Finite Group Databasebeta

← C2×#78737 #11664421

Subgroups — C2×#78737 #11664421

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×#5805C211yesmaximal, characteristic
2C2×#5500C211yesmaximal, characteristic, complemented
3C22×C3×#102C211yesmaximal, characteristic, complemented
4C2×#5215C211yesmaximal, characteristic, complemented
5C2×#5113C211yesmaximal, characteristic, complemented
6C2×#5101C211yesmaximal, characteristic, complemented
7C2×#4966C211yesmaximal, characteristic, complemented
8#78737C2116yesmaximal, direct product factor
9C2×#4764C212yesmaximal, complemented
10C2×#4765C212yesmaximal, complemented
11C2×#4768C212yesmaximal, complemented
12C2×#4760C212yesmaximal, complemented
13C2×#3136—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#953—22no—
19#5805—24no—
20#5805C2218yes—
21C2×#940C2212yes—
22C2×#938—21no—
23C2×#938—21no—
24C2×#936C2212yes—
25C2×#926C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C2×#924—22no—
28C22×#208C2211yescharacteristic
29#5500—24no—
30#5500C2218yescomplemented
31C2×#890—21no—
32C2×#890C2212yescomplemented
33C2×#888—21no—
34C2×#888C2212yescomplemented
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2218yescomplemented
38#5215C2218yescomplemented
39C2×#832C2212yes—
40C2×#832C2212yes—
41#5113C2218yescomplemented
42C2×#821C2211yescharacteristic, complemented
43C2×#821C2211yescharacteristic, complemented
44C2×#816C2211yescharacteristic, complemented
45C2×#816C2211yescharacteristic, complemented
46#5101C2218yescomplemented
47C2×#815C2212yescomplemented
48C2×C8×Dic3C2212yescomplemented
49C2×#810C2212yescomplemented
50#4966C2218yescomplemented
51C2×#809C2212yescomplemented
52#4768C22116yescomplemented
53#4765C22116yescomplemented
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×#416—31no—
66C22×#102D311yescharacteristic, complemented
67C2×#405—31no—
68C25×C3—21no—
69C22×C3×D4—21no—
70C22×C3×D4—22no—
71C22×C3×D4—24no—
72C22×C3×D4—24no—
73C22×C3×D4C2312yes—
74C22×C3×D4C2314yes—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C22×#67—22no—
79C22×#67—22no—
80C2×#235—22no—
81C2×#235—24no—
82C2×#235—24no—
83C2×#235C2314yes—
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×Dic6—22no—
87C22×Dic6C2311yescharacteristic
88C22×C3×C8—24no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2314yes—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2314yes—
96#953—216no—
97C2×#217—21no—
98C2×#217—22no—
99C2×#217—22no—
100C2×#217D412yes—
101#940—24no—
102#940C2318yes—
103#938—28no—
104#938—28no—
105#936—24no—
106#936C2318yes—
107#926—22no—
108#926D414yes—
109#926C2314yes—
110#926C2314yes—
111C2×C4⋊Dic3—22no—
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116#924—216no—
117C2×Dic3⋊C4—22no—
118C2×Dic3⋊C4—22no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3C2312yes—
121C2×#208—24no—
122C2×#208C2314yes—
123C22×C3⋊C8C2312yes—
124#890—28no—
125#890—24no—
126#890C2318yescomplemented
127#888—28no—
128#888—24no—
129#888C2318yescomplemented
130C3×#102C2318yes—
131C3×#102C2318yes—
132#832C2318yes—
133#832C2318yes—
134#821C2314yescomplemented
135#821C2314yescomplemented
136#816C2314yescomplemented
137#816C2314yescomplemented
138#815C2318yescomplemented
139C8×Dic3C2318yescomplemented
140#810C2318yescomplemented
141#809C2318yescomplemented
142C22×#41—31no—
143C23×D4C2×D311yescharacteristic
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×#117—31no—
162C22×C4⋊C4C2×D311yescharacteristic
163#490—316no—
164#489—316no—
165#488—316no—
166#486—316no—
167#418—38no—
168#417—38no—
169#416—38no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—32no—
173C2×#102—32no—
174C2×#102—32no—
175C2×#102—31no—
176C2×#102—31no—
177C2×#102—32no—
178C2×#102C2×D312yes—
179C2×#102C2×D312yes—
180C2×#102C2×D318yescomplemented
181#405—38no—
182C2×#97—32no—
183C2×C4×C8—32no—
184C24×C3—22no—
185C24×C3—24no—
186C24×C3—44no—
187C24×C3—82no—
188C24×C3—82no—
189C24×C3C2×D411yescharacteristic
190C23×D3—21no—
