Finite Group Databasebeta

← C2×#85832 #11703457

Subgroups — C2×#85832 #11703457

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×C4×#41C211yesmaximal, characteristic, complemented
2C2×#5831C212yesmaximal, complemented
3#85832C2132yesmaximal, direct product factor
4C2×Q8×D12C211yesmaximal, characteristic, complemented
5C2×#5572C212yesmaximal, complemented
6C2×Q8×Dic6C211yesmaximal, characteristic, complemented
7C2×#5561C216yesmaximal, complemented
8C2×#5542C213yesmaximal, complemented
9C2×#5563C213yesmaximal, complemented
10C2×#5533C216yesmaximal, complemented
11C2×#5479C213yesmaximal, complemented
12C2×#5470C213yesmaximal, complemented
13C2×#3518—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C22×D3×Q8C2212yes—
16C3×C4×#41C22116yescomplemented
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2213yes—
20C2×C3×C4×D4C2213yes—
21C2×C3×#117C2213yes—
22C22×C3×C42C2213yes—
23#5831C22132yescomplemented
24C2×Q8×Dic3C2212yes—
25C2×#938C2216yes—
26C2×#936C2216yes—
27C2×#930C2213yes—
28C2×#929C2216yes—
29C2×C4×#67C2216yes—
30C2×#926C2216yes—
31C2×#924C2213yes—
32C2×#924C2216yes—
33C2×#924C2216yes—
34Q8×D12C22116yescomplemented
35#5572C22132yescomplemented
36Q8×Dic6C22116yescomplemented
37#5563C22148yescomplemented
38#5561C22196yescomplemented
39#5542C22148yescomplemented
40#5533C22196yescomplemented
41C2×#897C2216yes—
42C2×#892C2216yes—
43C2×#888C2216yes—
44C2×#887C2216yes—
45C2×#885C22112yes—
46C2×#879C22112yes—
47#5479C22148yescomplemented
48C2×#876C2216yes—
49C2×#875C2211yescharacteristic
50C2×C4×D12C2213yes—
51C2×#872C2216yes—
52#5470C22148yescomplemented
53C2×#870C2213yes—
54C2×#869C2213yes—
55C2×#869C2213yes—
56C2×C4×Dic6C2213yes—
57C2×C4×Dic6C2216yes—
58C2×C4×Dic6C2216yes—
59#3518—332no—
60C2×Q82—31no—
61C2×#556—33no—
62C2×#556—36no—
63C2×#555—33no—
64C2×D4×Q8—31no—
65C2×#548—32no—
66C2×#542—36no—
67C2×#533—33no—
68C2×#529—33no—
69C2×#519—32no—
70C2×C4×#41D311yescharacteristic, complemented
71C2×C3×#41—28no—
72C2×C3×#41C2318yes—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4C2313yes—
75C23×C3×C4C2313yes—
76C23×C3×C4C2313yes—
77C23×C3×C4C2313yes—
78C22×#67C2316yes—
79C2×D3×Q8—216no—
80C2×D3×Q8C23116yes—
81C22×D12C2311yescharacteristic
82C22×C4×D3C2316yes—
83C22×Dic6C2316yes—
84C22×Dic6C2311yescharacteristic
85C22×Dic6C2316yes—
86C3×C4×Q8C2318yes—
87C3×C4×Q8C2318yes—
88C3×C4×D4C23124yes—
89C3×C4×D4C23124yes—
90C3×#117C23124yes—
91C2×C3×C4⋊C4C2313yes—
92C2×C3×C4⋊C4C2313yes—
93C2×C3×C22⋊C4C2316yes—
94C2×C3×C42C2313yes—
95C2×C3×C42C2313yes—
96C2×C3×C42C2313yes—
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C23124yes—
99C2×#217C2316yes—
100C2×#217C23112yes—
101Q8×Dic3C23116yes—
102#938C23148yes—
103#936C23148yes—
104#930C23124yes—
105#929C23148yes—
106C4×#67C23148yes—
107C2×(C2×D3)⋊C4C2312yes—
