Finite Group Databasebeta

← C2×#20257484 #27124501

Subgroups — C2×#20257484 #27124501

Order 12032.

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
1#20257484C2132yesmaximal, direct product factor
2C2×#15689852C211yesmaximal, characteristic, complemented
3C2×#15689851C213yesmaximal, complemented
4C2×#15689846C213yesmaximal, complemented
5C2×#15689808C216yesmaximal, complemented
6C2×Q8×D188C211yesmaximal, characteristic, complemented
7C2×Q8×C22⋊D47C212yesmaximal, complemented
8C2×#15689731C216yesmaximal, complemented
9C2×#15689680C213yesmaximal, complemented
10C2×#15689657C213yesmaximal
11C2×#15689599C212yesmaximal, complemented
12C22×C4×Q8×C47C211yesmaximal, characteristic, complemented
13C22×Q8×D47C2212yes—
14#15689852C22116yescomplemented
15#15689851C22148yescomplemented
16#15689846C22148yescomplemented
17#15689808C22196yescomplemented
18Q8×D188C22116yescomplemented
19Q8×C22⋊D47C22132yescomplemented
20#15689731C22196yescomplemented
21#15689680C22148yescomplemented
22C2×#13094024C2213yes—
23C2×#13094024C2213yes—
24C2×#13094022C2216yes—
25#15689657C22148yes—
26C2×#13094019C22112yes—
27C2×#13094018C2216yes—
28C2×#13094016C2213yes—
29C2×#13094016C2216yes—
30C2×#13094015C2216yes—
31C2×#13094014C22112yes—
32C2×#13094012C2213yes—
33C2×#13094007C2216yes—
34C2×#13094004C2216yes—
35C2×#13094003C2216yes—
36C2×#13094003C2213yes—
37C2×#13094000C2216yes—
38C2×#13093998C2211yescharacteristic
39#15689599C22132yescomplemented
40C2×Q8×Dic47C2212yes—
41C2×C4×Dic94C2216yes—
42C2×C4×Dic94C2213yes—
43C2×C4×Dic94C2216yes—
44C2×#13093989C2216yes—
45C2×C4×D188C2213yes—
46C2×C4×C22⋊D47C2216yes—
47C2×#13093983C2216yes—
48C2×#13093982C2216yes—
49C2×#13093981C2216yes—
50C23×Q8×C47C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2213yes—
54C2×C4×Q8×C47C2213yes—
55C2×C4×Q8×C47C22116yescomplemented
56C22×C4⋊C4×C47C2213yes—
57C22×C42×C47C2213yes—
58C22×Dic94C2316yes—
59C22×Dic94C2313yes—
60C2×Q8×D47—216no—
61C2×Q8×D47C23116yes—
62C22×D188C2311yescharacteristic
63C22×C22⋊D47C2312yes—
64C22×C4×D47C2316yes—
65#13094024C23124yes—
66#13094024C23124yes—
67#13094022C23148yes—
68#13094019C23196yes—
69#13094018C23148yes—
70#13094016C23148yes—
71#13094016C23124yes—
72#13094015C23148yes—
73#13094014C23196yes—
74#13094012C23124yes—
75#13094007C23148yes—
76#13094004C23148yes—
77#13094003C23148yes—
78#13094003C23124yes—
79#13094000C23148yes—
80#13093998C2318yes—
81Q8×Dic47C23116yes—
82C4×Dic94C23148yes—
83C4×Dic94C23148yes—
84C4×Dic94C23124yes—
