Finite Group Databasebeta

← C2×#86746 #11703599

Subgroups — C2×#86746 #11703599

Order 768.

11126 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1007C212yesmaximal, complemented
2C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
3C2×Q8×#67C212yesmaximal, complemented
4#86746C2132yesmaximal, direct product factor
5C2×#5726C213yesmaximal, complemented
6C2×D3×C4⋊Q8C213yesmaximal, complemented
7C2×#5631C216yesmaximal, complemented
8C2×#5628C216yesmaximal, complemented
9C2×Q8×D12C211yesmaximal, characteristic, complemented
10C2×C4×#239C211yesmaximal, characteristic, complemented
11C2×#5551C213yesmaximal, complemented
12C2×#5548C213yesmaximal, complemented
13C2×#3499—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C2×#1007—232no—
16C2×#1007C22132yescomplemented
17C22×#239C2216yes—
18C22×#239C2211yescharacteristic
19C22×D3×Q8C2212yes—
20C22×D3×Q8C2216yes—
21C22×#235C2216yes—
22C22×#235C2216yes—
23C3×D4×Q8C22116yescomplemented
24C2×C3×C4⋊Q8C2213yes—
25C2×C3×C22⋊Q8C2216yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×D4C2213yes—
28Q8×#67C22132yescomplemented
29C2×#946C2216yes—
30C2×Q8×Dic3C2211yescharacteristic
31C2×#944C2216yes—
32C2×#940C2211yescharacteristic
33C2×#938C2213yes—
34C2×C4×#67C2216yes—
35C2×#924C2216yes—
36#5726C22148yescomplemented
37D3×C4⋊Q8C22148yescomplemented
38#5631C22196yescomplemented
39#5628C22196yescomplemented
40Q8×D12C22116yescomplemented
41C4×#239C22116yescomplemented
42#5551C22148yescomplemented
43#5548C22148yescomplemented
44C2×#897C2216yes—
45C2×#895C2213yes—
46C2×#894C22112yes—
47C2×#893C2213yes—
48C2×#892C2213yes—
49C2×D3×C4⋊C4C2216yes—
50C2×D3×C4⋊C4C2216yes—
51C2×#888C2213yes—
52C2×#888C2216yes—
53C2×#885C2216yes—
54C2×#884C2216yes—
55C2×#883C22112yes—
56C2×C4×D12C2213yes—
57C2×C4×D12C2213yes—
58C2×C42×D3C2213yes—
59C2×#869C2213yes—
60C22×2−1+4—32no—
61#3499—332no—
62C2×D4×Q8—32no—
63C2×D4×Q8—31no—
64C2×D4×Q8D311yescharacteristic, complemented
65C2×#547—33no—
66C2×#547—36no—
67C2×#547—33no—
68C2×#536—36no—
69C2×#532—33no—
70C22×C4⋊Q8—33no—
71C2×C4×#41—31no—
72C22×C3×Q8—216no—
73C22×C3×Q8C2316yes—
74C22×C3×Q8C23116yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4C2316yes—
78C22×#67C2316yes—
79C22×#67C2312yes—
80C22×#67C2312yes—
81#1007—2256no—
82C2×#239—248no—
83C2×#239D4116yescomplemented
84C2×#239C23148yes—
85C2×#239C2318yes—
86C2×D3×Q8—216no—
87C2×D3×Q8—248no—
88C2×D3×Q8C23148yes—
89C2×D3×Q8C23116yes—
90C2×#235—248no—
91C2×#235—248no—
92C2×#235C23148yes—
93C2×#235C23148yes—
94C22×D12C2316yes—
95C22×D12C2311yescharacteristic
