Finite Group Databasebeta

← C2×#86750 #11703458

Subgroups — C2×#86750 #11703458

Order 768.

7118 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#5817C212yesmaximal, complemented
3C2×#5812C212yesmaximal, complemented
4#86750C2132yesmaximal, direct product factor
5C2×#5732C213yesmaximal, complemented
6C2×#5728C213yesmaximal, complemented
7C2×#5641C216yesmaximal, complemented
8C2×#5643C216yesmaximal, complemented
9C2×Q8×Dic6C211yesmaximal, characteristic, complemented
10C2×C4×#239C211yesmaximal, characteristic, complemented
11C2×#5557C213yesmaximal, complemented
12C2×#5563C213yesmaximal, complemented
13C2×#3518—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C22×#239C2211yescharacteristic
16C3×D4×Q8C22116yescomplemented
17C2×C3×C4⋊Q8C2213yes—
18C2×C3×C22⋊Q8C2216yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2213yes—
21#5817C22132yescomplemented
22C2×#947C2216yes—
23C2×#946C2216yes—
24#5812C22132yescomplemented
25C2×Q8×Dic3C2211yescharacteristic
26C2×Q8×Dic3C2212yes—
27C2×Q8×Dic3C2216yes—
28C2×Q8×Dic3C2216yes—
29C2×#938C2213yes—
30C2×#936C2211yescharacteristic
31C2×#930C2216yes—
32C2×C4×#67C2216yes—
33C2×#926C2216yes—
34C2×#926C2216yes—
35#5732C22148yescomplemented
36#5728C22148yescomplemented
37#5643C22196yescomplemented
38#5641C22196yescomplemented
39C4×#239C22116yescomplemented
40Q8×Dic6C22116yescomplemented
41#5563C22148yescomplemented
42#5557C22148yescomplemented
43C2×#898C22112yes—
44C2×#897C2213yes—
45C2×#897C2216yes—
46C2×#893C2213yes—
47C2×#892C2216yes—
48C2×#892C2213yes—
49C2×#890C2213yes—
50C2×#888C2216yes—
51C2×#887C2216yes—
52C2×#885C22112yes—
53C2×#883C2216yes—
54C2×#879C2216yes—
55C2×C4×D12C2213yes—
56C2×C42×D3C2213yes—
57C2×#869C2213yes—
58C2×C4×Dic6C2213yes—
59#3518—332no—
60C2×Q82—31no—
61C2×#556—33no—
62C2×#556—36no—
63C2×#555—33no—
64C2×D4×Q8D311yescharacteristic, complemented
65C2×#548—32no—
66C2×#542—36no—
67C2×#533—33no—
68C2×#529—33no—
69C2×#519—32no—
70C2×C4×#41—31no—
71C22×C3×Q8—216no—
72C22×C3×Q8C2316yes—
73C22×C3×Q8C23116yes—
74C22×C3×Q8C2311yescharacteristic
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4C2316yes—
77C22×#67C2316yes—
78C2×#239—28no—
79C2×#239C2318yes—
80C22×D12C2313yes—
81C22×C4×D3C2313yes—
82C22×C4×D3C2313yes—
83C22×C4×D3C2313yes—
84C22×Dic6C2311yescharacteristic
85C22×Dic6C2316yes—
86C3×C4⋊Q8C23124yes—
87C3×C22⋊Q8C23148yes—
88C3×C4×Q8C2318yes—
89C3×C4×D4C23124yes—
90C2×C3×C4⋊C4C2316yes—
91C2×C3×C4⋊C4C2313yes—
92C2×C3×C4⋊C4C2313yes—
93C2×C3×C22⋊C4C2316yes—
94C2×C3×C42C2313yes—
95C2×#217C2312yes—
96C2×#217C2316yes—
97C2×#217C2312yes—
98#947C23148yes—
99#946C23148yes—
100Q8×Dic3C23148yes—
101Q8×Dic3C23148yes—
102Q8×Dic3C2318yes—
103Q8×Dic3C23116yes—
104#938C23124yes—
105#936C2318yes—
106#930C23148yes—
107C4×#67C23148yes—
108C2×(C2×D3)⋊C4C2316yes—
109C2×(C2×D3)⋊C4C2316yes—
