Finite Group Databasebeta

← C22×#15689464 #27084023

Subgroups — C22×#15689464 #27084023

Order 12032.

2358 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094003C211yesmaximal, characteristic
2C2×#15689464C2124yesmaximal, direct product factor
3C22×#13093957C211yesmaximal, characteristic, complemented
4C22×#13093955C211yesmaximal, characteristic, complemented
5C22×#13093947C211yesmaximal, characteristic, complemented
6C22×#13093947C211yesmaximal, characteristic, complemented
7C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic
8C22×#117×C47C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C23×C47⋊C8C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12C2×#13094003—26no—
13C2×#13094003C22112yes—
14C22×C4⋊Dic47C2211yescharacteristic
15C22×Dic47⋊C4—21no—
16C22×C22⋊Dic47—21no—
17#15689464C22164yesdirect product factor
18C2×#13093957C22112yescomplemented
19C2×#13093955C22112yescomplemented
20C2×#13093947C22112yescomplemented
21C2×#13093947C22112yescomplemented
22C2×C22⋊(C47⋊C8)C22112yes—
23C24×C4×C47C2211yescharacteristic
24C2×#117×C47—26no—
25C2×#117×C47C22112yes—
26C22×C4⋊C4×C47C2211yescharacteristic
27C22×C4⋊C4×C47C2211yescharacteristic
28C22×C22⋊C4×C47—21no—
29C22×C42×C47—21no—
30C22×Dic94—24no—
31C22×Dic94—24no—
32C22×Dic94—23no—
33C22×Dic94C2316yes—
34C23×Dic47—21no—
35C23×Dic47—21no—
36C22×C47⋊C8—24no—
37C22×C47⋊C8—24no—
38C22×C47⋊C8C2316yes—
39C22×C47⋊C8C2316yes—
40#13094003—224no—
41#13094003C23116yes—
42C2×C4⋊Dic47—23no—
43C2×C4⋊Dic47C2316yes—
44C2×Dic47⋊C4—212no—
45C2×C22⋊Dic47—212no—
46#13093957C23116yescomplemented
47#13093955C23116yescomplemented
48#13093947C23116yescomplemented
49#13093947C23116yescomplemented
50C22⋊(C47⋊C8)C23116yes—
51C25×C47D411yescharacteristic
52C23×C4×C47—24no—
53C23×C4×C47—24no—
54C23×C4×C47—21no—
55C23×C4×C47—21no—
56C23×C4×C47D411yescharacteristic
57C23×C4×C47D416yes—
58C23×C4×C47C2316yes—
59C23×C4×C47C2311yescharacteristic
60#117×C47—224no—
61#117×C47C23116yes—
62C2×C4⋊C4×C47—23no—
63C2×C4⋊C4×C47—23no—
64C2×C4⋊C4×C47C2316yes—
65C2×C4⋊C4×C47C2316yes—
66C2×C22⋊C4×C47—212no—
67C2×C42×C47—212no—
68C24×C47—24no—
69C24×C47—24no—
70C24×C47—24no—
71C24×C47C2×D416yes—
72C24×C47C2×D411yescharacteristic
73C22×C4×C47—24no—
74C22×C4×C47—224no—
75C22×C4×C47—212no—
76C22×C4×C47—212no—
77C22×C4×C47—26no—
78C22×C4×C47—26no—
79C22×C4×C47—44no—
80C22×C4×C47—44no—
81C22×C4×C47C2×D416yes—
82C22×C4×C47C2×D4112yes—
83C22×C4×C47C2×D413yes—
84C22×C4×C47#4114yes—
85C22×C4×C47#4114yes—
86C22×C4×C47C2414yes—
87C22×C4×C47C2413yes—
88C22×Dic47—26no—
89C22×Dic47—26no—
90C22×Dic47—44no—
91C22×Dic47—44no—
92C2×Dic94—212no—
93C2×Dic94—212no—
94C2×Dic94—26no—
95C2×Dic94—412no—
96C2×Dic94C2414yes—
97C4⋊C4×C47—26no—
98C4⋊C4×C47—26no—
99C4⋊C4×C47C2414yes—
100C4⋊C4×C47C2414yes—
101C22⋊C4×C47—216no—
102C42×C47—216no—
103C22⋊Dic47—216no—