96C22×D12C2313yes—
97C22×C4×D3C2316yes—
98C22×C4×D3C23112yes—
99C22×C4×D3C2313yes—
100C22×C4×D3C2313yes—
101C22×C4×D3C2313yes—
102C22×Dic6C2316yes—
103C22×Dic6C2316yes—
104C3×C4⋊Q8C23124yes—
105C3×C22⋊Q8C23148yes—
106C3×C4×Q8C2318yes—
107C3×C4×D4C23124yes—
108C2×C3×C4⋊C4C2313yes—
109C2×C3×C4⋊C4C2316yes—
110C2×C3×C4⋊C4C2313yes—
111C2×C3×C22⋊C4C2316yes—
112C2×C3×C42C2313yes—
113C2×#217C2316yes—
114#946C23148yes—
115Q8×Dic3C2318yes—
116#944C23148yes—
117#940C2318yes—
118#938C23124yes—
119C4×#67C23148yes—
120C2×(C2×D3)⋊C4C2316yes—
121C2×(C2×D3)⋊C4C2316yes—
122C2×(C2×D3)⋊C4C2316yes—
123C2×C4⋊Dic3C2313yes—
124C2×C4⋊Dic3C2316yes—
125C2×C4⋊Dic3C2313yes—
126#924C23148yes—
127C2×Dic3⋊C4C23112yes—
128C2×Dic3⋊C4C2316yes—
129C2×C4×Dic3C2311yescharacteristic
130C2×C4×Dic3C2313yes—
131C2×C4×Dic3C2313yes—
132#897C23148yes—
133#895C23124yes—
134#894C23196yes—
135#893C23124yes—
136#892C23124yes—
137D3×C4⋊C4C23148yes—
138D3×C4⋊C4C23148yes—
139#888C23124yes—
140#888C23148yes—
141#885C23148yes—
142#884C23148yes—
143#883C23196yes—
144C4×D12C23124yes—
145C4×D12C23124yes—
146C42×D3C23124yes—
147#869C23124yes—
148C2×2−1+4—332no—
149C2×2−1+4—632no—
150C22×#41—31no—
151C22×#41—36no—
152C22×#41—36no—
153C22×#41—36no—
154C23×Q8—36no—
155C23×Q8—32no—
156C23×Q8C2×D312yes—
157D4×Q8—332no—
158D4×Q8—316no—
159D4×Q8C2×D3116yescomplemented
160#547—348no—
161#547—396no—
162#547—348no—
163#536—396no—
164#532—348no—
165C2×C4⋊Q8—36no—
166C2×C4⋊Q8—348no—
167C2×C4⋊Q8—33no—
168C2×C4⋊Q8—36no—
169C2×C4⋊Q8—33no—
170C2×C4⋊Q8C2×D313yes—
171C2×#127—31no—
172C2×#124—36no—
173C2×#123—312no—
174C2×#123—312no—
175C2×C22⋊Q8—36no—
176C2×C22⋊Q8—36no—
177C2×C22⋊Q8—36no—
178C2×C22⋊Q8C2×D316yes—
179C2×#121—33no—
180C2×#121—33no—
181C2×#121—36no—
182C4×#41—316no—
183C2×C4×Q8—31no—
184C2×C4×Q8C2×D311yescharacteristic
185C2×C4×D4—36no—
186C2×C4×D4—33no—
187C2×C4×D4—33no—
188C2×C4×D4—33no—
189C2×C4×D4C2×D313yes—
190C2×#117—33no—
191C22×C4⋊C4—36no—
192C22×C4⋊C4—36no—
193C22×C42—33no—
194C24×C3C2412yes—
195C23×D3C2412yes—
196C23×D3C2413yes—
197C2×C3×Q8—264no—
198C2×C3×Q8—224no—
199C2×C3×Q8C2×D418yes—
200C2×C3×Q8C24124yes—
201C2×C3×Q8C2414yes—
202C2×C3×D4—24no—
203C2×C3×D4C2414yes—
204C22×C3×C4—224no—
205C22×C3×C4C2413yes—
206C22×C3×C4C2416yes—
207C22×C3×C4C24124yes—
208C22×C3×C4C2413yes—
209C22×C3×C4C2411yescharacteristic
210C2×#67—224no—
211C2×#67—28no—
212C2×#67—28no—
213C2×#67C24124yes—
214C2×#67C2418yes—
215C2×#67C2418yes—
216C22×Dic3C2413yes—
217C22×Dic3C2416yes—
218C22×Dic3C2411yescharacteristic
219C22×Dic3C2411yescharacteristic
