Finite Group Databasebeta

← C2×#22573804 #31762894

Subgroups — C2×#22573804 #31762894

Order 17152.

7802 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22573804C2132yesmaximal, direct product factor
2C2×#16864131C213yesmaximal, complemented
3C2×#16864125C211yesmaximal, characteristic, complemented
4C2×#16864119C213yesmaximal, complemented
5C2×#16864094C216yesmaximal, complemented
6C2×#16864068C211yesmaximal, characteristic, complemented
7C2×#16864057C212yesmaximal, complemented
8C2×#16863946C213yesmaximal, complemented
9C2×#16863933C216yesmaximal, complemented
10C2×C4×#11761629C212yesmaximal, complemented
11C2×C4×#11761628C213yesmaximal, complemented
12C22×C4×Q8×C67C211yesmaximal, characteristic, complemented
13C22×#11761629C2212yes—
14C22×#11761628C2213yes—
15#16864131C22148yescomplemented
16#16864125C22116yescomplemented
17#16864119C22148yescomplemented
18#16864094C22196yescomplemented
19#16864068C22116yescomplemented
20#16864057C22132yescomplemented
21#16863946C22148yescomplemented
22C2×#14524043C2213yes—
23C2×#14524040C2213yes—
24#16863933C22196yescomplemented
25C2×#14524038C22112yes—
26C2×#14524034C2216yes—
27C2×#14524032C2216yes—
28C2×#14524031C2216yes—
29C2×#14524028C22112yes—
30C2×#14524019C2216yes—
31C2×#14524018C2216yes—
32C2×#14524017C2216yes—
33C2×#14524017C2216yes—
34C4×#11761629C22132yescomplemented
35C4×#11761628C22148yescomplemented
36C2×Q8×Dic67C2212yes—
37C2×C4×Dic134C2213yes—
38C2×C4×Dic134C2213yes—
39C2×C4×D268C2213yes—
40C2×C4×D268C2213yes—
41C2×C4×D268C2216yes—
42C2×C4×D268C2216yes—
43C2×#14524005C2216yes—
44C2×C4×C22⋊D67C2216yes—
45C2×C4×C22⋊D67C2216yes—
46C2×#14524002C2211yescharacteristic
47C2×#14524001C2216yes—
48C2×#14524000C2216yes—
49C2×C42×D67C2216yes—
50C23×Q8×C67C2211yescharacteristic
51C2×C4×Q8×C67C2211yescharacteristic
52C2×C4×Q8×C67C2211yescharacteristic
53C2×C4×Q8×C67C2213yes—
54C2×C4×Q8×C67C2213yes—
55C2×C4×Q8×C67C22116yescomplemented
56C22×C4⋊C4×C67C2213yes—
57C22×C42×C67C2213yes—
58C2×#11761629—216no—
59C2×#11761629C23116yes—
60C2×#11761628—224no—
61C2×#11761628C23124yes—
62C22×Dic134C2313yes—
63C22×D268C2316yes—
64C22×D268C2313yes—
65C22×D268C2316yes—
66C22×C22⋊D67C2316yes—
67C22×C4×D67C2316yes—
68C22×C4×D67C2316yes—
69C22×C4×D67C2316yes—
70#14524043C23124yes—
71#14524040C23124yes—
72#14524038C23196yes—
73#14524034C23148yes—
74#14524032C23148yes—
75#14524031C23148yes—
76#14524028C23196yes—
77#14524019C23148yes—
78#14524018C23148yes—
79#14524017C23148yes—
80#14524017C23148yes—
81Q8×Dic67C23116yes—
82C4×Dic134C23124yes—
83C4×Dic134C23124yes—
84C4×D268C23124yes—
85C4×D268C23124yes—
86C4×D268C23148yes—
87C4×D268C23148yes—
88#14524005C23148yes—
