Finite Group Databasebeta

← #26376347

Subgroups — #26376347

Order 12032.

2390 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246281C218yesmaximal, complemented
2#20246273C218yesmaximal, complemented
3#20246006C212yesmaximal, complemented
4#20245986C214yesmaximal
5#20245786C214yesmaximal, complemented
6#20245771C212yesmaximal
7#20245745C212yesmaximal, complemented
8C22×#117×C47C211yesmaximal, characteristic
9C2×#13093983C2214yes—
10#15689022C22116yescomplemented
11#15689016C22132yes—
12#15688991C2218yescomplemented
13#15688989C2218yescomplemented
14#15688986C2214yes—
15#15688983C2214yes—
16#15688983C2214yes—
17#15688983C2218yes—
18#15688979C2214yes—
19C4⋊C4×Dic47C22116yescomplemented
20C4×C4⋊Dic47C2218yescomplemented
21C4×C22⋊Dic47C2214yes—
22C4×C22⋊Dic47C2214yes—
23C4×C22⋊Dic47C2218yes—
24#15688969C2218yescomplemented
25C24×C4×C47C2211yescharacteristic
26C2×#117×C47C4116yescomplemented
27C2×#117×C47C2214yes—
28C2×#117×C47C2214yes—
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47C2212yes—
32C22×C42×C47C2212yes—
33#13093983—216no—
34C2×C4⋊Dic47C2318yes—
35C2×C4⋊Dic47C23116yes—
36C2×C22⋊Dic47C2318yes—
37C2×C22⋊Dic47C2318yes—
38C2×C22⋊Dic47C2318yes—
39C2×C4×Dic47C2318yes—
40C2×C4×Dic47C23116yes—
41C47⋊#95C2318yes—
42C47⋊#95C23116yes—
43C47⋊#95C2318yes—
44C47⋊#95C23116yes—
45C25×C47C2311yescharacteristic
46C23×C4×C47—28no—
47C23×C4×C47C2×C418yes—
48C23×C4×C47C2312yes—
49C23×C4×C47C2312yes—
50C23×C4×C47C2311yescharacteristic
51C23×C4×C47C2312yes—
52C23×C4×C47C2312yes—
53C23×C4×C47C2311yescharacteristic
54#117×C47C2×C4132yes—
55#117×C47C2×C4132yes—
56C2×C4⋊C4×C47C2×C418yes—
57C2×C4⋊C4×C47C2×C418yes—
58C2×C4⋊C4×C47C2314yes—
59C2×C4⋊C4×C47C2314yes—
60C2×C22⋊C4×C47C2×C4116yes—
61C2×C22⋊C4×C47C2318yes—
62C2×C42×C47C2×C4116yes—
63C2×C42×C47C2314yes—
64C2×C42×C47C2314yes—
65C24×C47—24no—
66C24×C47—42no—
67C24×C47—42no—
68C24×C47C22×C414yes—
69C24×C47C2412yes—
70C24×C47C2411yescharacteristic
71C22×C4×C47—24no—
72C22×C4×C47—232no—
73C22×C4×C47—28no—
74C22×C4×C47—28no—
75C22×C4×C47—44no—
76C22×C4×C47—44no—
77C22×C4×C47C22×C414yes—
78C22×C4×C47C22×C418yes—
79C22×C4×C47C22×C418yes—
80C22×C4×C47C22×C418yes—
81C22×C4×C47C22×C418yes—
82C22×C4×C47C22×C414yes—
83C22×C4×C47#4118yes—
84C22×C4×C47C2411yescharacteristic
85C22×C4×C47C2411yescharacteristic
86C22×C4×C47C2414yes—
87C22×C4×C47C2414yes—
88C22×C4×C47C2412yes—
89C22×Dic47C24116yes—
90C4⋊C4×C47C22×C4116yes—
91C4⋊C4×C47C22×C4116yes—
92C22⋊C4×C47C22×C4132yes—
93C42×C47C22×C4116yes—
94C42×C47C22×C4116yes—
95C22⋊Dic47—216no—
96C22⋊Dic47—48no—
97C22⋊Dic47—48no—
98C4⋊Dic47—216no—
99C4⋊Dic47—416no—
100C4×Dic47—216no—
