Finite Group Databasebeta

← C2×#20252026 #27080009

Subgroups — C2×#20252026 #27080009

Order 12032.

2198 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252026C2116yesmaximal, direct product factor
2C2×#15689184C218yesmaximal, complemented
3C2×#15689147C212yesmaximal, complemented
4C2×#15689087C212yesmaximal
5C2×C4×#10625329C212yesmaximal, complemented
6C22×#117×C47C211yesmaximal, characteristic
7C22×#10625329C2214yes—
8#15689184C22164yescomplemented
9#15689147C22116yescomplemented
10C2×C4⋊(C47⋊C8)C2218yescomplemented
11#15689087C22116yes—
12C2×C22⋊(C47⋊C8)C2218yes—
13C4×#10625329C22116yescomplemented
14C2×#13093935C2214yescomplemented
15C2×C4×C47⋊C8C2214yescomplemented
16C24×C4×C47C2211yescharacteristic
17C2×#117×C47C414yes—
18C2×#117×C47C418yes—
19C2×#117×C47C2214yes—
20C2×#117×C47C2218yes—
21C22×C4⋊C4×C47C412yes—
22C22×C22⋊C4×C47C412yes—
23C22×C42×C47C2211yescharacteristic
24C22×C42×C47C2211yescharacteristic
25C2×#10625329—216no—
26C2×#10625329C23116yes—
27C22×C47⋊C8C2318yes—
28C4⋊(C47⋊C8)C23132yescomplemented
29C22⋊(C47⋊C8)C23132yes—
30#13093935C23116yescomplemented
31C4×C47⋊C8C23116yescomplemented
32C25×C47C2×C411yescharacteristic
33C23×C4×C47—28no—
34C23×C4×C47C2×C411yescharacteristic
35C23×C4×C47C2×C412yes—
36C23×C4×C47C2×C414yes—
37C23×C4×C47C2×C412yes—
38C23×C4×C47C2×C412yes—
39C23×C4×C47C2312yes—
40C23×C4×C47C2314yes—
41C23×C4×C47C2311yescharacteristic
42#117×C47C2×C4116yes—
43#117×C47C2×C4132yes—
44#117×C47C23116yes—
45C2×C4⋊C4×C47C2×C418yes—
46C2×C4⋊C4×C47C2×C4116yes—
47C2×C22⋊C4×C47C2×C418yes—
48C2×C22⋊C4×C47C2×C4116yes—
49C2×C42×C47C2×C412yes—
50C2×C42×C47C2×C412yes—
51C2×C42×C47C2×C414yes—
52C2×C42×C47C2×C414yes—
53C2×C42×C47C2312yes—
54C2×C42×C47C2314yes—
55C2×C42×C47C2312yes—
56C2×C42×C47C2314yes—
57C24×C47—24no—
58C24×C47—44no—
59C24×C47C22×C412yes—
60C24×C47C22×C411yescharacteristic
61C24×C47C22×C414yes—
62C22×C4×C47—24no—
63C22×C4×C47—216no—
64C22×C4×C47—28no—
65C22×C4×C47—28no—
66C22×C4×C47—216no—
67C22×C4×C47—44no—
68C22×C4×C47—44no—
69C22×C4×C47C22×C414yes—
70C22×C4×C47C22×C414yes—
71C22×C4×C47C22×C414yes—
72C22×C4×C47C22×C412yes—
73C22×C4×C47C22×C414yes—
74C22×C4×C47C22×C418yes—
75C22×C4×C47C22×C418yes—
76C22×C4×C47C22×C418yes—
77C22×C4×C47C22×C412yes—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47#4118yes—
80C22×C4×C47C2414yes—
81C22×C4×C47C2412yes—
82C22×C4×C47C2411yescharacteristic
83C4⋊C4×C47C22×C4132yes—
84C22⋊C4×C47C22×C4132yes—
85C42×C47C22×C414yes—
86C42×C47C22×C418yes—
87C42×C47C22×C414yes—
88C42×C47C22×C418yes—
89C42×C47C2414yes—
90C42×C47C2414yes—
91#10625329—232no—
92C2×C47⋊C8—216no—
93C2×C47⋊C8C24116yes—
94C23×C47—24no—
