Finite Group Databasebeta

← C22×#15689484 #27085488

Subgroups — C22×#15689484 #27085488

Order 12032.

2486 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×#13094023C211yesmaximal, characteristic, complemented
2C22×#13094013C211yesmaximal, characteristic
3C2×#15689484C2124yesmaximal, direct product factor
4C22×#13093948C211yesmaximal, characteristic, complemented
5C22×#13093946C211yesmaximal, characteristic
6C22×#13093944C211yesmaximal, characteristic
7C22×#13093934C211yesmaximal, characteristic, complemented
8C22×#103×C47C211yesmaximal, characteristic, complemented
9C23×Dic94—21no—
10C23×D188C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12C2×#13094023—26no—
13C2×#13094023C22112yescomplemented
14C2×#13094013—26no—
15C2×#13094013C22112yes—
16C22×C4⋊Dic47C2211yescharacteristic
17C22×Dic47⋊C4—21no—
18C22×(C2×D47)⋊C4—21no—
19C22×(C2×D47)⋊C4—21no—
20C22×C4×Dic47C2211yescharacteristic
21#15689484C22164yesdirect product factor
22C2×#13093948C22112yescomplemented
23C2×#13093946C22112yes—
24C2×#13093944C22112yes—
25C2×#13093934C22112yescomplemented
26C23×Q8×C47C2211yescharacteristic
27C23×C8×C47C2211yescharacteristic
28C22×C4⋊C4×C47C2211yescharacteristic
29C2×#103×C47C22112yescomplemented
30C24×D47—21no—
31C22×Dic94—212no—
32C22×Dic94—48no—
33C22×D188—24no—
34C22×D188—24no—
35C22×D188—23no—
36C22×D188C2316yes—
37C23×Dic47—21no—
38C23×Dic47D411yescharacteristic
39C23×Dic47D411yescharacteristic
40C22×C47⋊C8—24no—
41C22×C47⋊C8C2316yes—
42#13094023—224no—
43#13094023C23116yescomplemented
44#13094013—224no—
45#13094013C23116yes—
46C2×C4⋊Dic47—23no—
47C2×C4⋊Dic47C2316yes—
48C2×Dic47⋊C4—212no—
49C2×(C2×D47)⋊C4—212no—
50C2×(C2×D47)⋊C4—212no—
51C2×C4×Dic47D416yes—
52C2×C4×Dic47C2316yes—
53#13093948C23116yescomplemented
54#13093946C23116yes—
55#13093944C23116yes—
56#13093934C23116yescomplemented
57C22×Q8×C47—23no—
58C22×Q8×C47—24no—
59C22×Q8×C47—24no—
60C22×Q8×C47C2316yes—
61C23×C4×C47—21no—
62C23×C4×C47—21no—
63C23×C4×C47C2311yescharacteristic
64C22×C8×C47—24no—
65C22×C8×C47C2316yes—
66C2×C4⋊C4×C47—23no—
67C2×C4⋊C4×C47C2316yes—
68#103×C47C23116yescomplemented
69C24×C47C2×D411yescharacteristic
70C23×D47—26no—
71C23×D47—44no—
72C23×D47—44no—
73C23×D47—44no—
74C2×Q8×C47—26no—
75C2×Q8×C47—212no—
76C2×Q8×C47—212no—
77C2×Q8×C47—412no—
78C2×Q8×C47C2414yes—
79C22×C4×C47—26no—
80C22×C4×C47—26no—
81C22×C4×C47—44no—
82C22×C4×C47—44no—
83C22×C4×C47C2×D413yes—
84C22×C4×C47#4114yes—
85C22×C4×C47#4114yes—
86C22×C4×C47C2413yes—
87C22×Dic47—24no—
88C22×Dic47—24no—
89C22×Dic47—26no—
90C22×Dic47—44no—
91C22×Dic47C2×D416yes—
92C22×Dic47C2×D416yes—
93C2×D188—212no—
94C2×D188—212no—
95C2×D188—26no—
96C2×D188—412no—
97C2×D188C2414yes—
98C2×Dic94—216no—
99C2×Dic94—448no—
100C2×C8×C47—212no—
101C2×C8×C47C2414yes—
102C4⋊C4×C47—26no—
103C4⋊C4×C47C2414yes—
104(C2×D47)⋊C4—216no—
105(C2×D47)⋊C4—216no—
106C4⋊Dic47—26no—
107C4⋊Dic47C2414yes—
