Finite Group Databasebeta

← C2×#22602425 #32905663

Subgroups — C2×#22602425 #32905663

Order 18176.

7758 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#22602425C2132yesmaximal, direct product factor
2C2×#17986734C213yesmaximal, complemented
3C2×Q8×Dic142C211yesmaximal, characteristic, complemented
4C2×#17986709C213yesmaximal, complemented
5C2×#17986671C211yesmaximal, characteristic, complemented
6C2×#17986660C212yesmaximal, complemented
7C2×#17986650C216yesmaximal, complemented
8C2×#17986621C216yesmaximal, complemented
9C2×#17986549C213yesmaximal, complemented
10C2×#17986533C213yesmaximal, complemented
11C2×#17986481C212yesmaximal, complemented
12C22×C4×Q8×C71C211yesmaximal, characteristic, complemented
13C22×#11765866C2212yes—
14C22×#11765865C2216yes—
15#17986734C22148yescomplemented
16Q8×Dic142C22116yescomplemented
17#17986709C22148yescomplemented
18#17986671C22116yescomplemented
19#17986660C22132yescomplemented
20#17986650C22196yescomplemented
21#17986621C22196yescomplemented
22#17986549C22148yescomplemented
23C2×#14556128C2213yes—
24C2×#14556127C2216yes—
25C2×#14556125C2213yes—
26#17986533C22148yescomplemented
27C2×#14556122C2216yes—
28C2×#14556120C2213yes—
29C2×#14556117C2216yes—
30C2×#14556116C2213yes—
31C2×#14556116C2216yes—
32C2×#14556113C22112yes—
33C2×#14556111C2216yes—
34C2×#14556107C2211yescharacteristic
35C2×#14556106C22112yes—
36C2×#14556104C2216yes—
37C2×#14556103C2216yes—
38C2×#14556102C2216yes—
39C2×#14556102C2213yes—
40#17986481C22132yescomplemented
41C2×Q8×Dic71C2212yes—
42C2×C4×Dic142C2213yes—
43C2×C4×D284C2216yes—
44C2×C4×D284C2213yes—
45C2×C4×D284C2216yes—
46C2×#14556090C2216yes—
47C2×C4×C22⋊D71C2216yes—
48C2×#14556085C2216yes—
49C2×C4⋊C4×D71C2216yes—
50C23×Q8×C71C2211yescharacteristic
51C2×C4×Q8×C71C2211yescharacteristic
52C2×C4×Q8×C71C2211yescharacteristic
53C2×C4×Q8×C71C2213yes—
54C2×C4×Q8×C71C2213yes—
55C2×C4×Q8×C71C22116yescomplemented
56C22×C4⋊C4×C71C2213yes—
57C22×C42×C71C2213yes—
58C2×#11765866—216no—
59C2×#11765866C23116yes—
60C2×#11765865—424no—
61C2×#11765865C23148yes—
62C22×Dic142C2311yescharacteristic
63C22×Dic142C2316yes—
64C22×D284C2316yes—
65C22×D284C2313yes—
66C22×D284C2316yes—
67C22×C22⋊D71C2316yes—
68C22×C4×D71C2316yes—
69C22×C4×D71C23112yes—
70#14556128C23124yes—
71#14556127C23148yes—
72#14556125C23124yes—
73#14556122C23148yes—
74#14556120C23124yes—
75#14556117C23148yes—
76#14556116C23148yes—
77#14556116C23124yes—
78#14556113C23196yes—
79#14556111C23148yes—
80#14556107C2318yes—
81#14556106C23196yes—
82#14556104C23148yes—
83#14556103C23148yes—
84#14556102C23148yes—
85#14556102C23124yes—