89C4×C22⋊D67C23148yes—
90C4×C22⋊D67C23148yes—
91#14524002C2318yes—
92#14524001C23148yes—
93#14524000C23148yes—
94C2×C4⋊Dic67C2316yes—
95C2×C4⋊Dic67C2311yescharacteristic
96C2×C4⋊Dic67C2311yescharacteristic
97C2×C4⋊Dic67C2316yes—
98C2×Dic67⋊C4C2316yes—
99C2×Dic67⋊C4C2316yes—
100C2×C22⋊Dic67C2312yes—
101C2×(C2×D67)⋊C4C2316yes—
102C2×(C2×D67)⋊C4C2316yes—
103C2×(C2×D67)⋊C4C23112yes—
104C2×(C2×D67)⋊C4C23112yes—
105C2×C4×Dic67C2312yes—
106C2×C4×Dic67C2316yes—
107C2×C4×Dic67C2316yes—
108C42×D67C23148yes—
109C22×Q8×C67—28no—
110C22×Q8×C67C2318yes—
111C22×Q8×C67C2313yes—
112C22×Q8×C67C2311yescharacteristic
113C23×C4×C67C2313yes—
114C23×C4×C67C2313yes—
115C23×C4×C67C2311yescharacteristic
116C4×Q8×C67C2318yes—
117C4×Q8×C67C23124yes—
118C4×Q8×C67C2318yes—
119C4×Q8×C67C23124yes—
120C2×C4⋊C4×C67C2313yes—
121C2×C4⋊C4×C67C2313yes—
122C2×C4⋊C4×C67C2316yes—
123C2×C4⋊C4×C67C23124yes—
124C2×C42×C67C2313yes—
125C2×C42×C67C2313yes—
126C2×C42×C67C2313yes—
127C2×C42×C67C2313yes—
128C2×C42×C67C23124yes—
129C23×D67C2416yes—
130#11761629—264no—
131#11761628—296no—
132C2×Dic134—212no—
133C2×Dic134C24112yes—
134C2×D268—212no—
135C2×D268—224no—
136C2×D268—412no—
137C2×D268C24124yes—
138C2×D268C24112yes—
139C2×D268C24124yes—
140C2×C22⋊D67—224no—
141C2×C22⋊D67C24124yes—
142C22×Dic67C2412yes—
143C22×Dic67C2412yes—
144C22×Dic67C2416yes—
145C2×C4×D67—224no—
146C2×C4×D67—224no—
147C2×C4×D67—412no—
148C2×C4×D67C24124yes—
149C2×C4×D67C24124yes—
150C2×C4×D67C24124yes—
151C4⋊Dic67C24124yes—
152C4⋊Dic67C2414yes—
153C4⋊Dic67C2414yes—
154C4⋊Dic67C24124yes—
155Dic67⋊C4C24124yes—
156Dic67⋊C4C24124yes—
157C22⋊Dic67C2418yes—
158(C2×D67)⋊C4C24124yes—
159(C2×D67)⋊C4C24124yes—
160(C2×D67)⋊C4C24148yes—
161(C2×D67)⋊C4C24148yes—
162C4×Dic67C2418yes—
163C4×Dic67C24124yes—
164C4×Dic67C24124yes—
165C24×C67C2411yescharacteristic
166C2×Q8×C67—232no—
167C2×Q8×C67—212no—
168C2×Q8×C67#4118yescomplemented
169C2×Q8×C67C24112yes—
170C2×Q8×C67C2414yes—
171C22×C4×C67—212no—
172C22×C4×C67—46no—
173C22×C4×C67#4118yescomplemented
174C22×C4×C67C2413yes—
175C22×C4×C67C2413yes—
176C22×C4×C67C2413yes—
177C22×C4×C67C24112yes—
178C22×C4×C67C24112yes—
179C22×C4×C67C2411yescharacteristic
180C22×C4×C67C2413yes—
181C22×C4×C67C2414yes—
182C22×C4×C67C2411yescharacteristic
183C4⋊C4×C67C24112yes—
184C4⋊C4×C67C24112yes—
185C4⋊C4×C67C24124yes—
186C42×C67C24112yes—
187C42×C67C24112yes—
188C42×C67C24112yes—
189C42×C67C24112yes—
190C23×C67—22no—
191C23×C67—22no—
192C23×C67C2×#4114yes—
193C23×C67C2512yes—
194C23×C67C2511yescharacteristic
195C22×D67—212no—
196C22×D67—212no—
197C22×D67—212no—
198C22×D67C25112yes—
199Q8×C67—224no—
200Q8×C67C2×#41116yescomplemented
