Finite Group Databasebeta

← C2×#81112 #11667235

Subgroups — C2×#81112 #11667235

Order 768.

2006 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic
2C2×#5578C211yesmaximal, characteristic
3C2×C3×#477C211yesmaximal, characteristic, complemented
4C2×#5091C211yesmaximal, characteristic, complemented
5C2×#5021C211yesmaximal, characteristic, complemented
6#81112C2116yesmaximal, direct product factor
7C2×#4884C212yesmaximal
8C2×#4874C212yesmaximal, complemented
9C2×#4850C211yesmaximal, characteristic, complemented
10C2×#4833C211yesmaximal, characteristic
11C2×#4811C212yesmaximal
12C2×#4792C212yesmaximal, complemented
13C2×#3230—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×#946C2211yescharacteristic
18C2×#944—21no—
19C2×#944C2211yescharacteristic
20D3×C4⋊Q8—24no—
21D3×C4⋊Q8C2218yes—
22#5578—24no—
23#5578C2218yes—
24C2×#897C2211yescharacteristic
25C2×#896—21no—
26C2×D3×C4⋊C4—21no—
27C2×D3×C4⋊C4C2212yes—
28C2×#889—21no—
29C2×#888—21no—
30C2×#888C2211yescharacteristic
31C2×#888C2212yes—
32C2×#872—21no—
33C2×C42×D3C2211yescharacteristic
34C2×#869C2211yescharacteristic
35C2×C4×Dic6—21no—
36C2×C4×Dic6C2211yescharacteristic
37C3×#477C2218yescomplemented
38C2×C3×#107C2212yes—
39C2×C3×C4⋊C8C2211yescharacteristic
40C2×C3×#103C2212yes—
41C2×#835C2212yes—
42C2×(C2×D3)⋊C8C2212yes—
43C2×#818C2212yes—
44C2×#816C2212yes—
45C2×C8×Dic3C2212yes—
46#5091C2218yescomplemented
47#5021C2218yescomplemented
48C2×#810C2212yes—
49C2×#807C2212yes—
50C2×#807C2212yes—
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4884C22116yes—
53#4874C22116yescomplemented
54#4850C2218yescomplemented
55#4833C2218yes—
56#4811C22116yes—
57#4792C22116yescomplemented
58C2×#533—31no—
59C22×C4⋊Q8—31no—
60#3230—316no—
61C2×#500—32no—
62C2×#487—32no—
63C2×#484—32no—
64C2×#484—32no—
65C2×#477—31no—
66C2×#477—31no—
67C2×#477—31no—
68C2×#477D311yescharacteristic, complemented
69C2×#431—31no—
70C22×C3×Q8—21no—
71C22×C3×Q8C2311yescharacteristic
72C2×D3×Q8—28no—
73C2×D3×Q8—48no—
74C22×C4×D3—22no—
75C22×C4×D3D411yescharacteristic
76C22×C4×D3D411yescharacteristic
77C22×C4×D3C2311yescharacteristic
78C22×Dic6—21no—
79C22×Dic6—22no—
80C22×Dic6—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2312yes—
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8C2314yes—
85C3×C4×Q8—22no—
86C3×C4×Q8C2314yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93#946—22no—
94#946C2314yes—
95#944—28no—
96#944—22no—
97#944C2314yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2312yes—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—22no—
110C2×Dic3⋊C4—22no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2312yes—
116#897—22no—
117#897C2314yes—
118#896—28no—
119D3×C4⋊C4—24no—
120D3×C4⋊C4—28no—
121D3×C4⋊C4C2318yes—
122#889—28no—
123#888—24no—
124#888—28no—
125#888—22no—
126#888C2314yes—
127#888C2318yes—
128#872—28no—
129C42×D3D414yes—
130C42×D3C2314yes—
131#869—22no—
132#869C2314yes—
133C4×Dic6—22no—
134C4×Dic6—28no—
135C4×Dic6C2314yes—
136C3×#107C2318yes—
137C3×C4⋊C8C2314yes—
138C3×#103C2318yes—
139#835C2318yes—
140(C2×D3)⋊C8C2318yes—
141#818C2318yes—
142#816C2318yes—
143C8×Dic3C2318yes—
144#810C2318yes—
145#807C2318yes—
146#807C2318yes—
147C4⋊(C3⋊C8)C2314yes—
148C23×Q8—61no—
149#533—38no—
150#533—64no—
151C2×C4⋊Q8—38no—
152C2×C4⋊Q8—32no—
153C2×C4⋊Q8—31no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8—61no—
157C2×C4⋊Q8—64no—
158C2×C4⋊Q8—61no—
159C2×C4⋊Q8C2×D311yescharacteristic
160C2×#125—61no—
161C2×C22⋊Q8—31no—
162C2×C22⋊Q8—31no—
163C2×C22⋊Q8—61no—
164C2×C4×Q8—31no—
165C2×C4×Q8—61no—
166C2×C4×Q8C2×D311yescharacteristic
167C2×#117—61no—
168C22×C4⋊C4—32no—
169C22×C4⋊C4—61no—
170C22×C42—31no—
171#500—316no—
172#487—316no—
173#484—316no—
174#484—316no—
175#477—38no—
176#477—38no—
