Finite Group Databasebeta

← C2×#22573831 #31762929

Subgroups — C2×#22573831 #31762929

Order 17152.

7706 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22573831C2132yesmaximal, direct product factor
2C2×#16864125C211yesmaximal, characteristic, complemented
3C2×#16864116C213yesmaximal, complemented
4C2×#16864085C216yesmaximal, complemented
5C2×Q8×D268C211yesmaximal, characteristic, complemented
6C2×#16864053C212yesmaximal, complemented
7C2×#16864051C213yesmaximal, complemented
8C2×#16863953C213yesmaximal, complemented
9C2×#16863928C216yesmaximal, complemented
10C2×C4×#11761628C213yesmaximal, complemented
11C2×C4×Q8×D67C212yesmaximal, complemented
12C2×C4×#41×C67C211yesmaximal, characteristic, complemented
13C22×#11761628C2213yes—
14C22×Q8×D67C2212yes—
15#16864125C22116yescomplemented
16#16864116C22148yescomplemented
17#16864085C22196yescomplemented
18Q8×D268C22116yescomplemented
19#16864053C22132yescomplemented
20#16864051C22148yescomplemented
21#16863953C22148yescomplemented
22C2×#14524043C2213yes—
23C2×#14524043C2213yes—
24#16863928C22196yescomplemented
25C2×#14524036C22112yes—
26C2×#14524035C2216yes—
27C2×#14524034C2216yes—
28C2×#14524030C2216yes—
29C2×#14524025C22112yes—
30C2×#14524023C2216yes—
31C2×#14524022C2216yes—
32C2×#14524022C2216yes—
33C2×#14524014C2216yes—
34C4×#11761628C22148yescomplemented
35C2×Q8×Dic67C2212yes—
36C2×C4×Dic134C2216yes—
37C2×C4×Dic134C2213yes—
38C2×C4×Dic134C2213yes—
39C2×#14524008C2216yes—
40C4×Q8×D67C22132yescomplemented
41C2×C4×D268C2211yescharacteristic
42C2×C4×D268C2213yes—
43C2×C4×C22⋊D67C2216yes—
44C2×C4×C22⋊D67C2216yes—
45C2×#14524002C2213yes—
46C2×#14524002C2216yes—
47C2×#14524000C2216yes—
48C2×C4⋊C4×D67C2216yes—
49C2×C42×D67C2216yes—
50C22×#41×C67C2211yescharacteristic
51C4×#41×C67C22116yescomplemented
52C2×C4×Q8×C67C2211yescharacteristic
53C2×C4×Q8×C67C2211yescharacteristic
54C2×C4×D4×C67C2213yes—
55C2×C4×D4×C67C2213yes—
56C2×#117×C67C2213yes—
57C22×C42×C67C2213yes—
58C2×#11761628—224no—
59C2×#11761628C23124yes—
60C22×Dic134C2316yes—
61C22×Dic134C2313yes—
62C2×Q8×D67—216no—
63C2×Q8×D67C23116yes—
64C22×D268C2311yescharacteristic
65C22×C22⋊D67C2316yes—
66C22×C4×D67C2312yes—
67C22×C4×D67C2316yes—
68C22×C4×D67C2316yes—
69#14524043C23124yes—
70#14524043C23124yes—
71#14524036C23196yes—
72#14524035C23148yes—
73#14524034C23148yes—
74#14524030C23148yes—
75#14524025C23196yes—
76#14524023C23148yes—
77#14524022C23148yes—
78#14524022C23148yes—
79#14524014C23148yes—
80Q8×Dic67C23116yes—
81C4×Dic134C23148yes—
82C4×Dic134C23124yes—
83C4×Dic134C23124yes—
84#14524008C23148yes—
85C4×D268C23124yes—
