Finite Group Databasebeta

← #27124838

Subgroups — #27124838

Order 12032.

6930 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257214C218yesmaximal, complemented
2#20257207C218yesmaximal, complemented
3#20257198C214yesmaximal, complemented
4#20257191C2112yesmaximal, complemented
5#41×Dic94C2116yesmaximal, complemented
6#20256793C216yesmaximal, complemented
7C2×#15689680C216yesmaximal, complemented
8#20256591C212yesmaximal
9C2×C4×#41×C47C211yesmaximal, characteristic
10Q8×Dic94C22116yescomplemented
11#15689852C22116yescomplemented
12#15689849C22132yescomplemented
13#15689832C22196yescomplemented
14#15689830C22148yescomplemented
15D4×Dic94C22148yescomplemented
16#15689754C22124yes—
17#15689751C22148yescomplemented
18#15689749C22124yes—
19#15689680C22112yes—
20#15689680C2214yes—
21#15689680C22124yes—
22#15689680C22112yescomplemented
23#15689680C22112yescomplemented
24#15689680C22124yescomplemented
25#15689676C22148yescomplemented
26C2×#13094024C22112yescomplemented
27C2×#13094016C22112yescomplemented
28C2×#13094003C22124yes—
29#15689606C22116yes—
30#15689601C2214yes—
31#15689598C22112yes—
32#41×Dic47C22116yes—
33C2×C4×Dic94C22124yescomplemented
34C2×#13093983C22112yes—
35C22×#41×C47C2211yescharacteristic
36C4×#41×C47C2214yes—
37C4×#41×C47C2214yes—
38C4×#41×C47C2218yes—
39C2×C4×Q8×C47C2212yes—
40C2×C4×D4×C47C2216yes—
41C2×#117×C47C2213yes—
42C22×C42×C47C2213yes—
43C22×Dic94C23124yes—
44#13094024C23124yes—
45#13094024C23124yes—
46#13094024C23124yes—
47#13094024C2318yes—
48#13094023C23196yes—
49#13094016C23124yes—
50#13094016C23148yes—
51#13094016C2318yes—
52#13094015C23196yes—
53#13094003C23148yes—
54#13094003C23148yes—
55#13094003C23148yes—
56#13094003C23116yes—
57#13094001C231192yes—
58Q8×Dic47C23132yes—
59C4×Dic94C23148yes—
60C4×Dic94C23148yes—
61C4×Dic94C23116yes—
62C4×Dic94C23148yes—
63D4×Dic47C23196yes—
64#13093983C23124yes—
65#13093983C23124yes—
66#13093983C2318yes—
67#13093983C23124yes—
68C2×C4⋊Dic47C23124yes—
69C2×C4⋊Dic47C23124yes—
70C2×Dic47⋊C4C23148yes—
71C2×C22⋊Dic47C23124yes—
72C2×C4×Dic47C23124yes—
73C2×#41×C47Q8116yescomplemented
74C2×#41×C47C2314yes—
75C2×#41×C47C2314yes—
76C22×Q8×C47C2311yescharacteristic
77C22×D4×C47C2313yes—
78C23×C4×C47C2316yes—
79C23×C4×C47C2313yes—
80C4×Q8×C47C2318yes—
81C4×Q8×C47C2318yes—
82C4×D4×C47C23124yes—
83C4×D4×C47C23124yes—
84#117×C47C23112yes—
85#117×C47C23112yes—
86C2×C4⋊C4×C47C2316yes—
87C2×C22⋊C4×C47C2316yes—
88C2×C42×C47C2311yescharacteristic
89C2×C42×C47C2313yes—
90C2×C42×C47C2316yes—
91C2×C42×C47C23112yes—
92C2×C42×C47C2316yes—
93C2×C42×C47C2316yes—
94C24×C47C2413yes—
95#41×C47C2×Q8132yescomplemented
96#41×C47C2×Q8132yescomplemented
97C2×Q8×C47C2×Q818yescomplemented
98C2×Q8×C47C2414yes—
99C2×D4×C47C2×Q8124yescomplemented
100C2×D4×C47C24112yes—
101C22×C4×C47—412no—
102C22×C4×C47C2×Q8124yescomplemented
103C22×C4×C47C2412yes—
104C22×C4×C47C24112yes—
105C22×C4×C47C2416yes—
106C22×C4×C47C24112yes—
107C22×C4×C47C2411yescharacteristic
108C22×C4×C47C2413yes—
109C22×C4×C47C2416yes—
110C22×C4×C47C2416yes—
111C22×Dic47C24148yes—
112C2×Dic94—448no—
113C2×Dic94C24116yes—
114C2×Dic94C24148yes—
115C4⋊C4×C47C24124yes—
116C22⋊C4×C47C24124yes—
117C42×C47C2412yes—