85#13093989C23148yes—
86C4×D188C23124yes—
87C4×C22⋊D47C23148yes—
88#13093983C23148yes—
89#13093982C23148yes—
90#13093981C23148yes—
91C2×C4⋊Dic47C2316yes—
92C2×C4⋊Dic47C2316yes—
93C2×C4⋊Dic47C2316yes—
94C2×C4⋊Dic47C2313yes—
95C2×C4⋊Dic47C2311yescharacteristic
96C2×Dic47⋊C4C2316yes—
97C2×Dic47⋊C4C2316yes—
98C2×Dic47⋊C4C2316yes—
99C2×Dic47⋊C4C23112yes—
100C2×Dic47⋊C4C2316yes—
101C2×C22⋊Dic47C2316yes—
102C2×(C2×D47)⋊C4C2312yes—
103C2×(C2×D47)⋊C4C2316yes—
104C2×(C2×D47)⋊C4C2316yes—
105C2×(C2×D47)⋊C4C2316yes—
106C2×C4×Dic47C2316yes—
107C2×C4×Dic47C23112yes—
108C22×Q8×C47—28no—
109C22×Q8×C47C2318yes—
110C22×Q8×C47C2313yes—
111C22×Q8×C47C2311yescharacteristic
112C23×C4×C47C2311yescharacteristic
113C23×C4×C47C2313yes—
114C23×C4×C47C2313yes—
115C4×Q8×C47C23124yes—
116C4×Q8×C47C2318yes—
117C4×Q8×C47C23124yes—
118C4×Q8×C47C2318yes—
119C2×C4⋊C4×C47C2313yes—
120C2×C4⋊C4×C47C2313yes—
121C2×C4⋊C4×C47C2316yes—
122C2×C4⋊C4×C47C23124yes—
123C2×C42×C47C2313yes—
124C2×C42×C47C2313yes—
125C2×C42×C47C2313yes—
126C2×C42×C47C2313yes—
127C2×C42×C47C23124yes—
128C24×C47C2411yescharacteristic
129C23×D47C2412yes—
130C2×Q8×C47—232no—
131C2×Q8×C47—212no—
132C2×Q8×C47#4118yes—
133C2×Q8×C47C24112yes—
134C2×Q8×C47C2414yes—
135C22×C4×C47—212no—
136C22×C4×C47—212no—
137C22×C4×C47—24no—
138C22×C4×C47C2411yescharacteristic
139C22×C4×C47C2414yes—
140C22×C4×C47C2413yes—
141C22×C4×C47C2413yes—
142C22×C4×C47C24112yes—
143C22×C4×C47C2413yes—
144C22×C4×C47C24112yes—
145C22×C4×C47C2413yes—
146C22×C4×C47C2411yescharacteristic
147C2×C22⋊D47—28no—
148C2×C22⋊D47C2418yes—
149C22×Dic47C2416yes—
150C22×Dic47C2412yes—
151C22×Dic47C2416yes—
152Q8×D47—264no—
153C2×D188—24no—
154C2×D188C2414yes—
155C2×C4×D47—224no—
156C2×C4×D47C24124yes—
157C2×Dic94—224no—
158C2×Dic94—212no—
159C2×Dic94C24124yes—
160C2×Dic94C24112yes—
161C4⋊C4×C47C24112yes—
162C4⋊C4×C47C24112yes—
163C4⋊C4×C47C24124yes—
164C42×C47C24112yes—
165C42×C47C24112yes—
166C42×C47C24112yes—
167C42×C47C24112yes—
168C22⋊Dic47C24124yes—
169(C2×D47)⋊C4C2418yes—
170(C2×D47)⋊C4C24124yes—
171(C2×D47)⋊C4C24124yes—
172(C2×D47)⋊C4C24124yes—
173C4⋊Dic47C24124yes—
174C4⋊Dic47C24124yes—
175C4⋊Dic47C24124yes—
176C4⋊Dic47C24112yes—
177C4⋊Dic47C2414yes—
178Dic47⋊C4C24124yes—
179Dic47⋊C4C24124yes—
180Dic47⋊C4C24124yes—
181Dic47⋊C4C24148yes—
182Dic47⋊C4C24124yes—
183C4×Dic47C24124yes—
184C4×Dic47C24148yes—
185C23×C47—22no—
186C23×C47—22no—
187C23×C472−1+414yes—
188C23×C47C2512yes—
189C23×C47C2511yescharacteristic
