Finite Group Databasebeta

← C2×#20251807 #27079677

Subgroups — C2×#20251807 #27079677

Order 12032.

3342 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625329C212yesmaximal
2#20251807C2116yesmaximal, direct product factor
3C2×#15689079C218yesmaximal, complemented
4C2×C4×#10625329C214yesmaximal, complemented
5C22×#117×C47C211yesmaximal, characteristic
6C22×#10625329C4132yescomplemented
7C22×#10625329C2214yes—
8C22×#10625329C2214yes—
9C22×#10625329C22116yes—
10C23×C47⋊C8C2214yes—
11#15689079C22164yescomplemented
12C4×#10625329C22132yescomplemented
13C2×#13093935C2218yescomplemented
14C2×C4×C47⋊C8C2218yescomplemented
15C24×C4×C47C2211yescharacteristic
16C2×#117×C47C414yes—
17C2×#117×C47C418yes—
18C2×#117×C47C2214yes—
19C2×#117×C47C2218yes—
20C22×C4⋊C4×C47C412yes—
21C22×C22⋊C4×C47C412yes—
22C22×C42×C47C2212yes—
23C2×#10625329C2×C4132yes—
24C2×#10625329C2×C4132yescomplemented
25C2×#10625329C2×C41128yescomplemented
26C2×#10625329C23116yes—
27C2×#10625329C23116yes—
28C22×C47⋊C8C2×C4132yescomplemented
29C22×C47⋊C8C2318yes—
30C22×C47⋊C8C23116yes—
31#13093935C23132yescomplemented
32C4×C47⋊C8C23132yescomplemented
33C25×C47C2×C411yescharacteristic
34C23×C4×C47C2×C414yes—
35C23×C4×C47C2×C411yescharacteristic
36C23×C4×C47C2×C412yes—
37C23×C4×C47C2×C414yes—
38C23×C4×C47C2×C4116yes—
39C23×C4×C47C2311yescharacteristic
40C23×C4×C47C2312yes—
41C23×C4×C47C2314yes—
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×C414yes—
50C2×C42×C47C2×C418yes—
51C2×C42×C47C2314yes—
52C2×C42×C47C2318yes—
53C24×C47—44no—
54C24×C47C4218yes—
55C24×C47C22×C412yes—
56C24×C47C22×C411yescharacteristic
57C24×C47C22×C414yes—
58C22×C4×C47—48no—
59C22×C4×C47C4218yes—
60C22×C4×C47C42116yes—
61C22×C4×C47C42132yes—
62C22×C4×C47C22×C418yes—
63C22×C4×C47C22×C4116yes—
64C22×C4×C47C22×C412yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C22×C414yes—
67C22×C4×C47C22×C412yes—
68C22×C4×C47C22×C414yes—
69C22×C4×C47C22×C418yes—
70C22×C4×C47C22×C418yes—
71C22×C4×C47C22×C418yes—
72C22×C4×C47C22×C418yes—
73C22×C4×C47C22×C4132yes—
74C22×C4×C47C2412yes—
75C22×C4×C47C2414yes—
76C22×C4×C47C2411yescharacteristic
77C4⋊C4×C47C22×C4132yes—
78C22⋊C4×C47C22×C4132yes—
79C42×C47C22×C418yes—
80C42×C47C22×C4116yes—
81C42×C47C2418yes—
82#10625329C22×C4164yes—
83#10625329C22×C4164yescomplemented
84C2×C47⋊C8C22×C4132yes—
85C2×C47⋊C8C22×C4164yescomplemented
86C2×C47⋊C8C24116yes—
87C23×C47—24no—
88C23×C47—28no—
89C23×C47—416no—
90C23×C47—48no—
91C23×C47C2×C4218yes—
92C23×C47C2×C4214yes—
93C23×C47C2×C42116yes—
94C23×C47C23×C414yes—