86Q8×Dic71C23116yes—
87C4×Dic142C23124yes—
88C4×D284C23148yes—
89C4×D284C23124yes—
90C4×D284C23148yes—
91#14556090C23148yes—
92C4×C22⋊D71C23148yes—
93#14556085C23148yes—
94C2×C4⋊Dic71C2316yes—
95C2×C4⋊Dic71C2311yescharacteristic
96C2×C4⋊Dic71C2313yes—
97C2×Dic71⋊C4C2316yes—
98C2×Dic71⋊C4C2312yes—
99C2×Dic71⋊C4C2316yes—
100C2×Dic71⋊C4C2316yes—
101C4⋊C4×D71C23148yes—
102C2×C22⋊Dic71C2312yes—
103C2×(C2×D71)⋊C4C2316yes—
104C2×(C2×D71)⋊C4C2316yes—
105C2×(C2×D71)⋊C4C2316yes—
106C2×(C2×D71)⋊C4C23112yes—
107C2×(C2×D71)⋊C4C2316yes—
108C2×C4×Dic71C2316yes—
109C22×Q8×C71—28no—
110C22×Q8×C71C2318yes—
111C22×Q8×C71C2313yes—
112C22×Q8×C71C2311yescharacteristic
113C23×C4×C71C2313yes—
114C23×C4×C71C2311yescharacteristic
115C23×C4×C71C2313yes—
116C4×Q8×C71C23124yes—
117C4×Q8×C71C2318yes—
118C4×Q8×C71C23124yes—
119C4×Q8×C71C2318yes—
120C2×C4⋊C4×C71C23124yes—
121C2×C4⋊C4×C71C2313yes—
122C2×C4⋊C4×C71C2313yes—
123C2×C4⋊C4×C71C2316yes—
124C2×C42×C71C2313yes—
125C2×C42×C71C2313yes—
126C2×C42×C71C2313yes—
127C2×C42×C71C2313yes—
128C2×C42×C71C23124yes—
129C23×D71C2416yes—
130#11765866—264no—
131#11765865—496no—
132C2×Dic142—24no—
133C2×Dic142—412no—
134C2×Dic142C2414yes—
135C2×Dic142C24124yes—
136C2×D284—224no—
137C2×D284—212no—
138C2×D284—412no—
139C2×D284C24124yes—
140C2×D284C24112yes—
141C2×D284C24124yes—
142C2×C22⋊D71—224no—
143C2×C22⋊D71C24124yes—
144C22×Dic71C2412yes—
145C22×Dic71C2416yes—
146C22×Dic71C2412yes—
147C2×C4×D71—224no—
148C2×C4×D71—424no—
149C2×C4×D71C24124yes—
150C2×C4×D71C24148yes—
151C4⋊Dic71C24124yes—
152C4⋊Dic71C2414yes—
153C4⋊Dic71C24112yes—
154Dic71⋊C4C24124yes—
155Dic71⋊C4C2418yes—
156Dic71⋊C4C24124yes—
157Dic71⋊C4C24124yes—
158C22⋊Dic71C2418yes—
159(C2×D71)⋊C4C24124yes—
160(C2×D71)⋊C4C24124yes—
161(C2×D71)⋊C4C24124yes—
162(C2×D71)⋊C4C24148yes—
163(C2×D71)⋊C4C24124yes—
164C4×Dic71C24124yes—
165C24×C71C2411yescharacteristic
166C2×Q8×C71—232no—
167C2×Q8×C71—212no—
168C2×Q8×C71#4118yescomplemented
169C2×Q8×C71C24112yes—
170C2×Q8×C71C2414yes—
171C22×C4×C71—212no—
172C22×C4×C71—212no—
173C22×C4×C71—24no—
174C22×C4×C71C2413yes—
175C22×C4×C71C2413yes—
176C22×C4×C71C24112yes—
177C22×C4×C71C2411yescharacteristic
178C22×C4×C71C2414yes—
179C22×C4×C71C2413yes—
180C22×C4×C71C24112yes—
181C22×C4×C71C2413yes—
182C22×C4×C71C2411yescharacteristic
183C4⋊C4×C71C24112yes—
184C4⋊C4×C71C24112yes—
185C4⋊C4×C71C24124yes—
186C42×C71C24112yes—
187C42×C71C24112yes—
188C42×C71C24112yes—
189C42×C71C24112yes—
190C23×C71—22no—
191C23×C71—22no—
192C23×C71H(5,𝔽2)14yes—
193C23×C71C2512yes—
194C23×C71C2511yescharacteristic
195C22×D71—212no—
196C22×D71—212no—
