Finite Group Databasebeta

← C2×#20252408 #27080780

Subgroups — C2×#20252408 #27080780

Order 12032.

2230 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252408C2116yesmaximal, direct product factor
2C2×Q8×C47⋊C8C212yesmaximal, complemented
3C2×#15689201C216yesmaximal, complemented
4C2×#15689180C213yesmaximal, complemented
5C2×C4×#10625329C213yesmaximal, complemented
6C2×C4×#41×C47C211yesmaximal, characteristic
7C22×#10625329C2216yes—
8Q8×C47⋊C8C22116yescomplemented
9#15689201C22148yescomplemented
10#15689180C22124yescomplemented
11C2×C4⋊(C47⋊C8)C2216yes—
12C2×C22⋊(C47⋊C8)C2216yes—
13C4×#10625329C22124yescomplemented
14C2×#13093935C2213yes—
15C2×#13093935C2211yescharacteristic
16C2×C4×C47⋊C8C2216yes—
17C22×#41×C47C411yescharacteristic
18C4×#41×C47C418yes—
19C4×#41×C47C2218yes—
20C2×C4×Q8×C47C411yescharacteristic
21C2×C4×Q8×C47C2211yescharacteristic
22C2×C4×D4×C47C413yes—
23C2×C4×D4×C47C2213yes—
24C2×#117×C47C413yes—
25C22×C42×C47C2213yes—
26C2×#10625329—224no—
27C2×#10625329C23124yes—
28C22×C47⋊C8C2316yes—
29C22×C47⋊C8C2312yes—
30C4⋊(C47⋊C8)C23124yes—
31C22⋊(C47⋊C8)C23124yes—
32#13093935C2314yes—
33#13093935C23112yes—
34C4×C47⋊C8C23124yes—
35C2×#41×C47—28no—
36C2×#41×C47C2×C418yes—
37C22×Q8×C47C2×C411yescharacteristic
38C22×D4×C47C2×C413yes—
39C23×C4×C47C2×C413yes—
40C23×C4×C47C2×C413yes—
41C23×C4×C47C2313yes—
42C4×Q8×C47C2×C418yes—
43C4×Q8×C47C2×C414yes—
44C4×Q8×C47C2314yes—
45C4×D4×C47C2×C4124yes—
46C4×D4×C47C2×C4112yes—
47C4×D4×C47C23112yes—
48#117×C47C2×C4124yes—
49C2×C4⋊C4×C47C2×C413yes—
50C2×C4⋊C4×C47C2×C413yes—
51C2×C22⋊C4×C47C2×C416yes—
52C2×C42×C47C2×C413yes—
53C2×C42×C47C2×C413yes—
54C2×C42×C47C2×C4112yes—
55C2×C42×C47C2311yescharacteristic
56C2×C42×C47C2313yes—
57C2×C42×C47C23112yes—
58C24×C47C22×C413yes—
59#41×C47—232no—
60C2×Q8×C47M1618yescomplemented
61C2×Q8×C47C22×C414yes—
62C2×D4×C47—212no—
63C2×D4×C47C22×C4112yes—
64C22×C4×C47—212no—
65C22×C4×C47—212no—
66C22×C4×C47—46no—
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C22×C413yes—
70C22×C4×C47C22×C416yes—
71C22×C4×C47C22×C413yes—
72C22×C4×C47C22×C4112yes—
73C22×C4×C47C22×C4112yes—
74C22×C4×C47C22×C413yes—
75C22×C4×C47C2416yes—
76C22×C4×C47C2411yescharacteristic
77C4⋊C4×C47C22×C4112yes—
78C4⋊C4×C47C22×C4112yes—
79C22⋊C4×C47C22×C4124yes—
80C42×C47C22×C412yes—
81C42×C47C22×C4112yes—
82C42×C47C22×C4112yes—
83C42×C47C22×C416yes—
84C42×C47C2412yes—
85C42×C47C2416yes—
86#10625329—248no—
87C2×C47⋊C8—212no—
88C2×C47⋊C8#4118yescomplemented