104C4⋊Dic47—26no—
105C4⋊Dic47C2414yes—
106Dic47⋊C4—216no—
107C2×C47⋊C8—212no—
108C2×C47⋊C8—212no—
109C2×C47⋊C8C2414yes—
110C2×C47⋊C8C2414yes—
111C23×C47—212no—
112C23×C47—212no—
113C23×C47—212no—
114C23×C47—416no—
115C23×C47#12314yes—
116C23×C47#13514yes—
117C23×C47#13714yes—
118C23×C47C22×D414yes—
119C23×C47C22×D413yes—
120C2×C4×C47—212no—
121C2×C4×C47—224no—
122C2×C4×C47—24no—
123C2×C4×C47—24no—
124C2×C4×C47—24no—
125C2×C4×C47—24no—
126C2×C4×C47—412no—
127C2×C4×C47—412no—
128C2×C4×C47#123112yes—
129C2×C4×C47C22×D414yes—
130C2×C4×C47C22×D413yes—
131C2×C4×C47C2×#4116yes—
132C2×C4×C47C2×#4116yes—
133C2×C4×C47C2511yescharacteristic
134C2×Dic47—24no—
135C2×Dic47—24no—
136C2×Dic47—412no—
137C2×Dic47—412no—
138Dic94—24no—
139Dic94—24no—
140Dic94—412no—
141C47⋊C8—24no—
142C47⋊C8—24no—
143C22×#485—471nomaximal, Sylow
144C22×C47—24no—
145C22×C47—24no—
146C22×C47—24no—
147C22×C47—424no—
148C22×C47#485116yescomplemented
149C22×C47C2×#12316yes—
150C22×C47C2×#13516yes—
151C22×C47C2×#13716yes—
152C22×C47C23×D411yescharacteristic
153C4×C47—24no—
154C4×C47—44no—
155C4×C47—44no—
156C4×C47C2×#12316yes—
157C4×C47C22×#4111yescharacteristic
158C4×C47C22×#4111yescharacteristic
159Dic47—44no—
160Dic47—44no—
161C22×C22⋊Q8—471no—
162C22×#117D4711yescharacteristic
163C2×#485—4724no—
164C22×#107—471no—
165C22×#106—471no—
166C22×#103—471no—
167C22×#103—471no—
168C22×C22⋊C8—471no—
169C2×C47—44no—
170C2×C47C2×#485112yescomplemented
171C2×C47C22×#12311yescharacteristic
172C2×C47C22×#13511yescharacteristic
173C2×C47C22×#13711yescharacteristic
174C23×Q8—471no—
175C24×C4C2×D4711yescharacteristic
176C23×C8—471no—
177C23×C8—471no—
178C2×C22⋊Q8—4712no—
179C2×C22⋊Q8—946no—
180C2×#117—26no—
181C2×#117C2×D47112yes—
182C22×C4⋊C4—471no—
183C22×C4⋊C4—941no—
184C22×C4⋊C4C2×D4711yescharacteristic
185C22×C4⋊C4C2×D4711yescharacteristic
186C22×C22⋊C4—21no—
187C22×C22⋊C4—941no—
188C22×C42—21no—
189#485—4764no—
190C2×#107—4712no—
191C2×#106—4712no—
192C2×#103—4712no—
193C2×#103—4712no—
194C2×C22⋊C8—4712no—
195C47C22×#48511yescharacteristic, complemented, Sylow
196C25C22⋊D4711yescharacteristic
197C22×Q8—476no—
198C22×Q8—943no—
199C22×Q8—944no—
200C22×Q8—944no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—941no—
206C23×C4—941no—
207C23×C4C22⋊D4716yes—
208C23×C4C22⋊D4711yescharacteristic
209C23×C4C22×D4716yes—
210C23×C4C22×D4711yescharacteristic
211C22×C8—476no—
212C22×C8—476no—
213C22×C8—944no—
214C22×C8—944no—
215C22⋊Q8—4716no—
216C22⋊Q8—9424no—
217#117—224no—
218#117C22×D47116yes—
219C2×C4⋊C4—23no—
220C2×C4⋊C4—23no—
221C2×C4⋊C4—476no—
222C2×C4⋊C4—943no—
223C2×C4⋊C4—9412no—
224C2×C4⋊C4C22×D4716yes—
225C2×C4⋊C4C22×D4716yes—
226C2×C22⋊C4—212no—
227C2×C22⋊C4—9412no—
228C2×C42—212no—
229#107—4716no—
230#106—4716no—
231#103—4716no—
232#103—4716no—
233C22⋊C8—4716no—
234C24—24no—
235C24—24no—