95C23×C47C23×C412yes—
96C23×C47C23×C411yescharacteristic
97C2×C4×C47—216no—
98C2×C4×C47—416no—
99C2×C4×C47C2×C4218yes—
100C2×C4×C47C2×C42116yes—
101C2×C4×C47C2×C42116yes—
102C2×C4×C47C2×C42132yes—
103C2×C4×C47C2×C4214yes—
104C2×C4×C47C2×C4218yes—
105C2×C4×C47C23×C414yes—
106C2×C4×C47C23×C418yes—
107C2×C4×C47C23×C418yes—
108C2×C4×C47C23×C418yes—
109C2×C4×C47C23×C4116yes—
110C2×C4×C47C23×C412yes—
111C2×C4×C47C23×C411yescharacteristic
112C2×C4×C47C23×C414yes—
113C2×C4×C47C2511yescharacteristic
114C47⋊C8C23×C4132yes—
115C2×#2873—471nomaximal, Sylow
116C22×C47—28no—
117C22×C47—216no—
118C22×C47—24no—
119C22×C47—28no—
120C22×C47—416no—
121C22×C47C22×C4218yes—
122C22×C47C22×C4214yes—
123C22×C47C22×C4212yes—
124C22×C47#56814yes—
125C22×C47C24×C411yescharacteristic
126C4×C47—216no—
127C4×C47C22×C4212yes—
128C4×C47C22×C4218yes—
129C4×C47C22×C4214yes—
130C4×C47C24×C412yes—
131C23×M16—472no—
132C22×#117D4711yescharacteristic
133#2873—4716no—
134C2×#405—478no—
135C2×C4×M16—474no—
136C2×C47—24no—
137C2×C47—28no—
138C2×C47#287314yescomplemented
139C2×C47C23×C4211yescharacteristic
140C2×C47C2×#56812yes—
141C24×C4C2×D4711yescharacteristic
142C22×M16—4732no—
143C22×M16—474no—
144C22×M16—474no—
145C22×M16—4716no—
146C23×C8—474no—
147C2×#117Dic4714yes—
148C2×#117Dic4718yes—
149C2×#117C2×D4714yes—
150C2×#117C2×D4718yes—
151C22×C4⋊C4Dic4712yes—
152C22×C22⋊C4Dic4712yes—
153C22×C42C2×D4712yes—
154#405—4764no—
155C4×M16—4732no—
156C2×#97—478no—
157C2×C4×C8—478no—
158C47C2×#287311yescharacteristic, complemented, Sylow
159C25C2×Dic4711yescharacteristic
160C23×C4C4×D47116yes—
161C23×C4C2×Dic4714yes—
162C23×C4C2×Dic4711yescharacteristic
163C23×C4C2×Dic4712yes—
164C23×C4C2×Dic4714yes—
165C23×C4C22×D4714yes—
166C23×C4C22×D4712yes—
167C23×C4C22×D4711yescharacteristic
168C2×M16—4732no—
169C2×M16—4732no—
170C2×M16—47128no—
171C2×M16—4716no—
172C2×M16—4716no—
173C22×C8—4732no—
174C22×C8—4716no—
175C22×C8—478no—
176#117C2×Dic47116yes—
177#117C2×Dic47132yes—
178#117C22×D47116yes—
179C2×C4⋊C4C2×Dic4718yes—
180C2×C4⋊C4C2×Dic47116yes—
181C2×C22⋊C4C2×Dic4718yes—
182C2×C22⋊C4C2×Dic47116yes—
183C2×C42C2×Dic4714yes—
184C2×C42C2×Dic4718yes—
185C2×C42C22×D4718yes—
186C2×C42C22×D4714yes—
187#97—4732no—
188C4×C8—4732no—
189C24—44no—
190C24C4×Dic4718yes—
191C24C22×Dic4712yes—
192C24C22×Dic4711yescharacteristic
193C24C22×Dic4714yes—
194C22×C4—48no—
195C22×C4C4×Dic4718yes—
196C22×C4C4×Dic47116yes—
197C22×C4C4×Dic47132yes—
198C22×C4C2×C4×D4718yes—
199C22×C4C2×C4×D47132yes—
200C22×C4C2×C4×D4718yes—