190C22×D47—24no—
191C22×D47—24no—
192C22×D47—24no—
193C22×D47C2514yes—
194Q8×C47—224no—
195Q8×C47C2×#41116yes—
196C2×C4×C47—26no—
197C2×C4×C47—224no—
198C2×C4×C47—26no—
199C2×C4×C47—26no—
200C2×C4×C47—22no—
201C2×C4×C47—224no—
202C2×C4×C47—28no—
203C2×C4×C47C2×#41112yes—
204C2×C4×C472−1+414yes—
205C2×C4×C47C2516yes—
206C2×C4×C47C2512yes—
207C2×C4×C47C2516yes—
208C2×C4×C47C2516yes—
209C2×C4×C47C2516yes—
210C2×C4×C47C2512yes—
211C22⋊D47—216no—
212C2×Dic47—212no—
213C2×Dic47—212no—
214C2×Dic47—24no—
215C2×Dic47C2514yes—
216C2×Dic47C25112yes—
217C2×Dic47C25112yes—
218D188—28no—
219C4×D47—248no—
220Dic94—248no—
221Dic94—224no—
222C2×#3523—471nomaximal, Sylow
223C22×C47—22no—
224C22×C47—22no—
225C22×C47—28no—
226C22×C47C22×#4112yes—
227C22×C47C2×2−1+414yes—
228C22×C47C2×2−1+412yes—
229C22×C47C2×2−1+412yes—
230C22×C47C2611yescharacteristic
231C2×D47—24no—
232C2×D47—24no—
233C2×D47—24no—
234C2×D47—48no—
235C4×C47—26no—
236C4×C47—26no—
237C4×C47—26no—
238C4×C47—22no—
239C4×C47C22×#41112yes—
240C4×C47C2×2−1+414yes—
241Dic47—212no—
242Dic47—212no—
243Dic47—24no—
244#3523—4732no—
245C2×#557—473no—
246C2×#557—471no—
247C2×#556—473no—
248C2×#552—476no—
249C2×D4×Q8—472no—
250C2×D4×Q8—471no—
251C2×#541—476no—
252C2×#533—473no—
253C2×#529—473no—
254C2×#519—472no—
255C22×C4×Q8D4711yescharacteristic, complemented
256C2×C47—24no—
257C2×C47#352314yescomplemented
258C2×C47C23×#4111yescharacteristic
259C2×C47C22×2−1+411yescharacteristic
260C2×C47C22×2−1+411yescharacteristic
261D47—44no—
262C23×Q8—472no—
263C23×Q8C2×D4711yescharacteristic
264#557—4748no—
265#557—4716no—
266#556—4748no—
267#552—4796no—
268D4×Q8—4732no—
269D4×Q8—4716no—
270#541—4796no—
271#533—4748no—
272C2×C4⋊Q8—476no—
273C2×C4⋊Q8—473no—
274C2×C4⋊Q8—473no—
275#529—4748no—
276C2×#126—4712no—
277C2×#126—476no—
278C2×#125—4712no—
279C2×#125—476no—
280C2×#125—473no—
281C2×#125—476no—
282C2×#124—473no—
283C2×#123—476no—
284C2×C22⋊Q8—476no—
285C2×C22⋊Q8—473no—
286C2×C22⋊Q8—476no—
287C2×C22⋊Q8—476no—
288C2×#121—471no—
289#519—4732no—
290C2×C4×Q8—472no—
291C2×C4×Q8—476no—
292C2×C4×Q8—476no—
293C2×C4×Q8—473no—
294C2×C4×Q8—476no—
295C2×C4×Q8C2×D4711yescharacteristic
296C2×C4×Q8C2×D4713yes—
297C2×C4×Q8C2×D4711yescharacteristic
298C2×C4×Q8C2×D4713yes—
299C2×C4×Q8C2×D47116yescomplemented
300C2×C4×D4—476no—
301C2×C4×D4—473no—
302C2×#117—476no—
303C2×#117—476no—
304C2×#117—476no—
305C22×C4⋊C4C2×D4713yes—
306C22×C42C2×D4713yes—
307C47C2×#352311yescharacteristic, complemented, Sylow
