Finite Group Databasebeta

← C22×#15688848 #26182066

Subgroups — C22×#15688848 #26182066

Order 12032.

1806 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
1C23×Dic47⋊C4C211yesmaximal, characteristic
2C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic
3C2×#15688848C2112yesmaximal, direct product factor
4C22×C22⋊C8×C47C211yesmaximal, characteristic
5C24×Dic47C411yescharacteristic
6C24×Dic47C411yescharacteristic
7C23×C47⋊C8—21no—
8C22×Dic47⋊C4—21no—
9C22×Dic47⋊C4—23no—
10C22×Dic47⋊C4—21no—
11C22×Dic47⋊C4—23no—
12C22×Dic47⋊C4C416yes—
13C22×Dic47⋊C4C2216yes—
14C2×C22⋊(C47⋊C8)—23no—
15C2×C22⋊(C47⋊C8)C2216yes—
16#15688848C22116yesdirect product factor
17C24×C4×C47C2211yescharacteristic
18C23×C8×C47—21no—
19C2×C22⋊C8×C47—23no—
20C2×C22⋊C8×C47C2216yes—
21C23×Dic47—23no—
22C23×Dic47—21no—
23C23×Dic47—24no—
24C23×Dic47—23no—
25C23×Dic47—21no—
26C23×Dic47—44no—
27C23×Dic47C818yescomplemented
28C23×Dic47C2×C416yes—
29C23×Dic47C2×C416yes—
30C22×C47⋊C8—26no—
31C22×C47⋊C8—44no—
32C2×Dic47⋊C4—26no—
33C2×Dic47⋊C4—26no—
34C2×Dic47⋊C4—212no—
35C2×Dic47⋊C4—26no—
36C2×Dic47⋊C4—26no—
37C2×Dic47⋊C4—26no—
38C2×Dic47⋊C4—26no—
39C2×Dic47⋊C4C2×C4112yes—
40C2×Dic47⋊C4C2314yes—
41C22⋊(C47⋊C8)—26no—
42C22⋊(C47⋊C8)C2314yes—
43C25×C47C2×C411yescharacteristic
44C23×C4×C47—44no—
45C23×C4×C47C2×C413yes—
46C23×C4×C47D411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D413yes—
49C23×C4×C47D413yes—
50C23×C4×C47C2313yes—
51C22×C8×C47—26no—
52C22×C8×C47—44no—
53C22⋊C8×C47—26no—
54C22⋊C8×C47C2314yes—
55C24×C47—24no—
56C24×C47—24no—
57C24×C47C22⋊C413yes—
58C24×C47C22⋊C411yescharacteristic
59C24×C47C2×C814yes—
60C24×C47M1614yes—
61C24×C47C22×C413yes—
62C22×C4×C47—24no—
63C22×C4×C47—24no—
64C22×C4×C47—46no—
65C22×C4×C47—46no—
66C22×C4×C47—46no—
67C22×C4×C47—46no—
68C22×C4×C47C22⋊C413yes—
69C22×C4×C47C22⋊C413yes—
70C22×C4×C47C22⋊C416yes—
71C22×C4×C47C22×C413yes—
72C22×C4×C47C2×D413yes—
73C22×C4×C47C2×D413yes—
74C22×C4×C47C2×D413yes—
75C22×C4×C47C2×D413yes—
76C22×C4×C47C2411yescharacteristic
77C22×Dic47—224no—
78C22×Dic47—28no—
79C22×Dic47—26no—
80C22×Dic47—26no—
81C22×Dic47—212no—
82C22×Dic47—26no—
83C22×Dic47—26no—
84C22×Dic47—412no—
85C22×Dic47—44no—
86C22×Dic47—412no—
87C22×Dic47C2×C8124yescomplemented
88C22×Dic47C22×C414yes—
89C22×Dic47C22×C414yes—
90C2×C8×C47—24no—
91C2×C8×C47—412no—
92Dic47⋊C4—212no—
93Dic47⋊C4—212no—
94Dic47⋊C4—24no—
95Dic47⋊C4—24no—
96C2×C47⋊C8—24no—
97C2×C47⋊C8—412no—
98C23×C47—216no—
99C23×C47—216no—
100C23×C47—26no—
101C23×C47—26no—
102C23×C47—26no—
103C23×C47—26no—
104C23×C47C22⋊C8112yes—
105C23×C47C22⋊C814yes—
106C23×C47#9914yes—
107C23×C47#10114yes—
108C23×C47C2×C22⋊C413yes—
109C23×C47C2×C22⋊C413yes—
110C23×C47C22×C816yes—