110C2×(C2×D3)⋊C4C23112yes—
111#926C23148yes—
112#926C23148yes—
113C2×C4⋊Dic3C2313yes—
114C2×C4⋊Dic3C2316yes—
115C2×C4⋊Dic3C2313yes—
116C2×C4⋊Dic3C2316yes—
117C2×C4⋊Dic3C2316yes—
118C2×Dic3⋊C4C2316yes—
119C2×Dic3⋊C4C2316yes—
120C2×C4×Dic3C2316yes—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2313yes—
123C2×C4×Dic3C2313yes—
124C2×C4×Dic3C23112yes—
125#898C23196yes—
126#897C23148yes—
127#897C23124yes—
128#893C23124yes—
129#892C23124yes—
130#892C23148yes—
131#890C23124yes—
132#888C23148yes—
133#887C23148yes—
134#885C23196yes—
135#883C23148yes—
136#879C23148yes—
137C4×D12C23124yes—
138C42×D3C23124yes—
139#869C23124yes—
140C4×Dic6C23124yes—
141C22×#41—31no—
142C23×Q8C2×D312yes—
143Q82—316no—
144#556—348no—
145#556—396no—
146#555—348no—
147D4×Q8C2×D3116yescomplemented
148#548—332no—
149#542—396no—
150#533—348no—
151C2×C4⋊Q8—36no—
152C2×C4⋊Q8—33no—
153C2×C4⋊Q8C2×D313yes—
154#529—348no—
155C2×#126—312no—
156C2×#126—36no—
157C2×#125—33no—
158C2×#124—36no—
159C2×#124—31no—
160C2×#124—312no—
161C2×#123—36no—
162C2×C22⋊Q8—36no—
163C2×C22⋊Q8—33no—
164C2×C22⋊Q8—36no—
165C2×C22⋊Q8C2×D316yes—
166C2×#121—33no—
167C2×#121—36no—
168#519—332no—
169C4×#41—316no—
170C2×C4×Q8—31no—
171C2×C4×Q8—36no—
172C2×C4×Q8—32no—
173C2×C4×Q8—36no—
174C2×C4×Q8—36no—
175C2×C4×Q8—33no—
176C2×C4×Q8C2×D311yescharacteristic
177C2×C4×D4—36no—
178C2×C4×D4—33no—
179C2×C4×D4—33no—
180C2×C4×D4C2×D313yes—
181C2×#117—36no—
182C2×#117—36no—
183C2×#117—33no—
184C2×#117—36no—
185C22×C42—33no—
186C24×C3C2412yes—
187C23×D3C2413yes—
188C2×C3×Q8—264no—
189C2×C3×Q8—224no—
190C2×C3×Q8#4118yes—
191C2×C3×Q8C24124yes—
192C2×C3×Q8C2414yes—
193C2×C3×D4—24no—
194C2×C3×D4C2414yes—
195C22×C3×C4—224no—
196C22×C3×C4C2416yes—
197C22×C3×C4C2413yes—
198C22×C3×C4C24124yes—
199C22×C3×C4C2413yes—
200C22×C3×C4C2411yescharacteristic
201C2×#67—412no—
202C2×#67C24124yes—
203C22×Dic3C2413yes—
204C22×Dic3C2411yescharacteristic
205C22×Dic3C2411yescharacteristic
206C22×Dic3C2412yes—
207C22×Dic3C2416yes—
208#239—232no—
209C2×D12—212no—
210C2×D12C24112yes—
211C2×C4×D3—212no—
212C2×C4×D3—212no—
213C2×C4×D3—46no—
214C2×C4×D3C24112yes—
215C2×C4×D3C24112yes—
216C2×C4×D3C24112yes—
217C2×Dic6—24no—
218C2×Dic6—412no—
219C2×Dic6C2414yes—
220C2×Dic6C24124yes—
221C3×C4⋊C4C24124yes—
222C3×C4⋊C4C24112yes—
223C3×C4⋊C4C24112yes—
224C3×C22⋊C4C24124yes—
225C3×C42C24112yes—
226#217C2418yes—
227#217C24124yes—
228#217C2418yes—
229(C2×D3)⋊C4C24124yes—
230(C2×D3)⋊C4C24124yes—
231(C2×D3)⋊C4C24148yes—
232C4⋊Dic3C24112yes—
233C4⋊Dic3C24124yes—
234C4⋊Dic3C24112yes—
235C4⋊Dic3C24124yes—
236C4⋊Dic3C24124yes—
237Dic3⋊C4C24124yes—
238Dic3⋊C4C24124yes—
239C4×Dic3C24124yes—
240C4×Dic3C2414yes—
241C4×Dic3C24112yes—
242C4×Dic3C24112yes—
243C4×Dic3C24148yes—
244C2×#41—38no—
245C2×#41—68no—
246C22×Q8—216no—
247C22×Q8—31no—
248C22×Q8—36no—
249C22×Q8C22×D316yes—