201C22×C4C2×C4×D4718yes—
202C22×C4C22×Dic4718yes—
203C22×C4C22×Dic47116yes—
204C22×C4C22×Dic4711yescharacteristic
205C22×C4C22×Dic4712yes—
206C22×C4C22×Dic4714yes—
207C22×C4C22×Dic4712yes—
208C22×C4C22×Dic4718yes—
209C22×C4C22×Dic4714yes—
210C22×C4C23×D4712yes—
211C22×C4C23×D4714yes—
212C22×C4C23×D4711yescharacteristic
213M16—4764no—
214M16—4764no—
215C2×C8—4732no—
216C2×C8—4764no—
217C2×C8—4716no—
218C4⋊C4C22×Dic47132yes—
219C22⋊C4C22×Dic47132yes—
220C42C22×Dic47116yes—
221C42C22×Dic4718yes—
222C42C23×D4718yes—
223C23—24no—
224C23—28no—
225C23—416no—
226C23—48no—
227C23C2×C4×Dic4718yes—
228C23C2×C4×Dic4714yes—
229C23C2×C4×Dic47116yes—
230C23C23×Dic4714yes—
231C23C23×Dic4712yes—
232C23C23×Dic4711yescharacteristic
233C2×C4—216no—
234C2×C4—416no—
235C2×C4C2×C4×Dic47116yes—
236C2×C4C2×C4×Dic4718yes—
237C2×C4C2×C4×Dic47116yes—
238C2×C4C2×C4×Dic47132yes—
239C2×C4C2×C4×Dic4718yes—
240C2×C4C2×C4×Dic4714yes—
241C2×C4C22×C4×D4718yes—
242C2×C4C22×C4×D4714yes—
243C2×C4C22×C4×D4718yes—
244C2×C4C22×C4×D4718yes—
245C2×C4C23×Dic47116yes—
246C2×C4C23×Dic4712yes—
247C2×C4C23×Dic4711yescharacteristic
248C2×C4C23×Dic4714yes—
249C2×C4C24×D4711yescharacteristic
250C8—4732no—
251C22—24no—
252C22—28no—
253C22—28no—
254C22—216no—
255C22—416no—
256C22C22×C4×Dic4718yes—
257C22C22×C4×Dic4714yes—
258C22C22×C4×Dic4712yes—
259C22#1568989014yes—
260C22C24×Dic4711yescharacteristic
261C4—216no—
262C4C22×C4×Dic4718yes—
263C4C22×C4×Dic4712yes—
264C4C22×C4×Dic4714yes—
265C4C23×C4×D4712yes—
266C2—24no—
267C2—28no—
268C2#2025180714yesdirect product factor
269C2C23×C4×Dic4711yescharacteristic
270C2C2×#1568989012yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC2×C47C23×C42
FrattiniC2×C4C24×D47
FittingC22×#117×C47C2
socleC23×C47C23×C4
prime radicalC25×C47C2×C4
cosocleC2×C4×C47C25
hypercenterC22×#117D47
hypocenterC47C2×#2873
LCSG ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. #20251807C2direct (N × H)
3. C2×#15689079C2semidirect
4. C2×C4×#10625329C2semidirect
6. C22×#10625329C4semidirect
11. #15689079C22semidirect
12. C4×#10625329C22semidirect
13. C2×#13093935C22semidirect
14. C2×C4×C47⋊C8C22semidirect
24. C2×#10625329C2×C4semidirect
25. C2×#10625329C2×C4semidirect
28. C22×C47⋊C8C2×C4semidirect
31. #13093935C23semidirect
32. C4×C47⋊C8C23semidirect
83. #10625329C22×C4semidirect
85. C2×C47⋊C8C22×C4semidirect
138. C2×C47#2873semidirect
158. C47C2×#2873semidirect
268. C2#20251807direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253292
#202518072
C2×#156890792
C2×C4×#106253292
C22×#117×C472
C2×#287347