no—
290#121—62no—
291#121—62no—
292#121—61no—
293#121—61no—
294#121—62no—
295#121—62no—
296#121—61no—
297#121—61no—
298#121—61no—
299#121—61no—
300#121—61no—
301#121—61no—
302#121D1211yescharacteristic
303#121C22×D311yescharacteristic
304#120—21no—
305#120—62no—
306#120—62no—
307#120—62no—
308#120—62no—
309#120—62no—
310#120—62no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—61no—
314C4×D4—61no—
315C4×D4—62no—
316C4×D4—62no—
317#117—61no—
318#117D1211yescharacteristic
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—32no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—64no—
331C2×C42—21no—
332#107—64no—
333#107—62no—
334#106—62no—
335#103—62no—
336#102—64no—
337#102—62no—
338C22⋊C8—21no—
339C22⋊C8—32no—
340C22⋊C8—61no—
341C22⋊C8C22×D312yes—
342#95—64no—
343#95C22×D312yes—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—21no—
347C23×C3—41no—
348C23×C3#12011yescharacteristic
349C23×C3#12011yescharacteristic
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—24no—
354C22×D3—42no—
355C22×D3—42no—
356C22×D3—42no—
357C22×D3—81no—
358C22×D3—81no—
359C22×D3—82no—
360C22×D3—82no—
361C22×D3—82no—
362C3×Q8—41no—
363C3×D4—41no—
364C3×D4—42no—
365C3×D4—42no—
366C3×D4—81no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4#12011yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384#67—44no—
385#67—44no—
386#67—82no—
387#67—82no—
388#67—82no—
389#67—82no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—44no—
395C2×Dic3—81no—
396C2×Dic3—82no—
397C2×Dic3—82no—
398D12—42no—
399D12—42no—
400D12—42no—
401D12—81no—
402D12—82no—
403D12—82no—
404D12—82no—
405C4×D3—82no—
406C4×D3—82no—
407C3×C8—42no—
408C3⋊C8—42no—
409C24—32no—
410C24—62no—
411C24—61no—
412C24D3×D411yescharacteristic
413#41—42no—
414C2×Q8C2×D1211yescharacteristic
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—32no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—122no—
432C2×D4—122no—
433C2×D4—124no—
434C2×D4—242no—
435C2×D4—244no—
436C2×D4—244no—
437C2×D4—242no—
438C2×D4C2×D1211yescharacteristic
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—32no—
442C22×C4—42no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—121no—
448C22×C4—121no—
449C22×C4—242no—
450C22×C4C2×D1211yescharacteristic
451C22×C4C2×D1211yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4C23×D311yescharacteristic
456M16—122no—
457C2×C8—22no—
458C2×C8—42no—
459C2×C8—62no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—32no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—121no—
471C4⋊C4—121no—
472C4⋊C4C2×D1211yescharacteristic
473C4⋊C4C2×D1211yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—42no—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—62no—
487C22⋊C4—62no—
488C22⋊C4—62no—
489C22⋊C4—244no—
490C42—21no—
491C42—42no—
492C42—62no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—44no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3#45512yes—
505C22×C3C2×#12011yescharacteristic
506C22×C3#57612yes—
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—84no—
514C2×D3—162no—
515C2×D3—164no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—82no—
523C3×C4#45812yes—
524Dic3—42no—
525Dic3—42no—
526Dic3—82no—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—41no—
531C23—62no—
532C23—62no—
533C23—64no—
534C23—122no—
535C23—122no—
536C23—122no—
537C23—241no—
538C23—241no—
539C23—242no—
540C23—242no—
541C23—242no—
542C23#88211yescharacteristic
543C23#88211yescharacteristic
544C23C2×D3×D411yescharacteristic
545Q8—41no—
546D4—41no—
547D4—42no—
548D4—42no—
549D4—81no—
550D4—122no—
551D4—124no—
552D4—124no—
553D4—122no—
554D4—122no—
555D4—241no—
556D4—242no—
557D4—242no—
558D4—242no—
559D4—242no—
560D4—242no—
561D4—242no—
562D4—242no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—124no—
581C2×C4—242no—
582C2×C4—242no—
583C2×C4—241no—
584C2×C4—242no—
585C2×C4—242no—
586C2×C4#88211yescharacteristic
587C2×C4#88211yescharacteristic
588C2×C4C22×D1211yescharacteristic
589C2×C4C2×D3×D411yescharacteristic
590C8—42no—
591C8—122no—
592C2×C3—22no—
593C2×C3—22no—
594C2×C3—41no—
595C2×C3—81no—
596C2×C3#300811yescharacteristic
597C2×C3C2×#45511yescharacteristic
598C2×C3C2×#45811yescharacteristic
599D3—82no—
600D3—82no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—44no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—122no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—122no—
617C22—122no—
618C22—244no—
619C22—482no—
620C22—484no—
621C22#482512yes—
622C22C2×#88211yescharacteristic
623C22#576912yes—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—82no—
631C4—122no—
632C4—122no—
633C4—242no—
634C4#474212yes—
635C3#1826411yescharacteristic, complemented, Sylow
636C2—22no—
637C2—22no—
638C2—41no—
639C2—81no—
640C2—242no—
641C2—242no—
642C2C2×#474211yescharacteristic
643C2#7883411yescharacteristic
644C2C2×#482511yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1501C2
socleC22×C3C2×#120
prime radical#86052C2
hypercenter#1501D3
hypocenterC3#18264
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1501

Complements

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

NH ≅ G/Ntype
2. #82914C2semidirect
3. C2×#4736C2semidirect
4. #73519C2semidirect
5. C3×#1501C2semidirect
6. #69921C2semidirect
7. #69444C2semidirect
8. #68321C2semidirect
9. #68288C2semidirect
15. C22×C4⋊Dic3C22semidirect
27. C2×C3×C22⋊C8C22semidirect
28. C3×C4×C22⋊C4C22semidirect
30. #5207C22semidirect
34. #5043C22semidirect
36. #5033C22semidirect
42. #4736C22semidirect
43. #4674C22semidirect
44. C3×#328C22semidirect
45. C3×#327C22semidirect
46. #4468C22semidirect
47. #4451C22semidirect
55. #1501D3semidirect
167. C2×C22⋊C8C2×D3semidirect
172. C4×C22⋊C4C2×D3semidirect
173. #328C2×D3semidirect
175. #327C2×D3semidirect
635. C3#18264semidirect

Maximal subgroups

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

subgroupindex
#860522
#829142
C2×#47362
#735192
C3×#15012
#699212
#694442
#683212
#682882
#182643