197C22×D71—212no—
198C22×D71C25112yes—
199Q8×C71—224no—
200Q8×C71C2×#41116yescomplemented
201C2×C4×C71—26no—
202C2×C4×C71—26no—
203C2×C4×C71—26no—
204C2×C4×C71—224no—
205C2×C4×C71—22no—
206C2×C4×C71—224no—
207C2×C4×C71—28no—
208C2×C4×C71C2×#41112yes—
209C2×C4×C712−1+414yes—
210C2×C4×C71C2512yes—
211C2×C4×C71C2516yes—
212C2×C4×C71C2516yes—
213C2×C4×C71C2516yes—
214C2×C4×C71C2516yes—
215C2×C4×C71C2512yes—
216C22⋊D71—248no—
217C2×Dic71—24no—
218C2×Dic71—212no—
219C2×Dic71—24no—
220C2×Dic71C2514yes—
221C2×Dic71C25112yes—
222C2×Dic71C2514yes—
223D284—248no—
224D284—224no—
225D284—424no—
226C4×D71—248no—
227C4×D71—448no—
228Dic142—28no—
229Dic142—424no—
230C22×C71—22no—
231C22×C71—22no—
232C22×C71—28no—
233C22×C71C22×#4112yes—
234C22×C71C2×H(5,𝔽2)14yes—
235C22×C71C2×H(5,𝔽2)12yes—
236C22×C71C2×2−1+412yes—
237C22×C71C2611yescharacteristic
238C2×D71—212no—
239C2×D71—212no—
240C2×D71—212no—
241C2×D71—424no—
242C4×C71—26no—
243C4×C71—26no—
244C4×C71—26no—
245C4×C71—22no—
246C4×C71C22×#41112yes—
247C4×C71C2×2−1+414yes—
248Dic71—24no—
249Dic71—212no—
250Dic71—24no—
251C2×#3522—711nomaximal, Sylow
252C2×C71—24no—
253C2×C71#352214yescomplemented
254C2×C71C23×#4111yescharacteristic
255C2×C71C22×H(5,𝔽2)11yescharacteristic
256C2×C71C22×2−1+411yescharacteristic
257D71—412no—
258#3522—7132no—
259C2×#559—713no—
260C2×Q82—711no—
261C2×#554—713no—
262C2×#550—711no—
263C2×#548—712no—
264C2×#547—716no—
265C2×#542—716no—
266C2×#532—713no—
267C2×#529—713no—
268C2×#520—712no—
269C22×C4×Q8D7111yescharacteristic, complemented
270C71C2×#352211yescharacteristic, complemented, Sylow
271C22×#41—712no—
272C22×#41—716no—
273C23×Q8C2×D7111yescharacteristic
274#559—7148no—
275Q82—7116no—
276#554—7148no—
277#550—7116no—
278#548—7132no—
279#547—7196no—
280#542—7196no—
281#532—7148no—
282C2×C4⋊Q8—716no—
283C2×C4⋊Q8—713no—
284C2×#127—713no—
285#529—7148no—
286C2×#126—716no—
287C2×#125—713no—
288C2×#124—716no—
289C2×#124—713no—
290C2×#124—716no—
291C2×#123—716no—
292C2×#123—7112no—
293C2×C22⋊Q8—716no—
294C2×C22⋊Q8—711no—
295C2×C22⋊Q8—7112no—
296C2×#121—713no—
297C2×#121—716no—
298C2×#121—716no—
299#520—7132no—
300C2×C4×Q8—712no—
301C2×C4×Q8—713no—
302C2×C4×Q8C2×D7111yescharacteristic
303C2×C4×Q8C2×D7113yes—
304C2×C4×Q8C2×D7111yescharacteristic
305C2×C4×Q8C2×D7113yes—
306C2×C4×Q8C2×D71116yescomplemented
307C2×C4×D4—716no—
308C2×C4×D4—716no—
309C2×C4×D4—716no—
310C2×C4×D4—713no—
311C2×C4×D4—716no—
312C2×#117—716no—
313C22×C4⋊C4—716no—
314C22×C4⋊C4C2×D7113yes—
315C22×C42C2×D7113yes—
316C2×#41—28424no—
317C2×#41—7116no—
318C2×#41—7148no—
319C2×#41—14216no—
320C22×Q8—28no—
321C22×Q8—716no—
