Finite Group Databasebeta

← C2×#86753 #11703411

Subgroups — C2×#86753 #11703411

Order 768.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×Q8×#67C212yesmaximal, complemented
3C2×#5812C212yesmaximal, complemented
4#86753C2132yesmaximal, direct product factor
5C2×#5721C213yesmaximal, complemented
6C2×#5728C213yesmaximal, complemented
7C2×#5639C216yesmaximal, complemented
8C2×#5625C216yesmaximal, complemented
9C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
10C2×#5570C211yesmaximal, characteristic, complemented
11C2×#5557C213yesmaximal
12C2×#5542C213yesmaximal, complemented
13C2×#3523—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C22×D3×Q8C2211yescharacteristic
16C3×D4×Q8C22116yescomplemented
17C2×C3×C4⋊Q8C2213yes—
18C2×C3×C22⋊Q8C2216yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2213yes—
21Q8×#67C22132yescomplemented
22C2×#946C2216yes—
23#5812C22132yescomplemented
24C2×Q8×Dic3C2212yes—
25C2×Q8×Dic3C2211yescharacteristic
26C2×Q8×Dic3C2216yes—
27C2×Q8×Dic3C2216yes—
28C2×#944C2216yes—
29C2×#938C2211yescharacteristic
30C2×#936C2213yes—
31C2×C4×#67C2216yes—
32C2×#926C2216yes—
33C2×#926C2216yes—
34C2×#924C2216yes—
35#5728C22148yescomplemented
36#5721C22148yescomplemented
37#5639C22196yescomplemented
38#5625C22196yescomplemented
39C4×D3×Q8C22116yescomplemented
40#5570C22116yescomplemented
41#5557C22148yes—
42#5542C22148yescomplemented
43C2×#897C2213yes—
44C2×#892C2216yes—
45C2×D3×C4⋊C4C2213yes—
46C2×#890C2213yes—
47C2×#890C2216yes—
48C2×#890C2216yes—
49C2×#889C22112yes—
50C2×#887C2216yes—
51C2×#887C2213yes—
52C2×#883C2216yes—
53C2×#879C22112yes—
54C2×#879C2216yes—
55C2×C42×D3C2213yes—
56C2×#869C2213yes—
57C2×C4×Dic6C2213yes—
58C2×C4×Dic6C2213yes—
59#3523—332no—
60C2×#557—33no—
61C2×#557—31no—
62C2×#556—33no—
63C2×#552—36no—
64C2×D4×Q8—32no—
65C2×D4×Q8D311yescharacteristic, complemented
66C2×#541—36no—
67C2×#533—33no—
68C2×#529—33no—
69C2×#519—32no—
70C22×C4×Q8—31no—
71C22×C3×Q8—216no—
72C22×C3×Q8C2316yes—
73C22×C3×Q8C23116yes—
74C22×C3×Q8C2311yescharacteristic
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4C2316yes—
77C22×#67C2312yes—
78C2×D3×Q8—28no—
79C2×D3×Q8C2318yes—
80C22×C4×D3C2313yes—
81C22×C4×D3C2311yescharacteristic
82C22×C4×D3C2313yes—
83C22×Dic6C2313yes—
84C22×Dic6C2313yes—
85C3×C4⋊Q8C23124yes—
86C3×C22⋊Q8C23148yes—
87C3×C4×Q8C2318yes—
88C3×C4×D4C23124yes—
89C2×C3×C4⋊C4C2313yes—
90C2×C3×C4⋊C4C2313yes—
91C2×C3×C4⋊C4C2316yes—
92C2×C3×C22⋊C4C2316yes—
93C2×C3×C42C2313yes—
94C2×#217C2316yes—
95C2×#217C2312yes—
96C2×#217C2316yes—
97#946C23148yes—
98Q8×Dic3C23148yes—
99Q8×Dic3C23148yes—
100Q8×Dic3C2318yes—
101Q8×Dic3C23116yes—
102#944C23148yes—
103#938C2318yes—
104#936C23124yes—
105C4×#67C23148yes—
106C2×(C2×D3)⋊C4C2316yes—
