Finite Group Databasebeta

← C2×#78827 #11664427

Subgroups — C2×#78827 #11664427

Order 768.

2262 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic
2C2×#5519C211yesmaximal, characteristic, complemented
3C2×C3×#416C211yesmaximal, characteristic, complemented
4C2×#5232C211yesmaximal, characteristic, complemented
5C22×#821C211yesmaximal, characteristic, complemented
6C2×#5101C211yesmaximal, characteristic, complemented
7C2×#5001C211yesmaximal, characteristic, complemented
8#78827C2116yesmaximal, direct product factor
9C2×#4794C212yesmaximal, complemented
10C2×#4800C212yesmaximal, complemented
11C2×#4764C212yesmaximal, complemented
12C2×#4760C212yesmaximal, complemented
13C2×#3136—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18#5830—24no—
19#5830C2218yes—
20C2×#947C2212yes—
21C2×#940C2212yes—
22C2×#937—22no—
23C2×#930—21no—
24C2×#930—21no—
25C2×#926C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C22×#208C2211yescharacteristic
28#5519—24no—
29#5519C2218yescomplemented
30C2×#890C2212yescomplemented
31C2×#888C2212yescomplemented
32C2×#878—22no—
33C2×#870—21no—
34C2×#869—21no—
35C3×#416C2218yescomplemented
36C2×C3×#103C2211yescharacteristic, complemented
37C2×C3×#103C2211yescharacteristic, complemented
38C2×C3×#102C2211yescharacteristic, complemented
39C2×C3×#102C2211yescharacteristic, complemented
40#5232C2218yescomplemented
41C2×#835C2212yescomplemented
42C2×#832C2212yescomplemented
43C2×#821C2212yes—
44C2×#821C2212yes—
45C2×#821C2218yescomplemented
46#5101C2218yescomplemented
47C2×#815C2212yescomplemented
48C2×C8×Dic3C2212yescomplemented
49#5001C2218yescomplemented
50C2×#809C2212yes—
51C2×#809C2212yes—
52#4800C22116yescomplemented
53#4794C22116yescomplemented
54#4764C22116yescomplemented
55#4760C22116yescomplemented
56C2×#544—31no—
57C2×#535—31no—
58#3136—316no—
59C2×#490—32no—
60C2×#489—32no—
61C2×#488—32no—
62C2×#486—32no—
63C2×#417—31no—
64C2×#418—31no—
65C2×#416D311yescharacteristic, complemented
66C22×#102—31no—
67C2×#405—31no—
68C24×D3—21no—
69C2×C3×#41—22no—
70C2×C3×#41—24no—
71C2×C3×#41—24no—
72C2×C3×#41C2314yes—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4—21no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—22no—
79C22×#67—22no—
80C23×Dic3—21no—
81C22×D12—24no—
82C22×D12—24no—
83C22×D12—21no—
84C22×D12—22no—
85C22×D12C2312yes—
86C22×D12C2314yes—
87C22×Dic6—22no—
88C22×C3×C8—24no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2314yes—
92C3×#117—22no—
93C3×#117C2314yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×#217D412yes—
99#947—24no—
100#947C2318yes—
101#940—24no—
102#940C2318yes—
103#937—216no—
104#930—28no—
105#930—28no—
106C2×(C2×D3)⋊C4—22no—
107C2×(C2×D3)⋊C4—22no—
108#926D414yes—
109#926C2314yes—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—22no—
112C2×C4⋊Dic3—22no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2312yes—
116C2×C4⋊Dic3C2314yes—
117C2×Dic3⋊C4—22no—
118C2×Dic3⋊C4—22no—
119C2×C4×Dic3C2312yes—
120C2×#208—24no—
121C2×#208C2314yes—
122C22×C3⋊C8C2312yes—
123#890—24no—
124#890C2318yescomplemented
125#888—24no—
126#888C2318yescomplemented
127#878—216no—
128#870—28no—
129#869—28no—
130C3×#103C2314yescomplemented
131C3×#103C2314yescomplemented
132C3×#102C2314yescomplemented
133C3×#102C2314yescomplemented
134#835C2318yescomplemented
135#832C2318yescomplemented
