Finite Group Databasebeta

← C2×#77254 #10965404

Subgroups — C2×#77254 #10965404

Order 768.

3374 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic, complemented
2C22×#893C211yesmaximal, characteristic
3C2×C3×#395C211yesmaximal, characteristic, complemented
4C2×#5231C211yesmaximal, characteristic, complemented
5C2×#4989C212yesmaximal, complemented
6C2×#4987C212yesmaximal, complemented
7C2×#4986C211yesmaximal, characteristic, complemented
8#77254C2116yesmaximal, direct product factor
9C2×#4677C211yesmaximal, characteristic, complemented
10C2×#4665C212yesmaximal
11C2×D3×#95C211yesmaximal, characteristic, complemented
12C2×#4654C212yesmaximal, complemented
13C2×#2564—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C23×C4×D3C2212yes—
17C23×C4×D3C2211yescharacteristic
18C23×C4×D3C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C2×#946—28no—
22C2×#946C2218yescomplemented
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×(C2×D3)⋊C4C2212yes—
28C22×C4⋊Dic3C2212yes—
29C22×C4⋊Dic3C2211yescharacteristic
30C22×Dic3⋊C4C2212yes—
31C22×C4×Dic3C2211yescharacteristic
32C2×#893—28no—
33C2×#893C2218yes—
34C3×#395C2218yescomplemented
35C2×C3×#95C2212yes—
36C2×C3×#95C2211yescharacteristic
37#5231C2218yescomplemented
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2211yescharacteristic
42C2×C3⋊#95C2212yes—
43#4989C22116yescomplemented
44#4987C22116yescomplemented
45#4986C2218yescomplemented
46#4677C2218yescomplemented
47#4665C22116yes—
48D3×#95C2218yescomplemented
49#4654C22116yescomplemented
50C22×C22⋊Q8—31no—
51C22×C4×D4—31no—
52#2564—316no—
53C2×#400—31no—
54C2×#399—31no—
55C2×#395D311yescharacteristic, complemented
56C2×#393—31no—
57C2×#387—32no—
58C2×#386—32no—
59C2×#385—32no—
60C2×#380—32no—
61C22×#95—31no—
62C24×D3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C22×C3×Q8—24no—
65C22×C3×Q8—44no—
66C22×C3×Q8C2314yes—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2312yes—
72C23×Dic3C2312yes—
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2312yes—
75C23×Dic3C2311yescharacteristic
76C22×D12—24no—
77C22×D12—24no—
78C22×D12Q818yescomplemented
79C22×D12C2314yes—
80C22×C4×D3—28no—
81C22×C4×D3—28no—
82C22×C4×D3—24no—
83C22×C4×D3—44no—
84C22×C4×D3—44no—
85C22×C4×D3—44no—
86C22×C4×D3D418yes—
87C22×C4×D3C2318yes—
88C22×C4×D3C2314yes—
89C22×C4×D3C2314yes—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4—44no—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C4⋊C4C2318yes—
94#946—232no—
95C2×(C2×D3)⋊C4—24no—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—28no—
99C2×(C2×D3)⋊C4—44no—
100C2×(C2×D3)⋊C4C2318yes—
101C2×(C2×D3)⋊C4C2318yes—
102C2×(C2×D3)⋊C4C2314yes—
103C2×(C2×D3)⋊C4C2314yes—
104C2×(C2×D3)⋊C4C2318yes—
105C2×C4⋊Dic3—24no—
106C2×C4⋊Dic3—28no—
107C2×C4⋊Dic3C2318yes—
108C2×C4⋊Dic3C2314yes—
109C2×Dic3⋊C4—28no—
110C2×Dic3⋊C4C2318yes—
