Finite Group Databasebeta

← #10967599

Subgroups — #10967599

Order 768.

2474 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#5479C211yesmaximal, characteristic
3#84066C211yesmaximal, characteristic
4#84065C211yesmaximal, characteristic, complemented
5#82491C214yesmaximal, complemented
6#82485C214yesmaximal, complemented
7#82477C212yesmaximal, complemented
8#82474C212yesmaximal, complemented
9C4×#944C212yesmaximal, complemented
10C4×#947C212yesmaximal, complemented
11#82179C211yesmaximal, characteristic, complemented
12#82177C211yesmaximal, characteristic
13#82170C214yesmaximal, complemented
14C4×#929C214yesmaximal, complemented
15C4×#926C211yesmaximal, characteristic, complemented
16#34972—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C2×C3×C4×Q8—28no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2212yes—
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C4⋊C4C2212yes—
25C22×C3×C42C2212yes—
26C22×C3×C42C2211yescharacteristic
27C2×#947C2212yes—
28C2×#944C2212yes—
29C2×#930C2211yescharacteristic
30C2×#929C2214yes—
31C2×#929C2212yes—
32C2×C4×#67C2212yes—
33C2×#926C2212yes—
34C2×#924C2211yescharacteristic
35#5479—28no—
36C2×#876C2212yes—
37C2×#875C2211yescharacteristic
38C2×C4×D12C2211yescharacteristic
39C2×#872C2212yes—
40C2×#870C2211yescharacteristic
41C2×C4×Dic6C2211yescharacteristic
42#5231C2214yes—
43#5230C2214yes—
44#5229C2212yes—
45#5229C2212yes—
46#5213C2214yes—
47#5211C2214yes—
48#5210C2214yes—
49C4×#217C2211yescharacteristic
50C4×#217C2211yescharacteristic
51C4×#217C2212yes—
52#4994C2218yes—
53#4993C2214yes—
54#4990C2218yes—
55#4988C2212yes—
56#4985C2214yes—
57#4982C2218yes—
58#4977C2212yes—
59#4976C2218yes—
60#4975C2212yes—
61#4973C2212yes—
62#4940C2218yes—
63C4×(C2×D3)⋊C4C2218yes—
64C4×C4⋊Dic3C2212yes—
65C4×C4⋊Dic3C2212yes—
66#4933C2218yes—
67#4932C2212yes—
68C4×Dic3⋊C4C2218yes—
69C42×Dic3C2212yes—
70C2×#529—31no—
71C22×C4×Q8D311yescharacteristic, complemented
72#2604—34no—
73#2565—32no—
74#2545—32no—
75#2503—31no—
76#2501—31no—
77#2436—31no—
78#2434—31no—
79#2413—34no—
80#2396—34no—
81C4×C4⋊Q8—32no—
82C4×#124—32no—
83C4×#123—34no—
84C4×#117—31no—
85C22×C3×Q8—28no—
86C22×C3×Q8—24no—
87C22×C3×Q8C2312yes—
88C22×C3×Q8C2312yes—
89C23×C3×C4C2312yes—
90C23×C3×C4C2312yes—
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C23×C3×C4C2311yescharacteristic
94C22×#67C2312yes—
95C22×D12C2311yescharacteristic
96C22×C4×D3C2312yes—
97C22×Dic6C2311yescharacteristic
98C3×C4×Q8—28no—
99C3×C4×Q8—28no—
100C3×C4×Q8—28no—
101C3×C4×Q8—28no—
102C2×C3×C4⋊C4—28no—
103C2×C3×C4⋊C4—24no—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C42—28no—
109C2×C3×C42D418yes—
110C2×C3×C42C2314yes—
111C2×C3×C42C2314yes—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×#217C2312yes—
117C2×#217C2312yes—
118C2×#217C2312yes—
119#947—28no—
120#944—28no—
121#930—24no—
122#929—28no—
123#929—216no—
124C4×#67—28no—
