Finite Group Databasebeta

← C2×#82248 #10963530

Subgroups — C2×#82248 #10963530

Order 768.

2854 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×C3×#117C211yesmaximal, characteristic
2C22×#888C211yesmaximal, characteristic, complemented
3C2×#5212C211yesmaximal, characteristic
4C2×C4×#217C211yesmaximal, characteristic
5#82248C2116yesmaximal, direct product factor
6C2×#4984C211yesmaximal, characteristic, complemented
7C2×#4972C211yesmaximal, characteristic, complemented
8C2×#4977C211yesmaximal, characteristic, complemented
9C2×#4948C212yesmaximal
10C2×#4947C212yesmaximal
11C2×#4929C212yesmaximal, complemented
12C2×#4933C212yesmaximal, complemented
13C2×#2439—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×Dic6C2212yes—
16C2×C3×#117—28no—
17C2×C3×#117C2218yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C42C2212yes—
22C22×#217C2211yescharacteristic
23C22×#217C2211yescharacteristic
24C22×#217C2212yes—
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C22×C4⋊Dic3C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×C4×Dic3C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C2×#888—28no—
33C2×#888C2218yescomplemented
34#5212C2218yes—
35C4×#217C2218yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4984C2218yescomplemented
44#4977C2218yescomplemented
45#4972C2218yescomplemented
46#4948C22116yes—
47#4947C22116yes—
48#4933C22116yescomplemented
49#4929C22116yescomplemented
50C22×C4⋊Q8—31no—
51C22×#117D311yescharacteristic
52#2439—316no—
53C2×#396—32no—
54C2×#393—32no—
55C2×#391—32no—
56C2×#389—31no—
57C2×#389—31no—
58C2×#387—32no—
59C2×#382—31no—
60C2×#380—31no—
61C2×C4×C22⋊C4—31no—
62C25×C3C2311yescharacteristic
63C23×C3×C4—24no—
64C23×C3×C4—24no—
65C23×C3×C4Q818yescomplemented
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2314yes—
70C23×C3×C4C2311yescharacteristic
71C23×Dic3C2312yes—
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2312yes—
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2311yescharacteristic
77C22×Dic6—28no—
78C22×Dic6—28no—
79C22×Dic6—44no—
80C22×Dic6C2318yes—
81C3×#117—232no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4D418yes—
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C22⋊C4—28no—
87C2×C3×C22⋊C4C2318yes—
88C2×C3×C42—28no—
89C2×C3×C42C2318yes—
90C2×#217—24no—
91C2×#217—24no—
92C2×#217—44no—
93C2×#217C2314yes—
94C2×#217C2314yes—
95C2×#217C2318yes—
96C2×C4⋊Dic3—24no—
97C2×C4⋊Dic3—24no—
98C2×C4⋊Dic3—44no—
99C2×C4⋊Dic3C2314yes—
100C2×C4⋊Dic3C2314yes—
101C2×C4⋊Dic3C2318yes—
102C2×Dic3⋊C4—28no—
103C2×Dic3⋊C4—28no—
104C2×Dic3⋊C4C2318yes—
105C2×Dic3⋊C4C2318yes—
106C2×C4×Dic3—24no—
107C2×C4×Dic3—24no—
108C2×C4×Dic3C2314yes—
109C2×C4×Dic3C2314yes—
110#888—232no—
111C3⋊#95C2314yes—
112C3⋊#95C2318yes—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C3⋊#95C2314yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C23×Q8—32no—
119C24×C4C2×D311yescharacteristic
120C2×C4⋊Q8—38no—
121C2×C4⋊Q8—68no—
122C2×#117—28no—
123C2×#117C2×D318yes—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4C2×D311yescharacteristic
130C22×C4⋊C4C2×D311yescharacteristic
131C22×C22⋊C4—31no—
132C22×C22⋊C4—31no—
133C22×C22⋊C4—32no—
134C22×C22⋊C4C2×D312yes—
135C22×C42—31no—
136C22×C42—31no—
137C22×C42C2×D312yes—
138#396—316no—
139#393—316no—
140#391—316no—
141#389—38no—
142#389—38no—
143#387—316no—
144#382—38no—
145#380—38no—
146C4×C22⋊C4—38no—
147C2×#95—32no—
148C2×#95—32no—
149C2×#95—32no—
150C2×#95—32no—
151C2×#95—31no—
152C2×#95—31no—
153C2×#95—32no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—22no—
159C24×C3—22no—
160C24×C3C2×Q814yes—
161C24×C3C2412yes—
162C24×C3C2411yescharacteristic
163C22×C3×C4—22no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—232no—
168C22×C3×C4—28no—
169C22×C3×C4—28no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4C2×D418yes—