89C2×C47⋊C8C24112yes—
90C2×C47⋊C8C2414yes—
91C23×C47—26no—
92C23×C47—26no—
93C23×C47—26no—
94C23×C47C23×C416yes—
95C23×C47C23×C411yescharacteristic
96Q8×C47C2×M16116yescomplemented
97D4×C47—224no—
98C2×C4×C47—26no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—26no—
102C2×C4×C47—212no—
103C2×C4×C47—224no—
104C2×C4×C47—212no—
105C2×C4×C47—412no—
106C2×C4×C47C2×M16112yes—
107C2×C4×C47C23×C412yes—
108C2×C4×C47C23×C412yes—
109C2×C4×C47C23×C416yes—
110C2×C4×C47C23×C416yes—
111C2×C4×C47C23×C411yescharacteristic
112C2×C4×C47C23×C416yes—
113C2×C4×C47C2×#4114yes—
114C2×C4×C47C2511yescharacteristic
115C47⋊C8—212no—
116C47⋊C8C2×#4118yescomplemented
117C2×#2971—471nomaximal, Sylow
118C22×C47—26no—
119C22×C47—26no—
120C22×C47—26no—
121C22×C47—412no—
122C22×C47C4×#4112yes—
123C22×C47C22×M1612yes—
124C22×C47#56812yes—
125C22×C47C24×C411yescharacteristic
126C4×C47—26no—
127C4×C47—22no—
128C4×C47—22no—
129C4×C47—26no—
130C4×C47C4×#4112yes—
131C4×C47C22×M16112yes—
132C4×C47C22×#4112yes—
133C2×C4×#41D4711yescharacteristic
134#2971—4716no—
135C2×C8×Q8—472no—
136C2×#436—476no—
137C2×#432—473no—
138C2×C4×M16—473no—
139C2×C47—46no—
140C2×C47#297114yescomplemented
141C2×C47C2×C4×#4111yescharacteristic
142C2×C47C23×M1611yescharacteristic
143C2×C47C2×#56811yescharacteristic
144C22×#41Dic4711yescharacteristic
145C22×M16—476no—
146C4×#41Dic4718yes—
147C4×#41C2×D4718yes—
148C2×C4×Q8Dic4711yescharacteristic
149C2×C4×Q8C2×D4711yescharacteristic
150C2×C4×D4Dic4713yes—
151C2×C4×D4C2×D4713yes—
152C2×#117Dic4713yes—
153C22×C42C2×D4713yes—
154C8×Q8—4716no—
155#436—4748no—
156#432—4724no—
157C2×C4⋊C8—476no—
158C2×C22⋊C8—476no—
159C4×M16—4724no—
160C2×#97—471no—
161C2×#97—473no—
162C2×C4×C8—476no—
163C47C2×#297111yescharacteristic, complemented, Sylow
164C2×#41—28no—
165C2×#41C2×Dic4718yes—
166C22×Q8C2×Dic4711yescharacteristic
167C22×D4C2×Dic4713yes—
168C23×C4C2×Dic4713yes—
169C23×C4C2×Dic4713yes—
170C23×C4C22×D4713yes—
171C2×M16—4724no—
172C2×M16—9424no—
173C22×C8—472no—
174C22×C8—476no—
175C4×Q8C2×Dic4718yes—
176C4×Q8C2×Dic4714yes—
177C4×Q8C22×D4714yes—
178C4×D4C2×Dic47124yes—
179C4×D4C2×Dic47112yes—
180C4×D4C22×D47112yes—
181#117C2×Dic47124yes—
182C2×C4⋊C4C2×Dic4713yes—
183C2×C4⋊C4C2×Dic4713yes—
184C2×C22⋊C4C2×Dic4716yes—
185C2×C42C2×Dic4713yes—
186C2×C42C2×Dic4713yes—
187C2×C42C2×Dic47112yes—
188C2×C42C22×D47112yes—
189C2×C42C22×D4713yes—
190C2×C42C22×D4711yescharacteristic
191C4⋊C8—4724no—
192C22⋊C8—4724no—
193#97—474no—
194#97—4712no—
195C4×C8—4724no—