95C23×C47—28no—
96C23×C47—24no—
97C23×C47—28no—
98C23×C47—48no—
99C23×C47—416no—
100C23×C47#11714yes—
101C23×C47C23×C414yes—
102C23×C47C23×C412yes—
103C23×C47C23×C411yescharacteristic
104C2×C4×C47—28no—
105C2×C4×C47—24no—
106C2×C4×C47—28no—
107C2×C4×C47—28no—
108C2×C4×C47—28no—
109C2×C4×C47—216no—
110C2×C4×C47—28no—
111C2×C4×C47—48no—
112C2×C4×C47—48no—
113C2×C4×C47#11718yes—
114C2×C4×C47#11714yes—
115C2×C4×C47C23×C418yes—
116C2×C4×C47C23×C418yes—
117C2×C4×C47C23×C414yes—
118C2×C4×C47C23×C412yes—
119C2×C4×C47C23×C411yescharacteristic
120C2×C4×C47C2×#4114yes—
121C2×C4×C47C2×#4118yes—
122C2×C4×C47C2511yescharacteristic
123C47⋊C8—216no—
124C2×#2921—471nomaximal, Sylow
125C22×C47—28no—
126C22×C47—24no—
127C22×C47—24no—
128C22×C47—28no—
129C22×C47—416no—
130C22×C47—48no—
131C22×C47C2×#11714yes—
132C22×C47C2×#11712yes—
133C22×C47#56814yes—
134C22×C47C24×C411yescharacteristic
135C4×C47—24no—
136C4×C47—28no—
137C4×C47—28no—
138C4×C47C2×#11712yes—
139C4×C47C2×#11714yes—
140C4×C47C22×#4112yes—
141C22×#117D4711yescharacteristic
142#2921—4716no—
143C2×#433—478no—
144C2×#423—472no—
145C2×#407—472no—
146C2×C4×M16—472no—
147C2×C47—24no—
148C2×C47—44no—
149C2×C47#292114yescomplemented
150C2×C47C22×#11711yescharacteristic
151C2×C47C2×#56812yes—
152C24×C4C2×D4711yescharacteristic
153C22×M16—474no—
154C2×#117Dic4714yes—
155C2×#117Dic4718yes—
156C2×#117C2×D4714yes—
157C2×#117C2×D4718yes—
158C22×C4⋊C4Dic4712yes—
159C22×C22⋊C4Dic4712yes—
160C22×C42C2×D4711yescharacteristic
161C22×C42C2×D4711yescharacteristic
162#433—4764no—
163#423—4716no—
164C2×C4⋊C8—478no—
165#407—4716no—
166C2×C22⋊C8—478no—
167C4×M16—4716no—
168C2×#97—474no—
169C2×C4×C8—474no—
170C47C2×#292111yescharacteristic, complemented, Sylow
171C25C2×Dic4711yescharacteristic
172C23×C4—28no—
173C23×C4C2×Dic4711yescharacteristic
174C23×C4C2×Dic4712yes—
175C23×C4C2×Dic4714yes—
176C23×C4C2×Dic4712yes—
177C23×C4C2×Dic4712yes—
178C23×C4C22×D4714yes—
179C23×C4C22×D4712yes—
180C23×C4C22×D4711yescharacteristic
181C2×M16—4716no—
182C2×M16—9416no—
183C22×C8—478no—
184#117C2×Dic47116yes—
185#117C2×Dic47132yes—
186#117C22×D47116yes—
187C2×C4⋊C4C2×Dic4718yes—
188C2×C4⋊C4C2×Dic47116yes—
189C2×C22⋊C4C2×Dic4718yes—
190C2×C22⋊C4C2×Dic47116yes—
191C2×C42C2×Dic4712yes—
192C2×C42C2×Dic4712yes—
193C2×C42C2×Dic4714yes—
194C2×C42C2×Dic4714yes—
195C2×C42C22×D4714yes—
196C2×C42C22×D4714yes—
197C2×C42C22×D4712yes—
198C2×C42C22×D4712yes—
199C4⋊C8—4732no—
200C22⋊C8—4732no—
201#97—4716no—
202C4×C8—4716no—
203C24—24no—
204C24—44no—
205C24C22×Dic4712yes—
206C24C22×Dic4711yescharacteristic
207C24C22×Dic4714yes—
208C22×C4—24no—
