Finite Group Databasebeta

← C22×#15689455 #27084010

Subgroups — C22×#15689455 #27084010

Order 12032.

2662 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094004C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic
3C2×#15689455C2124yesmaximal, direct product factor
4C22×#13093955C211yesmaximal, characteristic, complemented
5C22×#13093949C211yesmaximal, characteristic, complemented
6C22×#13093947C211yesmaximal, characteristic, complemented
7C22×(C2×D47)⋊C8C211yesmaximal, characteristic
8C22×#106×C47C211yesmaximal, characteristic, complemented
9C23×Dic94C2211yescharacteristic
10C23×C4×D47—21no—
11C23×C4×D47C2211yescharacteristic
12C23×C47⋊C8C2211yescharacteristic
13C2×#13094004—26no—
14C2×#13094004C22112yes—
15C22×C4⋊Dic47—21no—
16C22×C4⋊Dic47C2211yescharacteristic
17C22×Dic47⋊C4—21no—
18C2×C4⋊C4×D47—26no—
19C2×C4⋊C4×D47C22112yes—
20C22×(C2×D47)⋊C4—21no—
21#15689455C22164yesdirect product factor
22C2×#13093955C22112yescomplemented
23C2×#13093949C22112yescomplemented
24C2×#13093947C22112yescomplemented
25C2×(C2×D47)⋊C8C22112yes—
26C23×C8×C47C2211yescharacteristic
27C22×C4⋊C4×C47C2211yescharacteristic
28C22×C4⋊C4×C47C2211yescharacteristic
29C2×#106×C47C22112yescomplemented
30C24×D47D411yescharacteristic
31C22×Dic94—24no—
32C22×Dic94—24no—
33C22×Dic94—23no—
34C22×Dic94C2316yes—
35C23×Dic47—21no—
36C23×Dic47—21no—
37C23×Dic47D411yescharacteristic
38C22×C4×D47—24no—
39C22×C4×D47—24no—
40C22×C4×D47—212no—
41C22×C4×D47—48no—
42C22×C4×D47D416yes—
43C22×C4×D47C2316yes—
44C22×C47⋊C8—24no—
45C22×C47⋊C8C2316yes—
46#13094004—224no—
47#13094004C23116yes—
48C2×C4⋊Dic47—23no—
49C2×C4⋊Dic47—212no—
50C2×C4⋊Dic47C2316yes—
51C2×Dic47⋊C4—212no—
52C4⋊C4×D47—224no—
53C4⋊C4×D47C23116yes—
54C2×(C2×D47)⋊C4—212no—
55#13093955C23116yescomplemented
56#13093949C23116yescomplemented
57#13093947C23116yescomplemented
58(C2×D47)⋊C8C23116yes—
59C23×C4×C47—21no—
60C23×C4×C47—21no—
61C23×C4×C47C2311yescharacteristic
62C22×C8×C47—24no—
63C22×C8×C47C2316yes—
64C2×C4⋊C4×C47—23no—
65C2×C4⋊C4×C47—23no—
66C2×C4⋊C4×C47C2316yes—
67C2×C4⋊C4×C47C2316yes—
68#106×C47C23116yescomplemented
69C24×C47C2×D411yescharacteristic
70C23×D47—24no—
71C23×D47—24no—
72C23×D47SD1618yes—
73C23×D47C2×D416yes—
74C22×C4×C47—26no—
75C22×C4×C47—26no—
76C22×C4×C47—44no—
77C22×C4×C47—44no—
78C22×C4×C47C2×D413yes—
79C22×C4×C47#4114yes—
80C22×C4×C47#4114yes—
81C22×C4×C47C2413yes—
82C22×Dic47—24no—
83C22×Dic47—26no—
84C22×Dic47—26no—
85C22×Dic47—44no—
86C22×Dic47—44no—
87C22×Dic47C2×D416yes—
88C2×C4×D47—224no—
89C2×C4×D47—212no—
90C2×C4×D47—212no—
91C2×C4×D47—216no—
92C2×C4×D47—448no—
93C2×C4×D47C2×D4112yes—
94C2×C4×D47C2414yes—
95C2×Dic94—212no—
96C2×Dic94—212no—
97C2×Dic94—26no—
98C2×Dic94—412no—
99C2×Dic94C2414yes—
100C2×C8×C47—212no—
101C2×C8×C47C2414yes—
102C4⋊C4×C47—26no—
103C4⋊C4×C47—26no—
104C4⋊C4×C47C2414yes—
105C4⋊C4×C47C2414yes—
106(C2×D47)⋊C4—216no—
107C4⋊Dic47—26no—