136#821C2318yes—
137#821C2318yes—
138#815C2318yescomplemented
139C8×Dic3C2318yescomplemented
140#809C2318yes—
141#809C2318yes—
142C22×#41C2×D311yescharacteristic
143C23×D4—31no—
144C22×M16—31no—
145C23×C8C2×D311yescharacteristic
146#544—38no—
147#544—64no—
148#535—38no—
149#535—64no—
150C2×C4⋊Q8—32no—
151C2×C4⋊Q8—61no—
152C2×#127—32no—
153C2×#125—32no—
154C2×#125—61no—
155C2×#124—32no—
156C2×C22⋊Q8—62no—
157C2×#121—61no—
158C2×#121—61no—
159C2×#120—62no—
160C2×#117—31no—
161C2×#117C2×D311yescharacteristic
162C22×C4⋊C4—31no—
163#490—316no—
164#489—316no—
165#488—316no—
166#486—316no—
167#418—38no—
168#417—38no—
169#416C2×D318yescomplemented
170C2×#103—32no—
171C2×#103C2×D311yescharacteristic, complemented
172C2×#103C2×D311yescharacteristic, complemented
173C2×#102—32no—
174C2×#102—32no—
175C2×#102—32no—
176C2×#102—38no—
177C2×#102—32no—
178C2×#102—32no—
179C2×#102C2×D311yescharacteristic, complemented
180C2×#102C2×D311yescharacteristic, complemented
181#405—38no—
182C2×#97—32no—
183C2×C4×C8—32no—
184C24×C3—21no—
185C24×C3C2×D411yescharacteristic
186C23×D3—22no—
187C23×D3—24no—
188C23×D3—44no—
189C23×D3—82no—
190C23×D3—82no—
191C3×#41—28no—
192C3×#41—28no—
193C3×#41—48no—
194C2×C3×Q8—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8C2412yes—
198C2×C3×D4—24no—
199C2×C3×D4—21no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—44no—
203C2×C3×D4C2412yes—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—24no—
208C22×C3×C4—44no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D412yes—
211C22×C3×C4#4114yes—
212C22×C3×C4#4114yes—
213C22×C3×C4C2411yescharacteristic
214C22×C3×C4C2412yes—
215C2×#67—28no—
216C2×#67—28no—
217C2×#67—84no—
218C2×#67—84no—
219C22×Dic3—22no—
220C22×Dic3—24no—
221C22×Dic3—82no—
222C22×Dic3C2×D412yes—
223C22×Dic3C2×D412yes—
224C2×D12—24no—
225C2×D12—28no—
226C2×D12—24no—
227C2×D12—28no—
228C2×D12—24no—
229C2×D12—22no—
230C2×D12—44no—
231C2×D12—48no—
232C2×D12C2414yes—
233C2×Dic6—28no—
234C2×Dic6—84no—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C2×C3×C8—28no—
238C2×C3×C8C2412yes—
239C2×C3×C8C2412yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2412yes—
243C3×C4⋊C4C2412yes—
244C3×C22⋊C4—24no—
245C3×C42—24no—
246#217C2×D418yes—
247(C2×D3)⋊C4—28no—
248(C2×D3)⋊C4—28no—
249C4⋊Dic3—24no—
250C4⋊Dic3—22no—
251C4⋊Dic3—28no—
252C4⋊Dic3C2×D414yes—
253C4⋊Dic3C2×D414yes—
254C4⋊Dic3C2414yes—
255Dic3⋊C4—28no—
256Dic3⋊C4—28no—
257C4×Dic3C2×D414yes—
258C4×Dic3C2414yes—
259#208—28no—
260C2×C3⋊C8—24no—
261C2×C3⋊C8C2414yes—
262C25—61no—
263C2×#41—22no—
264C2×#41—24no—
265C2×#41—24no—
266C2×#41C22×D314yes—
267C22×Q8—62no—
268C22×Q8C22×D311yescharacteristic
269C22×D4—21no—
270C22×D4—32no—
271C22×D4—34no—
272C22×D4—62no—
273C22×D4—62no—
274C22×D4—64no—
275C22×D4—61no—
276C22×D4—62no—
277C22×D4—64no—
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C23×C4—61no—
281C23×C4C22×D311yescharacteristic
282C2×M16—34no—
283C2×M16—64no—
284C22×C8—24no—
285C22×C8—32no—
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D314yes—
289C4⋊Q8—38no—
290C4⋊Q8—68no—
291C4⋊Q8—64no—
292#127—38no—
293#127—64no—
294#125—38no—
295#125—68no—
296#125—64no—
297#124—38no—
298#124—64no—
299C22⋊Q8—616no—
300#121—68no—
301#121—68no—
302#120—616no—
303#117—22no—
304#117—34no—
305#117—34no—