107#926C23148yes—
108#926C23148yes—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2313yes—
111C2×C4⋊Dic3C2316yes—
112C2×C4⋊Dic3C2316yes—
113C2×C4⋊Dic3C2316yes—
114C2×C4⋊Dic3C2316yes—
115#924C23148yes—
116C2×Dic3⋊C4C2316yes—
117C2×Dic3⋊C4C2316yes—
118C2×Dic3⋊C4C23112yes—
119C2×Dic3⋊C4C2316yes—
120C2×C4×Dic3C2313yes—
121C2×C4×Dic3C2316yes—
122C2×C4×Dic3C2313yes—
123C2×C4×Dic3C2313yes—
124C2×C4×Dic3C23112yes—
125#897C23124yes—
126#892C23148yes—
127D3×C4⋊C4C23124yes—
128#890C23148yes—
129#890C23124yes—
130#890C23148yes—
131#889C23196yes—
132#887C23148yes—
133#887C23124yes—
134#883C23148yes—
135#879C23196yes—
136#879C23148yes—
137C42×D3C23124yes—
138#869C23124yes—
139C4×Dic6C23124yes—
140C4×Dic6C23124yes—
141C23×Q8—31no—
142C23×Q8C2×D312yes—
143#557—348no—
144#557—316no—
145#556—348no—
146#552—396no—
147D4×Q8—332no—
148D4×Q8C2×D3116yescomplemented
149#541—396no—
150#533—348no—
151C2×C4⋊Q8—36no—
152C2×C4⋊Q8—33no—
153C2×C4⋊Q8C2×D313yes—
154#529—348no—
155C2×#126—36no—
156C2×#126—312no—
157C2×#125—33no—
158C2×#125—36no—
159C2×#125—312no—
160C2×#125—36no—
161C2×#124—33no—
162C2×#123—36no—
163C2×C22⋊Q8—36no—
164C2×C22⋊Q8—36no—
165C2×C22⋊Q8—33no—
166C2×C22⋊Q8C2×D316yes—
167C2×#121—31no—
168#519—332no—
169C2×C4×Q8—31no—
170C2×C4×Q8—36no—
171C2×C4×Q8—32no—
172C2×C4×Q8—36no—
173C2×C4×Q8—316no—
174C2×C4×Q8—33no—
175C2×C4×Q8—36no—
176C2×C4×Q8—33no—
177C2×C4×Q8—33no—
178C2×C4×Q8C2×D311yescharacteristic
179C2×C4×D4—36no—
180C2×C4×D4C2×D313yes—
181C2×#117—36no—
182C2×#117—36no—
183C2×#117—36no—
184C22×C4⋊C4—33no—
185C22×C42—33no—
186C24×C3C2412yes—
187C23×D3C2411yescharacteristic
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—28no—
202C2×#67C2418yes—
203C22×Dic3C2413yes—
204C22×Dic3C2411yescharacteristic
205C22×Dic3C2412yes—
206C22×Dic3C2416yes—
207C22×Dic3C2413yes—
208D3×Q8—232no—
209C2×C4×D3—212no—
210C2×C4×D3—212no—
211C2×C4×D3—24no—
212C2×C4×D3C2414yes—
213C2×C4×D3C24112yes—
214C2×C4×D3C24112yes—
215C2×Dic6—212no—
216C2×Dic6—212no—
217C2×Dic6C24112yes—
218C2×Dic6C24112yes—
219C3×C4⋊C4C24112yes—
220C3×C4⋊C4C24112yes—
221C3×C4⋊C4C24124yes—
222C3×C22⋊C4C24124yes—
223C3×C42C24112yes—
224#217C24124yes—
225#217C2418yes—
226#217C24124yes—
227(C2×D3)⋊C4C24124yes—
228C4⋊Dic3C2414yes—
229C4⋊Dic3C24112yes—
230C4⋊Dic3C24124yes—
231C4⋊Dic3C24124yes—
232C4⋊Dic3C24124yes—
233C4⋊Dic3C24124yes—
234Dic3⋊C4C24124yes—
235Dic3⋊C4C24124yes—
236Dic3⋊C4C24148yes—
237Dic3⋊C4C24124yes—
238C4×Dic3C24112yes—
239C4×Dic3C24124yes—
240C4×Dic3C24112yes—
241C4×Dic3C24112yes—
242C4×Dic3C24148yes—
243C22×Q8—216no—
244C22×Q8—38no—
245C22×Q8—33no—
246C22×Q8—33no—
247C22×Q8—68no—
248C22×Q8C22×D316yes—
249C22×Q8C22×D3116yes—
250C22×Q8C22×D311yescharacteristic
251C22×D4—32no—
252C22×D4C22×D311yescharacteristic