220#239—2192no—
221#239C2×D4164yescomplemented
222D3×Q8—264no—
223D3×Q8—2192no—
224#235—2192no—
225#235—2192no—
226C2×D12—224no—
227C2×D12—24no—
228C2×D12C2×D4124yes—
229C2×D12C2414yes—
230C2×D12C24124yes—
231C2×D12C24112yes—
232C2×C4×D3—248no—
233C2×C4×D3—224no—
234C2×C4×D3—212no—
235C2×C4×D3—212no—
236C2×C4×D3C2×D4124yes—
237C2×C4×D3C24112yes—
238C2×C4×D3C24112yes—
239C2×C4×D3C24124yes—
240C2×C4×D3C24148yes—
241C2×C4×D3C24112yes—
242C2×Dic6—224no—
243C2×Dic6—224no—
244C2×Dic6C24124yes—
245C2×Dic6C24124yes—
246C3×C4⋊C4C24112yes—
247C3×C4⋊C4C24124yes—
248C3×C4⋊C4C24112yes—
249C3×C22⋊C4C24124yes—
250C3×C42C24112yes—
251#217C24124yes—
252(C2×D3)⋊C4C24124yes—
253(C2×D3)⋊C4C24124yes—
254(C2×D3)⋊C4C24124yes—
255C4⋊Dic3C24112yes—
256C4⋊Dic3C24124yes—
257C4⋊Dic3C24112yes—
258Dic3⋊C4C24148yes—
259Dic3⋊C4C24124yes—
260C4×Dic3C2414yes—
261C4×Dic3C24112yes—
262C4×Dic3C24112yes—
2632−1+4—6256no—
264C2×#41—316no—
265C2×#41—38no—
266C2×#41—348no—
267C2×#41—348no—
268C2×#41—348no—
269C2×#41—648no—
270C2×#41—648no—
271C2×#41—648no—
272C22×Q8—216no—
273C22×Q8—348no—
274C22×Q8—316no—
275C22×Q8—36no—
276C22×Q8—36no—
277C22×Q8—648no—
278C22×Q8—616no—
279C22×Q8C22×D316yes—
280C22×Q8C22×D3116yes—
281C22×Q8C22×D311yescharacteristic
282C22×D4—32no—
283C22×D4—36no—
284C22×D4—32no—
285C22×D4—33no—
286C22×D4—31no—
287C22×D4—36no—
288C22×D4C22×D311yescharacteristic
289C23×C4—33no—
290C23×C4—33no—
291C23×C4—33no—
292C23×C4—312no—
293C23×C4—36no—
294C23×C4C22×D316yes—
295C4⋊Q8—348no—
296C4⋊Q8—324no—
297C4⋊Q8—348no—
298C4⋊Q8—324no—
299C4⋊Q8C22×D3124yes—
300#127—38no—
301#124—348no—
302#123—396no—
303#123—396no—
304C22⋊Q8—348no—
305C22⋊Q8—348no—
306C22⋊Q8—348no—
307C22⋊Q8C22×D3148yes—
308#121—324no—
309#121—324no—
310#121—348no—
311C4×Q8—38no—
312C4×Q8C22×D318yes—
313C4×D4—348no—
314C4×D4—324no—
315C4×D4—324no—
316C4×D4—324no—
317C4×D4C22×D3124yes—
318#117—324no—
319C2×C4⋊C4—36no—
320C2×C4⋊C4—33no—
321C2×C4⋊C4—312no—
322C2×C4⋊C4—36no—
323C2×C4⋊C4—33no—
324C2×C4⋊C4—348no—
325C2×C4⋊C4—348no—
326C2×C4⋊C4C22×D313yes—
327C2×C4⋊C4C22×D316yes—
328C2×C4⋊C4C22×D313yes—
329C2×C22⋊C4—36no—
330C2×C22⋊C4—36no—
331C2×C22⋊C4—36no—
332C2×C22⋊C4—36no—
333C2×C22⋊C4C22×D316yes—
334C2×C42—33no—
335C2×C42—33no—
336C2×C42—31no—
337C2×C42—324no—
338C2×C42C22×D313yes—
339C23×C3—24no—
340C23×C3—24no—
341C23×C3—24no—
342C23×C3C2514yes—
343C23×C3C2511yescharacteristic
344C22×D3—26no—
345C22×D3—26no—
346C22×D3—24no—
347C22×D3—24no—
348C22×D3—24no—
349C22×D3C22×D4112yes—
350C22×D3C2514yes—
351C22×D3C2516yes—
352C3×Q8—248no—
