Finite Group Databasebeta

← C2×#81728 #11664963

Subgroups — C2×#81728 #11664963

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×#5728C211yesmaximal, characteristic, complemented
2C2×#5691C211yesmaximal, characteristic
3C2×C3×#491C211yesmaximal, characteristic, complemented
4C2×#5086C211yesmaximal, characteristic, complemented
5C2×#5064C211yesmaximal, characteristic, complemented
6#81728C2116yesmaximal, direct product factor
7C2×#4816C212yesmaximal, complemented
8C2×#4812C212yesmaximal
9C2×#4790C212yesmaximal, complemented
10C2×#4788C212yesmaximal, complemented
11C2×#4714C211yesmaximal, characteristic, complemented
12C2×#4707C211yesmaximal, characteristic, complemented
13C2×#3172—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×#946C2212yes—
17C2×Q8×Dic3C2212yes—
18C22×#216C2212yes—
19C22×#215C2212yes—
20C22×ΓL(1,𝔽25)C2212yes—
21#5728—24no—
22#5728C2218yescomplemented
23#5691—24no—
24#5691C2218yes—
25C2×#898—21no—
26C2×#897—21no—
27C2×#897C2211yescharacteristic
28C2×#895C2211yescharacteristic
29C2×#894—21no—
30C2×#893—21no—
31C2×#893C2211yescharacteristic
32C2×#892—21no—
33C2×#892—21no—
34C2×#890—21no—
35C2×#887—21no—
36C2×#887C2211yescharacteristic
37C2×#875C2211yescharacteristic
38C2×C42×D3C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C3×#491C2218yescomplemented
41C2×C3×#103C2212yes—
42C2×C3×#103C2212yes—
43C2×C3×#97C2211yescharacteristic
44C2×#821C2212yes—
45C2×#816C2212yes—
46C2×C8×Dic3C2212yes—
47#5086C2218yescomplemented
48#5064C2218yescomplemented
49C2×#808C2212yes—
50C2×#807C2212yes—
51C2×#803C2211yescharacteristic
52#4816C22116yescomplemented
53#4812C22116yes—
54#4790C22116yescomplemented
55#4788C22116yescomplemented
56#4714C2218yescomplemented
57#4707C2218yescomplemented
58C2×#533—31no—
59C2×#529—31no—
60#3172—316no—
61C2×#501—31no—
62C2×#497—31no—
63C2×#491D311yescharacteristic, complemented
64C2×#489—31no—
65C2×#470—32no—
66C2×#468—32no—
67C2×C4×Q16—32no—
68C2×C4×SD16—32no—
69C2×C4×M16—31no—
70C22×C3×Q8C2312yes—
71C22×D12—21no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3D411yescharacteristic
75C22×C4×D3D411yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×Dic6—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8C2314yes—
82C3×#125—22no—
83C3×#125C2314yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91#946—24no—
92#946C2318yes—
93Q8×Dic3—24no—
94Q8×Dic3C2318yes—
95C2×#216—28no—
96C2×#216C2318yes—
97C2×#215—28no—
98C2×#215C2318yes—
99C2×(C2×D3)⋊C4—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2312yes—
109C2×Dic3⋊C4—22no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—22no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2312yes—
117C2×ΓL(1,𝔽25)—28no—
118C2×ΓL(1,𝔽25)C2318yes—
119#898—28no—
120#897—22no—
121#897—28no—
122#897C2314yes—
123#895—22no—
124#895C2314yes—
125#894—28no—
126#893—28no—
127#893—22no—
128#893C2314yes—
129#892—28no—
130#892—28no—
131#890—28no—
132#887—22no—
133#887—28no—
134#887C2314yes—
135#875—22no—
136#875C2314yes—
137C42×D3D414yes—
138C42×D3C2314yes—
139#869—22no—
140#869C2314yes—
141C3×#103C2318yes—
142C3×#103C2318yes—
143C3×#97C2314yes—
144#821C2318yes—
145#816C2318yes—
146C8×Dic3C2318yes—
147#808C2318yes—
148#807C2318yes—
149#803C2314yes—
150C22×Q16—32no—
151C22×SD16—32no—
152C22×M16—32no—
153#533—38no—
154#533—64no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8C2×D311yescharacteristic
157#529—38no—
158#529—64no—
159C2×#126—61no—
160C2×#125—61no—
161C2×#125C2×D311yescharacteristic
162C2×#124—31no—
163C2×#123—61no—
164C2×C22⋊Q8—32no—
165C2×C22⋊Q8—31no—
166C2×C22⋊Q8—61no—
167C2×#121—31no—
168C2×C4×Q8—32no—
169C2×C4×Q8—31no—
170C2×C4×Q8—61no—
171C2×C4×D4—31no—
172C2×C4×D4—61no—