108Dic47⋊C4—216no—
109C4×Dic47C2×D4112yes—
110C4×Dic47C2414yes—
111C2×C47⋊C8—212no—
112C2×C47⋊C8C2414yes—
113C23×C47#12414yes—
114C23×C47AGL(1,ℤ/8)14yes—
115C23×C47#13714yes—
116C23×C47C22×D413yes—
117C22×D47—24no—
118C22×D47—412no—
119C22×D47—412no—
120C22×D47—412no—
121C22×D47—816no—
122Q8×C47—24no—
123Q8×C47—24no—
124Q8×C47—412no—
125C2×C4×C47—24no—
126C2×C4×C47—24no—
127C2×C4×C47—412no—
128C2×C4×C47—412no—
129C2×C4×C47#124112yes—
130C2×C4×C47C22×D413yes—
131C2×C4×C47C2×#4116yes—
132C2×C4×C47C2×#4116yes—
133C2×C4×C47C2511yescharacteristic
134C2×Dic47—212no—
135C2×Dic47—212no—
136C2×Dic47—24no—
137C2×Dic47—412no—
138C2×Dic47C22×D414yes—
139C2×Dic47C22×D414yes—
140D188—24no—
141D188—24no—
142D188—412no—
143Dic94—432no—
144C8×C47—24no—
145C47⋊C8—24no—
146C22×#489—471nomaximal, Sylow
147C22×C47#489116yescomplemented
148C22×C47C2×#12416yes—
149C22×C47C2×AGL(1,ℤ/8)16yes—
150C22×C47C2×#13716yes—
151C22×C47C23×D411yescharacteristic
152C2×D47—44no—
153C2×D47—44no—
154C2×D47—44no—
155C2×D47—824no—
156C4×C47—44no—
157C4×C47—44no—
158C4×C47C2×#12416yes—
159C4×C47C22×#4111yescharacteristic
160C4×C47C22×#4111yescharacteristic
161Dic47—24no—
162Dic47—24no—
163Dic47—44no—
164C22×C4⋊Q8—471no—
165C22×#124—471no—
166C2×#489—4724no—
167C22×#103—471no—
168C22×#103D4711yescharacteristic, complemented
169C22×#102—471no—
170C22×#102—471no—
171C22×#97—471no—
172C2×C47C2×#489112yescomplemented
173C2×C47C22×#12411yescharacteristic
174C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
175C2×C47C22×#13711yescharacteristic
176D47—84no—
177C23×Q8—941no—
178C23×Q8C2×D4711yescharacteristic
179C23×D4—471no—
180C23×C8—471no—
181C23×C8C2×D4711yescharacteristic
182C2×C4⋊Q8—4712no—
183C2×C4⋊Q8—946no—
184C2×#124—4712no—
185C2×#124—946no—
186C22×C4⋊C4—471no—
187C22×C4⋊C4—941no—
188C22×C4⋊C4C2×D4711yescharacteristic
189C22×C22⋊C4—941no—
190C22×C22⋊C4—941no—
191C22×C42—471no—
192#489—4764no—
193C2×#103—4712no—
194C2×#103C2×D47112yescomplemented
195C2×#102—4712no—
196C2×#102—4712no—
197C2×#97—4712no—
198C47C22×#48911yescharacteristic, complemented, Sylow
199C25—941no—
200C22×Q8—23no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—9412no—
204C22×Q8—1888no—
205C22×Q8C22×D4716yes—
206C22×D4—476no—
207C22×D4—944no—
208C22×D4—943no—
209C22×D4—944no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—471no—
213C23×C4—471no—
214C23×C4—941no—
215C23×C4C22×D4711yescharacteristic
216C22×C8—24no—
217C22×C8—476no—
218C22×C8—944no—
219C22×C8C22×D4716yes—
220C4⋊Q8—4716no—
221C4⋊Q8—9424no—
222#124—4716no—
223#124—9424no—
224C2×C4⋊C4—23no—
225C2×C4⋊C4—476no—
226C2×C4⋊C4—9412no—
227C2×C4⋊C4—943no—
228C2×C4⋊C4C22×D4716yes—
229C2×C22⋊C4—9412no—
230C2×C22⋊C4—9412no—
231C2×C42—476no—
232C2×C42—476no—
233#103—4716no—
234#103C22×D47116yescomplemented
235#102—4716no—
236#102—4716no—
237#97—4716no—
238C24—946no—
239C24—1884no—
240C24—1884no—
241C24—1884no—
