Finite Group Databasebeta

← C2×#76365 #10966532

Subgroups — C2×#76365 #10966532

Order 768.

2918 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#936C211yesmaximal, characteristic
2C22×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#390C211yesmaximal, characteristic
4C2×#5221C211yesmaximal, characteristic
5C2×#4973C211yesmaximal, characteristic, complemented
6C2×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
7C2×#4952C212yesmaximal, complemented
8C2×#4947C212yesmaximal, complemented
9C2×#4944C212yesmaximal, complemented
10C2×#4951C212yesmaximal, complemented
11C2×#4929C211yesmaximal, characteristic, complemented
12#76365C2116yesmaximal, direct product factor
13C2×#2592—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C23×Dic6C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C42C2211yescharacteristic
21C22×#217C2212yes—
22C22×#217C2212yes—
23C22×#217C2212yes—
24C22×#217C2212yes—
25C2×#936—28no—
26C2×#936C2218yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×Dic3⋊C4C2211yescharacteristic
30C22×Dic3⋊C4C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C22×C4×Dic3C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C2×C4×Dic6—28no—
35C2×C4×Dic6C2218yescomplemented
36C3×#390C2218yes—
37#5221C2218yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43Dic3×C4⋊C4C2218yescomplemented
44#4973C2218yescomplemented
45#4952C22116yescomplemented
46#4951C22116yescomplemented
47#4947C22116yescomplemented
48#4944C22116yescomplemented
49#4929C2218yescomplemented
50C22×#124—31no—
51C22×C4×Q8—31no—
52#2592—316no—
53C2×#397—32no—
54C2×#393—32no—
55C2×#391—31no—
56C2×#391—31no—
57C2×#390—31no—
58C2×#390D311yescharacteristic
59C2×#388—32no—
60C2×#388—32no—
61C2×C4×C4⋊C4—31no—
62C25×C3C2312yes—
63C22×C3×D4—24no—
64C22×C3×D4—24no—
65C22×C3×D4—24no—
66C22×C3×D4C2314yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×Dic3C2312yes—
72C23×Dic3C2312yes—
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2311yescharacteristic
77C22×Dic6—24no—
78C22×Dic6—24no—
79C22×Dic6—24no—
80C22×Dic6—24no—
81C22×Dic6—24no—
82C22×Dic6—24no—
83C22×Dic6C2314yes—
84C22×Dic6C2314yes—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4—44no—
89C2×C3×C22⋊C4C2318yes—
90C2×C3×C22⋊C4C2318yes—
91C2×C3×C42—24no—
92C2×C3×C42C2314yes—
93C2×#217—28no—
94C2×#217—28no—
95C2×#217—28no—
96C2×#217—44no—
97C2×#217C2318yes—
98C2×#217C2318yes—
99C2×#217C2318yes—
100C2×#217C2318yes—
101#936—232no—
102C2×C4⋊Dic3—24no—
103C2×C4⋊Dic3—24no—
104C2×C4⋊Dic3C2314yes—
105C2×C4⋊Dic3C2314yes—
106C2×Dic3⋊C4—24no—
107C2×Dic3⋊C4—24no—
108C2×Dic3⋊C4C2314yes—
109C2×Dic3⋊C4C2314yes—
110C2×C4×Dic3—24no—
111C2×C4×Dic3—44no—
112C2×C4×Dic3D418yes—
113C2×C4×Dic3C2314yes—
114C2×C4×Dic3C2318yes—
115C2×C4×Dic3C2314yes—
116C4×Dic6—232no—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C23×Q8—31no—
123C23×Q8—31no—
124C23×D4C2×D311yescharacteristic
125C2×#124—38no—
126C2×#124—68no—
127C2×C4×Q8—38no—
128C2×C4×Q8—68no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4C2×D311yescharacteristic
134C22×C22⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4C2×D312yes—
139C22×C22⋊C4C2×D312yes—
140C22×C42—31no—
141C22×C42—31no—
142C22×C42—32no—
143C22×C42C2×D311yescharacteristic
144#397—316no—
145#393—316no—
146#391—38no—
147#391—38no—
148#390—38no—
149#390C2×D318yes—
150#388—316no—
151#388—316no—
152C4×C4⋊C4—38no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95—32no—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—24no—
161C24×C3—24no—
162C24×C3—24no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3C2414yes—
166C24×C3C2411yescharacteristic
167C2×C3×D4—28no—