353C3×Q8C22×D4116yes—
354C3×D4—28no—
355C2×C3×C4—212no—
356C2×C3×C4—26no—
357C2×C3×C4—248no—
358C2×C3×C4C22×D4112yes—
359C2×C3×C42−1+414yescomplemented
360C2×C3×C4C2516yes—
361C2×C3×C4C25112yes—
362C2×C3×C4C2516yes—
363C2×C3×C4C2512yes—
364#67—248no—
365#67—216no—
366#67—216no—
367C2×Dic3—212no—
368C2×Dic3—26no—
369C2×Dic3C22×D414yes—
370C2×Dic32−1+414yescomplemented
371C2×Dic3C2516yes—
372C2×Dic3C25112yes—
373C2×Dic3C2512yes—
374C2×Dic3C2512yes—
375D12—248no—
376D12—28no—
377D12C22×D4148yes—
378C4×D3—296no—
379C4×D3—248no—
380C4×D3—224no—
381C4×D3—224no—
382C4×D3C22×D4148yes—
383Dic6—248no—
384Dic6—248no—
385C24—33no—
386C24—32no—
387C24C23×D312yes—
388#41—364no—
389#41—6192no—
390#41—6192no—
391#41—6192no—
392C2×Q8—264no—
393C2×Q8—224no—
394C2×Q8—324no—
395C2×Q8—324no—
396C2×Q8—624no—
397C2×Q8—6192no—
398C2×Q8—664no—
399C2×Q8—624no—
400C2×Q8D3×D418yes—
401C2×Q8C23×D3124yes—
402C2×Q8C23×D314yes—
403C2×D4—24no—
404C2×D4—38no—
405C2×D4—324no—
406C2×D4—38no—
407C2×D4—324no—
408C2×D4—312no—
409C2×D4—34no—
410C2×D4—324no—
411C2×D4—624no—
412C2×D4—68no—
413C2×D4—68no—
414C2×D4—624no—
415C2×D4—64no—
416C2×D4C23×D314yes—
417C22×C4—224no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—324no—
421C22×C4—312no—
422C22×C4—33no—
423C22×C4—36no—
424C22×C4—312no—
425C22×C4—312no—
426C22×C4—348no—
427C22×C4—324no—
428C22×C4—612no—
429C22×C4—648no—
430C22×C4—624no—
431C22×C4—612no—
432C22×C4C23×D3124yes—
433C22×C4C23×D313yes—
434C22×C4C23×D316yes—
435C22×C4C23×D313yes—
436C22×C4C23×D311yescharacteristic
437C4⋊C4—324no—
438C4⋊C4—312no—
439C4⋊C4—348no—
440C4⋊C4—324no—
441C4⋊C4—312no—
442C4⋊C4C23×D3112yes—
443C4⋊C4C23×D3124yes—
444C4⋊C4C23×D3112yes—
445C22⋊C4—324no—
446C22⋊C4—324no—
447C22⋊C4—324no—
448C22⋊C4—324no—
449C22⋊C4C23×D3124yes—
450C42—34no—
451C42—312no—
452C42—312no—
453C42C23×D3112yes—
454C22×C3—24no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—48no—
458C22×C3C23×D412yes—
459C22×C3C2×2−1+412yes—
460C22×C3C2×2−1+412yes—
461C22×C3C2611yescharacteristic
462C2×D3—26no—
463C2×D3—26no—
464C2×D3—224no—
465C2×D3—24no—
466C2×D3—24no—
467C2×D3—24no—
468C2×D3—48no—
469C2×D3C23×D4112yes—
470C3×C4—212no—
471C3×C4—26no—
472C3×C4C23×D4112yes—
473C3×C4C2×2−1+414yescomplemented
474Dic3—212no—
475Dic3—26no—
476Dic3C23×D414yes—
477Dic3C2×2−1+414yescomplemented
478C23—24no—
479C23—24no—
480C23—24no—
481C23—312no—
482C23—36no—
483C23—34no—
484C23—66no—
485C23—66no—
486C23—64no—
487C23—64no—
488C23—64no—
489C23C24×D314yes—
490C23C24×D311yescharacteristic
491Q8—248no—
492Q8—648no—
493Q8—648no—