111C2×C4×Dic3—24no—
112C2×C4×Dic3C2314yes—
113#893—232no—
114C3×#95C2318yes—
115C3×#95C2314yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2314yes—
120C3⋊#95C2318yes—
121C23×Q8C2×D311yescharacteristic
122C23×D4—31no—
123C24×C4—31no—
124C24×C4—32no—
125C24×C4—31no—
126C2×C22⋊Q8—38no—
127C2×C22⋊Q8—68no—
128C2×C4×D4—38no—
129C2×C4×D4—68no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4C2×D311yescharacteristic
134C22×C4⋊C4C2×D312yes—
135C22×C22⋊C4—31no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C42—31no—
141#400—38no—
142#399—38no—
143#395C2×D318yescomplemented
144#393—38no—
145#387—316no—
146#386—316no—
147#385—316no—
148#380—316no—
149C2×#95—31no—
150C2×#95—32no—
151C2×#95—32no—
152C2×#95—32no—
153C2×#95—32no—
154C2×#95—38no—
155C2×#95C2×D311yescharacteristic
156C2×#95C2×D312yes—
157C24×C3C2411yescharacteristic
158C23×D3—24no—
159C23×D3—24no—
160C23×D3—22no—
161C23×D3—24no—
162C23×D3—22no—
163C23×D3—24no—
164C23×D3C2×D414yes—
165C23×D3C2×Q814yes—
166C23×D3C2×Q814yes—
167C23×D3#4114yes—
168C23×D3C2412yes—
169C23×D3C2412yes—
170C2×C3×Q8—28no—
171C2×C3×Q8—48no—
172C2×C3×Q8—416no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×Q814yes—
185C22×C3×C4#4118yes—
186C22×C3×C4C2412yes—
187C22×C3×C4C2414yes—
188C22×C3×C4C2412yes—
189C22×C3×C4C2412yes—
190C22×C3×C4C2414yes—
191C22×Dic3—24no—
192C22×Dic3—22no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3C2×D414yes—
201C22×Dic3C2414yes—
202C22×Dic3C2412yes—
203C22×Dic3C2414yes—
204C22×Dic3C2412yes—
205C2×D12—232no—
206C2×D12—28no—
207C2×D12—28no—
208C2×D12C2×Q8116yescomplemented
209C2×C4×D3—216no—
210C2×C4×D3—216no—
211C2×C4×D3—28no—
212C2×C4×D3—416no—
213C2×C4×D3—48no—
214C2×C4×D3—48no—
215C2×C4×D3—48no—
216C2×C4×D3—416no—
217C2×C4×D3—432no—
218C2×C4×D3C2×D4116yes—
219C3×C4⋊C4—28no—
220C3×C4⋊C4—48no—
221(C2×D3)⋊C4—28no—
222(C2×D3)⋊C4—216no—
223(C2×D3)⋊C4—28no—
224(C2×D3)⋊C4—216no—
225(C2×D3)⋊C4—48no—
226C4⋊Dic3—28no—
227C4⋊Dic3—216no—
228Dic3⋊C4—216no—
229C4×Dic3—28no—
230C25—31no—
231C25—31no—
232C22×Q8—24no—
233C22×Q8—44no—
234C22×Q8C22×D314yes—
235C22×D4—38no—
236C22×D4—34no—
237C22×D4—64no—
238C22×D4—64no—
239C23×C4—31no—
240C23×C4—38no—
241C23×C4—34no—
242C23×C4—32no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4—38no—
246C23×C4—34no—
247C23×C4—64no—
248C23×C4—68no—
249C23×C4—68no—
250C23×C4—124no—
251C23×C4—124no—
252C23×C4—124no—
253C23×C4C22×D312yes—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D312yes—
258C22⋊Q8—632no—
259C4×D4—632no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—38no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—38no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—68no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4C22×D314yes—
269C2×C4⋊C4C22×D318yes—
270C2×C22⋊C4—34no—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—38no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—38no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—68no—