168C2×C3×D4—28no—
169C2×C3×D4—28no—
170C2×C3×D4—416no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D414yes—
180C22×C3×C4#4114yes—
181C22×C3×C4#4114yes—
182C22×C3×C4#4114yes—
183C22×C3×C4C2414yes—
184C22×C3×C4C2412yes—
185C22×C3×C4C2412yes—
186C22×C3×C4C2412yes—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3C2×D414yes—
203C22×Dic3C2×D414yes—
204C22×Dic3#4114yes—
205C22×Dic3C2414yes—
206C22×Dic3C2414yes—
207C22×Dic3C2412yes—
208C22×Dic3C2412yes—
209C22×Dic3C2412yes—
210C22×Dic3C2412yes—
211C2×Dic6—232no—
212C2×Dic6—28no—
213C2×Dic6—28no—
214C2×Dic6—28no—
215C2×Dic6—28no—
216C2×Dic6—28no—
217C2×Dic6—28no—
218C2×Dic6—416no—
219C3×C4⋊C4—28no—
220C3×C22⋊C4—216no—
221C3×C22⋊C4—48no—
222C3×C42—28no—
223#217—216no—
224#217—216no—
225#217—216no—
226#217—48no—
227C4⋊Dic3—28no—
228C4⋊Dic3—28no—
229Dic3⋊C4—28no—
230Dic3⋊C4—28no—
231C4×Dic3—28no—
232C4×Dic3—48no—
233C4×Dic3C2×D4116yes—
234C25C22×D312yes—
235C22×Q8—34no—
236C22×Q8—34no—
237C22×Q8—64no—
238C22×Q8—64no—
239C22×Q8—64no—
240C22×Q8—64no—
241C22×Q8—64no—
242C22×Q8—64no—
243C22×D4—24no—
244C22×D4—24no—
245C22×D4—24no—
246C22×D4C22×D314yes—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—31no—
251C23×C4—32no—
252C23×C4—32no—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D312yes—
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257#124—632no—
258C4×Q8—632no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—34no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4C22×D314yes—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—38no—
271C2×C22⋊C4—38no—
272C2×C22⋊C4—38no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—68no—
276C2×C22⋊C4—68no—
277C2×C22⋊C4—68no—
278C2×C22⋊C4—124no—
279C2×C22⋊C4C22×D318yes—
280C2×C22⋊C4C22×D318yes—
281C2×C42—24no—
282C2×C42—38no—
283C2×C42—34no—
284C2×C42—34no—
285C2×C42—38no—
286C2×C42—64no—
287C2×C42—124no—
288C2×C42C22×D314yes—
289#95—38no—
290#95—38no—
291#95—38no—
292#95—38no—
293#95—38no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—48no—
302C23×C3—48no—
303C23×C3—48no—
304C23×C3—48no—
305C23×C3—48no—
306C23×C3—48no—
307C23×C3—48no—
308C23×C3C22×D412yes—
309C23×C3C2×#4112yes—
310C23×C3C2×#4112yes—
311C23×C3C2×#4112yes—
312C23×C3C2×#4112yes—
313C23×C3H(5,𝔽2)12yes—
314C23×C32−1+412yes—
315C23×C3C2511yescharacteristic
316C3×D4—416no—
317C2×C3×C4—28no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—44no—
327C2×C3×C4—48no—
328C2×C3×C4#124116yes—
329C2×C3×C4C22×D414yes—
330C2×C3×C4C2×#4114yes—
331C2×C3×C4C2×#4114yes—
332C2×C3×C4C2×#4114yes—
333C2×Dic3—28no—
334C2×Dic3—28no—
335C2×Dic3—28no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—48no—
352C2×Dic3—44no—
353C2×Dic3—48no—
354C2×Dic3C22×D414yes—
355C2×Dic3C22×D414yes—
356C2×Dic3C2×#4114yes—
357Dic6—232no—
358Dic6—416no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24C23×D314yes—
367C24C23×D311yescharacteristic
368C2×Q8—68no—
369C2×Q8—68no—
370C2×Q8—632no—
371C2×Q8—68no—
372C2×Q8—68no—
373C2×Q8—68no—
374C2×Q8—68no—
375C2×Q8—1216no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—416no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—34no—
389C22×C4—34no—
390C22×C4—34no—
391C22×C4—32no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—34no—
395C22×C4—34no—
396C22×C4—32no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—62no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—62no—
410C22×C4—64no—
411C22×C4—62no—
412C22×C4#23514yes—
413C22×C4D3×D414yes—
414C22×C4#23714yes—
415C22×C4#23714yes—
416C22×C4C23×D314yes—
417C22×C4C23×D312yes—
418C22×C4C23×D312yes—
419C22×C4C23×D312yes—
420C4⋊C4—28no—
421C4⋊C4—68no—
422C4⋊C4—68no—