494Q8C2×D3×D4116yes—
495D4—28no—
496D4—348no—
497D4—648no—
498D4—616no—
499D4—616no—
500D4—648no—
501D4—68no—
502C2×C4—212no—
503C2×C4—26no—
504C2×C4—248no—
505C2×C4—34no—
506C2×C4—34no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—348no—
510C2×C4—36no—
511C2×C4—312no—
512C2×C4—624no—
513C2×C4—696no—
514C2×C4—612no—
515C2×C4—66no—
516C2×C4—648no—
517C2×C4—624no—
518C2×C4C2×D3×D4112yes—
519C2×C4#100714yescomplemented
520C2×C4C24×D316yes—
521C2×C4C24×D3112yes—
522C2×C4C24×D316yes—
523C2×C4C24×D312yes—
524C2×C3—44no—
525C2×C3#349914yescomplemented
526C2×C3C24×D411yescharacteristic
527C2×C3C22×2−1+411yescharacteristic
528C2×C3C22×2−1+411yescharacteristic
529D3—212no—
530D3—44no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—312no—
535C22—48no—
536C22—66no—
537C22—66no—
538C22—624no—
539C22—64no—
540C22—64no—
541C22—64no—
542C22—128no—
543C22C22×D3×D412yes—
544C22C2×#100712yes—
545C22D3×2−1+412yes—
546C22C25×D311yescharacteristic
547C4—212no—
548C4—26no—
549C4—34no—
550C4—34no—
551C4—612no—
552C4—66no—
553C4C22×D3×D4112yes—
554C4C2×#100714yescomplemented
555C3C2×#349911yescharacteristic, complemented, Sylow
556C2—44no—
557C2—612no—
558C2—124no—
559C2#8674614yesdirect product factor
560C2C23×D3×D411yescharacteristic
561C2C22×#100711yescharacteristic
562C2C2×D3×2−1+411yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C25
hypercenterC2×D4×Q8D3
hypocenterC3C2×#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#1007C2semidirect
2. C2×C3×D4×Q8C2semidirect
3. C2×Q8×#67C2semidirect
4. #86746C2direct (N × H)
5. C2×#5726C2semidirect
6. C2×D3×C4⋊Q8C2semidirect
7. C2×#5631C2semidirect
8. C2×#5628C2semidirect
9. C2×Q8×D12C2semidirect
10. C2×C4×#239C2semidirect
11. C2×#5551C2semidirect
12. C2×#5548C2semidirect
16. C2×#1007C22semidirect
23. C3×D4×Q8C22semidirect
28. Q8×#67C22semidirect
36. #5726C22semidirect
37. D3×C4⋊Q8C22semidirect
38. #5631C22semidirect
39. #5628C22semidirect
40. Q8×D12C22semidirect
41. C4×#239C22semidirect
42. #5551C22semidirect
43. #5548C22semidirect
64. C2×D4×Q8D3semidirect
83. C2×#239D4semidirect
159. D4×Q8C2×D3semidirect
221. #239C2×D4semidirect
359. C2×C3×C42−1+4semidirect
370. C2×Dic32−1+4semidirect
473. C3×C4C2×2−1+4semidirect
477. Dic3C2×2−1+4semidirect
519. C2×C4#1007semidirect
525. C2×C3#3499semidirect
554. C4C2×#1007semidirect
555. C3C2×#3499semidirect
559. C2#86746direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#10072
C2×C3×D4×Q82
C2×Q8×#672
#867462
C2×#57262
C2×D3×C4⋊Q82
C2×#56312
C2×#56282
C2×Q8×D122
C2×C4×#2392
C2×#55512
C2×#55482
C2×#34993