Finite Group Databasebeta

← C2×#76251 #10962253

Subgroups — C2×#76251 #10962253

Order 768.

2614 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#890C211yesmaximal, characteristic, complemented
2C22×#872C211yesmaximal, characteristic
3C2×C3×#389C211yesmaximal, characteristic, complemented
4C2×#4994C212yesmaximal
5C2×#4987C211yesmaximal, characteristic
6C2×#4987C211yesmaximal, characteristic
7C2×#4983C212yesmaximal, complemented
8C2×#4975C211yesmaximal, characteristic, complemented
9C2×#4928C211yesmaximal, characteristic, complemented
10#76251C2116yesmaximal, direct product factor
11C2×#4676C212yesmaximal
12C2×#4657C212yesmaximal, complemented
13C2×#2629—31nomaximal, Sylow
14C23×C4×D3C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C42C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2211yescharacteristic
29C22×Dic3⋊C4C2212yes—
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×C4×Dic3C2211yescharacteristic
33C2×#890—28no—
34C2×#890C2218yescomplemented
35C2×#872—28no—
36C2×#872C2218yes—
37C3×#389C2218yescomplemented
38C2×C3×#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42#4994C22116yes—
43#4987C2218yes—
44#4987C2218yes—
45#4983C22116yescomplemented
46#4975C2218yescomplemented
47#4928C2218yescomplemented
48#4676C22116yes—
49#4657C22116yescomplemented
50C22×#125—31no—
51C22×#117—31no—
52#2629—316no—
53C2×#398—32no—
54C2×#398—32no—
55C2×#396—32no—
56C2×#396—32no—
57C2×#389—31no—
58C2×#389D311yescharacteristic, complemented
59C2×#384—31no—
60C2×#380—31no—
61C2×#380—31no—
62C24×D3C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2312yes—
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C22×C4×D3—24no—
74C22×C4×D3—24no—
75C22×C4×D3Q818yes—
76C22×C4×D3C2314yes—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—44no—
80C2×C3×C4⋊C4C2314yes—
81C2×C3×C4⋊C4C2314yes—
82C2×C3×C4⋊C4C2318yes—
83C2×C3×C42—24no—
84C2×C3×C42C2314yes—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—44no—
88C2×(C2×D3)⋊C4—44no—
89C2×(C2×D3)⋊C4C2314yes—
90C2×(C2×D3)⋊C4C2314yes—
91C2×(C2×D3)⋊C4C2318yes—
92C2×(C2×D3)⋊C4C2318yes—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—28no—
96C2×C4⋊Dic3C2318yes—
97C2×C4⋊Dic3C2314yes—
98C2×C4⋊Dic3C2314yes—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—28no—
102C2×Dic3⋊C4—44no—
103C2×Dic3⋊C4—44no—
104C2×Dic3⋊C4—44no—
105C2×Dic3⋊C4C2314yes—
106C2×Dic3⋊C4C2318yes—
107C2×Dic3⋊C4C2314yes—
108C2×Dic3⋊C4C2318yes—
109C2×Dic3⋊C4C2318yes—
110C2×Dic3⋊C4C2318yes—
111C2×C4×Dic3D418yes—
112C2×C4×Dic3C2314yes—
113#890—232no—
114#872—232no—
115C3×#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C24×C4—31no—
120C2×#125—38no—
121C2×#125—68no—
122C2×#117—38no—
123C2×#117—68no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4C2×D311yescharacteristic
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D312yes—
136C22×C22⋊C4—31no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C42—31no—
141C22×C42C2×D311yescharacteristic
142#398—316no—
143#398—316no—
144#396—316no—
145#396—316no—
146#389—38no—
147#389C2×D318yescomplemented
148#384—38no—
149#380—38no—
150#380—38no—
151C2×#95—32no—
152C2×#95—32no—
153C2×#95—32no—
154C2×#95C2×D312yes—
155C24×C3C2411yescharacteristic
156C23×D3—22no—
157C23×D3—22no—
158C23×D3—22no—
159C23×D3—22no—
160C23×D3—22no—
161C23×D3—22no—
162C23×D3C2×Q814yes—
163C23×D3C2412yes—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×Q814yes—
178C22×C3×C4#4114yes—
179C22×C3×C4C2412yes—
180C22×C3×C4C2414yes—
181C22×C3×C4C2414yes—
182C22×C3×C4C2412yes—
