Finite Group Databasebeta

← #10967592

Subgroups — #10967592

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×Q8×Dic3C211yesmaximal, characteristic, complemented
2C2×#5624C211yesmaximal, characteristic
3C3×#2434C211yesmaximal, characteristic, complemented
4#84066C211yesmaximal, characteristic
5#83237C212yesmaximal, complemented
6Dic3×C4⋊Q8C212yesmaximal, complemented
7#82776C211yesmaximal, characteristic, complemented
8#77197C214yesmaximal, complemented
9#77170C214yesmaximal, complemented
10#76541C212yesmaximal, complemented
11#76480C212yesmaximal, complemented
12#74407C214yesmaximal, complemented
13#74380C214yesmaximal, complemented
14#73516C211yesmaximal, characteristic
15C4×#877C211yesmaximal, characteristic, complemented
16#34972—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C2×#947C2212yes—
22C2×Q8×Dic3—28no—
23C2×Q8×Dic3C2211yescharacteristic
24C2×Q8×Dic3C2212yes—
25C2×Q8×Dic3C2212yes—
26C2×Q8×Dic3C2212yes—
27C2×Q8×Dic3C2212yes—
28C2×#930C2211yescharacteristic
29C22×C4⋊Dic3C2211yescharacteristic
30C22×C4⋊Dic3C2212yes—
31C22×C4×Dic3C2212yes—
32C22×C4×Dic3C2211yescharacteristic
33#5624—28no—
34C2×#898C2212yes—
35C2×#893C2211yescharacteristic
36C2×#892C2212yes—
37C2×#890C2211yescharacteristic
38C2×#886C2214yes—
39C2×#886C2212yes—
40C2×#67⋊C4C2212yes—
41C2×#877C2212yes—
42C3×#395C2214yes—
43C3×#391C2212yes—
44C3×#385C2214yes—
45C3×C4×C22⋊C4C2211yescharacteristic
46#5231C2214yes—
47#5229C2212yes—
48#5229C2212yes—
49#5213C2214yes—
50C4×#217C2211yescharacteristic
51#4994C2218yes—
52#4990C2218yes—
53#4986C2214yes—
54#4984C2218yes—
55#4977C2212yes—
56Dic3×C4⋊C4C2218yes—
57#4943C2214yes—
58Dic3×C22⋊C4C2212yes—
59#4939C2212yes—
60#4935C2212yes—
61C4×Dic3⋊C4C2212yes—
62C42×Dic3C2212yes—
63#4677C2214yes—
64#4670C2218yes—
65#4666C2218yes—
66#4661C2218yes—
67#4654C2218yes—
68#4649C2212yes—
69#4648C2212yes—
70C2×#529—31no—
71C22×C4×Q8—31no—
72#2604—34no—
73#2565—32no—
74#2545—32no—
75#2503—31no—
76#2501—31no—
77#2436—31no—
78#2434D311yescharacteristic, complemented
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×Q8C2311yescharacteristic
89C22×C3×Q8C2312yes—
90C23×C3×C4C2312yes—
91C23×C3×C4C2311yescharacteristic
92C22×#67C2312yes—
93C23×Dic3C2312yes—
94C23×Dic3C2311yescharacteristic
95C23×Dic3C2311yescharacteristic
96C22×D12C2311yescharacteristic
97C22×C4×D3C2312yes—
98C3×C4⋊Q8—28no—
99C3×C22⋊Q8—24no—
100C2×C3×C4⋊C4C2318yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C22⋊C4C2312yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C42C2312yes—
106C2×#217C2312yes—
107#947—28no—
108#947—24no—
109Q8×Dic3—28no—
110Q8×Dic3—28no—
111Q8×Dic3—28no—
112Q8×Dic3—28no—
113Q8×Dic3—24no—
114#930—24no—
115C2×(C2×D3)⋊C4C2318yes—
116C2×(C2×D3)⋊C4C2318yes—
117C2×(C2×D3)⋊C4C2312yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×C4⋊Dic3—24no—
122C2×C4⋊Dic3—28no—
123C2×C4⋊Dic3C2318yes—
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2314yes—
126C2×C4⋊Dic3C2314yes—
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2311yescharacteristic
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2312yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2312yes—
133C2×C4×Dic3—28no—
134C2×C4×Dic3D418yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2314yes—
138C2×C4×Dic3C2314yes—
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2311yescharacteristic
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2311yescharacteristic
145#898—28no—
146#893—24no—
147#892—28no—
148#890—24no—
149#886—28no—
150#886—216no—
151#67⋊C4—28no—
152#877—28no—
153C3×#95C2312yes—
154C3×#95C2318yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2318yes—
158C3⋊#95C2318yes—
159C3⋊#95C2318yes—
160C23×Q8C2×D311yescharacteristic
161C2×C4⋊Q8C2×D312yes—
162#529—68no—
163C2×#126—32no—
164C2×#125—31no—
165C2×#124—31no—
166C2×#124—32no—
167C2×#123—32no—
168C2×#123—34no—
169C2×C22⋊Q8C2×D311yescharacteristic
