Finite Group Databasebeta

← C2×#83136 #11664965

Subgroups — C2×#83136 #11664965

Order 768.

1986 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×#5470C211yesmaximal, characteristic, complemented
3#83136C2116yesmaximal, direct product factor
4C2×#5064C211yesmaximal, characteristic, complemented
5C2×#5069C211yesmaximal, characteristic, complemented
6C2×#5060C211yesmaximal, characteristic, complemented
7C2×#5055C211yesmaximal, characteristic, complemented
8C2×#5052C212yesmaximal, complemented
9C2×#5042C212yesmaximal
10C2×C4×#216C212yesmaximal, complemented
11C2×C4×#214C212yesmaximal, complemented
12C2×C4×#208C211yesmaximal, characteristic, complemented
13C2×#3172—31nomaximal, Sylow
14C3×#529—24no—
15C3×#529C2218yes—
16C2×C3×#126—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C22×C3×C42C2211yescharacteristic
27C22×#216C2212yes—
28C22×#214C2212yes—
29C2×#926—21no—
30C2×#924—21no—
31C2×#924C2212yes—
32C22×#208C2212yes—
33#5470—24no—
34#5470C2218yescomplemented
35C2×#870—21no—
36C2×#869C2211yescharacteristic
37C2×#869C2211yescharacteristic
38C2×C4×Dic6—21no—
39C2×C4×Dic6C2212yes—
40C2×#835C2212yes—
41C2×#832C2212yes—
42#5069C2218yescomplemented
43#5064C2218yescomplemented
44#5060C2218yescomplemented
45#5055C2218yescomplemented
46#5052C22116yescomplemented
47#5042C22116yes—
48C4×#216C22116yescomplemented
49C4×#214C22116yescomplemented
50C2×#810C2212yes—
51C2×#810C2212yes—
52C2×#808C2212yes—
53C2×#807C2212yes—
54C4×#208C2218yescomplemented
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2212yes—
58C2×#533—31no—
59C2×#529D311yescharacteristic
60#3172—316no—
61C2×#501—31no—
62C2×#497—31no—
63C2×#491—31no—
64C2×#489—31no—
65C2×#470—32no—
66C2×#468—32no—
67C2×C4×Q16—32no—
68C2×C4×SD16—32no—
69C2×C4×M16—31no—
70C22×C3×Q8C2311yescharacteristic
71C22×C3×D4—21no—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4D411yescharacteristic
75C23×C3×C4D411yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C22×Dic6—21no—
78C22×Dic6C2312yes—
79C3×#126—28no—
80C3×#125—22no—
81C3×#125C2314yes—
82C3×#124—22no—
83C3×#124C2314yes—
84C3×#123—28no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8C2314yes—
87C3×#121—22no—
88C3×#121C2314yes—
89C3×C4×Q8—22no—
90C3×C4×Q8C2314yes—
91C3×C4×D4—28no—
92C3×C4×D4—22no—
93C3×C4×D4C2314yes—
94C3×#117—28no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C42—21no—
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D414yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2314yes—
113C2×#217—22no—
114C2×#216—28no—
115C2×#216C2318yes—
116C2×#214—28no—
117C2×#214C2318yes—
118#926—28no—
119C2×C4⋊Dic3—21no—
120C2×C4⋊Dic3—21no—
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3C2312yes—
123#924—24no—
124#924—28no—
125#924C2318yes—
126C2×Dic3⋊C4—22no—
127C2×Dic3⋊C4—22no—
128C2×C4×Dic3—22no—
129C2×#208—28no—
130C2×#208C2318yes—
131C22×C3⋊C8C2312yes—
132C22×C3⋊C8C2312yes—
133#870—28no—
134#869—22no—
135#869—22no—
136#869C2314yes—
137#869C2314yes—
138C4×Dic6—28no—
139C4×Dic6—24no—
140C4×Dic6C2318yes—
141#835C2318yes—
142#832C2318yes—
143#810C2318yes—
144#810C2318yes—
145#808C2318yes—
146#807C2318yes—
147#803C2314yes—
148#803C2314yes—
149C4×C3⋊C8C2318yes—
150C22×Q16—32no—
151C22×SD16—32no—
152C22×M16—32no—
153#533—38no—
154#533—64no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8—31no—
157#529—24no—
158#529C2×D318yes—
159C2×#126—21no—
160C2×#125—61no—
161C2×#125C2×D311yescharacteristic
162C2×#124C2×D311yescharacteristic
163C2×#123—21no—
164C2×C22⋊Q8—32no—
165C2×C22⋊Q8—61no—
166C2×C22⋊Q8C2×D311yescharacteristic
167C2×#121C2×D311yescharacteristic