118C42×C47C2411yescharacteristic
119C42×C47C2411yescharacteristic
120C42×C47C2416yes—
121C42×C47C2416yes—
122C42×C47C24112yes—
123C42×C47C24112yes—
124C22⋊Dic47C24116yes—
125C22⋊Dic47C24148yes—
126C4⋊Dic47C24116yes—
127C4⋊Dic47C24116yes—
128C4⋊Dic47C24148yes—
129C4⋊Dic47C24148yes—
130Dic47⋊C4C24132yes—
131Dic47⋊C4C24196yes—
132C4×Dic47C24116yes—
133C4×Dic47C24148yes—
134C23×C47—46no—
135C23×C47C22×Q8112yes—
136C23×C47C2511yescharacteristic
137C23×C47C2516yes—
138Q8×C47C22×Q8116yes—
139D4×C47C22×Q8148yes—
140C2×C4×C47—224no—
141C2×C4×C47—24no—
142C2×C4×C47—46no—
143C2×C4×C47—412no—
144C2×C4×C47C22×Q814yes—
145C2×C4×C47C22×Q8112yes—
146C2×C4×C47C22×Q8124yes—
147C2×C4×C47C22×Q8124yes—
148C2×C4×C47C2512yes—
149C2×C4×C47C25112yes—
150C2×C4×C47C2516yes—
151C2×C4×C47C2511yescharacteristic
152C2×C4×C47C2512yes—
153C2×C4×C47C2511yescharacteristic
154C2×Dic47—448no—
155C2×Dic47C25132yes—
156Dic94—232no—
157Dic94—448no—
158#62813—471nomaximal, Sylow
159C22×C47—212no—
160C22×C47—22no—
161C22×C47—412no—
162C22×C47C23×Q812yes—
163C22×C47C23×Q8112yes—
164C22×C47C2611yescharacteristic
165C4×C47—212no—
166C4×C47—22no—
167C4×C47C23×Q812yes—
168C4×C47C23×Q812yes—
169C4×C47C23×Q8112yes—
170C4×C47#58514yes—
171Dic47—232no—
172#3504—478no—
173#3503—478no—
174#3502—474no—
175#3501—4712no—
176Q8×#41—4716no—
177#3458—476no—
178C2×#533—476no—
179#3428—472no—
180C2×C4×#41D4711yescharacteristic
181C2×C47—22no—
182C2×C47—212no—
183C2×C47C24×Q811yescharacteristic
184C2×C47C2×#58512yes—
185C22×#41C2×D4711yescharacteristic
186Q82—4716no—
187#557—4732no—
188#557—4716no—
189#554—4748no—
190#554—4796no—
191D4×Q8—4748no—
192#544—4748no—
193#544—4724no—
194#543—4724no—
195#533—4748no—
196#533—474no—
197#533—4712no—
198#533—4724no—
199#533—4724no—
200#533—4712no—
201#533—4712no—
202C2×C4⋊Q8—4712no—
203C2×#125—4712no—
204C2×C22⋊Q8—4724no—
205#520—4716no—
206#519—474no—
207#518—4712no—
208C4×#41—4716no—
209C4×#41C2×D4718yes—
210C4×#41C2×D4714yes—
211C4×#41C2×D4714yes—
212C2×C4×Q8—4724no—
213C2×C4×Q8C2×D4712yes—
214C2×C4×D4C2×D4716yes—
215C2×#117—4712no—
216C2×#117C2×D4713yes—
217C22×C42C2×D4713yes—
218C47#6281311yescharacteristic, complemented, Sylow
219C2×#41Dic94116yescomplemented
220C2×#41C22×D4714yes—
221C2×#41C22×D4714yes—
222C22×Q8—4724no—
223C22×Q8C22×D4711yescharacteristic
224C22×D4C22×D4713yes—
225C23×C4C22×D4716yes—
226C23×C4C22×D4713yes—
227C4⋊Q8—4796no—
228C4⋊Q8—478no—
229C4⋊Q8—4724no—
230C4⋊Q8—4724no—
231C4⋊Q8—4724no—
232#125—4796no—
233#125—478no—
234#125—4724no—
235#125—4748no—
236C22⋊Q8—4716no—
237C22⋊Q8—4748no—
238C22⋊Q8—4748no—
239C22⋊Q8—4748no—
240C22⋊Q8—47192no—
241C4×Q8—4732no—
242C4×Q8—4716no—
243C4×Q8—4748no—
244C4×Q8—4748no—
245C4×Q8—4748no—
246C4×Q8C22×D4718yes—
247C4×Q8C22×D4718yes—
248C4×D4—4796no—
249C4×D4C22×D47124yes—
250C4×D4C22×D47124yes—
251#117—478no—
252#117—4724no—
253#117—4724no—
254#117—4724no—
255#117C22×D47112yes—
256#117C22×D47112yes—
257C2×C4⋊C4—4724no—
258C2×C4⋊C4—4748no—
259C2×C4⋊C4—4724no—
260C2×C4⋊C4C22×D4716yes—
261C2×C22⋊C4—4724no—
262C2×C22⋊C4C22×D4716yes—
263C2×C42—4724no—
264C2×C42C22×D4713yes—
265C2×C42C22×D4716yes—
266C2×C42C22×D47112yes—
267C2×C42C22×D4716yes—
268C2×C42C22×D4716yes—