253C23×C4—33no—
254C23×C4—33no—
255C23×C4—31no—
256C23×C4C22×D316yes—
257C4⋊Q8—348no—
258C4⋊Q8—324no—
259C4⋊Q8C22×D3124yes—
260#126—348no—
261#126—396no—
262#125—324no—
263#125—348no—
264#125—396no—
265#125—348no—
266#124—324no—
267#123—348no—
268C22⋊Q8—348no—
269C22⋊Q8—348no—
270C22⋊Q8—324no—
271C22⋊Q8C22×D3148yes—
272#121—38no—
273C4×Q8—38no—
274C4×Q8—348no—
275C4×Q8—316no—
276C4×Q8—348no—
277C4×Q8—324no—
278C4×Q8—348no—
279C4×Q8—324no—
280C4×Q8—324no—
281C4×Q8C22×D318yes—
282C4×D4—348no—
283C4×D4C22×D3124yes—
284#117—348no—
285#117—348no—
286#117—348no—
287C2×C4⋊C4—33no—
288C2×C4⋊C4—36no—
289C2×C4⋊C4—36no—
290C2×C4⋊C4—36no—
291C2×C4⋊C4—36no—
292C2×C4⋊C4—36no—
293C2×C4⋊C4—36no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—312no—
296C2×C4⋊C4—36no—
297C2×C4⋊C4—324no—
298C2×C4⋊C4C22×D313yes—
299C2×C4⋊C4C22×D316yes—
300C2×C4⋊C4C22×D313yes—
301C2×C22⋊C4—32no—
302C2×C22⋊C4—36no—
303C2×C22⋊C4—36no—
304C2×C22⋊C4—36no—
305C2×C22⋊C4C22×D316yes—
306C2×C42—33no—
307C2×C42—312no—
308C2×C42—36no—
309C2×C42—33no—
310C2×C42—33no—
311C2×C42—324no—
312C2×C42C22×D313yes—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3C2514yes—
317C23×C3C2511yescharacteristic
318C22×D3—22no—
319C22×D3—22no—
320C22×D32−1+414yes—
321C22×D3C2512yes—
322C3×Q8—248no—
323C3×Q8C2×#41116yes—
324C3×D4—28no—
325C2×C3×C4—212no—
326C2×C3×C4—26no—
327C2×C3×C4—248no—
328C2×C3×C4C2×#41112yes—
329C2×C3×C42−1+414yes—
330C2×C3×C4C25112yes—
331C2×C3×C4C2516yes—
332C2×C3×C4C2516yes—
333C2×C3×C4C2512yes—
334#67—216no—
335C2×Dic3—26no—
336C2×Dic3—212no—
337C2×Dic3—22no—
338C2×Dic3—26no—
339C2×Dic3—24no—
340C2×Dic3C2516yes—
341C2×Dic3C2512yes—
342C2×Dic3C2514yes—
343C2×Dic3C25112yes—
344C2×Dic3C2516yes—
345C4×D3—224no—
346C4×D3—224no—
347C4×D3—28no—
348Dic6—224no—
349Dic6—224no—
350C24—31no—
351C24C23×D312yes—
352C2×Q8—264no—
353C2×Q8—224no—
354C2×Q8—312no—
355C2×Q8—312no—
356C2×Q8—632no—
357C2×Q8—612no—
358C2×Q8—612no—
359C2×Q8#23718yes—
360C2×Q8C23×D3124yes—
361C2×Q8C23×D314yes—
362C2×D4—24no—
363C2×D4—38no—
364C2×D4—68no—
365C2×D4C23×D314yes—
366C22×C4—224no—
367C22×C4—31no—
368C22×C4—33no—
369C22×C4—36no—
370C22×C4—32no—
371C22×C4—33no—
372C22×C4—34no—
373C22×C4—312no—
374C22×C4—312no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—64no—
378C22×C4C23×D3124yes—
379C22×C4C23×D316yes—
380C22×C4C23×D313yes—
381C22×C4C23×D313yes—
382C22×C4C23×D311yescharacteristic
383C4⋊C4—312no—
384C4⋊C4—324no—
385C4⋊C4—324no—
386C4⋊C4—324no—
387C4⋊C4—324no—
388C4⋊C4—324no—
389C4⋊C4—324no—
390C4⋊C4—34no—
391C4⋊C4—348no—
392C4⋊C4—324no—
393C4⋊C4C23×D3112yes—
394C4⋊C4C23×D3124yes—
395C4⋊C4C23×D3112yes—
396C22⋊C4—38no—
397C22⋊C4—324no—
398C22⋊C4—324no—
399C22⋊C4—324no—
400C22⋊C4C23×D3124yes—
401C42—312no—