173C2×#117—61no—
174C2×#117—61no—
175C22×C42—31no—
176#501—38no—
177#497—38no—
178#491C2×D318yescomplemented
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×#103C2×D312yes—
188C2×#103C2×D312yes—
189C2×#102—32no—
190C4×M16—38no—
191C2×#97—31no—
192C2×#97C2×D311yescharacteristic
193C2×C4×C8—32no—
194C23×D3—21no—
195C23×D3C2×D411yescharacteristic
196C2×C3×Q8—22no—
197C2×C3×Q8—24no—
198C2×C3×Q8—24no—
199C2×C3×Q8C2414yes—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—22no—
207C22×Dic3—21no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C2×D12—24no—
212C2×D12—24no—
213C2×D12—21no—
214C2×D12—44no—
215C2×D12C2412yes—
216C2×C4×D3—24no—
217C2×C4×D3—24no—
218C2×C4×D3—24no—
219C2×C4×D3—42no—
220C2×C4×D3—44no—
221C2×C4×D3C2×D412yes—
222C2×C4×D3C2×D414yes—
223C2×C4×D3C2×D414yes—
224C2×C4×D3C2412yes—
225C2×Dic6—24no—
226C2×Dic6—24no—
227C2×Dic6—21no—
228C2×Dic6—44no—
229C2×Dic6C2412yes—
230C2×C3×C8—24no—
231C2×C3×C8C2414yes—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4C2412yes—
239C3×C4⋊C4C2412yes—
240C3×C42C2×D412yes—
241C3×C42C2412yes—
242#216—216no—
243#215—216no—
244(C2×D3)⋊C4—28no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—24no—
250C4⋊Dic3—22no—
251C4⋊Dic3—24no—
252C4⋊Dic3—28no—
253C4⋊Dic3—24no—
254C4⋊Dic3C2414yes—
255Dic3⋊C4—28no—
256Dic3⋊C4—24no—
257C4×Dic3—28no—
258C4×Dic3—24no—
259C4×Dic3C2×D412yes—
260C4×Dic3C2×D414yes—
261C4×Dic3C2×D414yes—
262C4×Dic3C2412yes—
263C2×C3⋊C8#4118yes—
264C2×C3⋊C8C2414yes—
265ΓL(1,𝔽25)—216no—
266C22×Q8—31no—
267C22×Q8—61no—
268C22×Q8C22×D312yes—
269C22×D4—31no—
270C22×D4—61no—
271C23×C4—31no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—61no—
275C2×Q16—38no—
276C2×Q16—68no—
277C2×SD16—38no—
278C2×SD16—68no—
279C2×M16—38no—
280C2×M16—68no—
281C22×C8—32no—
282C22×C8C22×D312yes—
283C4⋊Q8—22no—
284C4⋊Q8—34no—
285C4⋊Q8—62no—
286C4⋊Q8C22×D314yes—
287#126—68no—
288#125—22no—
289#125—68no—
290#125C22×D314yes—
291#124—34no—
292#124—62no—
293#123—68no—
294C22⋊Q8—38no—
295C22⋊Q8—34no—
296C22⋊Q8—62no—
297C22⋊Q8—68no—
298C22⋊Q8—64no—
299#121—34no—
300#121—62no—
301C4×Q8—38no—
302C4×Q8—34no—
303C4×Q8—62no—
304C4×Q8—68no—
305C4×Q8—64no—
306C4×D4—34no—
307C4×D4—62no—
308C4×D4—68no—
309#117—68no—
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—62no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—62no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—62no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—34no—
333C2×C42—34no—
334C2×C42—61no—
335C2×C42—62no—
336C2×C42C22×D311yescharacteristic
337#107—38no—
338#106—38no—
339#103—38no—
340#103C22×D318yes—
341#103C22×D318yes—
342#102—38no—
343#97—34no—
344#97C22×D314yes—
345C4×C8—38no—
346C23×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—22no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3—42no—
354C22×D3C22×D412yes—
355C3×Q8—24no—
356C3×Q8—24no—
357C3×Q8—44no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—44no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4C22×D412yes—
367C2×C3×C4C22×D412yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4114yes—
370C2×C3×C4C2511yescharacteristic
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—24no—
374C2×Dic3—22no—
375C2×Dic3—42no—
376C2×Dic3—44no—
377C2×Dic3#13714yes—
378C2×Dic3C22×D412yes—
379C2×Dic3C22×D412yes—
380C2×Dic3C22×D412yes—
381D12—24no—
382D12—48no—
383D12—42no—
384C4×D3—24no—
385C4×D3—24no—
386C4×D3—28no—
387C4×D3—44no—
388C4×D3—48no—
389Dic6—24no—
390Dic6—48no—
391Dic6—42no—
392C3×C8—24no—
393C3⋊C8C2×#4118yes—
394C24—31no—
395C24—61no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—32no—
400C2×Q8—61no—
401C2×Q8—64no—
402C2×Q8—64no—
403C2×Q8—124no—
404C2×Q8C23×D314yes—
405C2×D4—32no—