191C2×C3×D4—22no—
192C2×C3×D4—24no—
193C2×C3×D4—24no—
194C2×C3×D4—28no—
195C2×C3×D4—24no—
196C2×C3×D4—28no—
197C2×C3×D4—44no—
198C2×C3×D4—48no—
199C2×C3×D4C2414yes—
200C22×C3×C4—22no—
201C22×C3×C4—24no—
202C22×C3×C4—82no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D412yes—
205C22×C3×C4#4114yes—
206C22×C3×C4#4114yes—
207C22×C3×C4C2411yescharacteristic
208C22×C3×C4C2412yes—
209C2×#67—24no—
210C2×#67—28no—
211C2×#67—28no—
212C2×#67—44no—
213C2×#67—84no—
214C2×#67—84no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3C2×D412yes—
219C22×Dic3C2×D412yes—
220#235—28no—
221#235—28no—
222#235—48no—
223C2×D12—22no—
224C2×D12—22no—
225C2×D12—21no—
226C2×D12C2412yes—
227C2×C4×D3—24no—
228C2×C4×D3—44no—
229C2×Dic6—22no—
230C2×Dic6—22no—
231C2×Dic6—21no—
232C2×Dic6—28no—
233C2×Dic6—84no—
234C2×Dic6C2412yes—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C2×C3×C8—28no—
238C2×C3×C8C2412yes—
239C2×C3×C8C2412yes—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4C2414yes—
243#217—24no—
244#217—28no—
245#217—28no—
246#217C2×D418yes—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—28no—
250C4⋊Dic3C2×D414yes—
251C4⋊Dic3C2×D414yes—
252C4⋊Dic3C2412yes—
253C4⋊Dic3C2412yes—
254Dic3⋊C4—28no—
255Dic3⋊C4—28no—
256C4×Dic3—24no—
257C4×Dic3C2×D414yes—
258C4×Dic3C2414yes—
259#208—28no—
260C2×C3⋊C8—24no—
261C2×C3⋊C8C2414yes—
262C25—21no—
263C2×#41—34no—
264C2×#41—62no—
265C2×#41—64no—
266C2×#41—64no—
267C22×Q8—31no—
268C22×Q8—62no—
269C22×D4—24no—
270C22×D4—21no—
271C22×D4—22no—
272C22×D4—24no—
273C22×D4—31no—
274C22×D4—62no—
275C22×D4—62no—
276C22×D4—61no—
277C22×D4C22×D312yes—
278C22×D4C22×D314yes—
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—64no—
291C4⋊Q8—68no—
292#127—38no—
293#127—64no—
294#125—38no—
295#125—64no—
296#125—68no—
297#124—38no—
298#124—64no—
299C22⋊Q8—616no—
300#121—68no—
301#121—68no—
302#120—616no—
303#117—34no—
304#117—34no—
305#117—34no—
306#117—62no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4—62no—
316C2×C4⋊C4C22×D312yes—
317C2×C4⋊C4C22×D314yes—
318C2×C22⋊C4—32no—
319C2×C22⋊C4—62no—
320C2×C22⋊C4—62no—
321C2×C22⋊C4—61no—
322C2×C42—32no—
323C2×C42—61no—
324#103—34no—
325#103—34no—
326#103—38no—
327#102—38no—
328#102—38no—
329#102—34no—
330#102—34no—
331#102—38no—
332#102C22×D318yes—
333#102C22×D318yes—
334#97—38no—
335C4×C8—38no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—24no—
339C23×C3—48no—
340C23×C3—44no—
341C23×C3—44no—
342C23×C3—44no—
343C23×C3—84no—
344C23×C3—84no—
345C23×C3—816no—
346C23×C3#12414yes—
347C23×C3C22×D412yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—44no—
356C3×D4—48no—
357C2×C3×C4—24no—
358C2×C3×C4—44no—
359C2×C3×C4—84no—
360C2×C3×C4#12414yes—
361C2×C3×C4#12418yes—
362C2×C3×C4C22×D412yes—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4112yes—
365C2×C3×C4C2×#4114yes—
366C2×C3×C4C2×#4112yes—
367C2×C3×C4C2×#4114yes—
368C2×C3×C4C2511yescharacteristic
369#67—48no—
370#67—88no—
371#67—88no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3C22×D414yes—
381C2×Dic3C22×D414yes—
382D12—22no—
383D12—22no—
384D12—42no—
385C4×D3—48no—
386Dic6—22no—
387Dic6—22no—
388Dic6—42no—
389Dic6—88no—
390C3×C8—22no—
391C3×C8—22no—
392C3⋊C8—24no—
393C24—22no—
394C24—24no—
395C24—44no—
396C24—61no—
397C24—82no—
398C24—82no—
399C24D3×D411yescharacteristic
400#41—68no—
401#41—68no—
402#41—128no—
403C2×Q8—32no—
404C2×Q8—68no—
405C2×Q8—61no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—244no—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—32no—
416C2×D4—44no—
417C2×D4—48no—
418C2×D4—68no—
419C2×D4—68no—
420C2×D4—64no—
421C2×D4—62no—
422C2×D4—61no—
423C2×D4—62no—
424C2×D4—124no—
425C2×D4—244no—
426C2×D4—244no—
427C2×D4C23×D314yes—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—64no—
436C22×C4—82no—
437C22×C4—124no—
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—22no—
453C4⋊C4—24no—
454C4⋊C4—34no—
455C4⋊C4—34no—
456C4⋊C4—32no—
457C4⋊C4—32no—