108C4⋊Dic47—216no—
109C4⋊Dic47C2414yes—
110Dic47⋊C4—216no—
111C2×C47⋊C8—212no—
112C2×C47⋊C8C2414yes—
113C23×C47#12314yes—
114C23×C47C2×SD1614yes—
115C23×C47#13714yes—
116C23×C47C22×D413yes—
117C22×D47—212no—
118C22×D47—212no—
119C22×D47—232no—
120C22×D47C2×SD16124yes—
121C22×D47C22×D414yes—
122C2×C4×C47—24no—
123C2×C4×C47—24no—
124C2×C4×C47—412no—
125C2×C4×C47—412no—
126C2×C4×C47#123112yes—
127C2×C4×C47C22×D413yes—
128C2×C4×C47C2×#4116yes—
129C2×C4×C47C2×#4116yes—
130C2×C4×C47C2511yescharacteristic
131C2×Dic47—212no—
132C2×Dic47—24no—
133C2×Dic47—24no—
134C2×Dic47—412no—
135C2×Dic47—412no—
136C2×Dic47C22×D414yes—
137C4×D47—224no—
138C4×D47—24no—
139C4×D47—24no—
140C4×D47—432no—
141Dic94—24no—
142Dic94—24no—
143Dic94—412no—
144C8×C47—24no—
145C47⋊C8—24no—
146C22×#483—471nomaximal, Sylow
147C22×C47#483116yescomplemented
148C22×C47C2×#12316yes—
149C22×C47C22×SD1616yes—
150C22×C47C2×#13716yes—
151C22×C47C23×D411yescharacteristic
152C2×D47—248no—
153C2×D47—24no—
154C2×D47—24no—
155C2×D47C22×SD1618yes—
156C4×C47—44no—
157C4×C47—44no—
158C4×C47C2×#12316yes—
159C4×C47C22×#4111yescharacteristic
160C4×C47C22×#4111yescharacteristic
161Dic47—24no—
162Dic47—44no—
163Dic47—44no—
164C22×C22⋊Q8—471no—
165C23×C4⋊C4—471no—
166C2×#483—4724no—
167C22×#106—471no—
168C22×#106D4711yescharacteristic, complemented
169C22×#103—471no—
170C22×#103—471no—
171C22×C22⋊C8—471no—
172C2×C47C2×#483112yescomplemented
173C2×C47C22×#12311yescharacteristic
174C2×C47C23×SD1611yescharacteristic
175C2×C47C22×#13711yescharacteristic
176D47—28no—
177C23×Q8—471no—
178C24×C4—471no—
179C24×C4—941no—
180C23×C8—471no—
181C23×C8C2×D4711yescharacteristic
182C2×C22⋊Q8—4712no—
183C2×C22⋊Q8—946no—
184C22×C4⋊C4—471no—
185C22×C4⋊C4—4712no—
186C22×C4⋊C4—946no—
187C22×C4⋊C4—941no—
188C22×C4⋊C4—941no—
189C22×C4⋊C4C2×D4711yescharacteristic
190C22×C4⋊C4C2×D4711yescharacteristic
191C22×C22⋊C4—941no—
192#483—4764no—
193C2×#106—4712no—
194C2×#106C2×D47112yescomplemented
195C2×#103—4712no—
196C2×#103—4712no—
197C2×C22⋊C8—4712no—
198C47C22×#48311yescharacteristic, complemented, Sylow
199C25—471no—
200C22×Q8—476no—
201C22×Q8—943no—
202C22×Q8—944no—
203C22×Q8—944no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—471no—
207C23×C4—476no—
208C23×C4—476no—
209C23×C4—9412no—
210C23×C4—944no—
211C23×C4—944no—
212C23×C4—941no—
213C23×C4—941no—
214C23×C4—1888no—
215C23×C4C22×D4711yescharacteristic
216C22×C8—24no—
217C22×C8—476no—
218C22×C8—944no—
219C22×C8C22×D4716yes—
220C22⋊Q8—4716no—
221C22⋊Q8—9424no—
222C2×C4⋊C4—23no—
223C2×C4⋊C4—23no—
224C2×C4⋊C4—476no—
225C2×C4⋊C4—4716no—
226C2×C4⋊C4—9424no—
227C2×C4⋊C4—9412no—
228C2×C4⋊C4—9412no—
229C2×C4⋊C4—943no—
230C2×C4⋊C4C22×D4716yes—
231C2×C4⋊C4C22×D4716yes—
232C2×C22⋊C4—9412no—
233#106—4716no—
234#106C22×D47116yescomplemented
235#103—4716no—
236#103—4716no—
237C22⋊C8—4716no—
238C24—478no—
239C24—476no—
240C24—944no—
241C24—944no—
242C24D4×D4711yescharacteristic