86C4×D268C2318yes—
87C4×C22⋊D67C23148yes—
88C4×C22⋊D67C23148yes—
89#14524002C23124yes—
90#14524002C23148yes—
91#14524000C23148yes—
92C2×C4⋊Dic67C2316yes—
93C2×C4⋊Dic67C2311yescharacteristic
94C2×C4⋊Dic67C2313yes—
95C2×C4⋊Dic67C2316yes—
96C2×C4⋊Dic67C2316yes—
97C2×Dic67⋊C4C23112yes—
98C2×Dic67⋊C4C2316yes—
99C2×Dic67⋊C4C2316yes—
100C4⋊C4×D67C23148yes—
101C2×C22⋊Dic67C2316yes—
102C2×C22⋊Dic67C23112yes—
103C2×(C2×D67)⋊C4C2312yes—
104C2×(C2×D67)⋊C4C2316yes—
105C2×C4×Dic67C2316yes—
106C2×C4×Dic67C2316yes—
107C2×C4×Dic67C2316yes—
108C42×D67C23148yes—
109C2×#41×C67—28no—
110C2×#41×C67C2318yes—
111C22×Q8×C67C2311yescharacteristic
112C22×D4×C67C2313yes—
113C23×C4×C67C2313yes—
114C23×C4×C67C2313yes—
115C23×C4×C67C2313yes—
116C4×Q8×C67C2318yes—
117C4×Q8×C67C2318yes—
118C4×D4×C67C23124yes—
119C4×D4×C67C23124yes—
120#117×C67C23124yes—
121C2×C4⋊C4×C67C2313yes—
122C2×C4⋊C4×C67C2313yes—
123C2×C22⋊C4×C67C2316yes—
124C2×C42×C67C2311yescharacteristic
125C2×C42×C67C2313yes—
126C2×C42×C67C2313yes—
127C2×C42×C67C2313yes—
128C2×C42×C67C23124yes—
129C23×D67C2412yes—
130#11761628—296no—
131C2×Dic134—224no—
132C2×Dic134—212no—
133C2×Dic134C24124yes—
134C2×Dic134C24112yes—
135Q8×D67—264no—
136C2×D268#4118yescomplemented
137C2×D268C2414yes—
138C2×C22⋊D67—224no—
139C2×C22⋊D67C24124yes—
140C22×Dic67C2412yes—
141C22×Dic67C2416yes—
142C22×Dic67C2416yes—
143C2×C4×D67—224no—
144C2×C4×D67—224no—
145C2×C4×D67—28no—
146C2×C4×D67C2418yes—
147C2×C4×D67C24124yes—
148C2×C4×D67C24124yes—
149C4⋊Dic67C24124yes—
150C4⋊Dic67C2414yes—
151C4⋊Dic67C24112yes—
152C4⋊Dic67C24124yes—
153C4⋊Dic67C24124yes—
154Dic67⋊C4C24148yes—
155Dic67⋊C4C24124yes—
156Dic67⋊C4C24124yes—
157C22⋊Dic67C24124yes—
158C22⋊Dic67C24148yes—
159(C2×D67)⋊C4C2418yes—
160(C2×D67)⋊C4C24124yes—
161C4×Dic67C24124yes—
162C4×Dic67C24124yes—
163C4×Dic67C24124yes—
164C24×C67C2413yes—
165#41×C67—232no—
166C2×Q8×C67#4118yescomplemented
167C2×Q8×C67C2414yes—
168C2×D4×C67—212no—
169C2×D4×C67C24112yes—
170C22×C4×C67—212no—
171C22×C4×C67—212no—
172C22×C4×C67—212no—
173C22×C4×C67C2411yescharacteristic
174C22×C4×C67C2413yes—
175C22×C4×C67C2413yes—
176C22×C4×C67C24112yes—
177C22×C4×C67C24112yes—
178C22×C4×C67C24112yes—
179C22×C4×C67C2411yescharacteristic
180C22×C4×C67C2413yes—
181C22×C4×C67C2411yescharacteristic
182C4⋊C4×C67C24112yes—
183C4⋊C4×C67C24112yes—
184C22⋊C4×C67C24124yes—
185C42×C67C2414yes—
186C42×C67C24112yes—
187C42×C67C24112yes—
188C42×C67C24112yes—
189C23×C67—26no—
190C23×C67—26no—
191C23×C67—26no—
192C23×C67C2516yes—
193C23×C67C2511yescharacteristic
194C22×D67—24no—
195C22×D67—24no—
196C22×D67C2×#4118yes—
197C22×D67C2514yes—