277C2×C22⋊C4—64no—
278C2×C22⋊C4—68no—
279C2×C22⋊C4—124no—
280C2×C42—34no—
281C2×C42—64no—
282#95—38no—
283#95—34no—
284#95—38no—
285#95—38no—
286#95—38no—
287#95C22×D314yes—
288#95C22×D318yes—
289C23×C3C22×D412yes—
290C23×C3C22×Q812yes—
291C23×C3C2×#4114yes—
292C23×C3C2×#4112yes—
293C23×C3H(5,𝔽2)14yes—
294C23×C3C2511yescharacteristic
295C22×D3—28no—
296C22×D3—28no—
297C22×D3—216no—
298C22×D3—216no—
299C22×D3—24no—
300C22×D3—24no—
301C22×D3—28no—
302C22×D3—216no—
303C22×D3—22no—
304C22×D3—24no—
305C22×D3—22no—
306C22×D3—24no—
307C22×D3—44no—
308C22×D3—44no—
309C22×D3—48no—
310C22×D3C22⋊Q8116yes—
311C22×D3C22×D414yes—
312C22×D3C22×Q814yes—
313C22×D3C22×Q814yes—
314C22×D3C2×#4114yes—
315C3×Q8—416no—
316C2×C3×C4—28no—
317C2×C3×C4—28no—
318C2×C3×C4—216no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—44no—
330C2×C3×C4—48no—
331C2×C3×C4C22×D414yes—
332C2×C3×C4C22×Q814yes—
333C2×C3×C4C2×#4118yes—
334C2×Dic3—28no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—44no—
345C2×Dic3—48no—
346C2×Dic3—48no—
347C2×Dic3C22×D414yes—
348D12—232no—
349C4×D3—416no—
350C4×D3—416no—
351C4×D3—432no—
352C24—34no—
353C24—34no—
354C24—34no—
355C24—32no—
356C24—34no—
357C24—32no—
358C24—64no—
359C24—64no—
360C24—62no—
361C24—64no—
362C24—64no—
363C24—62no—
364C24C23×D311yescharacteristic
365C2×Q8—28no—
366C2×Q8—48no—
367C2×Q8—416no—
368C2×D4—316no—
369C2×D4—68no—
370C2×D4—68no—
371C2×D4—632no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—34no—
383C22×C4—32no—
384C22×C4—316no—
385C22×C4—34no—
386C22×C4—32no—
387C22×C4—34no—
388C22×C4—68no—
389C22×C4—616no—
390C22×C4—64no—
391C22×C4—62no—
392C22×C4—64no—
393C22×C4—64no—
394C22×C4—64no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—64no—
398C22×C4—64no—
399C22×C4—616no—
400C22×C4—128no—
401C22×C4—128no—
402C22×C4—1216no—
403C22×C4—128no—
404C22×C4—1232no—
405C22×C4—1216no—
406C22×C4D3×D414yes—
407C22×C4D3×Q814yes—
408C22×C4#23918yes—
409C22×C4C23×D312yes—
410C22×C4C23×D314yes—
411C22×C4C23×D312yes—
412C22×C4C23×D312yes—
413C22×C4C23×D314yes—
414C4⋊C4—28no—
415C4⋊C4—48no—
416C4⋊C4—616no—
417C4⋊C4—68no—
418C4⋊C4—616no—
419C22⋊C4—68no—
420C22⋊C4—616no—
421C22⋊C4—68no—
422C22⋊C4—616no—
423C22⋊C4—128no—
424C42—68no—
425C22×C3C2×C22⋊Q814yes—
426C22×C3#53214yes—
427C22×C3#53414yes—
428C22×C3#55118yes—
429C22×C3#55418yes—
430C22×C3C23×D411yescharacteristic
431C22×C3C23×Q811yescharacteristic
432C22×C3C22×#4112yes—
433C22×C3C22×#4111yescharacteristic
434C22×C3C2×H(5,𝔽2)12yes—
435C2×D3—232no—
436C2×D3—24no—
437C2×D3—24no—
438C2×D3—28no—
439C2×D3—28no—
440C2×D3—24no—
441C2×D3—28no—
442C2×D3—416no—
443C2×D3—42no—
444C2×D3—42no—
445C2×D3—44no—
446C2×D3C2×C22⋊Q818yes—
447C3×C4—24no—
448C3×C4—24no—
449C3×C4—28no—
450C3×C4—42no—
451C3×C4—44no—
452Dic3—24no—