101C4×Dic47—416no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—28no—
105C23×C47—24no—
106C23×C47—24no—
107C23×C47—24no—
108C23×C47—44no—
109C23×C47—44no—
110C23×C47—416no—
111C23×C47C23×C414yes—
112C23×C47C23×C412yes—
113C23×C47C2×#4114yes—
114C23×C47C2511yescharacteristic
115C2×C4×C47—28no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—28no—
119C2×C4×C47—28no—
120C2×C4×C47—216no—
121C2×C4×C47—216no—
122C2×C4×C47—48no—
123C2×C4×C47—48no—
124C2×C4×C47#117116yes—
125C2×C4×C47C23×C412yes—
126C2×C4×C47C23×C412yes—
127C2×C4×C47C23×C418yes—
128C2×C4×C47C23×C418yes—
129C2×C4×C47C23×C414yes—
130C2×C4×C47C2×#4118yes—
131C2×Dic47—216no—
132C2×Dic47—216no—
133C2×Dic47—216no—
134#34179—471nomaximal, Sylow
135C22×C47—22no—
136C22×C47—22no—
137C22×C47—28no—
138C22×C47—24no—
139C22×C47—24no—
140C22×C47—24no—
141C22×C47—416no—
142C22×C47—48no—
143C22×C47C2×#11714yes—
144C22×C47C24×C411yescharacteristic
145C22×C47C22×#4112yes—
146C22×C47#58512yes—
147C22×C47#58512yes—
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—28no—
151C4×C47—28no—
152C4×C47C2×#11718yes—
153Dic47—416no—
154C22×#117D4711yescharacteristic
155#2347—478no—
156#2346—478no—
157#2316—472no—
158#2312—474no—
159#2283—474no—
160#2280—472no—
161#2274—472no—
162C2×C47—24no—
163C2×C47—44no—
164C2×C47C22×#11711yescharacteristic
165C2×C47#342811yescharacteristic
166C2×C47#342811yescharacteristic
167C2×C47#349314yes—
168C24×C4C2×D4711yescharacteristic
169C2×#117—474no—
170C2×#117Dic47116yescomplemented
171C2×#117C2×D4714yes—
172C2×#117C2×D4714yes—
173C22×C4⋊C4C2×D4711yescharacteristic
174C22×C4⋊C4C2×D4711yescharacteristic
175C22×C22⋊C4C2×D4712yes—
176C22×C42C2×D4712yes—
177#389—4716no—
178#388—4732no—
179#383—478no—
180#382—478no—
181#381—474no—
182#380—478no—
183#380—474no—
184#380—474no—
185#379—474no—
186C4×C4⋊C4—4716no—
187C4×C4⋊C4—478no—
188C4×C22⋊C4—478no—
189C4×C22⋊C4—474no—
190C4×C22⋊C4—474no—
191#376—478no—
192C47#3417911yescharacteristic, complemented, Sylow
193C25C22×D4711yescharacteristic
194C23×C4—28no—
195C23×C4C2×Dic4718yes—
196C23×C4C22×D4712yes—
197C23×C4C22×D4712yes—
198C23×C4C22×D4711yescharacteristic
199C23×C4C22×D4712yes—
200C23×C4C22×D4712yes—
201C23×C4C22×D4711yescharacteristic
202#117—9416no—
203#117C2×Dic47132yes—
204#117C2×Dic47132yes—
205C2×C4⋊C4—478no—
206C2×C4⋊C4—4716no—
207C2×C4⋊C4C2×Dic4718yes—
208C2×C4⋊C4C2×Dic4718yes—
209C2×C4⋊C4C22×D4714yes—
210C2×C4⋊C4C22×D4714yes—
211C2×C22⋊C4—478no—
212C2×C22⋊C4—478no—
213C2×C22⋊C4—478no—
214C2×C22⋊C4C2×Dic47116yes—
215C2×C22⋊C4C22×D4718yes—
216C2×C42—478no—
217C2×C42—4716no—
218C2×C42C2×Dic47116yes—
219C2×C42C22×D4714yes—
220C2×C42C22×D4714yes—
221#95—478no—
