Finite Group Databasebeta

← C2×#78785 #11664433

Subgroups — C2×#78785 #11664433

Order 768.

2070 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic
2C2×#5500C211yesmaximal, characteristic, complemented
3C22×C3×#103C211yesmaximal, characteristic, complemented
4C2×#5225C211yesmaximal, characteristic, complemented
5C2×#5113C211yesmaximal, characteristic
6C2×#5101C211yesmaximal, characteristic, complemented
7C2×#4966C211yesmaximal, characteristic
8#78785C2116yesmaximal, direct product factor
9C2×#4797C212yesmaximal, complemented
10C2×#4794C212yesmaximal, complemented
11C2×#4800C212yesmaximal, complemented
12C2×#4792C212yesmaximal, complemented
13C2×#3137—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18#5816—24no—
19#5816C2218yes—
20C2×#947C2212yes—
21C2×#946—21no—
22C2×#946—21no—
23C2×#944C2212yes—
24C2×#929—22no—
25C2×#926C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C2×#924—22no—
28C22×#208C2211yescharacteristic
29#5500—24no—
30#5500C2218yescomplemented
31C2×#890—21no—
32C2×#890C2212yes—
33C2×#888—21no—
34C2×#888C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×#103C2218yescomplemented
38#5225C2218yescomplemented
39C2×#835C2212yes—
40C2×#835C2212yes—
41#5113C2218yes—
42C2×#821C2211yescharacteristic
43C2×#821C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C2×#816C2211yescharacteristic
46#5101C2218yescomplemented
47C2×#815C2212yes—
48C2×C8×Dic3C2212yes—
49C2×#810C2212yes—
50#4966C2218yes—
51C2×#809C2212yes—
52#4800C22116yescomplemented
53#4797C22116yescomplemented
54#4794C22116yescomplemented
55#4792C22116yescomplemented
56C2×#544—31no—
57C2×#536—31no—
58#3137—316no—
59C2×#491—32no—
60C2×#489—32no—
61C2×#488—32no—
62C2×#487—32no—
63C2×#417—31no—
64C2×#419—31no—
65C2×#416—31no—
66C22×#103D311yescharacteristic, complemented
67C2×#405—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8—22no—
70C22×C3×Q8—24no—
71C22×C3×Q8—24no—
72C22×C3×Q8C2312yes—
73C22×C3×Q8C2314yes—
74C23×C3×C4—21no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C2×#235—22no—
79C2×#235—24no—
80C2×#235—24no—
81C2×#235C2314yes—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C22×Dic6—22no—
85C22×Dic6C2311yescharacteristic
86C22×C3×C8—24no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2314yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2314yes—
94C2×#217—21no—
95C2×#217D412yes—
96#947—24no—
97#947C2318yes—
98#946—28no—
99#946—28no—
100#944—24no—
101#944C2318yes—
102#929—216no—
103C2×(C2×D3)⋊C4—22no—
104C2×(C2×D3)⋊C4—22no—
105#926—22no—
106#926D414yes—
107#926C2314yes—
108#926C2314yes—
109C2×C4⋊Dic3—22no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114#924—216no—
115C2×Dic3⋊C4—22no—
116C2×Dic3⋊C4—22no—
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#890C2318yes—
127#888—28no—
128#888—24no—
129#888C2318yes—
130C3×#103C2318yes—
131C3×#103C2318yes—
132#835C2318yes—
133#835C2318yes—
134#821C2314yes—
135#821C2314yes—
136#816C2314yes—
137#816C2314yes—
138#815C2318yes—
139C8×Dic3C2318yes—
140#810C2318yes—
141#809C2318yes—
142C22×#41—31no—
143C23×Q8C2×D311yescharacteristic
144C22×M16—31no—
145C23×C8C2×D311yescharacteristic
146#544—38no—
147#544—64no—
148#536—38no—
149#536—64no—
150C2×C4⋊Q8—32no—
151C2×C4⋊Q8—32no—
152C2×C4⋊Q8—61no—
153C2×#125—32no—
154C2×#125—61no—
155C2×#124—32no—
156C2×#123—62no—
157C2×C22⋊Q8—62no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8—61no—
160C2×#117—31no—
161C2×#117—31no—
162C22×C4⋊C4C2×D311yescharacteristic
163#491—316no—
164#489—316no—
165#488—316no—
166#487—316no—
167#419—38no—
168#417—38no—
169#416—38no—
170C2×#103—32no—
171C2×#103—32no—
172C2×#103—31no—
173C2×#103—31no—
174C2×#103—32no—
175C2×#103C2×D312yes—
176C2×#103C2×D312yes—
177C2×#103C2×D318yescomplemented
178C2×#102—31no—
179C2×#102—31no—