111C23×C47C2×M1616yes—
112C23×C47C23×C411yescharacteristic
113C2×C4×C47—26no—
114C2×C4×C47—26no—
115C2×C4×C47—26no—
116C2×C4×C47—26no—
117C2×C4×C47—46no—
118C2×C4×C47—43no—
119C2×C4×C47—43no—
120C2×C4×C47—43no—
121C2×C4×C47—41no—
122C2×C4×C47C2×C22⋊C413yes—
123C2×C4×C47C2×C22⋊C413yes—
124C2×C4×C47C22×D411yescharacteristic
125C2×C4×C47C22×D411yescharacteristic
126C2×Dic47—24no—
127C2×Dic47—24no—
128C2×Dic47—224no—
129C2×Dic47—224no—
130C2×Dic47—24no—
131C2×Dic47—412no—
132C2×Dic47—412no—
133C2×Dic47—44no—
134C2×Dic47C22×C818yescomplemented
135C8×C47—44no—
136C47⋊C8—44no—
137C22×#324—471nomaximal, Sylow
138C22×C47—23no—
139C22×C47—21no—
140C22×C47—23no—
141C22×C47—21no—
142C22×C47—224no—
143C22×C47—212no—
144C22×C47—212no—
145C22×C47#324116yescomplemented
146C22×C47C2×C22⋊C816yes—
147C22×C47C2×C22⋊C816yes—
148C22×C47C2×#9916yes—
149C22×C47C2×#10116yes—
150C22×C47C22×C22⋊C411yescharacteristic
151C22×C47C23×C811yescharacteristic
152C22×C47C22×M1611yescharacteristic
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—23no—
156C4×C47—23no—
157Dic47—28no—
158Dic47—44no—
159C23×C4⋊C4—471no—
160C22×C22⋊C8—471no—
161C22×C22⋊C8D4711yescharacteristic
162C2×#324—4712no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—26no—
166C2×C47C2×#324112yescomplemented
167C2×C47C22×C22⋊C811yescharacteristic
168C2×C47C22×#9911yescharacteristic
169C2×C47C22×#10111yescharacteristic
170C24×C4—471no—
171C24×C4—471no—
172C24×C4C2×D4711yescharacteristic
173C23×C8—21no—
174C23×C8—941no—
175C22×C4⋊C4—476no—
176C22×C4⋊C4—476no—
177C22×C4⋊C4—943no—
178C22×C4⋊C4—941no—
179C22×C4⋊C4—943no—
180C22×C4⋊C4—941no—
181C2×C22⋊C8—23no—
182C2×C22⋊C8—476no—
183C2×C22⋊C8—943no—
184C2×C22⋊C8C2×D4716yes—
185#324—4716no—
186C47C22×#32411yescharacteristic, complemented, Sylow
187C25C4×D4711yescharacteristic
188C23×C4—44no—
189C23×C4—478no—
190C23×C4—476no—
191C23×C4—476no—
192C23×C4—944no—
193C23×C4—943no—
194C23×C4—941no—
195C23×C4—943no—
196C23×C4—941no—
197C23×C4—1884no—
198C23×C4C4×D4713yes—
199C23×C4D18811yescharacteristic
200C23×C4D18813yes—
201C23×C4C22⋊D4711yescharacteristic
202C23×C4C22⋊D4713yes—
203C23×C4C22×D4713yes—
204C22×C8—26no—
205C22×C8—44no—
206C22×C8—946no—
207C22×C8—1884no—
208C2×C4⋊C4—4712no—
209C2×C4⋊C4—474no—
210C2×C4⋊C4—946no—
211C2×C4⋊C4—946no—
212C2×C4⋊C4—9412no—
213C2×C4⋊C4—946no—
214C2×C4⋊C4—946no—
215C2×C4⋊C4—946no—
216C2×C4⋊C4—946no—
217C22⋊C8—26no—
218C22⋊C8—474no—
219C22⋊C8—946no—
220C22⋊C8C22×D4714yes—
221C24—24no—
222C24—24no—
223C24C8×D4714yes—
224C24#1062532414yes—
225C24(C2×D47)⋊C413yes—
226C24(C2×D47)⋊C411yescharacteristic
227C24C2×C4×D4713yes—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—46no—
231C22×C4—46no—
232C22×C4—46no—