322C22×Q8—711no—
323C22×Q8C22×D7113yes—
324C22×Q8C22×D7118yes—
325C22×Q8C22×D7111yescharacteristic
326C22×D4—716no—
327C22×D4—716no—
328C22×D4—713no—
329C22×D4—716no—
330C23×C4—7112no—
331C23×C4—716no—
332C23×C4C22×D7113yes—
333C23×C4C22×D7113yes—
334C23×C4C22×D7111yescharacteristic
335C4⋊Q8—7148no—
336C4⋊Q8—7124no—
337#127—7124no—
338#126—7148no—
339#125—7124no—
340#124—7148no—
341#124—7124no—
342#124—7148no—
343#123—7148no—
344#123—7196no—
345C22⋊Q8—7148no—
346C22⋊Q8—718no—
347C22⋊Q8—7196no—
348#121—7124no—
349#121—7148no—
350#121—7148no—
351C4×Q8—7116no—
352C4×Q8—7124no—
353C4×Q8C22×D7118yes—
354C4×Q8C22×D71124yes—
355C4×Q8C22×D7118yes—
356C4×Q8C22×D71124yes—
357C4×D4—7148no—
358C4×D4—7148no—
359C4×D4—7148no—
360C4×D4—7124no—
361C4×D4—7148no—
362#117—7148no—
363C2×C4⋊C4—716no—
364C2×C4⋊C4—716no—
365C2×C4⋊C4—712no—
366C2×C4⋊C4—711no—
367C2×C4⋊C4—713no—
368C2×C4⋊C4—716no—
369C2×C4⋊C4—716no—
370C2×C4⋊C4—7148no—
371C2×C4⋊C4C22×D7113yes—
372C2×C4⋊C4C22×D7113yes—
373C2×C4⋊C4C22×D7116yes—
374C2×C4⋊C4C22×D71124yes—
375C2×C22⋊C4—712no—
376C2×C22⋊C4—716no—
377C2×C22⋊C4—716no—
378C2×C22⋊C4—716no—
379C2×C22⋊C4—7112no—
380C2×C22⋊C4—716no—
381C2×C42—716no—
382C2×C42C22×D7113yes—
383C2×C42C22×D7113yes—
384C2×C42C22×D7113yes—
385C2×C42C22×D7113yes—
386C2×C42C22×D71124yes—
387C24—716no—
388C24C23×D7111yescharacteristic
389#41—28496no—
390#41—14264no—
391C2×Q8—232no—
392C2×Q8—212no—
393C2×Q8—28412no—
394C2×Q8—7124no—
395C2×Q8—714no—
396C2×Q8—1424no—
397C2×Q8#1176586518yescomplemented
398C2×Q8C23×D71112yes—
399C2×Q8C23×D7114yes—
400C2×D4—28412no—
401C2×D4—7124no—
402C2×D4—7124no—
403C2×D4—7112no—
404C2×D4—7124no—
405C2×D4—14224no—
406C2×D4—14224no—
407C2×D4—14212no—
408C22×C4—212no—
409C22×C4—212no—
410C22×C4—24no—
411C22×C4—28424no—
412C22×C4—7148no—
413C22×C4—712no—
414C22×C4—716no—
415C22×C4—712no—
416C22×C4—7124no—
417C22×C4—14224no—
418C22×C4C23×D7114yes—
419C22×C4C23×D71112yes—
420C22×C4C23×D71112yes—
421C22×C4C23×D7111yescharacteristic
422C22×C4C23×D7113yes—
423C22×C4C23×D7113yes—
424C22×C4C23×D7113yes—
425C22×C4C23×D7113yes—
426C22×C4C23×D7111yescharacteristic
427C4⋊C4—7124no—
428C4⋊C4—718no—
429C4⋊C4—7124no—
430C4⋊C4—7124no—
431C4⋊C4—7124no—
432C4⋊C4—714no—
433C4⋊C4—7112no—
434C4⋊C4C23×D71112yes—
435C4⋊C4C23×D71112yes—
436C4⋊C4C23×D71124yes—
437C22⋊C4—718no—
438C22⋊C4—7124no—
439C22⋊C4—7124no—
440C22⋊C4—7124no—
441C22⋊C4—7148no—
442C22⋊C4—7124no—
443C42—7124no—
444C42C23×D71112yes—
445C42C23×D71112yes—
446C42C23×D71112yes—
447C42C23×D71112yes—
448C23—22no—
449C23—22no—
450C23—7112no—
451C23—14212no—
452C23—14212no—
453C23—14212no—