180C2×#102—32no—
181#405—38no—
182C2×#97—32no—
183C2×C4×C8—32no—
184C24×C3C2×D411yescharacteristic
185C23×D3—21no—
186C2×C3×Q8—22no—
187C2×C3×Q8—24no—
188C2×C3×Q8—24no—
189C2×C3×Q8—28no—
190C2×C3×Q8—24no—
191C2×C3×Q8—28no—
192C2×C3×Q8—44no—
193C2×C3×Q8—48no—
194C2×C3×Q8C2414yes—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—24no—
198C22×C3×C4—24no—
199C22×C3×C4—44no—
200C22×C3×C4—82no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4#4114yes—
204C22×C3×C4#4114yes—
205C22×C3×C4C2411yescharacteristic
206C22×C3×C4C2412yes—
207C2×#67—24no—
208C2×#67—44no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3C2×D412yes—
213C22×Dic3C2×D412yes—
214#235—28no—
215#235—28no—
216#235—48no—
217C2×D12—22no—
218C2×D12—22no—
219C2×D12—21no—
220C2×D12C2412yes—
221C2×C4×D3—24no—
222C2×C4×D3—44no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—21no—
226C2×Dic6—28no—
227C2×Dic6—84no—
228C2×Dic6C2412yes—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—28no—
232C2×C3×C8C2412yes—
233C2×C3×C8C2412yes—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4C2414yes—
237#217—24no—
238#217C2×D418yes—
239(C2×D3)⋊C4—28no—
240(C2×D3)⋊C4—28no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—28no—
244C4⋊Dic3C2×D414yes—
245C4⋊Dic3C2×D414yes—
246C4⋊Dic3C2412yes—
247C4⋊Dic3C2412yes—
248Dic3⋊C4—28no—
249Dic3⋊C4—28no—
250Dic3⋊C4—28no—
251Dic3⋊C4—28no—
252C4×Dic3—24no—
253C4×Dic3C2×D414yes—
254C4×Dic3C2414yes—
255#208—28no—
256C2×C3⋊C8—24no—
257C2×C3⋊C8C2414yes—
258C2×#41—34no—
259C2×#41—62no—
260C2×#41—64no—
261C2×#41—64no—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—24no—
265C22×Q8—24no—
266C22×Q8—31no—
267C22×Q8—62no—
268C22×Q8C22×D312yes—
269C22×Q8C22×D314yes—
270C22×D4—31no—
271C22×D4—61no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276C2×M16—34no—
277C2×M16—64no—
278C22×C8—24no—
279C22×C8—32no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D314yes—
283C4⋊Q8—38no—
284C4⋊Q8—38no—
285C4⋊Q8—64no—
286C4⋊Q8—64no—
287C4⋊Q8—68no—
288#125—38no—
289#125—64no—
290#125—68no—
291#124—38no—
292#124—64no—
293#123—616no—
294C22⋊Q8—616no—
295C22⋊Q8—68no—
296C22⋊Q8—68no—
297#117—34no—
298#117—34no—
299#117—34no—
300#117—62no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—62no—
312C2×C4⋊C4C22×D312yes—
313C2×C4⋊C4C22×D314yes—
314C2×C22⋊C4—32no—
315C2×C22⋊C4—62no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—61no—
318C2×C42—32no—
319C2×C42—61no—
320#103—38no—
321#103—38no—
322#103—34no—
323#103—34no—
324#103—38no—
325#103C22×D318yes—
326#103C22×D318yes—
327#102—34no—
328#102—34no—
329#102—38no—
330#97—38no—
331C4×C8—38no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3#12414yes—
335C23×C3C22×D412yes—
336C23×C3C22×D411yescharacteristic
337C22×D3—22no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C3×Q8—24no—
342C3×Q8—24no—
343C3×Q8—44no—
344C3×Q8—48no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—48no—
348C2×C3×C4—44no—
349C2×C3×C4—44no—
350C2×C3×C4—84no—
351C2×C3×C4#12414yes—
352C2×C3×C4#12418yes—
353C2×C3×C4C22×D412yes—
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4112yes—
356C2×C3×C4C2×#4114yes—
357C2×C3×C4C2×#4112yes—
358C2×C3×C4C2×#4114yes—
359C2×C3×C4C2511yescharacteristic
360#67—48no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3C22×D414yes—
370C2×Dic3C22×D414yes—
371D12—22no—
372D12—22no—
373D12—42no—
374C4×D3—48no—
375Dic6—22no—
376Dic6—22no—
377Dic6—42no—
378Dic6—88no—
379C3×C8—22no—
380C3×C8—22no—
381C3⋊C8—24no—
382C24—61no—
383C24D3×D411yescharacteristic
384#41—68no—
385#41—68no—
386#41—128no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—28no—
391C2×Q8—24no—
392C2×Q8—28no—
393C2×Q8—32no—
394C2×Q8—44no—
395C2×Q8—48no—
396C2×Q8—68no—
397C2×Q8—61no—
398C2×Q8—62no—
399C2×Q8—62no—
400C2×Q8—244no—
401C2×Q8C23×D314yes—
