Finite Group Databasebeta

← C22×#15689119 #27079796

Subgroups — C22×#15689119 #27079796

Order 12032.

5326 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C47⋊C8C211yesmaximal, characteristic, complemented
2C22×#13093983C211yesmaximal, characteristic, complemented
3C2×#15689119C2124yesmaximal, direct product factor
4C22×#13093948C212yesmaximal, complemented
5C22×#13093945C212yesmaximal, complemented
6C23×#41×C47C211yesmaximal, characteristic
7C23×C47⋊C8—28no—
8C23×C47⋊C8C2212yes—
9C23×C47⋊C8C22112yescomplemented
10C2×#13093983—26no—
11C2×#13093983C22112yescomplemented
12C22×C4⋊Dic47C2212yes—
13C22×C22⋊Dic47—21no—
14C22×C4×Dic47—21no—
15#15689119C22164yesdirect product factor
16C2×#13093948C22124yescomplemented
17C2×#13093945C22124yescomplemented
18C22×#41×C47—28no—
19C22×#41×C47—26no—
20C22×#41×C47C4116yescomplemented
21C22×#41×C47C22112yes—
22C23×Q8×C47C2211yescharacteristic
23C23×D4×C47—21no—
24C23×D4×C47C2211yescharacteristic
25C24×C4×C47—21no—
26C24×C4×C47C2211yescharacteristic
27C23×Dic47—22no—
28C22×C47⋊C8—28no—
29C22×C47⋊C8—248no—
30C22×C47⋊C8C23112yes—
31C22×C47⋊C8C23116yescomplemented
32#13093983—224no—
33#13093983C23116yescomplemented
34C2×C4⋊Dic47—26no—
35C2×C4⋊Dic47C23112yes—
36C2×C22⋊Dic47—212no—
37C2×C4×Dic47—212no—
38#13093948C23132yescomplemented
39#13093945C23132yescomplemented
40C25×C47—21no—
41C25×C47D411yescharacteristic
42C2×#41×C47—296no—
43C2×#41×C47—248no—
44C2×#41×C47—224no—
45C2×#41×C47C2×C4196yescomplemented
46C2×#41×C47C23116yes—
47C22×Q8×C47—24no—
48C22×Q8×C47—23no—
49C22×Q8×C47C2×C418yes—
50C22×Q8×C47C2316yes—
51C22×D4×C47—216no—
52C22×D4×C47—212no—
53C22×D4×C47—23no—
54C22×D4×C47—24no—
55C22×D4×C47C2×C418yes—
56C22×D4×C47C2316yes—
57C23×C4×C47—21no—
58C23×C4×C47—216no—
59C23×C4×C47—212no—
60C23×C4×C47C2×C418yes—
61C23×C4×C47D416yes—
62C23×C4×C47D411yescharacteristic
63C23×C4×C47D418yes—
64C23×C4×C47C2311yescharacteristic
65C23×C4×C47C2316yes—
66C24×C47—28no—
67C24×C47—28no—
68C24×C47—26no—
69C24×C47—48no—
70C24×C47C22⋊C418yes—
71C24×C47C2×D416yes—
72C24×C47C2×D411yescharacteristic
73#41×C47—2192no—
74#41×C47—232no—
75#41×C47C22×C4164yescomplemented
76C2×Q8×C47—224no—
77C2×Q8×C47—212no—
78C2×Q8×C47—26no—
79C2×Q8×C47C22×C4124yes—
80C2×Q8×C47C2414yes—
81C2×D4×C47—296no—
82C2×D4×C47—216no—
83C2×D4×C47—224no—
84C2×D4×C47—26no—
85C2×D4×C47—212no—
86C2×D4×C47C22×C4124yes—
87C2×D4×C47C2414yes—
88C22×C4×C47—28no—
89C22×C4×C47—26no—
90C22×C4×C47—296no—
91C22×C4×C47—216no—
92C22×C4×C47C22⋊C418yes—
93C22×C4×C47C22⋊C4148yes—
94C22×C4×C47C22×C414yes—
95C22×C4×C47C22×C4124yes—
96C22×C4×C47C2×D416yes—
97C22×C4×C47C2×D413yes—
98C22×C4×C47C2×D4112yes—
99C22×C4×C47C2×D414yes—
100C22×C4×C47C2×D4124yes—
101C22×C4×C47C2413yes—
102C22×C4×C47C2414yes—
103C22×Dic47—212no—
104C22×Dic47—48no—
105C22⋊Dic47—216no—
106C4⋊Dic47—212no—
107C4⋊Dic47C2418yes—
108C4×Dic47—216no—
109C2×C47⋊C8—224no—
110C2×C47⋊C8—232no—
111C2×C47⋊C8C2418yes—
112C23×C47—224no—
113C23×C47—232no—