108C2×(C2×D3)⋊C4C2316yes—
109C2×(C2×D3)⋊C4C2316yes—
110C2×(C2×D3)⋊C4C2312yes—
111#926C23148yes—
112C2×C4⋊Dic3C2313yes—
113C2×C4⋊Dic3C2316yes—
114C2×C4⋊Dic3C2316yes—
115C2×C4⋊Dic3C2313yes—
116#924C23148yes—
117#924C23148yes—
118#924C23124yes—
119C2×Dic3⋊C4C2316yes—
120C2×Dic3⋊C4C2316yes—
121C2×Dic3⋊C4C2316yes—
122C2×Dic3⋊C4C2316yes—
123C2×C4×Dic3C23112yes—
124C2×C4×Dic3C2316yes—
125#897C23148yes—
126#892C23148yes—
127#888C23148yes—
128#887C23148yes—
129#885C23196yes—
130#879C23196yes—
131#876C23148yes—
132#875C2318yes—
133C4×D12C23124yes—
134#872C23148yes—
135#870C23124yes—
136#869C23124yes—
137#869C23124yes—
138C4×Dic6C23148yes—
139C4×Dic6C23148yes—
140C4×Dic6C23124yes—
141C22×#41C2×D311yescharacteristic
142C23×Q8—32no—
143Q82—316no—
144#556—348no—
145#556—396no—
146#555—348no—
147D4×Q8—316no—
148#548—332no—
149#542—396no—
150#533—348no—
151C2×C4⋊Q8—36no—
152C2×C4⋊Q8—33no—
153C2×C4⋊Q8—33no—
154#529—348no—
155C2×#126—312no—
156C2×#126—36no—
157C2×#125—33no—
158C2×#124—36no—
159C2×#124—312no—
160C2×#124—31no—
161C2×#123—36no—
162C2×C22⋊Q8—33no—
163C2×C22⋊Q8—36no—
164C2×C22⋊Q8—36no—
165C2×C22⋊Q8—36no—
166C2×#121—36no—
167C2×#121—33no—
168#519—332no—
169C4×#41C2×D3116yescomplemented
170C2×C4×Q8—32no—
171C2×C4×Q8—36no—
172C2×C4×Q8—36no—
173C2×C4×Q8—33no—
174C2×C4×Q8—36no—
175C2×C4×Q8C2×D311yescharacteristic
176C2×C4×Q8C2×D311yescharacteristic
177C2×C4×D4—36no—
178C2×C4×D4—33no—
179C2×C4×D4C2×D313yes—
180C2×C4×D4C2×D313yes—
181C2×#117—36no—
182C2×#117—36no—
183C2×#117—36no—
184C2×#117C2×D313yes—
185C22×C42C2×D313yes—
186C24×C3C2413yes—
187C23×D3C2412yes—
188C3×#41—232no—
189C2×C3×Q8#4118yes—
190C2×C3×Q8C2414yes—
191C2×C3×D4—212no—
192C2×C3×D4C24112yes—
193C22×C3×C4—212no—
194C22×C3×C4—212no—
195C22×C3×C4—46no—
196C22×C3×C4C2413yes—
197C22×C3×C4C2411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×C3×C4C2413yes—
200C22×C3×C4C24112yes—
201C22×C3×C4C24112yes—
202C22×C3×C4C24112yes—
203C22×C3×C4C2413yes—
204C22×C3×C4C2411yescharacteristic
205C2×#67—412no—
206C2×#67C24124yes—
207C22×Dic3C2412yes—
208C22×Dic3C2416yes—
209C22×Dic3C2416yes—
210D3×Q8—264no—
211C2×D12—24no—
212C2×D12C2414yes—
213C2×C4×D3—224no—
214C2×C4×D3C24124yes—
215C2×Dic6—224no—
216C2×Dic6—24no—
217C2×Dic6—412no—
218C2×Dic6C2414yes—
219C2×Dic6C24124yes—
220C2×Dic6C24124yes—
221C3×C4⋊C4C24112yes—
222C3×C4⋊C4C24112yes—
223C3×C22⋊C4C24124yes—
224C3×C42C24112yes—
225C3×C42C24112yes—
226C3×C42C24112yes—
227C3×C42C2414yes—
228#217C24124yes—
229#217C24148yes—
230(C2×D3)⋊C4C2418yes—
231(C2×D3)⋊C4C24124yes—
232(C2×D3)⋊C4C24124yes—
233(C2×D3)⋊C4C2418yes—
234C4⋊Dic3C24112yes—
235C4⋊Dic3C24124yes—
236C4⋊Dic3C24124yes—
237C4⋊Dic3C24112yes—
238Dic3⋊C4C24124yes—
239Dic3⋊C4C24124yes—
240Dic3⋊C4C24124yes—
241Dic3⋊C4C24124yes—
242C4×Dic3C24148yes—
243C4×Dic3C24124yes—
244C2×#41—28no—
245C2×#41C22×D318yes—