250C22×Q8C22×D3116yes—
251C22×Q8C22×D311yescharacteristic
252C22×D4—36no—
253C22×D4—33no—
254C22×D4C22×D311yescharacteristic
255C23×C4—33no—
256C23×C4—33no—
257C23×C4—33no—
258C23×C4C22×D316yes—
259C4⋊Q8—348no—
260C4⋊Q8—324no—
261C4⋊Q8C22×D3124yes—
262#126—396no—
263#126—348no—
264#125—324no—
265#124—348no—
266#124—38no—
267#124—396no—
268#123—348no—
269C22⋊Q8—348no—
270C22⋊Q8—324no—
271C22⋊Q8—348no—
272C22⋊Q8C22×D3148yes—
273#121—324no—
274#121—348no—
275C4×Q8—38no—
276C4×Q8—348no—
277C4×Q8—316no—
278C4×Q8—348no—
279C4×Q8—348no—
280C4×Q8—324no—
281C4×Q8C22×D318yes—
282C4×D4—348no—
283C4×D4—324no—
284C4×D4—324no—
285C4×D4C22×D3124yes—
286#117—348no—
287#117—348no—
288#117—324no—
289#117—348no—
290C2×C4⋊C4—33no—
291C2×C4⋊C4—33no—
292C2×C4⋊C4—36no—
293C2×C4⋊C4—36no—
294C2×C4⋊C4—36no—
295C2×C4⋊C4—36no—
296C2×C4⋊C4—36no—
297C2×C4⋊C4C22×D313yes—
298C2×C4⋊C4C22×D313yes—
299C2×C4⋊C4C22×D316yes—
300C2×C22⋊C4—36no—
301C2×C22⋊C4—32no—
302C2×C22⋊C4—32no—
303C2×C22⋊C4—36no—
304C2×C22⋊C4—312no—
305C2×C22⋊C4—36no—
306C2×C22⋊C4C22×D316yes—
307C2×C42—33no—
308C2×C42—312no—
309C2×C42—33no—
310C2×C42—36no—
311C2×C42—31no—
312C2×C42—324no—
313C2×C42C22×D313yes—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3C2514yes—
318C23×C3C2511yescharacteristic
319C22×D3—26no—
320C22×D3—26no—
321C22×D3—26no—
322C22×D3C2516yes—
323C3×Q8—248no—
324C3×Q8C2×#41116yes—
325C3×D4—28no—
326C2×C3×C4—212no—
327C2×C3×C4—26no—
328C2×C3×C4—248no—
329C2×C3×C4C2×#41112yes—
330C2×C3×C42−1+414yes—
331C2×C3×C4C25112yes—
332C2×C3×C4C2516yes—
333C2×C3×C4C2516yes—
334C2×C3×C4C2512yes—
335#67—424no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—26no—
339C2×Dic3—212no—
340C2×Dic3—22no—
341C2×Dic3C2516yes—
342C2×Dic3C2512yes—
343C2×Dic3C2512yes—
344C2×Dic3C2514yes—
345C2×Dic3C25112yes—
346D12—224no—
347C4×D3—224no—
348C4×D3—224no—
349C4×D3—412no—
350Dic6—28no—
351Dic6—424no—
352C24—33no—
353C24C23×D312yes—
354#41—632no—
355C2×Q8—264no—
356C2×Q8—224no—
357C2×Q8—34no—
358C2×Q8—324no—
359C2×Q8—64no—
360C2×Q8—1212no—
361C2×Q8#23718yes—
362C2×Q8C23×D3124yes—
363C2×Q8C23×D314yes—
364C2×D4—24no—
365C2×D4—324no—
366C2×D4—312no—
367C2×D4—612no—
368C2×D4—1212no—
369C2×D4C23×D314yes—
370C22×C4—224no—
371C22×C4—33no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—32no—
375C22×C4—36no—
376C22×C4—312no—
377C22×C4—312no—
378C22×C4—312no—
379C22×C4—612no—
380C22×C4—612no—
381C22×C4—126no—
382C22×C4C23×D3124yes—
383C22×C4C23×D316yes—
384C22×C4C23×D313yes—
385C22×C4C23×D313yes—
386C22×C4C23×D311yescharacteristic
387C4⋊C4—312no—
388C4⋊C4—312no—
389C4⋊C4—324no—
390C4⋊C4—324no—
391C4⋊C4—324no—
392C4⋊C4—324no—
393C4⋊C4—324no—
394C4⋊C4C23×D3112yes—
395C4⋊C4C23×D3112yes—
396C4⋊C4C23×D3124yes—
397C22⋊C4—324no—
398C22⋊C4—38no—
399C22⋊C4—38no—
400C22⋊C4—324no—
401C22⋊C4—348no—
402C22⋊C4—324no—