308C22×Q8—28no—
309C22×Q8—4716no—
310C22×Q8—473no—
311C22×Q8—476no—
312C22×Q8—9416no—
313C22×Q8C22×D4713yes—
314C22×Q8C22×D4718yes—
315C22×Q8C22×D4711yescharacteristic
316C22×D4—472no—
317C22×D4—471no—
318C23×C4—476no—
319C23×C4C22×D4713yes—
320C23×C4C22×D4713yes—
321C23×C4C22×D4711yescharacteristic
322C4⋊Q8—4748no—
323C4⋊Q8—4724no—
324C4⋊Q8—4724no—
325#126—4796no—
326#126—4748no—
327#125—4796no—
328#125—4748no—
329#125—4724no—
330#125—4748no—
331#124—4724no—
332#123—4748no—
333C22⋊Q8—4748no—
334C22⋊Q8—4724no—
335C22⋊Q8—4748no—
336C22⋊Q8—4748no—
337#121—478no—
338C4×Q8—4716no—
339C4×Q8—4748no—
340C4×Q8—4748no—
341C4×Q8—4724no—
342C4×Q8—4748no—
343C4×Q8C22×D4718yes—
344C4×Q8C22×D47124yes—
345C4×Q8C22×D4718yes—
346C4×Q8C22×D47124yes—
347C4×D4—4748no—
348C4×D4—4724no—
349#117—4748no—
350#117—4748no—
351#117—4748no—
352C2×C4⋊C4—476no—
353C2×C4⋊C4—476no—
354C2×C4⋊C4—476no—
355C2×C4⋊C4—476no—
356C2×C4⋊C4—476no—
357C2×C4⋊C4—473no—
358C2×C4⋊C4—476no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—4712no—
361C2×C4⋊C4—476no—
362C2×C4⋊C4C22×D4713yes—
363C2×C4⋊C4C22×D4713yes—
364C2×C4⋊C4C22×D4716yes—
365C2×C4⋊C4C22×D47124yes—
366C2×C22⋊C4—476no—
367C2×C22⋊C4—472no—
368C2×C22⋊C4—476no—
369C2×C22⋊C4—476no—
370C2×C22⋊C4—476no—
371C2×C42—4712no—
372C2×C42—476no—
373C2×C42C22×D4713yes—
374C2×C42C22×D4713yes—
375C2×C42C22×D4713yes—
376C2×C42C22×D4713yes—
377C2×C42C22×D47124yes—
378C24—472no—
379C24C23×D4711yescharacteristic
380C2×Q8—232no—
381C2×Q8—212no—
382C2×Q8—4712no—
383C2×Q8—4724no—
384C2×Q8—9464no—
385C2×Q8—9424no—
386C2×Q8—9412no—
387C2×Q8#1062535018yes—
388C2×Q8C23×D47112yes—
389C2×Q8C23×D4714yes—
390C2×D4—478no—
391C2×D4—474no—
392C2×D4—948no—
393C2×D4—944no—
394C22×C4—212no—
395C22×C4—212no—
396C22×C4—24no—
397C22×C4—476no—
398C22×C4—472no—
399C22×C4—476no—
400C22×C4—4724no—
401C22×C4—9424no—
402C22×C4C23×D4714yes—
403C22×C4C23×D47112yes—
404C22×C4C23×D47112yes—
405C22×C4C23×D4713yes—
406C22×C4C23×D4711yescharacteristic
407C22×C4C23×D4713yes—
408C22×C4C23×D4713yes—
409C22×C4C23×D4713yes—
410C22×C4C23×D4711yescharacteristic
411C4⋊C4—4724no—
412C4⋊C4—4724no—
413C4⋊C4—4724no—
414C4⋊C4—4748no—
415C4⋊C4—4724no—
416C4⋊C4—4724no—
417C4⋊C4—4724no—
418C4⋊C4—4724no—
419C4⋊C4—4712no—
420C4⋊C4—474no—
421C4⋊C4C23×D47112yes—
422C4⋊C4C23×D47112yes—
423C4⋊C4C23×D47124yes—
424C22⋊C4—4724no—
425C22⋊C4—478no—
426C22⋊C4—4724no—
427C22⋊C4—4724no—
428C22⋊C4—4724no—
429C42—4748no—
430C42—4724no—
431C42C23×D47112yes—