269C2×C42C22×D4711yescharacteristic
270C24C23×D4713yes—
271#41C2×Dic94132yescomplemented
272#41C2×Dic94132yescomplemented
273C2×Q8—4716no—
274C2×Q8—4748no—
275C2×Q8—18848no—
276C2×Q8C2×Dic9418yescomplemented
277C2×Q8C23×D4714yes—
278C2×D4C2×Dic94124yescomplemented
279C2×D4C23×D47112yes—
280C22×C4—412no—
281C22×C4—4748no—
282C22×C4C2×Dic94124yescomplemented
283C22×C4C23×D4712yes—
284C22×C4C23×D47112yes—
285C22×C4C23×D4716yes—
286C22×C4C23×D47112yes—
287C22×C4C23×D4716yes—
288C22×C4C23×D4713yes—
289C22×C4C23×D4716yes—
290C22×C4C23×D4711yescharacteristic
291C4⋊C4—4732no—
292C4⋊C4—4796no—
293C4⋊C4—4716no—
294C4⋊C4—4716no—
295C4⋊C4—4748no—
296C4⋊C4—4748no—
297C4⋊C4C23×D47124yes—
298C22⋊C4—4716no—
299C22⋊C4—4748no—
300C22⋊C4C23×D47124yes—
301C42—4748no—
302C42—4716no—
303C42C23×D4711yescharacteristic
304C42C23×D4711yescharacteristic
305C42C23×D4716yes—
306C42C23×D47112yes—
307C42C23×D47112yes—
308C42C23×D4712yes—
309C42C23×D4716yes—
310C23—46no—
311C23C22×Dic94112yes—
312C23C24×D4711yescharacteristic
313C23C24×D4716yes—
314Q8—9432no—
315Q8—18848no—
316Q8C22×Dic94116yes—
317D4C22×Dic94148yes—
318C2×C4—24no—
319C2×C4—224no—
320C2×C4—412no—
321C2×C4—46no—
322C2×C4—4732no—
323C2×C4—18848no—
324C2×C4C22×Dic9414yes—
325C2×C4C22×Dic94112yes—
326C2×C4C22×Dic94124yes—
327C2×C4C22×Dic94124yes—
328C2×C4C24×D4712yes—
329C2×C4C24×D47112yes—
330C2×C4C24×D4716yes—
331C2×C4C24×D4711yescharacteristic
332C2×C4C24×D4712yes—
333C2×C4C24×D4711yescharacteristic
334C22—22no—
335C22—212no—
336C22—412no—
337C22C23×Dic9412yes—
338C22C23×Dic94112yes—
339C22C25×D4711yescharacteristic
340C4—22no—
341C4—212no—
342C4—9432no—
343C4C23×Dic9412yes—
344C4C23×Dic9412yes—
345C4C23×Dic94112yes—
346C4#1568995914yes—
347C2—22no—
348C2—212no—
349C2C24×Dic9411yescharacteristic
350C2C2×#1568995912yes—
subgroupG/H
centerC2×C4C24×D47
commutator, socle, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C4×#41×C47C2
prime radicalC22×#41×C47C22
hypercenterC2×C4×#41D47
hypocenterC47#62813
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #20257214C2semidirect
2. #20257207C2semidirect
3. #20257198C2semidirect
4. #20257191C2semidirect
5. #41×Dic94C2semidirect
6. #20256793C2semidirect
7. C2×#15689680C2semidirect
10. Q8×Dic94C22semidirect
11. #15689852C22semidirect
12. #15689849C22semidirect
13. #15689832C22semidirect
14. #15689830C22semidirect
15. D4×Dic94C22semidirect
17. #15689751C22semidirect
22. #15689680C22semidirect
23. #15689680C22semidirect
24. #15689680C22semidirect
25. #15689676C22semidirect
26. C2×#13094024C22semidirect
27. C2×#13094016C22semidirect
33. C2×C4×Dic94C22semidirect
73. C2×#41×C47Q8semidirect
95. #41×C47C2×Q8semidirect
96. #41×C47C2×Q8semidirect
97. C2×Q8×C47C2×Q8semidirect
99. C2×D4×C47C2×Q8semidirect
102. C22×C4×C47C2×Q8semidirect
218. C47#62813semidirect
219. C2×#41Dic94semidirect
271. #41C2×Dic94semidirect
272. #41C2×Dic94semidirect
276. C2×Q8C2×Dic94semidirect
278. C2×D4C2×Dic94semidirect
282. C22×C4C2×Dic94semidirect

Maximal subgroups

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

subgroupindex
#202572142
#202572072
#202571982
#202571912
#41×Dic942
#202567932
C2×#156896802
#202565912
C2×C4×#41×C472
#6281347