402C2×D4—32no—
403C2×D4—64no—
404C2×D4—62no—
405C2×D4—61no—
406C2×D4—62no—
407C2×D4—124no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—44no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—64no—
419C22×C4—82no—
420C22×C4—124no—
421C22×C4#23514yes—
422C22×C4D3×D412yes—
423C22×C4D3×D411yescharacteristic
424C22×C4#23714yes—
425C22×C4C23×D312yes—
426C22×C4C23×D311yescharacteristic
427M16—68no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—28no—
431C2×C8—34no—
432C2×C8—64no—
433C2×C8C23×D312yes—
434C2×C8C23×D312yes—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—34no—
438C4⋊C4—34no—
439C4⋊C4—32no—
440C4⋊C4—32no—
441C4⋊C4—68no—
442C4⋊C4—68no—
443C4⋊C4—68no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4C23×D314yes—
449C22⋊C4—38no—
450C22⋊C4—68no—
451C22⋊C4—68no—
452C22⋊C4—64no—
453C42—34no—
454C42—34no—
455C42—64no—
456C22×C3—28no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3C2×#12414yes—
460C22×C3C2×#12412yes—
461C22×C3#57712yes—
462C22×C3#57812yes—
463C22×C3C23×D411yescharacteristic
464C2×D3—42no—
465C2×D3—42no—
466C2×D3—42no—
467C2×D3—84no—
468C3×C4—44no—
469C3×C4—44no—
470C3×C4C2×#12412yes—
471C3×C4C2×#12414yes—
472C3×C4C22×#4111yescharacteristic
473C3×C4C22×#4111yescharacteristic
474Dic3—24no—
475Dic3—24no—
476Dic3—42no—
477Dic3—42no—
478Dic3—42no—
479C23—22no—
480C23—22no—
481C23—62no—
482C23—122no—
483C23—122no—
484C23—122no—
485C23#88514yes—
486C23C2×D3×D412yes—
487C23C2×D3×D411yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—48no—
491Q8—44no—
492Q8—62no—
493Q8—62no—
494Q8—122no—
495Q8—248no—
496D4—62no—
497D4—62no—
498D4—128no—
499D4—122no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—34no—
503C2×C4—34no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—84no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—128no—
517C2×C4#88514yes—
518C2×C4#88518yes—
519C2×C4C2×#23512yes—
520C2×C4C2×#23514yes—
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×D3×D412yes—
523C2×C4C2×#23712yes—
524C2×C4C2×#23714yes—
525C2×C4C24×D311yescharacteristic
526C8—22no—
527C8—22no—
528C8—64no—
529C2×C3—24no—
530C2×C3#313714yescomplemented
531C2×C3C22×#12411yescharacteristic
532C2×C3C2×#57711yescharacteristic
533C2×C3C2×#57811yescharacteristic
534D3—82no—
535C22—22no—
536C22—22no—
537C22—28no—
538C22—122no—
539C22—122no—
540C22—122no—
541C22—244no—
542C22C2×#88514yes—
543C22C2×#88512yes—
544C22#576212yes—
545C22#576612yes—
546C22C22×D3×D411yescharacteristic
547C4—44no—
548C4—44no—
549C4—64no—
550C4—64no—
551C4—122no—
552C4—122no—
553C4—122no—
554C4C2×#88514yes—
555C4C2×#88512yes—
556C4C22×#23511yescharacteristic
557C4C22×#23711yescharacteristic
558C3C2×#313711yescharacteristic, complemented, Sylow
559C2—24no—
560C2—242no—
561C2#7878514yesdirect product factor
562C2C22×#88511yescharacteristic
563C2C2×#576211yescharacteristic
564C2C2×#576611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×#103C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC22×#103D3
hypocenterC3C2×#3137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C2×#5500C2semidirect
3. C22×C3×#103C2semidirect
4. C2×#5225C2semidirect
6. C2×#5101C2semidirect
8. #78785C2direct (N × H)
9. C2×#4797C2semidirect
10. C2×#4794C2semidirect
11. C2×#4800C2semidirect
12. C2×#4792C2semidirect
30. #5500C22semidirect
37. C2×C3×#103C22semidirect
38. #5225C22semidirect
46. #5101C22semidirect
52. #4800C22semidirect
53. #4797C22semidirect
54. #4794C22semidirect
55. #4792C22semidirect
66. C22×#103D3semidirect
177. C2×#103C2×D3semidirect
530. C2×C3#3137semidirect
558. C3C2×#3137semidirect
561. C2#78785direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#58162
C2×#55002
C22×C3×#1032
C2×#52252
C2×#51132
C2×#51012
C2×#49662
#787852
C2×#47972
C2×#47942
C2×#48002
C2×#47922
C2×#31373