432C42C23×D47112yes—
433C42C23×D47112yes—
434C42C23×D47112yes—
435C23—22no—
436C23—22no—
437C23—474no—
438C23—944no—
439C23—944no—
440C23—944no—
441C23#1309407114yes—
442C23C24×D4712yes—
443C23C24×D4711yescharacteristic
444Q8—224no—
445Q8—9448no—
446Q8—9424no—
447Q8C2×#10625350116yes—
448D4—9416no—
449D4—948no—
450C2×C4—26no—
451C2×C4—224no—
452C2×C4—26no—
453C2×C4—26no—
454C2×C4—22no—
455C2×C4—224no—
456C2×C4—28no—
457C2×C4—4712no—
458C2×C4—474no—
459C2×C4—4712no—
460C2×C4—9412no—
461C2×C4—944no—
462C2×C4—9412no—
463C2×C4—9448no—
464C2×C4C2×#10625350112yes—
465C2×C4#1309407214yes—
466C2×C4C24×D4716yes—
467C2×C4C24×D4712yes—
468C2×C4C24×D4716yes—
469C2×C4C24×D4716yes—
470C2×C4C24×D4716yes—
471C2×C4C24×D4712yes—
472C22—22no—
473C22—22no—
474C22—28no—
475C22—944no—
476C22—944no—
477C22—944no—
478C22—1888no—
479C22C22×#1062535012yes—
480C22C2×#1309407114yes—
481C22C2×#1309407112yes—
482C22C2×#1309407212yes—
483C22C25×D4711yescharacteristic
484C4—26no—
485C4—26no—
486C4—26no—
487C4—22no—
488C4—9412no—
489C4—944no—
490C4—9412no—
491C4C22×#10625350112yes—
492C4C2×#1309407214yes—
493C2—24no—
494C2—1884no—
495C2#2025748414yesdirect product factor
496C2C23×#1062535011yescharacteristic
497C2C22×#1309407111yescharacteristic
498C2C22×#1309407211yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC22×C4×Q8×C47C2
socleC23×C47C25
prime radicalC2×#13093998C22
hypercenterC22×C4×Q8D47
hypocenterC47C2×#3523
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20257484C2direct (N × H)
2. C2×#15689852C2semidirect
3. C2×#15689851C2semidirect
4. C2×#15689846C2semidirect
5. C2×#15689808C2semidirect
6. C2×Q8×D188C2semidirect
7. C2×Q8×C22⋊D47C2semidirect
8. C2×#15689731C2semidirect
9. C2×#15689680C2semidirect
11. C2×#15689599C2semidirect
12. C22×C4×Q8×C47C2semidirect
14. #15689852C22semidirect
15. #15689851C22semidirect
16. #15689846C22semidirect
17. #15689808C22semidirect
18. Q8×D188C22semidirect
19. Q8×C22⋊D47C22semidirect
20. #15689731C22semidirect
21. #15689680C22semidirect
39. #15689599C22semidirect
55. C2×C4×Q8×C47C22semidirect
255. C22×C4×Q8D47semidirect
257. C2×C47#3523semidirect
299. C2×C4×Q8C2×D47semidirect
307. C47C2×#3523semidirect
495. C2#20257484direct (N × H)

Maximal subgroups

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

subgroupindex
#202574842
C2×#156898522
C2×#156898512
C2×#156898462
C2×#156898082
C2×Q8×D1882
C2×Q8×C22⋊D472
C2×#156897312
C2×#156896802
C2×#156896572
C2×#156895992
C22×C4×Q8×C472
C2×#352347