174C22×C3×C4C2×D414yes—
175C22×C3×C4C2×Q814yes—
176C22×C3×C4C2×Q814yes—
177C22×C3×C4C2×Q8116yescomplemented
178C22×C3×C4#4114yes—
179C22×C3×C4C2414yes—
180C22×C3×C4C2414yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2412yes—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—24no—
191C22×Dic3—24no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—24no—
195C22×Dic3—22no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—22no—
199C22×Dic3#4114yes—
200C22×Dic3#4114yes—
201C22×Dic3C2412yes—
202C22×Dic3C2412yes—
203C22×Dic3C2414yes—
204C22×Dic3C2414yes—
205C22×Dic3C2412yes—
206C22×Dic3C2412yes—
207C2×Dic6—216no—
208C2×Dic6—216no—
209C2×Dic6—48no—
210C2×Dic6—432no—
211C3×C4⋊C4—28no—
212C3×C4⋊C4C2×D4116yes—
213C3×C22⋊C4—216no—
214C3×C42—216no—
215#217—28no—
216#217—28no—
217#217—48no—
218C4⋊Dic3—28no—
219C4⋊Dic3—28no—
220C4⋊Dic3—48no—
221Dic3⋊C4—216no—
222Dic3⋊C4—216no—
223C4×Dic3—28no—
224C4×Dic3—28no—
225C25C22×D311yescharacteristic
226C22×Q8—38no—
227C22×Q8—68no—
228C22×Q8—68no—
229C22×Q8—124no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—32no—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4Dic618yescomplemented
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D314yes—
242C23×C4C22×D312yes—
243C23×C4C22×D311yescharacteristic
244C4⋊Q8—632no—
245#117—232no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—38no—
249C2×C4⋊C4—34no—
250C2×C4⋊C4—38no—
251C2×C4⋊C4—38no—
252C2×C4⋊C4—68no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—68no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—124no—
257C2×C4⋊C4#6718yes—
258C2×C4⋊C4C22×D314yes—
259C2×C4⋊C4C22×D314yes—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—34no—
262C2×C22⋊C4—34no—
263C2×C22⋊C4—38no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—124no—
267C2×C22⋊C4C22×D318yes—
268C2×C42—28no—
269C2×C42—34no—
270C2×C42—34no—
271C2×C42—64no—
272C2×C42—64no—
273C2×C42C22×D318yes—
274#95—34no—
275#95—34no—
276#95—38no—
277#95—38no—
278#95—38no—
279#95—38no—
280#95—38no—
281C23×C3—28no—
282C23×C3—28no—
283C23×C3—28no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—44no—
291C23×C3—44no—
292C23×C3—44no—
293C23×C3—44no—
294C23×C3C22×D412yes—
295C23×C3C22×Q814yes—
296C23×C3C22×Q812yes—
297C23×C3C2×#4112yes—
298C23×C3C2×#4112yes—
299C23×C3C2×#4112yes—
300C23×C3H(5,𝔽2)12yes—
301C23×C32−1+412yes—
302C23×C3C2511yescharacteristic
303C2×C3×C4—28no—
304C2×C3×C4—216no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—232no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—48no—
314C2×C3×C4C22⋊Q8116yes—
315C2×C3×C4C22×D418yes—
316C2×C3×C4C22×D414yes—
317C2×C3×C4C22×Q814yes—
318C2×C3×C4C22×Q814yes—
319C2×C3×C4C2×#4114yes—
320C2×Dic3—28no—
321C2×Dic3—28no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—24no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—48no—
339C2×Dic3—44no—
340C2×Dic3—48no—
341C2×Dic3—44no—
342C2×Dic3C2×#4114yes—
343C2×Dic3C2×#4114yes—
344Dic6—432no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24C2×Dic614yes—
352C24C23×D312yes—
353C24C23×D311yescharacteristic
354C2×Q8—616no—
355C2×Q8—616no—
356C2×Q8—128no—
357C2×Q8—1232no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—232no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—34no—
369C22×C4—34no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—32no—
373C22×C4—32no—
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—64no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—64no—
386C22×C4—64no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—64no—
392C22×C4C2×Dic614yes—
393C22×C4C2×Dic614yes—
394C22×C4C2×Dic6116yescomplemented
395C22×C4#23514yes—
396C22×C4C2×#6718yes—
397C22×C4C2×#6714yes—
398C22×C4C23×D314yes—
399C22×C4C23×D312yes—
400C22×C4C23×D314yes—