177#477—38no—
178#477C2×D318yescomplemented
179#431—38no—
180C2×#107—32no—
181C2×#107—32no—
182C2×#107—32no—
183C2×#107C2×D312yes—
184C2×C4⋊C8—31no—
185C2×C4⋊C8C2×D311yescharacteristic
186C2×#103—32no—
187C2×#103—32no—
188C2×#103—32no—
189C2×#103C2×D312yes—
190C2×C22⋊C8—32no—
191C2×C4×C8—32no—
192C23×D3C2×D411yescharacteristic
193C2×C3×Q8—21no—
194C2×C3×Q8—24no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—44no—
198C2×C3×Q8C2412yes—
199C22×C3×C4—21no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—22no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211D3×Q8—432no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—24no—
215C2×C4×D3—28no—
216C2×C4×D3—48no—
217C2×C4×D3Q1618yes—
218C2×C4×D3C2×D412yes—
219C2×C4×D3C2×D414yes—
220C2×C4×D3C2×D414yes—
221C2×C4×D3C2412yes—
222C2×Dic6—22no—
223C2×Dic6—22no—
224C2×Dic6—21no—
225C2×Dic6—24no—
226C2×Dic6—24no—
227C2×Dic6—28no—
228C2×Dic6—44no—
229C2×Dic6—44no—
230C2×Dic6—48no—
231C2×Dic6C2412yes—
232C2×C3×C8—24no—
233C2×C3×C8C2414yes—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4C2412yes—
239C3×C4⋊C4C2414yes—
240C3×C42—24no—
241C3×C42C2×D412yes—
242C3×C42C2412yes—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245C4⋊Dic3—21no—
246C4⋊Dic3—22no—
247C4⋊Dic3—24no—
248C4⋊Dic3—24no—
249C4⋊Dic3C2412yes—
250C4⋊Dic3C2414yes—
251Dic3⋊C4—24no—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254Dic3⋊C4—24no—
255Dic3⋊C4—24no—
256Dic3⋊C4—28no—
257Dic3⋊C4—28no—
258C4×Dic3—24no—
259C4×Dic3C2×D414yes—
260C4×Dic3C2×D412yes—
261C4×Dic3C2×D414yes—
262C4×Dic3C2412yes—
263C2×C3⋊C8—24no—
264C2×C3⋊C8C2414yes—
265C22×Q8—21no—
266C22×Q8—31no—
267C22×Q8—61no—
268C22×Q8—62no—
269C22×Q8—68no—
270C22×Q8—61no—
271C22×Q8—128no—
272C22×Q8C22×D311yescharacteristic
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—31no—
276C23×C4—62no—
277C22×C8—32no—
278C22×C8C22×D312yes—
279C4⋊Q8—22no—
280C4⋊Q8—38no—
281C4⋊Q8—34no—
282C4⋊Q8—34no—
283C4⋊Q8—34no—
284C4⋊Q8—68no—
285C4⋊Q8—64no—
286C4⋊Q8—62no—
287C4⋊Q8—68no—
288C4⋊Q8—62no—
289C4⋊Q8—62no—
290C4⋊Q8C22×D314yes—
291#125—68no—
292C22⋊Q8—34no—
293C22⋊Q8—34no—
294C22⋊Q8—62no—
295C22⋊Q8—68no—
296C22⋊Q8—62no—
297C4×Q8—22no—
298C4×Q8—34no—
299C4×Q8—68no—
300C4×Q8—62no—
301C4×Q8C22×D314yes—
302#117—68no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—38no—
308C2×C4⋊C4—64no—
309C2×C4⋊C4—68no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—62no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4C22×D312yes—
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—21no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—34no—
328C2×C42—34no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#107—38no—
332#107—38no—
333#107—38no—
334#107C22×D318yes—
335C4⋊C8—34no—
336C4⋊C8C22×D314yes—
337#103—38no—
338#103—38no—
339#103—38no—
340#103C22×D318yes—
341C22⋊C8—38no—
342C4×C8—38no—
343C23×C3C22×D411yescharacteristic
344C22×D3—22no—
345C22×D3—22no—
346C22×D3C2×Q1614yes—
347C22×D3C22×D412yes—
348C3×Q8—22no—
349C3×Q8—22no—
350C3×Q8—48no—
351C3×Q8—42no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4C2×Q1614yes—
360C2×C3×C4C22×D412yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D412yes—
363C2×C3×C42−1+412yes—
364C2×C3×C42−1+412yes—
365C2×C3×C4C2511yescharacteristic
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—24no—
370C2×Dic3—42no—
371C2×Dic3—44no—
372C2×Dic3—42no—
373C2×Dic3C2×Q1614yes—
374C2×Dic3AGL(1,ℤ/8)14yes—
375C2×Dic3C22×D412yes—
376C2×Dic3C22×D412yes—
377C2×Dic3C22×D412yes—
378C4×D3—24no—
379C4×D3—22no—
380C4×D3—22no—
381C4×D3—28no—
382C4×D3—416no—
383C4×D3C2×Q16116yes—
384Dic6—22no—
385Dic6—22no—
386Dic6—48no—
387Dic6—42no—
388Dic6—48no—
389Dic6—416no—
390C3×C8—24no—
391C3⋊C8—24no—
392C24—31no—
393C2×Q8—21no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—32no—