236C24—24no—
237C24C2×C22⋊D4716yes—
238C24C2×C22⋊D4711yescharacteristic
239C2×Q8—474no—
240C2×Q8—946no—
241C2×Q8—9412no—
242C2×Q8—9412no—
243C2×Q8—18812no—
244C22×C4—24no—
245C22×C4—224no—
246C22×C4—212no—
247C22×C4—212no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—946no—
253C22×C4—946no—
254C22×C4—1884no—
255C22×C4—1884no—
256C22×C4#1062535014yes—
257C22×C4#1062535014yes—
258C22×C4C2×C22⋊D4716yes—
259C22×C4C2×C22⋊D47112yes—
260C22×C4C2×C22⋊D4713yes—
261C22×C4C23×D4714yes—
262C22×C4C23×D4713yes—
263C2×C8—474no—
264C2×C8—474no—
265C2×C8—9412no—
266C2×C8—9412no—
267C4⋊C4—26no—
268C4⋊C4—26no—
269C4⋊C4—474no—
270C4⋊C4—9416no—
271C4⋊C4—946no—
272C4⋊C4C23×D4714yes—
273C4⋊C4C23×D4714yes—
274C22⋊C4—216no—
275C22⋊C4—9416no—
276C42—216no—
277C23—212no—
278C23—212no—
279C23—212no—
280C23—416no—
281C23#1309400714yes—
282C23#1309404414yes—
283C23#1309405414yes—
284C23C22×C22⋊D4714yes—
285C23C22×C22⋊D4713yes—
286Q8—944no—
287Q8—944no—
288Q8—18812no—
289C2×C4—212no—
290C2×C4—224no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—412no—
296C2×C4—412no—
297C2×C4—944no—
298C2×C4—944no—
299C2×C4—18812no—
300C2×C4—18812no—
301C2×C4#13094007112yes—
302C2×C4C22×C22⋊D4714yes—
303C2×C4C22×C22⋊D4713yes—
304C2×C4C2×#1062535016yes—
305C2×C4C2×#1062535016yes—
306C2×C4C24×D4711yescharacteristic
307C8—944no—
308C8—944no—
309C22—24no—
310C22—24no—
311C22—24no—
312C22—424no—
313C22#15689464116yesdirect product factor
314C22C2×#1309400716yes—
315C22C2×#1309404416yes—
316C22C2×#1309405416yes—
317C22C23×C22⋊D4711yescharacteristic
318C4—24no—
319C4—44no—
320C4—44no—
321C4—1884no—
322C4—1884no—
323C4C2×#1309400716yes—
324C4C22×#1062535011yescharacteristic
325C4C22×#1062535011yescharacteristic
326C2—44no—
327C2C2×#15689464112yesdirect product factor
328C2C22×#1309400711yescharacteristic
329C2C22×#1309404411yescharacteristic
330C2C22×#1309405411yescharacteristic
subgroupG/H
centerC24C2×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC22×#117×C47C2
socleC24×C47C2×D4
prime radicalC25×C47D4
hypercenterC22×#117D47
hypocenterC47C22×#485
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#15689464C2direct (N × H)
3. C22×#13093957C2semidirect
4. C22×#13093955C2semidirect
5. C22×#13093947C2semidirect
6. C22×#13093947C2semidirect
17. #15689464C22direct (N × H)
18. C2×#13093957C22semidirect
19. C2×#13093955C22semidirect
20. C2×#13093947C22semidirect
21. C2×#13093947C22semidirect
46. #13093957C23semidirect
47. #13093955C23semidirect
48. #13093947C23semidirect
49. #13093947C23semidirect
148. C22×C47#485semidirect
170. C2×C47C2×#485semidirect
195. C47C22×#485semidirect
313. C22#15689464direct (N × H)
327. C2C2×#15689464direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940032
C2×#156894642
C22×#130939572
C22×#130939552
C22×#130939472
C22×C22⋊(C47⋊C8)2
C22×#117×C472
C22×#48547