201C2×C4×C67—26no—
202C2×C4×C67—26no—
203C2×C4×C67—26no—
204C2×C4×C67—224no—
205C2×C4×C67—412no—
206C2×C4×C67C2×#41112yes—
207C2×C4×C67C2×#4114yes—
208C2×C4×C67C2×#4114yes—
209C2×C4×C67C2×#41116yescomplemented
210C2×C4×C67C2516yes—
211C2×C4×C67C2516yes—
212C2×C4×C67C2516yes—
213C2×C4×C67C2512yes—
214C2×C4×C67C2512yes—
215C2×C4×C67C2516yes—
216C22⋊D67—248no—
217C2×Dic67—24no—
218C2×Dic67—212no—
219C2×Dic67—24no—
220C2×Dic67C2514yes—
221C2×Dic67C2514yes—
222C2×Dic67C25112yes—
223D268—224no—
224D268—248no—
225D268—424no—
226C4×D67—248no—
227C4×D67—248no—
228C4×D67—424no—
229Dic134—224no—
230C22×C67—22no—
231C22×C67—22no—
232C22×C67—28no—
233C22×C67C22×#4112yes—
234C22×C67C22×#4114yes—
235C22×C67C22×#4112yes—
236C22×C67#58512yes—
237C22×C67C2611yescharacteristic
238C2×D67—212no—
239C2×D67—212no—
240C2×D67—212no—
241C2×D67—424no—
242C4×C67—26no—
243C4×C67—26no—
244C4×C67—26no—
245C4×C67C22×#41112yes—
246C4×C67C22×#4114yes—
247C4×C67C22×#4114yes—
248Dic67—212no—
249Dic67—24no—
250Dic67—24no—
251C2×#3480—671nomaximal, Sylow
252C2×C67—24no—
253C2×C67#348014yescomplemented
254C2×C67C23×#4111yescharacteristic
255C2×C67C23×#4111yescharacteristic
256C2×C67C2×#58511yescharacteristic
257D67—412no—
258#3480—6732no—
259C2×#559—673no—
260C2×#557—671no—
261C2×#556—673no—
262C2×#553—676no—
263C2×#550—671no—
264C2×#548—672no—
265C2×#532—673no—
266C2×#529—676no—
267C2×C4×#41—673no—
268C2×C4×#41—672no—
269C22×C4×Q8D6711yescharacteristic, complemented
270C67C2×#348011yescharacteristic, complemented, Sylow
271C22×#41—672no—
272C22×#41—673no—
273C23×Q8C2×D6711yescharacteristic
274#559—6748no—
275#557—6716no—
276#556—6748no—
277#553—6796no—
278#550—6716no—
279#548—6732no—
280#532—6748no—
281C2×C4⋊Q8—673no—
282C2×#127—673no—
283#529—6796no—
284C2×#126—6712no—
285C2×#125—676no—
286C2×#124—676no—
287C2×#124—676no—
288C2×#123—6712no—
289C2×C22⋊Q8—676no—
290C2×#121—676no—
291C2×#121—676no—
292C2×#121—676no—
293C4×#41—6732no—
294C4×#41—6748no—
295C2×C4×Q8—672no—
296C2×C4×Q8—673no—
297C2×C4×Q8—673no—
298C2×C4×Q8C2×D6711yescharacteristic
299C2×C4×Q8C2×D6713yes—
300C2×C4×Q8C2×D6711yescharacteristic
301C2×C4×Q8C2×D6713yes—
302C2×C4×Q8C2×D67116yescomplemented
303C2×C4×D4—676no—
304C2×C4×D4—676no—
305C2×C4×D4—676no—
306C2×C4×D4—676no—
307C2×C4×D4—673no—
308C2×C4×D4—673no—
309C2×C4×D4—676no—
310C2×#117—671no—
311C2×#117—676no—
312C2×#117—676no—
313C22×C4⋊C4C2×D6713yes—
314C22×C42—676no—
315C22×C42C2×D6713yes—
316C2×#41—6716no—
317C2×#41—6724no—
318C2×#41—13416no—
319C2×#41—13424no—
320C22×Q8—28no—
321C22×Q8—673no—
322C22×Q8C22×D6713yes—
323C22×Q8C22×D6718yes—
324C22×Q8C22×D6711yescharacteristic
325C22×D4—676no—