170C2×#121—31no—
171C2×C4×Q8—31no—
172C2×C4×Q8—32no—
173C2×C4×Q8—32no—
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×Q8—68no—
177C2×C4×D4—31no—
178C2×C4×D4—32no—
179C2×#117—32no—
180C2×#117—32no—
181C22×C4⋊C4—32no—
182C22×C4⋊C4—31no—
183C22×C42—31no—
184C22×C42—32no—
185#400—34no—
186#399—34no—
187#398—38no—
188#396—38no—
189#395C2×D314yes—
190#393—34no—
191#391—32no—
192#391—32no—
193#391C2×D312yes—
194#390—32no—
195#389—38no—
196#388—38no—
197#388—38no—
198#387—38no—
199#386—34no—
200#385C2×D314yes—
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—32no—
209C4×C22⋊C4—38no—
210C4×C22⋊C4C2×D311yescharacteristic
211#376—32no—
212C43—32no—
213C24×C3C2411yescharacteristic
214C23×D3C2412yes—
215C2×C3×Q8—232no—
216C2×C3×Q8—24no—
217C2×C3×Q8—24no—
218C2×C3×Q8—22no—
219C2×C3×Q8—28no—
220C2×C3×Q8—24no—
221C2×C3×Q8—24no—
222C2×C3×Q8#4118yes—
223C22×C3×C4—24no—
224C22×C3×C4—28no—
225C22×C3×C4—24no—
226C22×C3×C4C2×D414yes—
227C22×C3×C4C2412yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2414yes—
231C22×C3×C4C2414yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—24no—
234C2×#67—28no—
235C22×Dic3—24no—
236C22×Dic3—24no—
237C22×Dic3—24no—
238C22×Dic3—28no—
239C22×Dic3C2×D414yes—
240C22×Dic3C2×D414yes—
241C22×Dic3C2414yes—
242C22×Dic3C2414yes—
243C22×Dic3C2412yes—
244C22×Dic3C2411yescharacteristic
245C22×Dic3C2411yescharacteristic
246C22×Dic3C2411yescharacteristic
247C22×Dic3C2411yescharacteristic
248C2×D12—22no—
249C2×D12—24no—
250C2×C4×D3—24no—
251C2×C4×D3—28no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—216no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4#4118yes—
257C3×C42—24no—
258#217—24no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—216no—
261(C2×D3)⋊C4—216no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265C4⋊Dic3—22no—
266C4⋊Dic3—24no—
267C4⋊Dic3—24no—
268C4⋊Dic3—216no—
269C4⋊Dic3—24no—
270C4⋊Dic3—24no—
271C4⋊Dic3—28no—
272C4⋊Dic3—28no—
273Dic3⋊C4—28no—
274Dic3⋊C4—24no—
275C4×Dic3—28no—
276C4×Dic3—24no—
277C4×Dic3—24no—
278C4×Dic3—28no—
279C4×Dic3—28no—
280C4×Dic3—28no—
281C4×Dic3C2×D414yes—
282C4×Dic3C2×D414yes—
283C4×Dic3C2×D414yes—
284C4×Dic3C2×D414yes—
285C22×Q8—28no—
286C22×Q8—24no—
287C22×Q8C22×D312yes—
288C22×Q8C22×D311yescharacteristic
289C22×Q8C22×D312yes—
290C22×D4—32no—
291C22×D4—31no—
292C23×C4—31no—
293C23×C4—31no—
294C23×C4—32no—
295C23×C4—32no—
296C23×C4C22×D312yes—
297C23×C4C22×D311yescharacteristic
298C4⋊Q8—28no—
299#126—68no—
300#125—64no—
301#124—64no—
302#124—68no—
303#123—68no—
304#123—616no—
305C22⋊Q8—24no—
306#121—64no—
307C4×Q8—68no—
308C4×Q8—68no—
309C4×Q8—68no—
310C4×Q8—68no—
311C4×Q8—64no—
312C4×D4—64no—
313C4×D4—68no—
314#117—68no—
315#117—68no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—34no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—32no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—38no—
326C2×C4⋊C4—68no—
327C2×C4⋊C4—64no—
328C2×C4⋊C4C22×D312yes—
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C4⋊C4C22×D318yes—
331C2×C22⋊C4—32no—
332C2×C22⋊C4—32no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—38no—
337C2×C22⋊C4—38no—
338C2×C22⋊C4C22×D312yes—
339C2×C22⋊C4C22×D312yes—
340C2×C42—38no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—31no—
344C2×C42—31no—
345C2×C42—32no—
346C2×C42—34no—
347C2×C42—34no—
348C2×C42—34no—
349C2×C42—32no—
350C2×C42—34no—
351C2×C42—68no—
352C2×C42C22×D312yes—
353#95—34no—
354#95—32no—
355#95—38no—
356#95—38no—
357#95—38no—
358#95C22×D312yes—
359#95C22×D318yes—
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—
371C3×Q8—44no—
372C2×C3×C4—216no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—24no—
377C2×C3×C4—24no—
378C2×C3×C4—28no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4C22×D412yes—
384C2×C3×C4C22×D412yes—
385C2×C3×C4C2×#4118yes—
386C2×C3×C4C2×#4114yes—
387C2×C3×C4C2×#4118yes—
388#67—48no—