198Q8×C67C2×#41116yescomplemented
199D4×C67—224no—
200C2×C4×C67—22no—
201C2×C4×C67—26no—
202C2×C4×C67—26no—
203C2×C4×C67—224no—
204C2×C4×C67—224no—
205C2×C4×C67—224no—
206C2×C4×C67C2×#4114yes—
207C2×C4×C67C2×#41112yes—
208C2×C4×C672−1+414yes—
209C2×C4×C67C2512yes—
210C2×C4×C67C2516yes—
211C2×C4×C67C2516yes—
212C2×C4×C67C2512yes—
213C2×C4×C67C2516yes—
214C2×C4×C67C2512yes—
215C22⋊D67—248no—
216C2×Dic67—212no—
217C2×Dic67—212no—
218C2×Dic67—24no—
219C2×Dic67C2514yes—
220C2×Dic67C25112yes—
221C2×Dic67C25112yes—
222D268C2×#41116yescomplemented
223C4×D67—248no—
224C4×D67—248no—
225C4×D67—216no—
226Dic134—248no—
227Dic134—224no—
228C22×C67—26no—
229C22×C67—26no—
230C22×C67—26no—
231C22×C67—412no—
232C22×C67C22×#4112yes—
233C22×C67C22×#4112yes—
234C22×C67C2×2−1+412yes—
235C22×C67C2611yescharacteristic
236C2×D67—24no—
237C2×D67—24no—
238C2×D67—216no—
239C2×D67C22×#4118yes—
240C4×C67—26no—
241C4×C67—22no—
242C4×C67—26no—
243C4×C67C22×#4114yes—
244C4×C67C22×#41112yes—
245C4×C67C2×2−1+414yes—
246Dic67—212no—
247Dic67—212no—
248Dic67—24no—
249C2×#3482—671nomaximal, Sylow
250C2×C67—46no—
251C2×C67#348214yescomplemented
252C2×C67C23×#4111yescharacteristic
253C2×C67C23×#4111yescharacteristic
254C2×C67C22×2−1+411yescharacteristic
255D67—28no—
256#3482—6732no—
257C2×#557—671no—
258C2×#556—673no—
259C2×#552—676no—
260C2×D4×Q8—671no—
261C2×#548—672no—
262C2×#547—673no—
263C2×#533—673no—
264C2×#529—676no—
265C2×C4×#41—673no—
266C2×C4×#41D6711yescharacteristic, complemented
267C22×C4×Q8—672no—
268C67C2×#348211yescharacteristic, complemented, Sylow
269C22×#41—673no—
270C22×#41C2×D6711yescharacteristic
271C23×Q8—672no—
272#557—6716no—
273#556—6748no—
274#552—6796no—
275D4×Q8—6716no—
276#548—6732no—
277#547—6748no—
278#533—6748no—
279C2×C4⋊Q8—673no—
280C2×C4⋊Q8—673no—
281#529—6796no—
282C2×#126—6712no—
283C2×#125—676no—
284C2×#125—676no—
285C2×#124—676no—
286C2×#123—6712no—
287C2×C22⋊Q8—676no—
288C2×C22⋊Q8—676no—
289C2×C22⋊Q8—676no—
290C2×#121—676no—
291C4×#41—6748no—
292C4×#41C2×D67116yescomplemented
293C2×C4×Q8—672no—
294C2×C4×Q8—6732no—
295C2×C4×Q8—676no—
296C2×C4×Q8—676no—
297C2×C4×Q8—673no—
298C2×C4×Q8—673no—
299C2×C4×Q8C2×D6711yescharacteristic
300C2×C4×Q8C2×D6711yescharacteristic
301C2×C4×D4—676no—
302C2×C4×D4—676no—
303C2×C4×D4—671no—
304C2×C4×D4—673no—
305C2×C4×D4C2×D6713yes—
306C2×C4×D4C2×D6713yes—
307C2×#117—673no—
308C2×#117—676no—
309C2×#117—676no—
310C2×#117C2×D6713yes—
311C22×C4⋊C4—676no—
312C22×C42—676no—
313C22×C42C2×D6713yes—
314C2×#41—28no—
315C2×#41—6724no—
316C2×#41—13424no—
317C2×#41C22×D6718yes—
318C22×Q8—6716no—
319C22×Q8—676no—