453Dic3—42no—
454Dic3—44no—
455Dic3—44no—
456C23—34no—
457C23—34no—
458C23—34no—
459C23—316no—
460C23—34no—
461C23—616no—
462C23—64no—
463C23—64no—
464C23—68no—
465C23—616no—
466C23—68no—
467C23—62no—
468C23—64no—
469C23—64no—
470C23—62no—
471C23—616no—
472C23—68no—
473C23—128no—
474C23—124no—
475C23—124no—
476C23C2×D3×D412yes—
477C23#100414yes—
478C23C2×D3×Q812yes—
479C23C2×#23914yes—
480C23D3×#4112yes—
481C23C24×D311yescharacteristic
482Q8—416no—
483D4—632no—
484C2×C4—216no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—34no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—64no—
501C2×C4—68no—
502C2×C4—62no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—1216no—
511C2×C4—1232no—
512C2×C4—1216no—
513C2×C4—128no—
514C2×C4—124no—
515C2×C4—128no—
516C2×C4C2×D3×D414yes—
517C2×C4C2×D3×Q814yes—
518C2×C4C2×#23918yes—
519C2×C3#256418yescomplemented
520C2×C3C22×C22⋊Q811yescharacteristic
521C2×C3C2×#53211yescharacteristic
522C2×C3C2×#53411yescharacteristic
523C2×C3C2×#55112yes—
524C2×C3C2×#55412yes—
525D3—28no—
526D3—44no—
527C22—38no—
528C22—64no—
529C22—64no—
530C22—68no—
531C22—632no—
532C22—68no—
533C22—68no—
534C22—64no—
535C22—122no—
536C22—122no—
537C22—1216no—
538C22—124no—
539C22#558914yes—
540C22D3×C22⋊Q814yes—
541C22#563718yes—
542C22#572514yes—
543C22#573018yes—
544C22C22×D3×D411yescharacteristic
545C22C2×#100412yes—
546C22C22×D3×Q811yescharacteristic
547C22C22×#23912yes—
548C22C2×D3×#4111yescharacteristic
549C4—24no—
550C4—24no—
551C4—28no—
552C4—42no—
553C4—44no—
554C4—64no—
555C4—122no—
556C4—124no—
557C4—124no—
558C3C2×#256411yescharacteristic, complemented, Sylow
559C2—68no—
560C2—124no—
561C2#7725418yesdirect product factor
562C2C2×#558911yescharacteristic
563C2C2×D3×C22⋊Q811yescharacteristic
564C2C2×#563712yes—
565C2C2×#572511yescharacteristic
566C2C2×#573012yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#395C2
socleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#395D3
hypocenterC3C2×#2564
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
1. C22×#946C2semidirect
3. C2×C3×#395C2semidirect
4. C2×#5231C2semidirect
5. C2×#4989C2semidirect
6. C2×#4987C2semidirect
7. C2×#4986C2semidirect
8. #77254C2direct (N × H)
9. C2×#4677C2semidirect
11. C2×D3×#95C2semidirect
12. C2×#4654C2semidirect
22. C2×#946C22semidirect
34. C3×#395C22semidirect
37. #5231C22semidirect
43. #4989C22semidirect
44. #4987C22semidirect
45. #4986C22semidirect
46. #4677C22semidirect
48. D3×#95C22semidirect
49. #4654C22semidirect
55. C2×#395D3semidirect
78. C22×D12Q8semidirect
143. #395C2×D3semidirect
208. C2×D12C2×Q8semidirect
519. C2×C3#2564semidirect
558. C3C2×#2564semidirect
561. C2#77254direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9462
C22×#8932
C2×C3×#3952
C2×#52312
C2×#49892
C2×#49872
C2×#49862
#772542
C2×#46772
C2×#46652
C2×D3×#952
C2×#46542
C2×#25643