403C22⋊C4C23×D3124yes—
404C42—34no—
405C42—312no—
406C42—312no—
407C42—348no—
408C42—324no—
409C42C23×D3112yes—
410C22×C3—24no—
411C22×C3—24no—
412C22×C3—24no—
413C22×C3—48no—
414C22×C3C22×#4112yes—
415C22×C3C2×2−1+412yes—
416C22×C3#58512yes—
417C22×C3C2611yescharacteristic
418C2×D3—26no—
419C2×D3—26no—
420C2×D3—26no—
421C2×D3—412no—
422C3×C4—212no—
423C3×C4—26no—
424C3×C4C22×#41112yes—
425C3×C4C2×2−1+414yes—
426Dic3—22no—
427Dic3—24no—
428Dic3—26no—
429Dic3—212no—
430Dic3—22no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—36no—
435C23—66no—
436C23—66no—
437C23—66no—
438C23C24×D314yes—
439C23C24×D311yescharacteristic
440Q8—248no—
441Q8—68no—
442Q8—1224no—
443Q8C2×#237116yes—
444D4—28no—
445D4—624no—
446D4—1224no—
447C2×C4—212no—
448C2×C4—26no—
449C2×C4—248no—
450C2×C4—36no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—34no—
454C2×C4—312no—
455C2×C4—62no—
456C2×C4—66no—
457C2×C4—64no—
458C2×C4—62no—
459C2×C4—612no—
460C2×C4—624no—
461C2×C4—624no—
462C2×C4—1212no—
463C2×C4C2×#237112yes—
464C2×C4#100714yes—
465C2×C4C24×D3112yes—
466C2×C4C24×D316yes—
467C2×C4C24×D316yes—
468C2×C4C24×D312yes—
469C2×C3—44no—
470C2×C3#351814yescomplemented
471C2×C3C23×#4111yescharacteristic
472C2×C3C22×2−1+411yescharacteristic
473C2×C3C2×#58511yescharacteristic
474D3—46no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—48no—
479C22—66no—
480C22—66no—
481C22—66no—
482C22—1212no—
483C22C22×#23712yes—
484C22C2×#100712yes—
485C22#596812yes—
486C22C25×D311yescharacteristic
487C4—212no—
488C4—26no—
489C4—62no—
490C4—66no—
491C4—64no—
492C4—62no—
493C4—612no—
494C4C22×#237112yes—
495C4C2×#100714yes—
496C3C2×#351811yescharacteristic, complemented, Sylow
497C2—44no—
498C2—126no—
499C2#8675014yesdirect product factor
500C2C23×#23711yescharacteristic
501C2C22×#100711yescharacteristic
502C2C2×#596811yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C25
hypercenterC2×D4×Q8D3
hypocenterC3C2×#3518
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C2×#5817C2semidirect
3. C2×#5812C2semidirect
4. #86750C2direct (N × H)
5. C2×#5732C2semidirect
6. C2×#5728C2semidirect
7. C2×#5641C2semidirect
8. C2×#5643C2semidirect
9. C2×Q8×Dic6C2semidirect
10. C2×C4×#239C2semidirect
11. C2×#5557C2semidirect
12. C2×#5563C2semidirect
16. C3×D4×Q8C22semidirect
21. #5817C22semidirect
24. #5812C22semidirect
35. #5732C22semidirect
36. #5728C22semidirect
37. #5643C22semidirect
38. #5641C22semidirect
39. C4×#239C22semidirect
40. Q8×Dic6C22semidirect
41. #5563C22semidirect
42. #5557C22semidirect
64. C2×D4×Q8D3semidirect
147. D4×Q8C2×D3semidirect
470. C2×C3#3518semidirect
496. C3C2×#3518semidirect
499. C2#86750direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×#58172
C2×#58122
#867502
C2×#57322
C2×#57282
C2×#56412
C2×#56432
C2×Q8×Dic62
C2×C4×#2392
C2×#55572
C2×#55632
C2×#35183