454C23#1455617414yes—
455C23C24×D7112yes—
456C23C24×D7111yescharacteristic
457Q8—224no—
458Q8—28424no—
459Q8—1428no—
460Q8C2×#11765865116yescomplemented
461D4—28424no—
462D4—14248no—
463D4—14248no—
464D4—14224no—
465C2×C4—26no—
466C2×C4—26no—
467C2×C4—26no—
468C2×C4—224no—
469C2×C4—22no—
470C2×C4—224no—
471C2×C4—28no—
472C2×C4—28448no—
473C2×C4—714no—
474C2×C4—7112no—
475C2×C4—714no—
476C2×C4—1424no—
477C2×C4—14212no—
478C2×C4—1424no—
479C2×C4—14248no—
480C2×C4C2×#11765865112yes—
481C2×C4#1455617614yes—
482C2×C4C24×D7112yes—
483C2×C4C24×D7116yes—
484C2×C4C24×D7116yes—
485C2×C4C24×D7116yes—
486C2×C4C24×D7116yes—
487C2×C4C24×D7112yes—
488C22—22no—
489C22—22no—
490C22—28no—
491C22—28424no—
492C22—14212no—
493C22—14212no—
494C22—14212no—
495C22C22×#1176586512yes—
496C22C2×#1455617414yes—
497C22C2×#1455617412yes—
498C22C2×#1455617612yes—
499C22C25×D7111yescharacteristic
500C4—26no—
501C4—26no—
502C4—26no—
503C4—22no—
504C4—1424no—
505C4—14212no—
506C4—1424no—
507C4C22×#11765865112yes—
508C4C2×#1455617614yes—
509C2—24no—
510C2—28412no—
511C2#2260242514yesdirect product factor
512C2C23×#1176586511yescharacteristic
513C2C22×#1455617411yescharacteristic
514C2C22×#1455617611yescharacteristic
subgroupG/H
centerC23C24×D71
commutator, cosocleC22×C71C26
FrattiniC22C25×D71
FittingC22×C4×Q8×C71C2
socleC23×C71C25
hypercenterC22×C4×Q8D71
hypocenterC71C2×#3522
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22602425C2direct (N × H)
2. C2×#17986734C2semidirect
3. C2×Q8×Dic142C2semidirect
4. C2×#17986709C2semidirect
5. C2×#17986671C2semidirect
6. C2×#17986660C2semidirect
7. C2×#17986650C2semidirect
8. C2×#17986621C2semidirect
9. C2×#17986549C2semidirect
10. C2×#17986533C2semidirect
11. C2×#17986481C2semidirect
12. C22×C4×Q8×C71C2semidirect
15. #17986734C22semidirect
16. Q8×Dic142C22semidirect
17. #17986709C22semidirect
18. #17986671C22semidirect
19. #17986660C22semidirect
20. #17986650C22semidirect
21. #17986621C22semidirect
22. #17986549C22semidirect
26. #17986533C22semidirect
40. #17986481C22semidirect
55. C2×C4×Q8×C71C22semidirect
168. C2×Q8×C71#41semidirect
200. Q8×C71C2×#41semidirect
253. C2×C71#3522semidirect
269. C22×C4×Q8D71semidirect
270. C71C2×#3522semidirect
306. C2×C4×Q8C2×D71semidirect
397. C2×Q8#11765865semidirect
460. Q8C2×#11765865semidirect
511. C2#22602425direct (N × H)

Maximal subgroups

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

subgroupindex
#226024252
C2×#179867342
C2×Q8×Dic1422
C2×#179867092
C2×#179866712
C2×#179866602
C2×#179866502
C2×#179866212
C2×#179865492
C2×#179865332
C2×#179864812
C22×C4×Q8×C712
C2×#352271