398C2×Q8—44no—
399C2×Q8—64no—
400C2×Q8—68no—
401C2×Q8—61no—
402C2×Q8—64no—
403C2×Q8—62no—
404C2×Q8—62no—
405C2×Q8—124no—
406C2×Q8—128no—
407C2×Q8—124no—
408C2×Q8—1232no—
409C2×Q8C23×D312yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—38no—
416C22×C4—34no—
417C22×C4—31no—
418C22×C4—32no—
419C22×C4—34no—
420C22×C4—32no—
421C22×C4—64no—
422C22×C4—68no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—128no—
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4C23×D311yescharacteristic
432C2×C8—24no—
433C2×C8—34no—
434C2×C8—64no—
435C2×C8C23×D314yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—34no—
441C4⋊C4—32no—
442C4⋊C4—68no—
443C4⋊C4—68no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—62no—
450C4⋊C4—61no—
451C4⋊C4—64no—
452C4⋊C4—64no—
453C4⋊C4C23×D314yes—
454C4⋊C4C23×D312yes—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C42—24no—
458C42—34no—
459C42—34no—
460C42—32no—
461C42—32no—
462C42—64no—
463C42D3×D412yes—
464C42C23×D312yes—
465C22×C3#53612yes—
466C22×C3C22×Q1612yes—
467C22×C3C2×AGL(1,ℤ/8)12yes—
468C22×C3C23×D411yescharacteristic
469C2×D3—22no—
470C2×D3—28no—
471C2×D3—22no—
472C2×D3C22×Q1614yes—
473C3×C4—22no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4—42no—
477C3×C4#53612yes—
478C3×C4C22×Q1614yes—
479C3×C4C2×2−1+411yescharacteristic
480C3×C4C2×2−1+411yescharacteristic
481Dic3—22no—
482Dic3—42no—
483Dic3—44no—
484Dic3—42no—
485Dic3C22×Q1614yes—
486Dic3C2×AGL(1,ℤ/8)14yes—
487C23—32no—
488C23—34no—
489C23—62no—
490C23—62no—
491C23C2×D3×D411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—48no—
495Q8—42no—
496Q8—62no—
497Q8—62no—
498Q8—128no—
499Q8—1216no—
500Q8—128no—
501Q8—122no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—34no—
509C2×C4—34no—
510C2×C4—316no—
511C2×C4—32no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—68no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—64no—
523C2×C4—1216no—
524C2×C4—124no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4D3×Q1614yes—
528C2×C4C2×D3×D412yes—
529C2×C4C2×D3×D412yes—
530C2×C4C2×D3×D411yescharacteristic
531C2×C4#100712yes—
532C2×C4#101012yes—
533C2×C4C24×D311yescharacteristic
534C8—24no—
535C8—64no—
536C2×C3#323014yescomplemented
537C2×C3C2×#53611yescharacteristic
538C2×C3C23×Q1611yescharacteristic
539C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
540D3—24no—
541C22—34no—
542C22—62no—
543C22—62no—
544C22—68no—
545C22#562812yes—
546C22C2×D3×Q1612yes—
547C22D3×AGL(1,ℤ/8)12yes—
548C22C22×D3×D411yescharacteristic
549C4—22no—
550C4—34no—
551C4—34no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—62no—
556C4—124no—
557C4—122no—
558C4—122no—
559C4#562812yes—
560C4C2×D3×Q1614yes—
561C4C2×#100711yescharacteristic
562C4C2×#101011yescharacteristic
563C3C2×#323011yescharacteristic, complemented, Sylow
564C2—64no—
565C2#8111214yesdirect product factor
566C2C2×#562811yescharacteristic
567C2C22×D3×Q1611yescharacteristic
568C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#477C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#477D3
hypocenterC3C2×#3230
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. C2×C3×#477C2semidirect
4. C2×#5091C2semidirect
5. C2×#5021C2semidirect
6. #81112C2direct (N × H)
8. C2×#4874C2semidirect
9. C2×#4850C2semidirect
12. C2×#4792C2semidirect
37. C3×#477C22semidirect
46. #5091C22semidirect
47. #5021C22semidirect
53. #4874C22semidirect
54. #4850C22semidirect
57. #4792C22semidirect
68. C2×#477D3semidirect
178. #477C2×D3semidirect
536. C2×C3#3230semidirect
563. C3C2×#3230semidirect
565. C2#81112direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#55782
C2×C3×#4772
C2×#50912
C2×#50212
#811122
C2×#48842
C2×#48742
C2×#48502
C2×#48332
C2×#48112
C2×#47922
C2×#32303