458C4⋊C4—68no—
459C4⋊C4—68no—
460C4⋊C4—68no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4C23×D314yes—
464C22⋊C4—38no—
465C22⋊C4—68no—
466C22⋊C4—68no—
467C22⋊C4—64no—
468C42—34no—
469C42—34no—
470C42—64no—
471C22×C3—28no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—816no—
478C22×C3—88no—
479C22×C3C2×#12414yes—
480C22×C3C2×#12412yes—
481C22×C3#57612yes—
482C22×C3#57712yes—
483C22×C3C23×D411yescharacteristic
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—84no—
488C3×C4—44no—
489C3×C4C2×#12412yes—
490C3×C4C2×#12414yes—
491C3×C4C22×#4111yescharacteristic
492C3×C4C22×#4111yescharacteristic
493Dic3—24no—
494Dic3—24no—
495Dic3—42no—
496Dic3—42no—
497Dic3—42no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—48no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—62no—
506C23—84no—
507C23—84no—
508C23—816no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23#88514yes—
513C23C2×D3×D412yes—
514C23C2×D3×D411yescharacteristic
515Q8—62no—
516Q8—62no—
517Q8—122no—
518Q8—248no—
519D4—24no—
520D4—24no—
521D4—48no—
522D4—44no—
523D4—62no—
524D4—62no—
525D4—128no—
526D4—122no—
527D4—248no—
528D4—248no—
529C2×C4—24no—
530C2×C4—34no—
531C2×C4—34no—
532C2×C4—44no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—84no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—128no—
543C2×C4#88514yes—
544C2×C4#88518yes—
545C2×C4C2×#23512yes—
546C2×C4C2×#23514yes—
547C2×C4C2×D3×D411yescharacteristic
548C2×C4C2×D3×D412yes—
549C2×C4C2×#23712yes—
550C2×C4C2×#23714yes—
551C2×C4C24×D311yescharacteristic
552C8—22no—
553C8—22no—
554C8—64no—
555C2×C3—24no—
556C2×C3—84no—
557C2×C3#313614yescomplemented
558C2×C3C22×#12411yescharacteristic
559C2×C3C2×#57611yescharacteristic
560C2×C3C2×#57711yescharacteristic
561D3—82no—
562C22—22no—
563C22—22no—
564C22—28no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—816no—
569C22—88no—
570C22—122no—
571C22—122no—
572C22—122no—
573C22—244no—
574C22C2×#88514yes—
575C22C2×#88512yes—
576C22#575712yes—
577C22#576212yes—
578C22C22×D3×D411yescharacteristic
579C4—44no—
580C4—64no—
581C4—64no—
582C4—122no—
583C4—122no—
584C4—122no—
585C4C2×#88514yes—
586C4C2×#88512yes—
587C4C22×#23511yescharacteristic
588C4C22×#23711yescharacteristic
589C3C2×#313611yescharacteristic, complemented, Sylow
590C2—24no—
591C2—84no—
592C2—242no—
593C2#7873714yesdirect product factor
594C2C22×#88511yescharacteristic
595C2C2×#575711yescharacteristic
596C2C2×#576211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×#102C2
socleC23×C3C22×D4
prime radicalC2×#5805C2
hypercenterC22×#102D3
hypocenterC3C2×#3136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C2×#5500C2semidirect
3. C22×C3×#102C2semidirect
4. C2×#5215C2semidirect
5. C2×#5113C2semidirect
6. C2×#5101C2semidirect
7. C2×#4966C2semidirect
8. #78737C2direct (N × H)
9. C2×#4764C2semidirect
10. C2×#4765C2semidirect
11. C2×#4768C2semidirect
12. C2×#4760C2semidirect
30. #5500C22semidirect
32. C2×#890C22semidirect
34. C2×#888C22semidirect
37. C2×C3×#102C22semidirect
38. #5215C22semidirect
41. #5113C22semidirect
42. C2×#821C22semidirect
43. C2×#821C22semidirect
44. C2×#816C22semidirect
45. C2×#816C22semidirect
46. #5101C22semidirect
47. C2×#815C22semidirect
48. C2×C8×Dic3C22semidirect
49. C2×#810C22semidirect
50. #4966C22semidirect
51. C2×#809C22semidirect
52. #4768C22semidirect
53. #4765C22semidirect
54. #4764C22semidirect
55. #4760C22semidirect
66. C22×#102D3semidirect
126. #890C23semidirect
129. #888C23semidirect
134. #821C23semidirect
135. #821C23semidirect
136. #816C23semidirect
137. #816C23semidirect
138. #815C23semidirect
139. C8×Dic3C23semidirect
140. #810C23semidirect
141. #809C23semidirect
180. C2×#102C2×D3semidirect
557. C2×C3#3136semidirect
589. C3C2×#3136semidirect
593. C2#78737direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#58052
C2×#55002
C22×C3×#1022
C2×#52152
C2×#51132
C2×#51012
C2×#49662
#787372
C2×#47642
C2×#47652
C2×#47682
C2×#47602
C2×#31363