183C22×C3×C4C2412yes—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—24no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3C2×D414yes—
197C22×Dic3C2×D414yes—
198C22×Dic3C2×Q814yes—
199C22×Dic3C2414yes—
200C22×Dic3C2414yes—
201C22×Dic3C2412yes—
202C22×Dic3C2412yes—
203C22×Dic3C2412yes—
204C2×C4×D3—232no—
205C2×C4×D3—28no—
206C2×C4×D3—28no—
207C2×C4×D3C2×Q8116yes—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—28no—
210C3×C4⋊C4—48no—
211C3×C42—28no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—48no—
215(C2×D3)⋊C4—48no—
216C4⋊Dic3—28no—
217C4⋊Dic3—28no—
218C4⋊Dic3—216no—
219Dic3⋊C4—28no—
220Dic3⋊C4—28no—
221Dic3⋊C4—216no—
222Dic3⋊C4—48no—
223Dic3⋊C4—48no—
224Dic3⋊C4—48no—
225C4×Dic3C2×D4116yes—
226C25—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—38no—
231C23×C4—34no—
232C23×C4—32no—
233C23×C4—32no—
234C23×C4—64no—
235C23×C4—64no—
236C23×C4C22×D311yescharacteristic
237C23×C4C22×D312yes—
238C23×C4C22×D312yes—
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D311yescharacteristic
241#125—632no—
242#117—632no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—38no—
249C2×C4⋊C4—38no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—38no—
252C2×C4⋊C4—38no—
253C2×C4⋊C4—38no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—68no—
258C2×C4⋊C4—68no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—124no—
262C2×C4⋊C4—124no—
263C2×C4⋊C4—124no—
264C2×C4⋊C4C22×D314yes—
265C2×C4⋊C4C22×D314yes—
266C2×C4⋊C4C22×D318yes—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—38no—
270C2×C22⋊C4—38no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—124no—
274C2×C22⋊C4—124no—
275C2×C42—24no—
276C2×C42—38no—
277C2×C42—34no—
278C2×C42C22×D314yes—
279#95—38no—
280#95—38no—
281#95—38no—
282#95C22×D318yes—
283C23×C3C22×D412yes—
284C23×C3C22×Q812yes—
285C23×C3C2×#4112yes—
286C23×C3H(5,𝔽2)12yes—
287C23×C3H(5,𝔽2)12yes—
288C23×C32−1+412yes—
289C23×C32−1+412yes—
290C23×C3C2511yescharacteristic
291C22×D3—28no—
292C22×D3—28no—
293C22×D3—28no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—44no—
301C22×D3—44no—
302C22×D3—44no—
303C22×D3—44no—
304C22×D3C22×Q814yes—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—48no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4—48no—
321C2×C3×C4C22⋊Q8116yes—
322C2×C3×C4C22×D414yes—
323C2×C3×C4C22×Q814yes—
324C2×C3×C4C2×#4114yes—
325C2×Dic3—28no—
326C2×Dic3—28no—
327C2×Dic3—28no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—48no—
341C2×Dic3—48no—
342C2×Dic3C22×D414yes—
343C2×Dic3C22×D414yes—
344C2×Dic3C22×Q814yes—
345C4×D3—232no—
346C24—34no—
347C24—32no—
348C24—62no—
349C24—62no—
350C24—62no—
351C24—62no—
352C24—62no—
353C24—62no—
354C24C23×D311yescharacteristic
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—34no—
368C22×C4—34no—
369C22×C4—34no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—32no—
373C22×C4—316no—
374C22×C4—34no—
375C22×C4—34no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—632no—
383C22×C4—68no—
384C22×C4—68no—
385C22×C4—64no—
386C22×C4—64no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4—64no—
390C22×C4—64no—
391C22×C4#23514yes—
392C22×C4D3×D414yes—
393C22×C4D3×Q814yes—
394C22×C4C23×D312yes—
395C22×C4C23×D314yes—
396C22×C4C23×D314yes—
397C22×C4C23×D312yes—
398C22×C4C23×D312yes—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—48no—
402C4⋊C4—68no—
403C4⋊C4—68no—
404C4⋊C4—616no—
405C4⋊C4—616no—
406C4⋊C4—68no—
407C4⋊C4—68no—
408C4⋊C4—128no—
409C4⋊C4—128no—
410C4⋊C4—128no—
411C22⋊C4—68no—
412C22⋊C4—68no—
413C22⋊C4—128no—
414C22⋊C4—128no—