389C2×Dic3—28no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—28no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—24no—
397C2×Dic3—24no—
398C2×Dic3—24no—
399C2×Dic3—28no—
400C2×Dic3—24no—
401C2×Dic3—48no—
402C2×Dic3C22×D412yes—
403C2×Dic3C22×D412yes—
404C2×Dic3C22×D412yes—
405C2×Dic3C22×D412yes—
406C2×Dic32−1+414yes—
407D12—44no—
408C4×D3—48no—
409C24—32no—
410C24C23×D311yescharacteristic
411C2×Q8—232no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—28no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8#23718yes—
419C2×D4—64no—
420C2×D4—68no—
421C2×D4—62no—
422C2×D4—64no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—31no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—34no—
433C22×C4—34no—
434C22×C4—32no—
435C22×C4—64no—
436C22×C4—68no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—68no—
441C22×C4D3×D414yes—
442C22×C4C23×D312yes—
443C22×C4C23×D312yes—
444C22×C4C23×D311yescharacteristic
445C22×C4C23×D314yes—
446C22×C4C23×D314yes—
447C22×C4C23×D311yescharacteristic
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—216no—
451C4⋊C4—68no—
452C4⋊C4—68no—
453C4⋊C4—64no—
454C4⋊C4—64no—
455C4⋊C4—64no—
456C4⋊C4—64no—
457C4⋊C4—64no—
458C4⋊C4—62no—
459C4⋊C4—68no—
460C4⋊C4—616no—
461C22⋊C4—24no—
462C22⋊C4—64no—
463C22⋊C4—64no—
464C22⋊C4—64no—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C22⋊C4—616no—
468C22⋊C4—616no—
469C22⋊C4#23918yes—
470C42—24no—
471C42—34no—
472C42—34no—
473C42—34no—
474C42—34no—
475C42—68no—
476C42—68no—
477C42—64no—
478C42—68no—
479C42—64no—
480C42—68no—
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—28no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4#52514yes—
499Dic3—22no—
500Dic3—22no—
501Dic3—22no—
502Dic3—44no—
503Dic3—42no—
504Dic3—44no—
505Dic3#53614yes—
506C23—22no—
507C23—22no—
508C23—62no—
509C23—64no—
510C23—64no—
511C23—64no—
512C23C2×D3×D412yes—
513C23C2×#23914yes—
514C23C24×D311yescharacteristic
515Q8—216no—
516Q8—216no—
517Q8—44no—
518D4—128no—
519D4—124no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—32no—
532C2×C4—34no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—32no—
536C2×C4—68no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—68no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—68no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—128no—
549C2×C4—128no—
550C2×C4C2×D3×D412yes—
551C2×C4C2×D3×D412yes—
552C2×C4C2×#23718yes—
553C2×C4C2×#23714yes—
554C2×C4C2×#23918yes—
555C2×C3—24no—
556C2×C3C2×#52511yescharacteristic
557C2×C3C2×#53211yescharacteristic
558C2×C3C2×#53611yescharacteristic
559C2×C3#348214yes—
560D3—82no—
561C22—22no—
562C22—22no—
563C22—28no—
564C22—122no—
565C22—122no—
566C22—122no—
567C22—128no—
568C22#572514yes—
569C22C22×D3×D411yescharacteristic
570C22C22×#23712yes—
571C22C22×#23912yes—
572C22D3×2−1+412yes—
573C4—22no—
574C4—28no—
575C4—28no—
576C4—34no—
577C4—42no—
578C4—42no—
579C4—62no—
580C4—62no—
581C4—62no—
582C4—124no—
583C4—122no—
584C4—124no—
585C4#558614yes—
586C3#3497211yescharacteristic, complemented, Sylow
587C2—24no—
588C2—242no—
589C2C2×#558611yescharacteristic
590C2C2×#572511yescharacteristic
591C2D3×#53611yescharacteristic
592C2#8674414yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2434C2
prime radicalC2×#930C22
hypercenter#2434D3
hypocenterC3#34972
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2434

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic3C2semidirect
3. C3×#2434C2semidirect
5. #83237C2semidirect
6. Dic3×C4⋊Q8C2semidirect
7. #82776C2semidirect
8. #77197C2semidirect
9. #77170C2semidirect
10. #76541C2semidirect
11. #76480C2semidirect
12. #74407C2semidirect
13. #74380C2semidirect
15. C4×#877C2semidirect
78. #2434D3semidirect
586. C3#34972semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×#56242
C3×#24342
#840662
#832372
Dic3×C4⋊Q82
#827762
#771972
#771702
#765412
#764802
#744072
#743802
#735162
C4×#8772
#349723