125C2×(C2×D3)⋊C4C2318yes—
126C2×(C2×D3)⋊C4C2318yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2312yes—
131#926—28no—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2311yescharacteristic
135#924—24no—
136C2×Dic3⋊C4C2318yes—
137C2×Dic3⋊C4C2318yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2314yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2312yes—
147#876—28no—
148#875—24no—
149C4×D12—24no—
150#872—28no—
151#870—24no—
152C4×Dic6—24no—
153C3⋊#95C2318yes—
154C3⋊#95C2318yes—
155C3⋊#95C2314yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2318yes—
159C3⋊#95C2318yes—
160C23×Q8C2×D311yescharacteristic
161C2×C4⋊Q8—32no—
162#529—68no—
163C2×#126—32no—
164C2×#125—31no—
165C2×#124—32no—
166C2×#124—31no—
167C2×#123—32no—
168C2×#123—34no—
169C2×C22⋊Q8—31no—
170C2×#121—31no—
171C2×C4×Q8—28no—
172C2×C4×Q8—31no—
173C2×C4×Q8C2×D312yes—
174C2×C4×Q8C2×D312yes—
175C2×C4×Q8C2×D312yes—
176C2×C4×Q8C2×D312yes—
177C2×C4×D4—32no—
178C2×C4×D4—31no—
179C2×#117—32no—
180C2×#117—32no—
181C22×C4⋊C4C2×D312yes—
182C22×C4⋊C4C2×D311yescharacteristic
183C22×C42C2×D312yes—
184C22×C42C2×D311yescharacteristic
185#400—34no—
186#399—34no—
187#398—38no—
188#396—38no—
189#395—34no—
190#393—34no—
191#391—32no—
192#391—32no—
193#391—32no—
194#390—32no—
195#389—38no—
196#388—38no—
197#388—38no—
198#387—38no—
199#386—34no—
200#385—34no—
201#384—32no—
202#382—32no—
203#381—34no—
204C4×C4⋊C4—38no—
205C4×C4⋊C4—32no—
206C4×C4⋊C4—32no—
207C4×C22⋊C4—31no—
208C4×C22⋊C4—31no—
209C4×C22⋊C4—32no—
210C4×C22⋊C4—38no—
211#376—32no—
212C43—32no—
213C24×C3C2411yescharacteristic
214C23×D3C2412yes—
215C2×C3×Q8—232no—
216C2×C3×Q8—24no—
217C2×C3×Q8—24no—
218C2×C3×Q8—28no—
219C2×C3×Q8—24no—
220C2×C3×Q8#4118yes—
221C22×C3×C4—24no—
222C22×C3×C4—24no—
223C22×C3×C4—28no—
224C22×C3×C4—24no—
225C22×C3×C4—24no—
226C22×C3×C4—24no—
227C22×C3×C4—28no—
228C22×C3×C4C2×D414yes—
229C22×C3×C4C2×D414yes—
230C22×C3×C4C2×D414yes—
231C22×C3×C4C2414yes—
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C22×C3×C4C2414yes—
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2411yescharacteristic
239C2×#67—24no—
240C2×#67—28no—
241C22×Dic3C2414yes—
242C22×Dic3C2412yes—
243C22×Dic3C2412yes—
244C22×Dic3C2412yes—
245C22×Dic3C2414yes—
246C2×D12—22no—
247C2×D12—24no—
248C2×C4×D3—24no—
249C2×C4×D3—28no—
250C2×Dic6—22no—
251C2×Dic6—24no—
252C3×C4⋊C4—28no—
253C3×C4⋊C4—28no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C42—28no—
257C3×C42—28no—
258C3×C42C2×D414yes—
259C3×C42C2×D414yes—
260C3×C42C2×D414yes—
261C3×C42C2×D414yes—
262#217—24no—
263#217—24no—
264#217#4118yes—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—216no—
270(C2×D3)⋊C4—216no—
271C4⋊Dic3—22no—
272C4⋊Dic3—22no—
273C4⋊Dic3—28no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—216no—
279Dic3⋊C4—216no—
280C4×Dic3—24no—
281C4×Dic3—24no—
282C4×Dic3—24no—
283C4×Dic3—28no—
284C4×Dic3—28no—
285C22×Q8—28no—