246C22×Q8—316no—
247C22×Q8—31no—
248C22×Q8—36no—
249C22×Q8—36no—
250C22×Q8—616no—
251C22×Q8C22×D311yescharacteristic
252C22×D4—36no—
253C22×D4—31no—
254C22×D4C22×D313yes—
255C23×C4—36no—
256C23×C4C22×D313yes—
257C23×C4C22×D313yes—
258C23×C4C22×D313yes—
259C4⋊Q8—348no—
260C4⋊Q8—324no—
261C4⋊Q8—324no—
262#126—396no—
263#126—348no—
264#125—324no—
265#124—348no—
266#124—396no—
267#124—38no—
268#123—348no—
269C22⋊Q8—324no—
270C22⋊Q8—348no—
271C22⋊Q8—348no—
272C22⋊Q8—348no—
273#121—348no—
274#121—324no—
275C4×Q8—316no—
276C4×Q8—348no—
277C4×Q8—348no—
278C4×Q8—324no—
279C4×Q8—348no—
280C4×Q8C22×D318yes—
281C4×Q8C22×D318yes—
282C4×D4—348no—
283C4×D4—324no—
284C4×D4C22×D3124yes—
285C4×D4C22×D3124yes—
286#117—348no—
287#117—348no—
288#117—348no—
289#117C22×D3124yes—
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⋊C4—36no—
298C2×C4⋊C4C22×D313yes—
299C2×C4⋊C4C22×D313yes—
300C2×C22⋊C4—312no—
301C2×C22⋊C4—36no—
302C2×C22⋊C4—36no—
303C2×C22⋊C4—32no—
304C2×C22⋊C4—32no—
305C2×C22⋊C4—36no—
306C2×C22⋊C4C22×D316yes—
307C2×C42—312no—
308C2×C42—36no—
309C2×C42C22×D313yes—
310C2×C42C22×D313yes—
311C2×C42C22×D313yes—
312C2×C42C22×D3124yes—
313C2×C42C22×D311yescharacteristic
314C23×C3—26no—
315C23×C3—26no—
316C23×C3—26no—
317C23×C3C2516yes—
318C23×C3C2511yescharacteristic
319C22×D3—24no—
320C22×D3—24no—
321C22×D3—24no—
322C22×D3C2514yes—
323C3×Q8C2×#41116yes—
324C3×D4—224no—
325C2×C3×C4—22no—
326C2×C3×C4—26no—
327C2×C3×C4—26no—
328C2×C3×C4—22no—
329C2×C3×C4—224no—
330C2×C3×C4—224no—
331C2×C3×C4—412no—
332C2×C3×C4C2×#41112yes—
333C2×C3×C42−1+414yes—
334C2×C3×C4C2516yes—
335C2×C3×C4C2512yes—
336C2×C3×C4C2512yes—
337C2×C3×C4C2516yes—
338C2×C3×C4C2516yes—
339C2×C3×C4C2512yes—
340#67—424no—
341C2×Dic3—212no—
342C2×Dic3—24no—
343C2×Dic3—212no—
344C2×Dic3C2514yes—
345C2×Dic3C25112yes—
346C2×Dic3C25112yes—
347D12—28no—
348C4×D3—248no—
349Dic6—248no—
350Dic6—28no—
351Dic6—424no—
352C24—32no—
353C24C23×D313yes—
354#41—232no—
355C2×Q8—34no—
356C2×Q8—324no—
357C2×Q8—324no—
358C2×Q8—664no—
359C2×Q8—624no—
360C2×Q8—64no—
361C2×Q8—1212no—
362C2×Q8#23518yes—
363C2×Q8C23×D314yes—
364C2×D4—212no—
365C2×D4—324no—
366C2×D4—34no—
367C2×D4—64no—
368C2×D4—1212no—
369C2×D4C23×D3112yes—
370C22×C4—212no—
371C22×C4—212no—
372C22×C4—32no—
373C22×C4—36no—
374C22×C4—36no—
375C22×C4—324no—
376C22×C4—46no—
377C22×C4—624no—
378C22×C4C23×D3112yes—
379C22×C4C23×D3112yes—
380C22×C4C23×D313yes—
381C22×C4C23×D311yescharacteristic
382C22×C4C23×D311yescharacteristic
383C22×C4C23×D313yes—
384C22×C4C23×D3112yes—
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⋊C4—324no—
395C4⋊C4C23×D3112yes—
396C4⋊C4C23×D3112yes—
397C22⋊C4—348no—
398C22⋊C4—324no—
399C22⋊C4—324no—
400C22⋊C4—38no—
401C22⋊C4—38no—
402C22⋊C4—324no—
403C22⋊C4C23×D3124yes—
404C42—348no—
405C42—324no—
406C42C23×D314yes—