326C22×D4—676no—
327C22×D4—676no—
328C22×D4—673no—
329C23×C4—676no—
330C23×C4—676no—
331C23×C4—676no—
332C23×C4C22×D6713yes—
333C23×C4C22×D6713yes—
334C23×C4C22×D6711yescharacteristic
335C4⋊Q8—6724no—
336#127—6724no—
337#126—6796no—
338#125—6748no—
339#124—6748no—
340#124—6748no—
341#123—6796no—
342C22⋊Q8—6748no—
343#121—6748no—
344#121—6748no—
345#121—6748no—
346C4×Q8—6716no—
347C4×Q8—6724no—
348C4×Q8—6724no—
349C4×Q8C22×D6718yes—
350C4×Q8C22×D67124yes—
351C4×Q8C22×D6718yes—
352C4×Q8C22×D67124yes—
353C4×D4—6748no—
354C4×D4—6748no—
355C4×D4—6748no—
356C4×D4—6748no—
357C4×D4—6724no—
358C4×D4—6724no—
359C4×D4—6748no—
360#117—678no—
361#117—6748no—
362#117—6748no—
363C2×C4⋊C4—671no—
364C2×C4⋊C4—671no—
365C2×C4⋊C4—676no—
366C2×C4⋊C4—676no—
367C2×C4⋊C4—676no—
368C2×C4⋊C4—676no—
369C2×C4⋊C4C22×D6713yes—
370C2×C4⋊C4C22×D6713yes—
371C2×C4⋊C4C22×D6716yes—
372C2×C4⋊C4C22×D67124yes—
373C2×C22⋊C4—672no—
374C2×C22⋊C4—676no—
375C2×C22⋊C4—6712no—
376C2×C22⋊C4—676no—
377C2×C22⋊C4—6712no—
378C2×C42—676no—
379C2×C42—676no—
380C2×C42—672no—
381C2×C42—6748no—
382C2×C42C22×D6713yes—
383C2×C42C22×D6713yes—
384C2×C42C22×D67124yes—
385C2×C42C22×D6713yes—
386C2×C42C22×D6713yes—
387C24—676no—
388C24C23×D6711yescharacteristic
389#41—13464no—
390#41—13496no—
391C2×Q8—232no—
392C2×Q8—212no—
393C2×Q8—6712no—
394C2×Q8—13412no—
395C2×Q8#1176162818yescomplemented
396C2×Q8C23×D67112yes—
397C2×Q8C23×D6714yes—
398C2×D4—26812no—
399C2×D4—6724no—
400C2×D4—6724no—
401C2×D4—6724no—
402C2×D4—6712no—
403C2×D4—13424no—
404C2×D4—13424no—
405C2×D4—13412no—
406C22×C4—212no—
407C22×C4—46no—
408C22×C4—26812no—
409C22×C4—6724no—
410C22×C4—672no—
411C22×C4—672no—
412C22×C4—676no—
413C22×C4—6724no—
414C22×C4—6724no—
415C22×C4—13424no—
416C22×C4—13424no—
417C22×C4#1176162918yescomplemented
418C22×C4C23×D6713yes—
419C22×C4C23×D6713yes—
420C22×C4C23×D6713yes—
421C22×C4C23×D67112yes—
422C22×C4C23×D67112yes—
423C22×C4C23×D6713yes—
424C22×C4C23×D6711yescharacteristic
425C22×C4C23×D6714yes—
426C22×C4C23×D6711yescharacteristic
427C4⋊C4—674no—
428C4⋊C4—674no—
429C4⋊C4—6724no—
430C4⋊C4—6724no—
431C4⋊C4—6724no—
432C4⋊C4—6724no—
433C4⋊C4C23×D67112yes—
434C4⋊C4C23×D67112yes—
435C4⋊C4C23×D67124yes—
436C22⋊C4—678no—
437C22⋊C4—6724no—
438C22⋊C4—6748no—
439C22⋊C4—6724no—
440C22⋊C4—6748no—
441C42—678no—
442C42—6724no—
443C42—6724no—
444C42C23×D67112yes—
445C42C23×D67112yes—
446C42C23×D67112yes—
447C42C23×D67112yes—
448C23—22no—
449C23—22no—
450C23—6712no—
451C23—13412no—
452C23—13412no—
453C23—13412no—
454C23C2×#1176162914yes—
455C23C24×D6712yes—
456C23C24×D6711yescharacteristic
457Q8—224no—
458Q8—13424no—
459Q8C2×#11761628116yescomplemented
460D4—26824no—