320C22×Q8—673no—
321C22×Q8—13416no—
322C22×Q8C22×D6711yescharacteristic
323C22×D4—676no—
324C22×D4—671no—
325C22×D4C22×D6713yes—
326C23×C4—676no—
327C23×C4—676no—
328C23×C4—672no—
329C23×C4C22×D6713yes—
330C23×C4C22×D6713yes—
331C23×C4C22×D6713yes—
332C4⋊Q8—6724no—
333C4⋊Q8—6724no—
334#126—6796no—
335#125—6748no—
336#125—6748no—
337#124—6748no—
338#123—6796no—
339C22⋊Q8—6748no—
340C22⋊Q8—6748no—
341C22⋊Q8—6748no—
342#121—6748no—
343C4×Q8—6716no—
344C4×Q8—6748no—
345C4×Q8—6748no—
346C4×Q8—6724no—
347C4×Q8—6724no—
348C4×Q8C22×D6718yes—
349C4×Q8C22×D6718yes—
350C4×D4—6748no—
351C4×D4—6748no—
352C4×D4—678no—
353C4×D4—6724no—
354C4×D4C22×D67124yes—
355C4×D4C22×D67124yes—
356#117—6724no—
357#117—6748no—
358#117—6748no—
359#117C22×D67124yes—
360C2×C4⋊C4—676no—
361C2×C4⋊C4—6712no—
362C2×C4⋊C4—671no—
363C2×C4⋊C4—676no—
364C2×C4⋊C4—673no—
365C2×C4⋊C4—676no—
366C2×C4⋊C4—676no—
367C2×C4⋊C4—676no—
368C2×C4⋊C4—6748no—
369C2×C4⋊C4C22×D6713yes—
370C2×C4⋊C4C22×D6713yes—
371C2×C22⋊C4—676no—
372C2×C22⋊C4—6712no—
373C2×C22⋊C4—672no—
374C2×C22⋊C4—676no—
375C2×C22⋊C4C22×D6716yes—
376C2×C42—676no—
377C2×C42—676no—
378C2×C42—676no—
379C2×C42—6748no—
380C2×C42C22×D6713yes—
381C2×C42C22×D6713yes—
382C2×C42C22×D67124yes—
383C2×C42C22×D6711yescharacteristic
384C2×C42C22×D6713yes—
385C24—672no—
386C24C23×D6713yes—
387#41—232no—
388#41—13496no—
389C2×Q8—6724no—
390C2×Q8—6712no—
391C2×Q8—13464no—
392C2×Q8—13412no—
393C2×Q8—13424no—
394C2×Q8#1176162818yescomplemented
395C2×Q8C23×D6714yes—
396C2×D4—212no—
397C2×D4—6724no—
398C2×D4—678no—
399C2×D4—674no—
400C2×D4—13424no—
401C2×D4C23×D67112yes—
402C22×C4—212no—
403C22×C4—212no—
404C22×C4—212no—
405C22×C4—672no—
406C22×C4—676no—
407C22×C4—676no—
408C22×C4—678no—
409C22×C4—6724no—
410C22×C4—6724no—
411C22×C4—13424no—
412C22×C4—13424no—
413C22×C4—1348no—
414C22×C4C23×D6713yes—
415C22×C4C23×D6711yescharacteristic
416C22×C4C23×D6713yes—
417C22×C4C23×D67112yes—
418C22×C4C23×D67112yes—
419C22×C4C23×D67112yes—
420C22×C4C23×D6711yescharacteristic
421C22×C4C23×D6713yes—
422C22×C4C23×D6711yescharacteristic
423C4⋊C4—6724no—
424C4⋊C4—6748no—
425C4⋊C4—674no—
426C4⋊C4—6724no—
427C4⋊C4—6712no—
428C4⋊C4—6724no—
429C4⋊C4—6724no—
430C4⋊C4—6724no—
431C4⋊C4C23×D67112yes—
432C4⋊C4C23×D67112yes—
433C22⋊C4—6724no—
434C22⋊C4—6748no—
435C22⋊C4—678no—
436C22⋊C4—6724no—
437C22⋊C4C23×D67124yes—
438C42—6724no—
439C42—6724no—
440C42—6724no—
441C42C23×D67112yes—
442C42C23×D6714yes—
443C42C23×D67112yes—
444C42C23×D67112yes—
445C23—26no—
446C23—26no—
447C23—26no—
448C23—678no—
449C23—674no—
450C23—1344no—
451C23—1344no—
452C23C24×D6716yes—