242C24D4×D4711yescharacteristic
243C2×Q8—26no—
244C2×Q8—212no—
245C2×Q8—212no—
246C2×Q8—412no—
247C2×Q8—9416no—
248C2×Q8—18848no—
249C2×Q8C23×D4714yes—
250C2×D4—474no—
251C2×D4—946no—
252C2×D4—9412no—
253C2×D4—9412no—
254C2×D4—18812no—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—476no—
260C22×C4—476no—
261C22×C4—944no—
262C22×C4—944no—
263C22×C4—946no—
264C22×C4—1884no—
265C22×C4#1062535014yes—
266C22×C4D4×D4713yes—
267C22×C4#1062535214yes—
268C22×C4C23×D4713yes—
269C2×C8—212no—
270C2×C8—474no—
271C2×C8—9412no—
272C2×C8C23×D4714yes—
273C4⋊C4—26no—
274C4⋊C4—474no—
275C4⋊C4—9416no—
276C4⋊C4—946no—
277C4⋊C4C23×D4714yes—
278C22⋊C4—9416no—
279C22⋊C4—9416no—
280C42—4712no—
281C42—474no—
282C23—944no—
283C23—37616no—
284C23—18812no—
285C23—18812no—
286C23—18812no—
287C23#1309401014yes—
288C23#1309405114yes—
289C23#1309405614yes—
290C23C2×D4×D4713yes—
291Q8—24no—
292Q8—24no—
293Q8—412no—
294Q8—18832no—
295D4—944no—
296D4—944no—
297D4—18812no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—412no—
301C2×C4—412no—
302C2×C4—474no—
303C2×C4—474no—
304C2×C4—9412no—
305C2×C4—9412no—
306C2×C4—944no—
307C2×C4—18812no—
308C2×C4#13094010112yes—
309C2×C4C2×D4×D4713yes—
310C2×C4C2×#1062535216yes—
311C2×C4C2×#1062535016yes—
312C2×C4C24×D4711yescharacteristic
313C8—24no—
314C8—944no—
315C22—37624no—
316C22—1884no—
317C22—1884no—
318C22—1884no—
319C22#15689484116yesdirect product factor
320C22C2×#1309401016yes—
321C22C2×#1309405116yes—
322C22C2×#1309405616yes—
323C22C22×D4×D4711yescharacteristic
324C4—44no—
325C4—44no—
326C4—944no—
327C4—944no—
328C4—1884no—
329C4C2×#1309401016yes—
330C4C22×#1062535211yescharacteristic
331C4C22×#1062535011yescharacteristic
332C2—3764no—
333C2C2×#15689484112yesdirect product factor
334C2C22×#1309401011yescharacteristic
335C2C22×#1309405111yescharacteristic
336C2C22×#1309405611yescharacteristic
subgroupG/H
centerC24D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC22×#103×C47C2
socleC24×C47C2×D4
prime radicalC23×D188C22
hypercenterC22×#103D47
hypocenterC47C22×#489
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
3. C2×#15689484C2direct (N × H)
4. C22×#13093948C2semidirect
7. C22×#13093934C2semidirect
8. C22×#103×C47C2semidirect
13. C2×#13094023C22semidirect
21. #15689484C22direct (N × H)
22. C2×#13093948C22semidirect
25. C2×#13093934C22semidirect
29. C2×#103×C47C22semidirect
43. #13094023C23semidirect
53. #13093948C23semidirect
56. #13093934C23semidirect
68. #103×C47C23semidirect
147. C22×C47#489semidirect
168. C22×#103D47semidirect
172. C2×C47C2×#489semidirect
194. C2×#103C2×D47semidirect
198. C47C22×#489semidirect
234. #103C22×D47semidirect
319. C22#15689484direct (N × H)
333. C2C2×#15689484direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940232
C22×#130940132
C2×#156894842
C22×#130939482
C22×#130939462
C22×#130939442
C22×#130939342
C22×#103×C472
C22×#48947