415C42—28no—
416C42—316no—
417C22×C3C2×C22⋊Q814yes—
418C22×C3#53714yes—
419C22×C3#53714yes—
420C22×C3#54014yes—
421C22×C3#54114yes—
422C22×C3#54414yes—
423C22×C3#54414yes—
424C22×C3C23×D411yescharacteristic
425C22×C3C23×Q811yescharacteristic
426C22×C3C22×#4111yescharacteristic
427C22×C3C2×H(5,𝔽2)11yescharacteristic
428C22×C3C2×H(5,𝔽2)11yescharacteristic
429C22×C3C2×2−1+411yescharacteristic
430C22×C3C2×2−1+411yescharacteristic
431C2×D3—24no—
432C2×D3—24no—
433C2×D3—24no—
434C2×D3—416no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C3×C4—44no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—44no—
443C3×C4C2×C22⋊Q818yes—
444Dic3—24no—
445Dic3—24no—
446Dic3—24no—
447Dic3—44no—
448Dic3—44no—
449C23—34no—
450C23—68no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23—62no—
455C23—62no—
456C23—62no—
457C23—68no—
458C23—68no—
459C23—124no—
460C23—124no—
461C23—124no—
462C23—124no—
463C23C2×#23512yes—
464C23C2×D3×D412yes—
465C23#100412yes—
466C23C2×D3×Q812yes—
467C23#100712yes—
468C23#100912yes—
469C23#101012yes—
470C23C24×D311yescharacteristic
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—34no—
484C2×C4—34no—
485C2×C4—34no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—44no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—68no—
497C2×C4—68no—
498C2×C4—68no—
499C2×C4—632no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—128no—
507C2×C4—128no—
508C2×C4#896116yes—
509C2×C4C2×#23514yes—
510C2×C4C2×D3×D414yes—
511C2×C4C2×D3×Q814yes—
512C2×C3#262918yescomplemented
513C2×C3C22×C22⋊Q811yescharacteristic
514C2×C3C2×#53711yescharacteristic
515C2×C3C2×#53711yescharacteristic
516C2×C3C2×#54011yescharacteristic
517C2×C3C2×#54111yescharacteristic
518C2×C3C2×#54411yescharacteristic
519C2×C3C2×#54411yescharacteristic
520D3—44no—
521C22—64no—
522C22—64no—
523C22—64no—
524C22—122no—
525C22—122no—
526C22—122no—
527C22—122no—
528C22—1216no—
529C22C2×#89614yes—
530C22#556214yes—
531C22#558314yes—
532C22#560814yes—
533C22#562914yes—
534C22#568914yes—
535C22#572914yes—
536C22C22×#23511yescharacteristic
537C22C22×D3×D411yescharacteristic
538C22C2×#100411yescharacteristic
539C22C22×D3×Q811yescharacteristic
540C22C2×#100711yescharacteristic
541C22C2×#100911yescharacteristic
542C22C2×#101011yescharacteristic
543C4—44no—
544C4—42no—
545C4—44no—
546C4—42no—
547C4—64no—
548C4—64no—
549C4—64no—
550C4—124no—
551C4—124no—
552C4C2×#89618yes—
553C3C2×#262911yescharacteristic, complemented, Sylow
554C2—124no—
555C2#7625118yesdirect product factor
556C2C22×#89611yescharacteristic
557C2C2×#556211yescharacteristic
558C2C2×#558311yescharacteristic
559C2C2×#560811yescharacteristic
560C2C2×#562911yescharacteristic
561C2C2×#568911yescharacteristic
562C2C2×#572911yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#389C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#389D3
hypocenterC3C2×#2629
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#890C2semidirect
3. C2×C3×#389C2semidirect
7. C2×#4983C2semidirect
8. C2×#4975C2semidirect
9. C2×#4928C2semidirect
10. #76251C2direct (N × H)
12. C2×#4657C2semidirect
34. C2×#890C22semidirect
37. C3×#389C22semidirect
45. #4983C22semidirect
46. #4975C22semidirect
47. #4928C22semidirect
49. #4657C22semidirect
58. C2×#389D3semidirect
147. #389C2×D3semidirect
512. C2×C3#2629semidirect
553. C3C2×#2629semidirect
555. C2#76251direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8902
C22×#8722
C2×C3×#3892
C2×#49942
C2×#49872
C2×#49832
C2×#49752
C2×#49282
#762512
C2×#46762
C2×#46572
C2×#26293