168C2×C4×Q8—32no—
169C2×C4×Q8—61no—
170C2×C4×Q8C2×D311yescharacteristic
171C2×C4×D4—21no—
172C2×C4×D4C2×D311yescharacteristic
173C2×#117—21no—
174C2×#117—61no—
175C22×C42C2×D311yescharacteristic
176#501—38no—
177#497—38no—
178#491—38no—
179#489—38no—
180#470—316no—
181#468—316no—
182C4×Q16—316no—
183C4×SD16—316no—
184C2×#107—32no—
185C2×#106—32no—
186C2×#103—32no—
187C2×#103—32no—
188C2×#103—32no—
189C2×#102—32no—
190C4×M16—38no—
191C2×#97—31no—
192C2×#97—31no—
193C2×C4×C8—32no—
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C2×C3×Q8—21no—
197C2×C3×Q8—24no—
198C2×C3×Q8C2412yes—
199C2×C3×D4—24no—
200C2×C3×D4—21no—
201C2×C3×D4—24no—
202C2×C3×D4—44no—
203C2×C3×D4C2412yes—
204C22×C3×C4—24no—
205C22×C3×C4—24no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—42no—
211C22×C3×C4—44no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D414yes—
218C22×C3×C4C2×D414yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2412yes—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—22no—
223C22×Dic3—22no—
224C2×Dic6—24no—
225C2×Dic6—24no—
226C2×Dic6—22no—
227C2×Dic6—24no—
228C2×Dic6—44no—
229C2×Dic6C2414yes—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4C2412yes—
237C3×C4⋊C4C2412yes—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—24no—
243C3×C42—24no—
244C3×C42C2×D414yes—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247C3×C42C2×D414yes—
248C3×C42C2412yes—
249C3×C42C2412yes—
250#217—28no—
251#216—216no—
252#214—216no—
253C4⋊Dic3—22no—
254C4⋊Dic3—24no—
255C4⋊Dic3—24no—
256C4⋊Dic3—24no—
257C4⋊Dic3C2414yes—
258Dic3⋊C4—28no—
259Dic3⋊C4—28no—
260C4×Dic3—28no—
261#208—216no—
262C2×C3⋊C8—24no—
263C2×C3⋊C8#4118yes—
264C2×C3⋊C8C2414yes—
265C2×C3⋊C8C2414yes—
266C22×Q8—32no—
267C22×Q8—61no—
268C22×Q8C22×D311yescharacteristic
269C22×D4—21no—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4#6711yescharacteristic
273C23×C4#6711yescharacteristic
274C23×C4C22×D311yescharacteristic
275C2×Q16—38no—
276C2×Q16—68no—
277C2×SD16—38no—
278C2×SD16—68no—
279C2×M16—38no—
280C2×M16—68no—
281C22×C8—32no—
282C22×C8—32no—
283C4⋊Q8—34no—
284C4⋊Q8—34no—
285C4⋊Q8—62no—
286C4⋊Q8—62no—
287#126—28no—
288#125—22no—
289#125—68no—
290#125C22×D314yes—
291#124—22no—
292#124C22×D314yes—
293#123—28no—
294C22⋊Q8—22no—
295C22⋊Q8—38no—
296C22⋊Q8—64no—
297C22⋊Q8—68no—
298C22⋊Q8C22×D314yes—
299#121—22no—
300#121C22×D314yes—
301C4×Q8—22no—
302C4×Q8—38no—
303C4×Q8—64no—
304C4×Q8—68no—
305C4×Q8C22×D314yes—
306C4×D4—28no—
307C4×D4—22no—
308C4×D4C22×D314yes—
309#117—28no—
310#117—68no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—62no—
319C2×C4⋊C4—62no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—62no—
329C2×C42—21no—
330C2×C42—62no—
331C2×C42#6711yescharacteristic
332C2×C42#6714yes—
333C2×C42#6711yescharacteristic
334C2×C42C22×D311yescharacteristic
335C2×C42C22×D314yes—
336C2×C42C22×D311yescharacteristic
337#107—38no—
338#106—38no—
339#103—38no—
340#103—38no—
341#103—38no—
342#102—38no—
343#97—34no—
344#97—34no—
345C4×C8—38no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3—22no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—42no—
353C23×C3C22×D412yes—
354C23×C3C22×D411yescharacteristic
355C3×Q8—24no—
356C3×Q8—42no—
357C3×D4—24no—
358C3×D4—48no—
359C3×D4—42no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—28no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—44no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—48no—
375C2×C3×C4#13714yes—
376C2×C3×C4C22×D412yes—
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D412yes—
382C2×C3×C4C2×#4114yes—
383C2×C3×C4C2511yescharacteristic