243C2×Q8—474no—
244C2×Q8—946no—
245C2×Q8—9412no—
246C2×Q8—9412no—
247C2×Q8—18812no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—476no—
253C22×C4—4712no—
254C22×C4—474no—
255C22×C4—9416no—
256C22×C4—944no—
257C22×C4—9424no—
258C22×C4—9412no—
259C22×C4—9412no—
260C22×C4—946no—
261C22×C4—946no—
262C22×C4—18848no—
263C22×C4—1884no—
264C22×C4—1884no—
265C22×C4#1062535014yes—
266C22×C4D4×D4713yes—
267C22×C4#1062535414yes—
268C22×C4C23×D4713yes—
269C2×C8—212no—
270C2×C8—474no—
271C2×C8—9412no—
272C2×C8C23×D4714yes—
273C4⋊C4—26no—
274C4⋊C4—26no—
275C4⋊C4—474no—
276C4⋊C4—9416no—
277C4⋊C4—9416no—
278C4⋊C4—946no—
279C4⋊C4C23×D4714yes—
280C4⋊C4C23×D4714yes—
281C22⋊C4—9416no—
282C23—4724no—
283C23—474no—
284C23—9412no—
285C23—9412no—
286C23—9432no—
287C23#1309400914yes—
288C23SD16×D4714yes—
289C23#1309405314yes—
290C23C2×D4×D4713yes—
291Q8—944no—
292Q8—944no—
293Q8—18812no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—412no—
297C2×C4—412no—
298C2×C4—474no—
299C2×C4—9412no—
300C2×C4—9424no—
301C2×C4—944no—
302C2×C4—944no—
303C2×C4—944no—
304C2×C4—944no—
305C2×C4—18832no—
306C2×C4—18812no—
307C2×C4—18812no—
308C2×C4#13094009112yes—
309C2×C4C2×D4×D4713yes—
310C2×C4C2×#1062535016yes—
311C2×C4C2×#1062535416yes—
312C2×C4C24×D4711yescharacteristic
313C8—24no—
314C8—944no—
315C22—478no—
316C22—944no—
317C22—9448no—
318C22—944no—
319C22#15689455116yesdirect product factor
320C22C2×#1309400916yes—
321C22C2×SD16×D4716yes—
322C22C2×#1309405316yes—
323C22C22×D4×D4711yescharacteristic
324C4—44no—
325C4—44no—
326C4—944no—
327C4—1884no—
328C4—1884no—
329C4C2×#1309400916yes—
330C4C22×#1062535011yescharacteristic
331C4C22×#1062535411yescharacteristic
332C2—948no—
333C2C2×#15689455112yesdirect product factor
334C2C22×#1309400911yescharacteristic
335C2C22×SD16×D4711yescharacteristic
336C2C22×#1309405311yescharacteristic
subgroupG/H
centerC24D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC22×#106×C47C2
socleC24×C47C2×D4
prime radicalC24×D47D4
hypercenterC22×#106D47
hypocenterC47C22×#483
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
3. C2×#15689455C2direct (N × H)
4. C22×#13093955C2semidirect
5. C22×#13093949C2semidirect
6. C22×#13093947C2semidirect
8. C22×#106×C47C2semidirect
21. #15689455C22direct (N × H)
22. C2×#13093955C22semidirect
23. C2×#13093949C22semidirect
24. C2×#13093947C22semidirect
29. C2×#106×C47C22semidirect
55. #13093955C23semidirect
56. #13093949C23semidirect
57. #13093947C23semidirect
68. #106×C47C23semidirect
147. C22×C47#483semidirect
168. C22×#106D47semidirect
172. C2×C47C2×#483semidirect
194. C2×#106C2×D47semidirect
198. C47C22×#483semidirect
234. #106C22×D47semidirect
319. C22#15689455direct (N × H)
333. C2C2×#15689455direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940042
C22×C4⋊C4×D472
C2×#156894552
C22×#130939552
C22×#130939492
C22×#130939472
C22×(C2×D47)⋊C82
C22×#106×C472
C22×#48347