406C2×D4—64no—
407C2×D4—61no—
408C2×D4—64no—
409C2×D4—124no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—34no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—34no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—64no—
423C22×C4—61no—
424C22×C4—62no—
425C22×C4—122no—
426C22×C4—124no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430Q16—616no—
431SD16—616no—
432M16—616no—
433C2×C8—24no—
434C2×C8—38no—
435C2×C8—34no—
436C2×C8C23×D314yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—34no—
444C4⋊C4—64no—
445C4⋊C4—68no—
446C4⋊C4—64no—
447C4⋊C4—68no—
448C4⋊C4—62no—
449C4⋊C4—64no—
450C4⋊C4C23×D312yes—
451C4⋊C4C23×D312yes—
452C22⋊C4—64no—
453C22⋊C4—64no—
454C22⋊C4—68no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C42—34no—
459C42—32no—
460C42—34no—
461C42—32no—
462C42—64no—
463C42—68no—
464C42D3×D412yes—
465C42C23×D312yes—
466C22×C3#53212yes—
467C22×C3C2×#13712yes—
468C22×C3#57512yes—
469C22×C3C23×D411yescharacteristic
470C2×D3—22no—
471C2×D3—22no—
472C2×D3—22no—
473C2×D3—44no—
474C2×D3—42no—
475C2×D3—42no—
476C2×D3—42no—
477C2×D3—84no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—44no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4#53212yes—
484C3×C4C22×#4112yes—
485Dic3—22no—
486Dic3—22no—
487Dic3—42no—
488Dic3—44no—
489Dic3C2×#13714yes—
490C23—32no—
491C23—62no—
492C23—62no—
493C23—62no—
494C23—62no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23C2×D3×D411yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—44no—
502Q8—64no—
503Q8—128no—
504Q8—122no—
505D4—64no—
506D4—128no—
507D4—122no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—32no—
514C2×C4—34no—
515C2×C4—32no—
516C2×C4—32no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—68no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—64no—
527C2×C4—124no—
528C2×C4—128no—
529C2×C4—122no—
530C2×C4—124no—
531C2×C4C2×D3×D412yes—
532C2×C4C2×D3×D412yes—
533C2×C4C2×D3×D411yescharacteristic
534C2×C4D3×#4114yes—
535C2×C4C24×D311yescharacteristic
536C8—24no—
537C8—38no—
538C2×C3#317214yescomplemented
539C2×C3C2×#53211yescharacteristic
540C2×C3C22×#13711yescharacteristic
541C2×C3C2×#57511yescharacteristic
542D3—42no—
543D3—82no—
544C22—62no—
545C22—62no—
546C22—62no—
547C22—124no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—244no—
552C22#566812yes—
553C22D3×#13712yes—
554C22#577712yes—
555C22C22×D3×D411yescharacteristic
556C4—22no—
557C4—22no—
558C4—34no—
559C4—44no—
560C4—42no—
561C4—42no—
562C4—62no—
563C4—62no—
564C4—122no—
565C4—124no—
566C4#566812yes—
567C4C2×D3×#4112yes—
568C3C2×#317211yescharacteristic, complemented, Sylow
569C2—122no—
570C2—242no—
571C2#8172814yesdirect product factor
572C2C2×#566811yescharacteristic
573C2C2×D3×#13711yescharacteristic
574C2C2×#577711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#491C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenterC2×#491D3
hypocenterC3C2×#3172
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
3. C2×C3×#491C2semidirect
4. C2×#5086C2semidirect
5. C2×#5064C2semidirect
6. #81728C2direct (N × H)
7. C2×#4816C2semidirect
9. C2×#4790C2semidirect
10. C2×#4788C2semidirect
11. C2×#4714C2semidirect
12. C2×#4707C2semidirect
22. #5728C22semidirect
40. C3×#491C22semidirect
47. #5086C22semidirect
48. #5064C22semidirect
52. #4816C22semidirect
54. #4790C22semidirect
55. #4788C22semidirect
56. #4714C22semidirect
57. #4707C22semidirect
63. C2×#491D3semidirect
178. #491C2×D3semidirect
538. C2×C3#3172semidirect
568. C3C2×#3172semidirect
571. C2#81728direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#56912
C2×C3×#4912
C2×#50862
C2×#50642
#817282
C2×#48162
C2×#48122
C2×#47902
C2×#47882
C2×#47142
C2×#47072
C2×#31723