384C2×Dic3—24no—
385C2×Dic3—24no—
386C2×Dic3—44no—
387C2×Dic3—44no—
388Dic6—24no—
389Dic6—24no—
390Dic6—48no—
391Dic6—44no—
392C3⋊C8—24no—
393C3⋊C8C2×#4118yes—
394C24—21no—
395C24C2×#6711yescharacteristic
396C2×Q8—21no—
397C2×Q8—24no—
398C2×Q8—34no—
399C2×Q8—64no—
400C2×Q8—62no—
401C2×Q8—64no—
402C2×Q8—64no—
403C2×Q8—124no—
404C2×Q8C23×D312yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—44no—
409C2×D4C23×D312yes—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6712yes—
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C2×#6714yes—
426C22×C4C2×#6714yes—
427C22×C4C2×#6711yescharacteristic
428C22×C4C23×D312yes—
429C22×C4C23×D311yescharacteristic
430Q16—616no—
431SD16—616no—
432M16—616no—
433C2×C8—38no—
434C2×C8—34no—
435C2×C8—34no—
436C2×C8—64no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—34no—
444C4⋊C4—62no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—68no—
448C4⋊C4—68no—
449C4⋊C4—64no—
450C4⋊C4C23×D312yes—
451C4⋊C4C23×D312yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—68no—
458C42—24no—
459C42—68no—
460C42C2×#6714yes—
461C42C2×#6712yes—
462C42C2×#6712yes—
463C42C2×#6714yes—
464C42C23×D312yes—
465C42C23×D312yes—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—44no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—84no—
474C22×C3#53212yes—
475C22×C3C2×#13712yes—
476C22×C3#57512yes—
477C22×C3C23×D411yescharacteristic
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4#53212yes—
486C3×C4C2×#13714yes—
487C3×C4C22×#4112yes—
488Dic3—44no—
489Dic3—44no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23C22×#6712yes—
498C23C22×#6711yescharacteristic
499Q8—24no—
500Q8—42no—
501Q8—64no—
502Q8—64no—
503Q8—128no—
504Q8—124no—
505D4—24no—
506D4—48no—
507D4—42no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—124no—
526C2×C4—124no—
527C2×C4#95114yes—
528C2×C4D3×#4114yes—
529C2×C4C22×#6712yes—
530C2×C4C22×#6712yes—
531C2×C4C22×#6712yes—
532C2×C4C22×#6712yes—
533C2×C4C22×#6712yes—
534C2×C4C22×#6711yescharacteristic
535C2×C4C24×D311yescharacteristic
536C8—38no—
537C8—64no—
538C2×C3—42no—
539C2×C3—82no—
540C2×C3#317214yescomplemented
541C2×C3C2×#53211yescharacteristic
542C2×C3C22×#13711yescharacteristic
543C2×C3C2×#57511yescharacteristic
544C22—22no—
545C22—22no—
546C22—22no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—84no—
552C22#582212yes—
553C22C2×#95112yes—
554C22#583212yes—
555C22C23×#6711yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—124no—
564C4—124no—
565C4C2×#95114yes—
566C4#583212yes—
567C4C2×D3×#4112yes—
568C3C2×#317211yescharacteristic, complemented, Sylow
569C2—42no—
570C2—82no—
571C2#8313614yesdirect product factor
572C2C2×#582211yescharacteristic
573C2C22×#95111yescharacteristic
574C2C2×#583211yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#529D3
hypocenterC3C2×#3172
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#5470C2semidirect
3. #83136C2direct (N × H)
4. C2×#5064C2semidirect
5. C2×#5069C2semidirect
6. C2×#5060C2semidirect
7. C2×#5055C2semidirect
8. C2×#5052C2semidirect
10. C2×C4×#216C2semidirect
11. C2×C4×#214C2semidirect
12. C2×C4×#208C2semidirect
34. #5470C22semidirect
42. #5069C22semidirect
43. #5064C22semidirect
44. #5060C22semidirect
45. #5055C22semidirect
46. #5052C22semidirect
48. C4×#216C22semidirect
49. C4×#214C22semidirect
54. C4×#208C22semidirect
540. C2×C3#3172semidirect
568. C3C2×#3172semidirect
571. C2#83136direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#54702
#831362
C2×#50642
C2×#50692
C2×#50602
C2×#50552
C2×#50522
C2×#50422
C2×C4×#2162
C2×C4×#2142
C2×C4×#2082
C2×#31723