233C22×C4—46no—
234C22×C4—4724no—
235C22×C4—474no—
236C22×C4—474no—
237C22×C4—9424no—
238C22×C4—948no—
239C22×C4—946no—
240C22×C4—946no—
241C22×C4—9412no—
242C22×C4—946no—
243C22×C4—946no—
244C22×C4—18812no—
245C22×C4—1884no—
246C22×C4—18812no—
247C22×C4(C2×D47)⋊C413yes—
248C22×C4(C2×D47)⋊C413yes—
249C22×C4(C2×D47)⋊C416yes—
250C22×C4C2×C4×D4713yes—
251C22×C4C2×D18813yes—
252C22×C4C2×D18813yes—
253C22×C4C2×C22⋊D4713yes—
254C22×C4C2×C22⋊D4713yes—
255C22×C4C23×D4711yescharacteristic
256C2×C8—24no—
257C2×C8—412no—
258C2×C8—944no—
259C2×C8—18812no—
260C4⋊C4—9412no—
261C4⋊C4—9412no—
262C4⋊C4—944no—
263C4⋊C4—944no—
264C23—26no—
265C23—26no—
266C23—26no—
267C23—26no—
268C23—216no—
269C23—216no—
270C23(C2×D47)⋊C8112yes—
271C23(C2×D47)⋊C814yes—
272C23#1309393814yes—
273C23#1309394214yes—
274C23C2×(C2×D47)⋊C413yes—
275C23C2×(C2×D47)⋊C413yes—
276C23C2×C8×D4716yes—
277C23C2×#1062532416yes—
278C23C22×C4×D4711yescharacteristic
279C2×C4—26no—
280C2×C4—26no—
281C2×C4—26no—
282C2×C4—26no—
283C2×C4—46no—
284C2×C4—43no—
285C2×C4—43no—
286C2×C4—43no—
287C2×C4—41no—
288C2×C4—478no—
289C2×C4—944no—
290C2×C4—9424no—
291C2×C4—9424no—
292C2×C4—944no—
293C2×C4—944no—
294C2×C4—18812no—
295C2×C4—18812no—
296C2×C4—1884no—
297C2×C4C2×(C2×D47)⋊C413yes—
298C2×C4C2×(C2×D47)⋊C413yes—
299C2×C4C22×C22⋊D4711yescharacteristic
300C2×C4C22×D18811yescharacteristic
301C8—44no—
302C8—1884no—
303C22—224no—
304C22—212no—
305C22—212no—
306C22—23no—
307C22—21no—
308C22—23no—
309C22—21no—
310C22#15688848116yesdirect product factor
311C22C2×(C2×D47)⋊C816yes—
312C22C2×(C2×D47)⋊C816yes—
313C22C2×#1309393816yes—
314C22C2×#1309394216yes—
315C22C22×(C2×D47)⋊C411yescharacteristic
316C22C22×C8×D4711yescharacteristic
317C22C22×#1062532411yescharacteristic
318C4—21no—
319C4—21no—
320C4—23no—
321C4—23no—
322C4—948no—
323C4—1884no—
324C2—26no—
325C2—21no—
326C2—21no—
327C2C2×#15688848112yesdirect product factor
328C2C22×(C2×D47)⋊C811yescharacteristic
329C2C22×#1309393811yescharacteristic
330C2C22×#1309394211yescharacteristic
subgroupG/H
centerC24(C2×D47)⋊C4
commutatorC22×C47C23×C8
FrattiniC22×C4C23×D47
FittingC22×C22⋊C8×C47C2
socleC24×C47C22⋊C4
prime radicalC25×C47C2×C4
cosocleC22×C4×C47C24
hypercenterC22×C22⋊C8D47
hypocenterC47C22×#324
LCSG ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×#15688848C2direct (N × H)
16. #15688848C22direct (N × H)
27. C23×Dic47C8semidirect
87. C22×Dic47C2×C8semidirect
134. C2×Dic47C22×C8semidirect
145. C22×C47#324semidirect
166. C2×C47C2×#324semidirect
186. C47C22×#324semidirect
310. C22#15688848direct (N × H)
327. C2C2×#15688848direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Dic47⋊C42
C22×C22⋊(C47⋊C8)2
C2×#156888482
C22×C22⋊C8×C472
C22×#32447