114C23×C47—224no—
115C23×C47—24no—
116C23×C47—424no—
117C23×C47—432no—
118C23×C47C2×C22⋊C4124yes—
119C23×C47C2×C22⋊C414yes—
120C23×C47#13518yes—
121C23×C47C22×D414yes—
122C23×C47C22×D413yes—
123Q8×C47—224no—
124Q8×C47—24no—
125Q8×C47C23×C418yes—
126D4×C47—264no—
127D4×C47—224no—
128D4×C47—24no—
129D4×C47C23×C418yes—
130C2×C4×C47—224no—
131C2×C4×C47—24no—
132C2×C4×C47—264no—
133C2×C4×C47C2×C22⋊C4124yes—
134C2×C4×C47C2×C22⋊C4112yes—
135C2×C4×C47C2×C22⋊C4148yes—
136C2×C4×C47C23×C416yes—
137C2×C4×C47C23×C418yes—
138C2×C4×C47C22×D414yes—
139C2×C4×C47C22×D413yes—
140C2×C4×C47C22×D416yes—
141C2×C4×C47C22×D418yes—
142C2×C4×C47C2511yescharacteristic
143C2×Dic47—28no—
144C2×Dic47—424no—
145C47⋊C8—28no—
146C22×#416—471nomaximal, Sylow
147C22×C47—248no—
148C22×C47—28no—
149C22×C47—28no—
150C22×C47—448no—
151C22×C47—48no—
152C22×C47#416116yescomplemented
153C22×C47C22×C22⋊C418yes—
154C22×C47C22×C22⋊C416yes—
155C22×C47C2×#135112yes—
156C22×C47C23×D411yescharacteristic
157C4×C47—28no—
158C4×C47C22×C22⋊C418yes—
159C4×C47C22×C22⋊C416yes—
160C4×C47C24×C411yescharacteristic
161C4×C47C23×D411yescharacteristic
162Dic47—48no—
163C23×#41D4711yescharacteristic
164C24×C8—471no—
165C22×#117—471no—
166C2×#416—4724no—
167C22×#103—472no—
168C22×#102—472no—
169C2×C47—28no—
170C2×C47—48no—
171C2×C47C2×#416112yescomplemented
172C2×C47C23×C22⋊C411yescharacteristic
173C2×C47C22×#13512yes—
174C22×#41—28no—
175C22×#41—26no—
176C22×#41Dic47116yescomplemented
177C22×#41C2×D47112yes—
178C23×Q8C2×D4711yescharacteristic
179C23×D4—21no—
180C23×D4C2×D4711yescharacteristic
181C24×C4—21no—
182C24×C4C2×D4711yescharacteristic
183C23×C8—472no—
184C23×C8—4712no—
185C23×C8—948no—
186C2×#117—4712no—
187C2×#117—946no—
188C22×C4⋊C4—472no—
189C22×C22⋊C4—941no—
190C22×C42—941no—
191#416—4764no—
192C2×#103—4724no—
193C2×#102—4724no—
194C47C22×#41611yescharacteristic, complemented, Sylow
195C25—21no—
196C25C22⋊D4711yescharacteristic
197C2×#41—296no—
198C2×#41—248no—
199C2×#41—224no—
200C2×#41C2×Dic47196yescomplemented
201C2×#41C22×D47116yes—
202C22×Q8—24no—
203C22×Q8—23no—
204C22×Q8C2×Dic4718yes—
205C22×Q8C22×D4716yes—
206C22×D4—216no—
207C22×D4—212no—
208C22×D4—23no—
209C22×D4—24no—
210C22×D4C2×Dic4718yes—
211C22×D4C22×D4716yes—
212C23×C4—21no—
213C23×C4—216no—
214C23×C4—212no—
215C23×C4—942no—
216C23×C4C2×Dic4718yes—
217C23×C4C22⋊D4716yes—
218C23×C4C22⋊D4718yes—
219C23×C4C22⋊D4711yescharacteristic
220C23×C4C22×D4711yescharacteristic
221C23×C4C22×D4716yes—
222C22×C8—4712no—
223C22×C8—4716no—
224C22×C8—948no—
225C22×C8—9448no—
226#117—4716no—
227#117—9424no—
228C2×C4⋊C4—4712no—
229C2×C4⋊C4—946no—
230C2×C22⋊C4—9412no—
231C2×C42—9412no—
232#103—4732no—
233#102—4732no—
234C24—28no—
235C24—28no—
236C24—26no—
237C24—48no—
238C24C22⋊Dic4718yes—
239C24C2×C22⋊D4716yes—
240C24C2×C22⋊D4711yescharacteristic
241#41—2192no—
242#41—232no—
243#41C22×Dic47164yescomplemented
244C2×Q8—224no—
245C2×Q8—212no—
246C2×Q8—26no—
247C2×Q8C22×Dic47124yes—