423C4⋊C4—68no—
424C4⋊C4—68no—
425C22⋊C4—216no—
426C22⋊C4—48no—
427C22⋊C4—616no—
428C22⋊C4—616no—
429C22⋊C4—616no—
430C22⋊C4—128no—
431C42—28no—
432C42—316no—
433C42—68no—
434C42—128no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—816no—
443C22×C3#52514yes—
444C22×C3C2×#12414yes—
445C22×C3#53714yes—
446C22×C3#55614yes—
447C22×C3#55614yes—
448C22×C3#55914yes—
449C22×C3#55914yes—
450C22×C3C23×D411yescharacteristic
451C22×C3C22×#4111yescharacteristic
452C22×C3C22×#4111yescharacteristic
453C22×C3C22×#4111yescharacteristic
454C22×C3C22×#4111yescharacteristic
455C22×C3C2×H(5,𝔽2)11yescharacteristic
456C22×C3C2×2−1+411yescharacteristic
457C3×C4—24no—
458C3×C4—42no—
459C3×C4—44no—
460C3×C4C2×#12418yes—
461Dic3—24no—
462Dic3—24no—
463Dic3—24no—
464Dic3—44no—
465Dic3—42no—
466Dic3—44no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—48no—
475C23—48no—
476C23—48no—
477C23—48no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23C2×#23512yes—
482C23C2×D3×D412yes—
483C23C2×#23712yes—
484C23C2×#23712yes—
485C23#100412yes—
486C23D3×#4112yes—
487C23#101012yes—
488C23C24×D311yescharacteristic
489Q8—632no—
490Q8—1216no—
491D4—416no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—34no—
502C2×C4—34no—
503C2×C4—34no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—68no—
513C2×C4—68no—
514C2×C4—68no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—62no—
522C2×C4—64no—
523C2×C4—62no—
524C2×C4—128no—
525C2×C4—128no—
526C2×C4—124no—
527C2×C4#885116yes—
528C2×C4C2×#23514yes—
529C2×C4C2×D3×D414yes—
530C2×C4C2×#23714yes—
531C2×C4C2×#23714yes—
532C2×C3—84no—
533C2×C3#259218yescomplemented
534C2×C3C2×#52511yescharacteristic
535C2×C3C22×#12411yescharacteristic
536C2×C3C2×#53711yescharacteristic
537C2×C3C2×#55611yescharacteristic
538C2×C3C2×#55611yescharacteristic
539C2×C3C2×#55911yescharacteristic
540C2×C3C2×#55911yescharacteristic
541C22—44no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—816no—
549C22C2×#88514yes—
550C22#554214yes—
551C22#555914yes—
552C22#560614yes—
553C22#560814yes—
554C22#567114yes—
555C22#571314yes—
556C22C22×#23511yescharacteristic
557C22C22×D3×D411yescharacteristic
558C22C22×#23711yescharacteristic
559C22C22×#23711yescharacteristic
560C22C2×#100411yescharacteristic
561C22C2×D3×#4111yescharacteristic
562C22C2×#101011yescharacteristic
563C4—24no—
564C4—44no—
565C4—42no—
566C4—64no—
567C4—64no—
568C4—64no—
569C4—124no—
570C4—124no—
571C4—122no—
572C4C2×#88518yes—
573C3C2×#259211yescharacteristic, complemented, Sylow
574C2—84no—
575C2#7636518yesdirect product factor
576C2C22×#88511yescharacteristic
577C2C2×#554211yescharacteristic
578C2C2×#555911yescharacteristic
579C2C2×#560611yescharacteristic
580C2C2×#560811yescharacteristic
581C2C2×#567111yescharacteristic
582C2C2×#571311yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#390C2
socleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3C2×#2592
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic6C2semidirect
5. C2×#4973C2semidirect
6. C2×Dic3×C4⋊C4C2semidirect
7. C2×#4952C2semidirect
8. C2×#4947C2semidirect
9. C2×#4944C2semidirect
10. C2×#4951C2semidirect
11. C2×#4929C2semidirect
12. #76365C2direct (N × H)
35. C2×C4×Dic6C22semidirect
43. Dic3×C4⋊C4C22semidirect
44. #4973C22semidirect
45. #4952C22semidirect
46. #4951C22semidirect
47. #4947C22semidirect
48. #4944C22semidirect
49. #4929C22semidirect
533. C2×C3#2592semidirect
573. C3C2×#2592semidirect
575. C2#76365direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9362
C22×C4×Dic62
C2×C3×#3902
C2×#52212
C2×#49732
C2×Dic3×C4⋊C42
C2×#49522
C2×#49472
C2×#49442
C2×#49512
C2×#49292
#763652
C2×#25923