306#117C22×D314yes—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—62no—
312C2×C4⋊C4—62no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—62no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—62no—
321C2×C22⋊C4—62no—
322C2×C42—21no—
323C2×C42—32no—
324#103—38no—
325#103C22×D314yescomplemented
326#103C22×D314yescomplemented
327#102—38no—
328#102—38no—
329#102—38no—
330#102—38no—
331#102—38no—
332#102C22×D314yescomplemented
333#102C22×D314yescomplemented
334#97—38no—
335C4×C8—38no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3#12414yes—
343C23×C3C22×D412yes—
344C23×C3C22×D411yescharacteristic
345C22×D3—24no—
346C22×D3—48no—
347C22×D3—44no—
348C22×D3—44no—
349C22×D3—44no—
350C22×D3—84no—
351C22×D3—84no—
352C22×D3—816no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—42no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—48no—
359C3×D4—42no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—48no—
367C2×C3×C4#12414yes—
368C2×C3×C4#12418yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D412yes—
371C2×C3×C4C2×#4112yes—
372C2×C3×C4C2×#4114yes—
373C2×C3×C4C2×#4112yes—
374C2×C3×C4C2×#4114yes—
375C2×C3×C4C2511yescharacteristic
376#67—88no—
377#67—88no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—44no—
382C2×Dic3—84no—
383C2×Dic3C22×D414yes—
384C2×Dic3C22×D414yes—
385D12—24no—
386D12—24no—
387D12—48no—
388D12—44no—
389Dic6—88no—
390C3×C8—22no—
391C3×C8—22no—
392C3⋊C8—24no—
393C24—21no—
394C24—62no—
395C24—64no—
396C24—124no—
397C24—242no—
398C24—242no—
399C24D3×D411yescharacteristic
400#41—28no—
401#41—28no—
402#41—48no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—68no—
407C2×Q8—244no—
408C2×Q8C23×D312yes—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—34no—
414C2×D4—44no—
415C2×D4—68no—
416C2×D4—68no—
417C2×D4—64no—
418C2×D4—68no—
419C2×D4—64no—
420C2×D4—62no—
421C2×D4—64no—
422C2×D4—68no—
423C2×D4—124no—
424C2×D4—128no—
425C2×D4—244no—
426C2×D4—244no—
427C2×D4C23×D312yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—32no—
433C22×C4—32no—
434C22×C4—44no—
435C22×C4—62no—
436C22×C4—64no—
437C22×C4—242no—
438C22×C4#23514yes—
439C22×C4D3×D412yes—
440C22×C4D3×D411yescharacteristic
441C22×C4#23714yes—
442C22×C4C23×D312yes—
443C22×C4C23×D311yescharacteristic
444M16—68no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—28no—
448C2×C8—34no—
449C2×C8—64no—
450C2×C8C23×D312yes—
451C2×C8C23×D312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—34no—
455C4⋊C4—34no—
456C4⋊C4—34no—
457C4⋊C4—68no—
458C4⋊C4—68no—
459C4⋊C4—68no—
460C4⋊C4—64no—
461C4⋊C4—62no—
462C4⋊C4C23×D312yes—
463C4⋊C4C23×D312yes—
464C22⋊C4—24no—
465C22⋊C4—38no—
466C22⋊C4—68no—
467C22⋊C4—68no—
468C42—24no—
469C42—34no—
470C42—34no—
471C22×C3—28no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—84no—
478C22×C3C2×#12414yes—
479C22×C3C2×#12412yes—
480C22×C3#57612yes—
481C22×C3#57712yes—
482C22×C3C23×D411yescharacteristic
483C2×D3—44no—
484C2×D3—44no—
485C2×D3—44no—
486C2×D3—816no—
487C2×D3—88no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4C2×#12414yes—
492C3×C4C2×#12412yes—
493C3×C4C22×#4111yescharacteristic
494C3×C4C22×#4111yescharacteristic
495Dic3—24no—
496Dic3—24no—
497Dic3—44no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—64no—
505C23—128no—
506C23—124no—
507C23—124no—
508C23—124no—
509C23—244no—
510C23—244no—