286C22×Q8—24no—
287C22×Q8—31no—
288C22×Q8C22×D312yes—
289C22×Q8C22×D312yes—
290C22×D4—32no—
291C22×D4—31no—
292C23×C4—32no—
293C23×C4C22×D312yes—
294C23×C4C22×D312yes—
295C23×C4C22×D311yescharacteristic
296C23×C4C22×D311yescharacteristic
297C23×C4C22×D311yescharacteristic
298C4⋊Q8—68no—
299#126—68no—
300#125—64no—
301#124—68no—
302#124—64no—
303#123—616no—
304#123—68no—
305C22⋊Q8—64no—
306#121—64no—
307C4×Q8—28no—
308C4×Q8—28no—
309C4×Q8—28no—
310C4×Q8—28no—
311C4×Q8—64no—
312C4×D4—68no—
313C4×D4—64no—
314#117—68no—
315#117—68no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—38no—
326C2×C4⋊C4—38no—
327C2×C4⋊C4C22×D312yes—
328C2×C4⋊C4C22×D314yes—
329C2×C4⋊C4C22×D314yes—
330C2×C4⋊C4C22×D312yes—
331C2×C22⋊C4—32no—
332C2×C22⋊C4—32no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—32no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—38no—
339C2×C22⋊C4—38no—
340C2×C42—28no—
341C2×C42—34no—
342C2×C42—32no—
343C2×C42—32no—
344C2×C42—32no—
345C2×C42—34no—
346C2×C42#6718yes—
347C2×C42C22×D314yes—
348C2×C42C22×D314yes—
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D311yescharacteristic
351C2×C42C22×D311yescharacteristic
352C2×C42C22×D311yescharacteristic
353#95—34no—
354#95—32no—
355#95—32no—
356#95—38no—
357#95—38no—
358#95—38no—
359#95—38no—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3C22×D412yes—
363C23×C3C2×#4114yes—
364C23×C3C2511yescharacteristic
365C22×D3—22no—
366C22×D3—24no—
367C22×D3—24no—
368C22×D3—24no—
369C3×Q8—216no—
370C3×Q8—216no—
371C2×C3×C4—216no—
372C2×C3×C4—28no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4—28no—
380C2×C3×C4—28no—
381C2×C3×C4—24no—
382C2×C3×C4—28no—
383C2×C3×C4—48no—
384C2×C3×C4C22×D412yes—
385C2×C3×C4C22×D412yes—
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C2×#4118yes—
391C2×C3×C4C2×#4114yes—
392C2×C3×C42−1+414yes—
393#67—48no—
394C2×Dic3—24no—
395C2×Dic3—24no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—24no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3—24no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3C2×#4118yes—
406D12—44no—
407C4×D3—48no—
408Dic6—44no—
409C24—32no—
410C24C23×D311yescharacteristic
411C2×Q8—232no—
412C2×Q8—28no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—64no—
417C2×Q8—62no—
418C2×Q8#23518yes—
419C2×D4—68no—
420C2×D4—64no—
421C2×D4—64no—
422C2×D4—62no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—34no—
431C22×C4—34no—
432C22×C4—32no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—68no—
436C22×C4—64no—
437C22×C4C2×#6714yes—
438C22×C4C2×#6714yes—
439C22×C4C2×#6714yes—
440C22×C4C23×D314yes—
441C22×C4C23×D314yes—
442C22×C4C23×D311yescharacteristic
443C22×C4C23×D311yescharacteristic
444C22×C4C23×D311yescharacteristic
445C22×C4C23×D311yescharacteristic
446C22×C4C23×D311yescharacteristic
447C22×C4C23×D311yescharacteristic
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—68no—