222#95—4716no—
223#95—478no—
224#95—4716no—
225C24—24no—
226C24—42no—
227C24—42no—
228C24C22×Dic4714yes—
229C24C23×D4712yes—
230C24C23×D4711yescharacteristic
231C22×C4—24no—
232C22×C4—232no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—4716no—
238C22×C4#1062535018yes—
239C22×C4C22×Dic4714yes—
240C22×C4C22×Dic4718yes—
241C22×C4C22×Dic4718yes—
242C22×C4C22×Dic4718yes—
243C22×C4C22×Dic4718yes—
244C22×C4C22×Dic4714yes—
245C22×C4C23×D4711yescharacteristic
246C22×C4C23×D4711yescharacteristic
247C22×C4C23×D4714yes—
248C22×C4C23×D4714yes—
249C22×C4C23×D4712yes—
250C4⋊C4—9416no—
251C4⋊C4—18816no—
252C4⋊C4C22×Dic47116yes—
253C4⋊C4C22×Dic47116yes—
254C22⋊C4—9416no—
255C22⋊C4—1888no—
256C22⋊C4—1888no—
257C22⋊C4C22×Dic47132yes—
258C42—9416no—
259C42—18816no—
260C42C22×Dic47116yes—
261C42C22×Dic47116yes—
262C23—22no—
263C23—22no—
264C23—28no—
265C23—24no—
266C23—24no—
267C23—24no—
268C23—44no—
269C23—44no—
270C23—416no—
271C23C23×Dic4714yes—
272C23C23×Dic4712yes—
273C23C2×#1062535014yes—
274C23C24×D4711yescharacteristic
275C2×C4—28no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—216no—
281C2×C4—216no—
282C2×C4—48no—
283C2×C4—48no—
284C2×C4—9416no—
285C2×C4—9416no—
286C2×C4—9416no—
287C2×C4#13093983116yes—
288C2×C4C23×Dic4712yes—
289C2×C4C23×Dic4712yes—
290C2×C4C23×Dic4718yes—
291C2×C4C23×Dic4718yes—
292C2×C4C23×Dic4714yes—
293C2×C4C2×#1062535018yes—
294C22—22no—
295C22—22no—
296C22—28no—
297C22—24no—
298C22—24no—
299C22—24no—
300C22—416no—
301C22—48no—
302C22C2×#1309398314yes—
303C22C24×Dic4711yescharacteristic
304C22C22×#1062535012yes—
305C22#1568995912yes—
306C22#1568995912yes—
307C4—22no—
308C4—22no—
309C4—28no—
310C4—28no—
311C4—18816no—
312C4C2×#1309398318yes—
313C2—24no—
314C2—44no—
315C2C22×#1309398311yescharacteristic
316C2#2025659111yescharacteristic
317C2#2025659111yescharacteristic
318C2#2025710814yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC22×#117×C47C2
socle, cosocleC23×C47C25
prime radicalC25×C47C23
hypercenterC22×#117D47
hypocenterC47#34179
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. #20246281C2semidirect
2. #20246273C2semidirect
3. #20246006C2semidirect
5. #20245786C2semidirect
7. #20245745C2semidirect
10. #15689022C22semidirect
12. #15688991C22semidirect
13. #15688989C22semidirect
19. C4⋊C4×Dic47C22semidirect
20. C4×C4⋊Dic47C22semidirect
24. #15688969C22semidirect
26. C2×#117×C47C4semidirect
170. C2×#117Dic47semidirect
192. C47#34179semidirect

Maximal subgroups

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

subgroupindex
#202462812
#202462732
#202460062
#202459862
#202457862
#202457712
#202457452
C22×#117×C472
#3417947