511C23—2416no—
512C23#88514yes—
513C23C2×D3×D412yes—
514C23C2×D3×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—42no—
518Q8—248no—
519D4—22no—
520D4—22no—
521D4—48no—
522D4—42no—
523D4—64no—
524D4—64no—
525D4—128no—
526D4—124no—
527D4—248no—
528D4—248no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—34no—
533C2×C4—34no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—48no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—124no—
542C2×C4—244no—
543C2×C4#88514yes—
544C2×C4#88518yes—
545C2×C4C2×#23512yes—
546C2×C4C2×#23514yes—
547C2×C4C2×D3×D412yes—
548C2×C4C2×D3×D411yescharacteristic
549C2×C4C2×#23712yes—
550C2×C4C2×#23714yes—
551C2×C4C24×D311yescharacteristic
552C8—22no—
553C8—22no—
554C8—64no—
555C2×C3—24no—
556C2×C3—82no—
557C2×C3#313614yescomplemented
558C2×C3C22×#12411yescharacteristic
559C2×C3C2×#57611yescharacteristic
560C2×C3C2×#57711yescharacteristic
561D3—84no—
562C22—22no—
563C22—22no—
564C22—28no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—84no—
569C22—124no—
570C22—124no—
571C22—124no—
572C22—2416no—
573C22—248no—
574C22C2×#88514yes—
575C22C2×#88512yes—
576C22#576912yes—
577C22#577012yes—
578C22C22×D3×D411yescharacteristic
579C4—42no—
580C4—42no—
581C4—42no—
582C4—64no—
583C4—64no—
584C4—124no—
585C4C2×#88514yes—
586C4C2×#88512yes—
587C4C22×#23511yescharacteristic
588C4C22×#23711yescharacteristic
589C3C2×#313611yescharacteristic, complemented, Sylow
590C2—24no—
591C2—82no—
592C2—244no—
593C2#7882714yesdirect product factor
594C2C22×#88511yescharacteristic
595C2C2×#576911yescharacteristic
596C2C2×#577011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#416C2
socleC23×C3C22×D4
prime radicalC2×#5830C2
hypercenterC2×#416D3
hypocenterC3C2×#3136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×#5519C2semidirect
3. C2×C3×#416C2semidirect
4. C2×#5232C2semidirect
5. C22×#821C2semidirect
6. C2×#5101C2semidirect
7. C2×#5001C2semidirect
8. #78827C2direct (N × H)
9. C2×#4794C2semidirect
10. C2×#4800C2semidirect
11. C2×#4764C2semidirect
12. C2×#4760C2semidirect
29. #5519C22semidirect
30. C2×#890C22semidirect
31. C2×#888C22semidirect
35. C3×#416C22semidirect
36. C2×C3×#103C22semidirect
37. C2×C3×#103C22semidirect
38. C2×C3×#102C22semidirect
39. C2×C3×#102C22semidirect
40. #5232C22semidirect
41. C2×#835C22semidirect
42. C2×#832C22semidirect
45. C2×#821C22semidirect
46. #5101C22semidirect
47. C2×#815C22semidirect
48. C2×C8×Dic3C22semidirect
49. #5001C22semidirect
52. #4800C22semidirect
53. #4794C22semidirect
54. #4764C22semidirect
55. #4760C22semidirect
65. C2×#416D3semidirect
124. #890C23semidirect
126. #888C23semidirect
130. C3×#103C23semidirect
131. C3×#103C23semidirect
132. C3×#102C23semidirect
133. C3×#102C23semidirect
134. #835C23semidirect
135. #832C23semidirect
138. #815C23semidirect
139. C8×Dic3C23semidirect
169. #416C2×D3semidirect
171. C2×#103C2×D3semidirect
172. C2×#103C2×D3semidirect
179. C2×#102C2×D3semidirect
180. C2×#102C2×D3semidirect
325. #103C22×D3semidirect
326. #103C22×D3semidirect
332. #102C22×D3semidirect
333. #102C22×D3semidirect
557. C2×C3#3136semidirect
589. C3C2×#3136semidirect
593. C2#78827direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#58302
C2×#55192
C2×C3×#4162
C2×#52322
C22×#8212
C2×#51012
C2×#50012
#788272
C2×#47942
C2×#48002
C2×#47642
C2×#47602
C2×#31363