453C4⋊C4—616no—
454C4⋊C4—616no—
455C4⋊C4—64no—
456C4⋊C4—64no—
457C4⋊C4—62no—
458C4⋊C4—64no—
459C4⋊C4—62no—
460C4⋊C4—64no—
461C22⋊C4—38no—
462C22⋊C4—616no—
463C22⋊C4—64no—
464C22⋊C4—64no—
465C22⋊C4—616no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—64no—
469C22⋊C4—64no—
470C42—28no—
471C42—28no—
472C42—68no—
473C42—64no—
474C42—64no—
475C42—68no—
476C42—64no—
477C42C2×#6714yes—
478C42C2×#6714yes—
479C42C2×#6714yes—
480C42C2×#6714yes—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—28no—
484C22×C3#53214yes—
485C22×C3C23×D411yescharacteristic
486C22×C3C22×#4112yes—
487C22×C3C22×#4112yes—
488C22×C3C2×2−1+412yes—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—48no—
493C3×C4—22no—
494C3×C4—28no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—44no—
499C3×C4#52514yes—
500C3×C4#53614yes—
501Dic3—28no—
502Dic3—42no—
503Dic3—42no—
504Dic3—42no—
505Dic3—44no—
506C23—22no—
507C23—22no—
508C23—64no—
509C23—64no—
510C23—64no—
511C23—62no—
512C23D3×#4114yes—
513C23C22×#6712yes—
514C23C24×D311yescharacteristic
515Q8—216no—
516Q8—216no—
517Q8—124no—
518D4—128no—
519D4—124no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—38no—
533C2×C4—48no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—128no—
546C2×C4C2×#23518yes—
547C2×C4C2×#23514yes—
548C2×C4#100714yes—
549C2×C4C22×#6712yes—
550C2×C4C22×#6712yes—
551C2×C4C22×#6712yes—
552C2×C4C22×#6712yes—
553C2×C4C22×#6712yes—
554C2×C4C22×#6712yes—
555C2×C3—24no—
556C2×C3C2×#52511yescharacteristic
557C2×C3C2×#53211yescharacteristic
558C2×C3C2×#53611yescharacteristic
559C2×C3#348214yes—
560D3—82no—
561C22—28no—
562C22—22no—
563C22—22no—
564C22—128no—
565C22—122no—
566C22—122no—
567C22—122no—
568C22#583214yes—
569C22C22×#23512yes—
570C22C2×#100712yes—
571C22C2×D3×#4112yes—
572C22C23×#6711yescharacteristic
573C4—28no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—44no—
579C4—68no—
580C4—124no—
581C4—122no—
582C4—122no—
583C4—122no—
584C4#579114yes—
585C4#581614yes—
586C3#3497211yescharacteristic, complemented, Sylow
587C2—24no—
588C2—242no—
589C2#8581714yes—
590C2C2×#579111yescharacteristic
591C2C2×#581611yescharacteristic
592C2C2×#583211yescharacteristic
subgroupG/H
centerC22×C4C23×D3
commutator, socle, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×C4×Q8C2
prime radicalC2×#930C22
hypercenterC22×C4×Q8D3
hypocenterC3#34972
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×Q8C2semidirect
4. #84065C2semidirect
5. #82491C2semidirect
6. #82485C2semidirect
7. #82477C2semidirect
8. #82474C2semidirect
9. C4×#944C2semidirect
10. C4×#947C2semidirect
11. #82179C2semidirect
13. #82170C2semidirect
14. C4×#929C2semidirect
15. C4×#926C2semidirect
71. C22×C4×Q8D3semidirect
586. C3#34972semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×#54792
#840662
#840652
#824912
#824852
#824772
#824742
C4×#9442
C4×#9472
#821792
#821772
#821702
C4×#9292
C4×#9262
#349723