461D4—13448no—
462D4—13448no—
463D4—13424no—
464C2×C4—26no—
465C2×C4—26no—
466C2×C4—26no—
467C2×C4—224no—
468C2×C4—412no—
469C2×C4—26824no—
470C2×C4—674no—
471C2×C4—674no—
472C2×C4—6712no—
473C2×C4—1344no—
474C2×C4—13412no—
475C2×C4—1344no—
476C2×C4—13448no—
477C2×C4—13448no—
478C2×C4C2×#11761628112yes—
479C2×C4C2×#1176162914yes—
480C2×C4C2×#1176162914yes—
481C2×C4C2×#11761629116yescomplemented
482C2×C4C24×D6716yes—
483C2×C4C24×D6716yes—
484C2×C4C24×D6716yes—
485C2×C4C24×D6716yes—
486C2×C4C24×D6712yes—
487C2×C4C24×D6712yes—
488C22—22no—
489C22—22no—
490C22—28no—
491C22—26824no—
492C22—13412no—
493C22—13412no—
494C22—13412no—
495C22C22×#1176162812yes—
496C22C22×#1176162914yes—
497C22C22×#1176162912yes—
498C22#1686423212yes—
499C22C25×D6711yescharacteristic
500C4—26no—
501C4—26no—
502C4—26no—
503C4—1344no—
504C4—13412no—
505C4—1344no—
506C4C22×#11761628112yes—
507C4C22×#1176162914yes—
508C4C22×#1176162914yes—
509C2—24no—
510C2—26812no—
511C2#2257380414yesdirect product factor
512C2C23×#1176162811yescharacteristic
513C2C23×#1176162911yescharacteristic
514C2C2×#1686423211yescharacteristic
subgroupG/H
centerC22×C4C23×D67
commutator, cosocleC22×C67C26
FrattiniC22C25×D67
FittingC22×C4×Q8×C67C2
socleC23×C67C25
hypercenterC22×C4×Q8D67
hypocenterC67C2×#3480
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22573804C2direct (N × H)
2. C2×#16864131C2semidirect
3. C2×#16864125C2semidirect
4. C2×#16864119C2semidirect
5. C2×#16864094C2semidirect
6. C2×#16864068C2semidirect
7. C2×#16864057C2semidirect
8. C2×#16863946C2semidirect
9. C2×#16863933C2semidirect
10. C2×C4×#11761629C2semidirect
11. C2×C4×#11761628C2semidirect
12. C22×C4×Q8×C67C2semidirect
15. #16864131C22semidirect
16. #16864125C22semidirect
17. #16864119C22semidirect
18. #16864094C22semidirect
19. #16864068C22semidirect
20. #16864057C22semidirect
21. #16863946C22semidirect
24. #16863933C22semidirect
34. C4×#11761629C22semidirect
35. C4×#11761628C22semidirect
55. C2×C4×Q8×C67C22semidirect
168. C2×Q8×C67#41semidirect
173. C22×C4×C67#41semidirect
200. Q8×C67C2×#41semidirect
209. C2×C4×C67C2×#41semidirect
253. C2×C67#3480semidirect
269. C22×C4×Q8D67semidirect
270. C67C2×#3480semidirect
302. C2×C4×Q8C2×D67semidirect
395. C2×Q8#11761628semidirect
417. C22×C4#11761629semidirect
459. Q8C2×#11761628semidirect
481. C2×C4C2×#11761629semidirect
511. C2#22573804direct (N × H)

Maximal subgroups

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

subgroupindex
#225738042
C2×#168641312
C2×#168641252
C2×#168641192
C2×#168640942
C2×#168640682
C2×#168640572
C2×#168639462
C2×#168639332
C2×C4×#117616292
C2×C4×#117616282
C22×C4×Q8×C672
C2×#348067