402C42—312no—
403C42—312no—
404C42—348no—
405C42—324no—
406C42C23×D3112yes—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3—48no—
411C22×C3C22×#4112yes—
412C22×C3C2×2−1+412yes—
413C22×C3C2×2−1+412yes—
414C22×C3C2611yescharacteristic
415C2×D3—22no—
416C2×D3—22no—
417C2×D3—28no—
418C2×D3C2×2−1+414yes—
419C3×C4—212no—
420C3×C4—26no—
421C3×C4C22×#41112yes—
422C3×C4C2×2−1+414yes—
423Dic3—26no—
424Dic3—212no—
425Dic3—22no—
426Dic3—26no—
427Dic3—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—34no—
432C23—32no—
433C23—62no—
434C23—62no—
435C23C24×D314yes—
436C23C24×D311yescharacteristic
437Q8—248no—
438Q8—624no—
439Q8—624no—
440Q8C2×#237116yes—
441D4—28no—
442D4—616no—
443C2×C4—212no—
444C2×C4—26no—
445C2×C4—248no—
446C2×C4—32no—
447C2×C4—36no—
448C2×C4—312no—
449C2×C4—34no—
450C2×C4—36no—
451C2×C4—624no—
452C2×C4—66no—
453C2×C4—62no—
454C2×C4—612no—
455C2×C4—66no—
456C2×C4—64no—
457C2×C4—624no—
458C2×C4—68no—
459C2×C4C2×#237112yes—
460C2×C4#100714yes—
461C2×C4C24×D3112yes—
462C2×C4C24×D316yes—
463C2×C4C24×D316yes—
464C2×C4C24×D312yes—
465C2×C3—44no—
466C2×C3#352314yescomplemented
467C2×C3C23×#4111yescharacteristic
468C2×C3C22×2−1+411yescharacteristic
469C2×C3C22×2−1+411yescharacteristic
470D3—24no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22—34no—
475C22—48no—
476C22—62no—
477C22—62no—
478C22—68no—
479C22C22×#23712yes—
480C22C2×#100712yes—
481C22D3×2−1+412yes—
482C22C25×D311yescharacteristic
483C4—212no—
484C4—26no—
485C4—66no—
486C4—62no—
487C4—612no—
488C4—66no—
489C4—64no—
490C4C22×#237112yes—
491C4C2×#100714yes—
492C3C2×#352311yescharacteristic, complemented, Sylow
493C2—44no—
494C2—64no—
495C2#8675314yesdirect product factor
496C2C23×#23711yescharacteristic
497C2C22×#100711yescharacteristic
498C2C2×D3×2−1+411yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C25
prime radicalC2×#938C22
hypercenterC2×D4×Q8D3
hypocenterC3C2×#3523
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×Q8×#67C2semidirect
3. C2×#5812C2semidirect
4. #86753C2direct (N × H)
5. C2×#5721C2semidirect
6. C2×#5728C2semidirect
7. C2×#5639C2semidirect
8. C2×#5625C2semidirect
9. C2×C4×D3×Q8C2semidirect
10. C2×#5570C2semidirect
12. C2×#5542C2semidirect
16. C3×D4×Q8C22semidirect
21. Q8×#67C22semidirect
23. #5812C22semidirect
35. #5728C22semidirect
36. #5721C22semidirect
37. #5639C22semidirect
38. #5625C22semidirect
39. C4×D3×Q8C22semidirect
40. #5570C22semidirect
42. #5542C22semidirect
65. C2×D4×Q8D3semidirect
148. D4×Q8C2×D3semidirect
466. C2×C3#3523semidirect
492. C3C2×#3523semidirect
495. C2#86753direct (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×Q8×#672
C2×#58122
#867532
C2×#57212
C2×#57282
C2×#56392
C2×#56252
C2×C4×D3×Q82
C2×#55702
C2×#55572
C2×#55422
C2×#35233