248C2×Q8C23×D4714yes—
249C2×D4—296no—
250C2×D4—216no—
251C2×D4—224no—
252C2×D4—26no—
253C2×D4—212no—
254C2×D4C22×Dic47124yes—
255C2×D4C23×D4714yes—
256C22×C4—28no—
257C22×C4—26no—
258C22×C4—296no—
259C22×C4—216no—
260C22×C4—9412no—
261C22×C4—1888no—
262C22×C4C22⋊Dic4718yes—
263C22×C4C22⋊Dic47148yes—
264C22×C4C22×Dic4714yes—
265C22×C4C22×Dic47124yes—
266C22×C4C2×C22⋊D4716yes—
267C22×C4C2×C22⋊D4713yes—
268C22×C4C2×C22⋊D47112yes—
269C22×C4C2×C22⋊D4714yes—
270C22×C4C2×C22⋊D47124yes—
271C22×C4C23×D4713yes—
272C22×C4C23×D4714yes—
273C2×C8—478no—
274C2×C8—9424no—
275C2×C8—9432no—
276C4⋊C4—478no—
277C4⋊C4—9412no—
278C22⋊C4—9416no—
279C42—9416no—
280C23—232no—
281C23—224no—
282C23—224no—
283C23—24no—
284C23—432no—
285C23—424no—
286C23C2×C22⋊Dic47124yes—
287C23C2×C22⋊Dic4714yes—
288C23#1309404418yes—
289C23C22×C22⋊D4714yes—
290C23C22×C22⋊D4713yes—
291Q8—224no—
292Q8—24no—
293Q8C23×Dic4718yes—
294D4—264no—
295D4—224no—
296D4—24no—
297D4C23×Dic4718yes—
298C2×C4—224no—
299C2×C4—24no—
300C2×C4—264no—
301C2×C4—948no—
302C2×C4—18824no—
303C2×C4C2×C22⋊Dic47124yes—
304C2×C4C2×C22⋊Dic47112yes—
305C2×C4C2×C22⋊Dic47148yes—
306C2×C4C23×Dic4716yes—
307C2×C4C23×Dic4718yes—
308C2×C4C22×C22⋊D4714yes—
309C2×C4C22×C22⋊D4713yes—
310C2×C4C22×C22⋊D4716yes—
311C2×C4C22×C22⋊D4718yes—
312C2×C4C24×D4711yescharacteristic
313C8—948no—
314C22—28no—
315C22—248no—
316C22—28no—
317C22—48no—
318C22—448no—
319C22#15689119116yesdirect product factor
320C22C22×C22⋊Dic4718yes—
321C22C22×C22⋊Dic4716yes—
322C22C2×#13094044112yes—
323C22C23×C22⋊D4711yescharacteristic
324C4—28no—
325C4—1888no—
326C4C22×C22⋊Dic4718yes—
327C4C22×C22⋊Dic4716yes—
328C4C24×Dic4711yescharacteristic
329C4C23×C22⋊D4711yescharacteristic
330C2—28no—
331C2—48no—
332C2C2×#15689119112yesdirect product factor
333C2C23×C22⋊Dic4711yescharacteristic
334C2C22×#1309404412yes—
subgroupG/H
centerC23×C4C22⋊D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
Fitting, prime radicalC23×#41×C47C2
socleC24×C47C2×D4
cosocleC2×C4×C47C25
hypercenterC23×#41D47
hypocenterC47C22×#416
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C24×C47⋊C8C2semidirect
2. C22×#13093983C2semidirect
3. C2×#15689119C2direct (N × H)
4. C22×#13093948C2semidirect
5. C22×#13093945C2semidirect
9. C23×C47⋊C8C22semidirect
11. C2×#13093983C22semidirect
15. #15689119C22direct (N × H)
16. C2×#13093948C22semidirect
17. C2×#13093945C22semidirect
20. C22×#41×C47C4semidirect
31. C22×C47⋊C8C23semidirect
33. #13093983C23semidirect
38. #13093948C23semidirect
39. #13093945C23semidirect
45. C2×#41×C47C2×C4semidirect
75. #41×C47C22×C4semidirect
152. C22×C47#416semidirect
171. C2×C47C2×#416semidirect
176. C22×#41Dic47semidirect
194. C47C22×#416semidirect
200. C2×#41C2×Dic47semidirect
243. #41C22×Dic47semidirect
319. C22#15689119direct (N × H)
332. C2C2×#15689119direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C47⋊C82
C22×#130939832
C2×#156891192
C22×#130939482
C22×#130939452
C23×#41×C472
C22×#41647