453C23C24×D6711yescharacteristic
454Q8—13424no—
455Q8—13448no—
456Q8C2×#11761628116yescomplemented
457D4—224no—
458D4—6716no—
459D4—13448no—
460C2×C4—26no—
461C2×C4—22no—
462C2×C4—26no—
463C2×C4—224no—
464C2×C4—224no—
465C2×C4—224no—
466C2×C4—674no—
467C2×C4—6712no—
468C2×C4—6712no—
469C2×C4—13448no—
470C2×C4—13412no—
471C2×C4—13412no—
472C2×C4—1344no—
473C2×C4—13448no—
474C2×C4—13416no—
475C2×C4#41×D6714yes—
476C2×C4C2×#11761628112yes—
477C2×C4#1452409014yes—
478C2×C4C24×D6716yes—
479C2×C4C24×D6712yes—
480C2×C4C24×D6716yes—
481C2×C4C24×D6712yes—
482C2×C4C24×D6716yes—
483C2×C4C24×D6712yes—
484C22—26no—
485C22—26no—
486C22—26no—
487C22—412no—
488C22—678no—
489C22—13416no—
490C22—1344no—
491C22—1344no—
492C22C2×#41×D6712yes—
493C22C22×#1176162812yes—
494C22C2×#1452409012yes—
495C22C25×D6711yescharacteristic
496C4—26no—
497C4—22no—
498C4—26no—
499C4—13412no—
500C4—13412no—
501C4—1344no—
502C4C2×#41×D6714yes—
503C4C22×#11761628112yes—
504C4C2×#1452409014yes—
505C2—46no—
506C2—1348no—
507C2#2257383114yesdirect product factor
508C2C22×#41×D6711yescharacteristic
509C2C23×#1176162811yescharacteristic
510C2C22×#1452409011yescharacteristic
subgroupG/H
centerC22×C4C23×D67
commutator, cosocleC22×C67C26
FrattiniC22C25×D67
FittingC2×C4×#41×C67C2
socleC23×C67C25
hypercenterC2×C4×#41D67
hypocenterC67C2×#3482
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #22573831C2direct (N × H)
2. C2×#16864125C2semidirect
3. C2×#16864116C2semidirect
4. C2×#16864085C2semidirect
5. C2×Q8×D268C2semidirect
6. C2×#16864053C2semidirect
7. C2×#16864051C2semidirect
8. C2×#16863953C2semidirect
9. C2×#16863928C2semidirect
10. C2×C4×#11761628C2semidirect
11. C2×C4×Q8×D67C2semidirect
12. C2×C4×#41×C67C2semidirect
15. #16864125C22semidirect
16. #16864116C22semidirect
17. #16864085C22semidirect
18. Q8×D268C22semidirect
19. #16864053C22semidirect
20. #16864051C22semidirect
21. #16863953C22semidirect
24. #16863928C22semidirect
34. C4×#11761628C22semidirect
40. C4×Q8×D67C22semidirect
51. C4×#41×C67C22semidirect
136. C2×D268#41semidirect
166. C2×Q8×C67#41semidirect
198. Q8×C67C2×#41semidirect
222. D268C2×#41semidirect
251. C2×C67#3482semidirect
266. C2×C4×#41D67semidirect
268. C67C2×#3482semidirect
292. C4×#41C2×D67semidirect
394. C2×Q8#11761628semidirect
456. Q8C2×#11761628semidirect
507. C2#22573831direct (N × H)

Maximal subgroups

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

subgroupindex
#225738312
C2×#168641252
C2×#168641162
C2×#168640852
C2×Q8×D2682
C2×#168640532
C2×#168640512
C2×#168639532
C2×#168639282
C2×C4×#117616282
C2×C4×Q8×D672
C2×C4×#41×C672
C2×#348267