407C42C23×D3112yes—
408C42C23×D3112yes—
409C42C23×D3112yes—
410C22×C3—26no—
411C22×C3—26no—
412C22×C3—26no—
413C22×C3—412no—
414C22×C3C22×#4112yes—
415C22×C3C2×2−1+412yes—
416C22×C3#58512yes—
417C22×C3C2611yescharacteristic
418C2×D3—24no—
419C2×D3—24no—
420C2×D3—24no—
421C2×D3—48no—
422C3×C4—22no—
423C3×C4—26no—
424C3×C4—26no—
425C3×C4—22no—
426C3×C4C22×#41112yes—
427C3×C4C2×2−1+414yes—
428Dic3—212no—
429Dic3—24no—
430Dic3—212no—
431C23—26no—
432C23—26no—
433C23—26no—
434C23—34no—
435C23—64no—
436C23—64no—
437C23—64no—
438C23C24×D316yes—
439C23C24×D311yescharacteristic
440Q8—648no—
441Q8—68no—
442Q8—1224no—
443Q8C2×#235116yes—
444D4—224no—
445D4—68no—
446D4—1224no—
447C2×C4—22no—
448C2×C4—26no—
449C2×C4—22no—
450C2×C4—26no—
451C2×C4—224no—
452C2×C4—224no—
453C2×C4—34no—
454C2×C4—312no—
455C2×C4—312no—
456C2×C4—412no—
457C2×C4—612no—
458C2×C4—64no—
459C2×C4—612no—
460C2×C4—648no—
461C2×C4C2×#235112yes—
462C2×C4#101014yes—
463C2×C4C24×D316yes—
464C2×C4C24×D312yes—
465C2×C4C24×D312yes—
466C2×C4C24×D316yes—
467C2×C4C24×D316yes—
468C2×C4C24×D312yes—
469C2×C3—46no—
470C2×C3#351814yescomplemented
471C2×C3C23×#4111yescharacteristic
472C2×C3C22×2−1+411yescharacteristic
473C2×C3C2×#58511yescharacteristic
474D3—44no—
475C22—26no—
476C22—26no—
477C22—26no—
478C22—412no—
479C22—64no—
480C22—64no—
481C22—64no—
482C22—128no—
483C22C22×#23512yes—
484C22C2×#101012yes—
485C22#596412yes—
486C22C25×D311yescharacteristic
487C4—22no—
488C4—26no—
489C4—22no—
490C4—26no—
491C4—612no—
492C4—64no—
493C4—612no—
494C4C22×#235112yes—
495C4C2×#101014yes—
496C3C2×#351811yescharacteristic, complemented, Sylow
497C2—46no—
498C2—124no—
499C2#8583214yesdirect product factor
500C2C23×#23511yescharacteristic
501C2C22×#101011yescharacteristic
502C2C2×#596411yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×C4×#41C2
socleC23×C3C25
hypercenterC2×C4×#41D3
hypocenterC3C2×#3518
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×#41C2semidirect
2. C2×#5831C2semidirect
3. #85832C2direct (N × H)
4. C2×Q8×D12C2semidirect
5. C2×#5572C2semidirect
6. C2×Q8×Dic6C2semidirect
7. C2×#5561C2semidirect
8. C2×#5542C2semidirect
9. C2×#5563C2semidirect
10. C2×#5533C2semidirect
11. C2×#5479C2semidirect
12. C2×#5470C2semidirect
16. C3×C4×#41C22semidirect
23. #5831C22semidirect
34. Q8×D12C22semidirect
35. #5572C22semidirect
36. Q8×Dic6C22semidirect
37. #5563C22semidirect
38. #5561C22semidirect
39. #5542C22semidirect
40. #5533C22semidirect
47. #5479C22semidirect
52. #5470C22semidirect
70. C2×C4×#41D3semidirect
169. C4×#41C2×D3semidirect
470. C2×C3#3518semidirect
496. C3C2×#3518semidirect
499. C2#85832direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C2×#58312
#858322
C2×Q8×D122
C2×#55722
C2×Q8×Dic62
C2×#55612
C2×#55422
C2×#55632
C2×#55332
C2×#54792
C2×#54702
C2×#35183