401C22×C4C23×D312yes—
402C4⋊C4—28no—
403C4⋊C4—616no—
404C4⋊C4—68no—
405C4⋊C4—616no—
406C4⋊C4—68no—
407C4⋊C4—128no—
408C4⋊C4C2×#67116yes—
409C22⋊C4—216no—
410C22⋊C4—68no—
411C22⋊C4—68no—
412C22⋊C4—128no—
413C42—216no—
414C42—68no—
415C42—68no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—416no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3C2×C22⋊Q814yes—
425C22×C3#53314yes—
426C22×C3#54414yes—
427C22×C3#54614yes—
428C22×C3#54814yes—
429C22×C3#55514yes—
430C22×C3#55614yes—
431C22×C3C23×D411yescharacteristic
432C22×C3C23×Q811yescharacteristic
433C22×C3C22×#4111yescharacteristic
434C22×C3C22×#4111yescharacteristic
435C22×C3C22×#4111yescharacteristic
436C22×C3C2×H(5,𝔽2)11yescharacteristic
437C22×C3C2×2−1+411yescharacteristic
438C3×C4—24no—
439C3×C4—28no—
440C3×C4—44no—
441C3×C4C2×C22⋊Q818yes—
442Dic3—24no—
443Dic3—24no—
444Dic3—44no—
445Dic3—42no—
446Dic3—44no—
447Dic3—42no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23C22×Dic614yes—
462C23C22×Dic612yes—
463C23C2×#23512yes—
464C23D3×#4112yes—
465C23D3×#4112yes—
466C23#100912yes—
467C23#101012yes—
468C23C22×#6712yes—
469C23C24×D311yescharacteristic
470Q8—1232no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—216no—
477C2×C4—232no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—34no—
482C2×C4—34no—
483C2×C4—48no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—68no—
489C2×C4—68no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—64no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—64no—
502C2×C4—124no—
503C2×C4—128no—
504C2×C4—124no—
505C2×C4—128no—
506C2×C4#924116yes—
507C2×C4C22×Dic614yes—
508C2×C4C22×Dic614yes—
509C2×C4C2×#23514yes—
510C2×C4C22×#6718yes—
511C2×C4C22×#6714yes—
512C2×C3—44no—
513C2×C3#243918yescomplemented
514C2×C3C22×C22⋊Q811yescharacteristic
515C2×C3C2×#53311yescharacteristic
516C2×C3C2×#54411yescharacteristic
517C2×C3C2×#54611yescharacteristic
518C2×C3C2×#54811yescharacteristic
519C2×C3C2×#55511yescharacteristic
520C2×C3C2×#55611yescharacteristic
521C22—24no—
522C22—24no—
523C22—24no—
524C22—416no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22#551714yes—
530C22#551914yes—
531C22#553214yes—
532C22#553314yes—
533C22C2×#92414yes—
534C22#582914yes—
535C22#583114yes—
536C22C23×Dic611yescharacteristic
537C22C22×#23511yescharacteristic
538C22C2×D3×#4111yescharacteristic
539C22C2×D3×#4111yescharacteristic
540C22C2×#100911yescharacteristic
541C22C2×#101011yescharacteristic
542C22C23×#6711yescharacteristic
543C4—24no—
544C4—28no—
545C4—44no—
546C4—64no—
547C4—64no—
548C4—124no—
549C4—122no—
550C4—124no—
551C4—122no—
552C4C2×#92418yes—
553C3C2×#243911yescharacteristic, complemented, Sylow
554C2—44no—
555C2#8224818yesdirect product factor
556C2C2×#551711yescharacteristic
557C2C2×#551911yescharacteristic
558C2C2×#553211yescharacteristic
559C2C2×#553311yescharacteristic
560C2C22×#92411yescharacteristic
561C2C2×#582911yescharacteristic
562C2C2×#583111yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×#117C2
socleC24×C3C24
prime radicalC25×C3C23
hypercenterC22×#117D3
hypocenterC3C2×#2439
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C22×#888C2semidirect
5. #82248C2direct (N × H)
6. C2×#4984C2semidirect
7. C2×#4972C2semidirect
8. C2×#4977C2semidirect
11. C2×#4929C2semidirect
12. C2×#4933C2semidirect
33. C2×#888C22semidirect
43. #4984C22semidirect
44. #4977C22semidirect
45. #4972C22semidirect
48. #4933C22semidirect
49. #4929C22semidirect
65. C23×C3×C4Q8semidirect
177. C22×C3×C4C2×Q8semidirect
238. C23×C4Dic6semidirect
394. C22×C4C2×Dic6semidirect
513. C2×C3#2439semidirect
553. C3C2×#2439semidirect
555. C2#82248direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C22×#8882
C2×#52122
C2×C4×#2172
#822482
C2×#49842
C2×#49722
C2×#49772
C2×#49482
C2×#49472
C2×#49292
C2×#49332
C2×#24393