209C22×C4—216no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—216no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4#1062535018yes—
216C22×C4C22×Dic4714yes—
217C22×C4C22×Dic4714yes—
218C22×C4C22×Dic4714yes—
219C22×C4C22×Dic4712yes—
220C22×C4C22×Dic4718yes—
221C22×C4C22×Dic4714yes—
222C22×C4C22×Dic4718yes—
223C22×C4C22×Dic4718yes—
224C22×C4C22×Dic4711yescharacteristic
225C22×C4C22×Dic4712yes—
226C22×C4C23×D4714yes—
227C22×C4C23×D4712yes—
228C22×C4C23×D4711yescharacteristic
229M16—9432no—
230C2×C8—4716no—
231C2×C8—9416no—
232C4⋊C4C22×Dic47132yes—
233C22⋊C4C22×Dic47132yes—
234C42C22×Dic4718yes—
235C42C22×Dic4718yes—
236C42C22×Dic4714yes—
237C42C22×Dic4714yes—
238C42C23×D4714yes—
239C42C23×D4714yes—
240C23—24no—
241C23—28no—
242C23—24no—
243C23—28no—
244C23—48no—
245C23—416no—
246C23#1309398314yes—
247C23C23×Dic4714yes—
248C23C23×Dic4712yes—
249C23C23×Dic4711yescharacteristic
250C2×C4—28no—
251C2×C4—24no—
252C2×C4—28no—
253C2×C4—28no—
254C2×C4—216no—
255C2×C4—28no—
256C2×C4—28no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4#1309398318yes—
260C2×C4#1309398314yes—
261C2×C4C23×Dic4718yes—
262C2×C4C23×Dic4718yes—
263C2×C4C23×Dic4714yes—
264C2×C4C23×Dic4712yes—
265C2×C4C23×Dic4711yescharacteristic
266C2×C4C2×#1062535014yes—
267C2×C4C2×#1062535018yes—
268C2×C4C24×D4711yescharacteristic
269C8—9416no—
270C22—24no—
271C22—28no—
272C22—24no—
273C22—28no—
274C22—416no—
275C22—48no—
276C22C2×#1309398314yes—
277C22C2×#1309398312yes—
278C22#1568989014yes—
279C22C24×Dic4711yescharacteristic
280C4—24no—
281C4—28no—
282C4—28no—
283C4C2×#1309398314yes—
284C4C2×#1309398312yes—
285C4C22×#1062535012yes—
286C2—24no—
287C2—44no—
288C2#2025202614yesdirect product factor
289C2C22×#1309398311yescharacteristic
290C2C2×#1568989012yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC2×C4C24×D47
FittingC22×#117×C47C2
socleC23×C47C23×C4
prime radicalC25×C47C2×C4
cosocleC2×C4×C47C25
hypercenterC22×#117D47
hypocenterC47C2×#2921
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. #20252026C2direct (N × H)
2. C2×#15689184C2semidirect
3. C2×#15689147C2semidirect
5. C2×C4×#10625329C2semidirect
8. #15689184C22semidirect
9. #15689147C22semidirect
10. C2×C4⋊(C47⋊C8)C22semidirect
13. C4×#10625329C22semidirect
14. C2×#13093935C22semidirect
15. C2×C4×C47⋊C8C22semidirect
28. C4⋊(C47⋊C8)C23semidirect
30. #13093935C23semidirect
31. C4×C47⋊C8C23semidirect
149. C2×C47#2921semidirect
170. C47C2×#2921semidirect
288. C2#20252026direct (N × H)

Maximal subgroups

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

subgroupindex
#202520262
C2×#156891842
C2×#156891472
C2×#156890872
C2×C4×#106253292
C22×#117×C472
C2×#292147