196C24C22×Dic4713yes—
197#41—232no—
198C2×Q8#1062532918yescomplemented
199C2×Q8C22×Dic4714yes—
200C2×D4—212no—
201C2×D4C22×Dic47112yes—
202C22×C4—212no—
203C22×C4—212no—
204C22×C4—46no—
205C22×C4C22×Dic4711yescharacteristic
206C22×C4C22×Dic4711yescharacteristic
207C22×C4C22×Dic4713yes—
208C22×C4C22×Dic4716yes—
209C22×C4C22×Dic4713yes—
210C22×C4C22×Dic47112yes—
211C22×C4C22×Dic47112yes—
212C22×C4C22×Dic4713yes—
213C22×C4C23×D4716yes—
214C22×C4C23×D4711yescharacteristic
215M16—9448no—
216C2×C8—478no—
217C2×C8—474no—
218C2×C8—4712no—
219C2×C8—9412no—
220C4⋊C4C22×Dic47112yes—
221C4⋊C4C22×Dic47112yes—
222C22⋊C4C22×Dic47124yes—
223C42C22×Dic4712yes—
224C42C22×Dic47112yes—
225C42C22×Dic47112yes—
226C42C22×Dic4716yes—
227C42C23×D4716yes—
228C42C23×D4712yes—
229C23—26no—
230C23—26no—
231C23—26no—
232C23C23×Dic4716yes—
233C23C23×Dic4711yescharacteristic
234Q8C2×#10625329116yescomplemented
235D4—224no—
236C2×C4—26no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—26no—
240C2×C4—212no—
241C2×C4—224no—
242C2×C4—212no—
243C2×C4—412no—
244C2×C4C2×#10625329112yes—
245C2×C4C23×Dic4712yes—
246C2×C4C23×Dic4712yes—
247C2×C4C23×Dic4716yes—
248C2×C4C23×Dic4716yes—
249C2×C4C23×Dic4711yescharacteristic
250C2×C4C23×Dic4716yes—
251C2×C4#41×D4714yes—
252C2×C4C24×D4711yescharacteristic
253C8—478no—
254C8—9412no—
255C22—26no—
256C22—26no—
257C22—26no—
258C22—412no—
259C22#41×Dic4712yes—
260C22C22×#1062532912yes—
261C22#1568989012yes—
262C22C24×Dic4711yescharacteristic
263C4—22no—
264C4—22no—
265C4—26no—
266C4—26no—
267C4#41×Dic4712yes—
268C4C22×#10625329112yes—
269C4C2×#41×D4712yes—
270C2—46no—
271C2#2025240814yesdirect product factor
272C2C2×#41×Dic4711yescharacteristic
273C2C23×#1062532911yescharacteristic
274C2C2×#1568989011yescharacteristic
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC2×C4C24×D47
FittingC2×C4×#41×C47C2
socleC23×C47C23×C4
prime radicalC22×#41×C47C4
cosocleC2×C4×C47C25
hypercenterC2×C4×#41D47
hypocenterC47C2×#2971
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #20252408C2direct (N × H)
2. C2×Q8×C47⋊C8C2semidirect
3. C2×#15689201C2semidirect
4. C2×#15689180C2semidirect
5. C2×C4×#10625329C2semidirect
8. Q8×C47⋊C8C22semidirect
9. #15689201C22semidirect
10. #15689180C22semidirect
13. C4×#10625329C22semidirect
60. C2×Q8×C47M16semidirect
88. C2×C47⋊C8#41semidirect
96. Q8×C47C2×M16semidirect
116. C47⋊C8C2×#41semidirect
140. C2×C47#2971semidirect
163. C47C2×#2971semidirect
198. C2×Q8#10625329semidirect
234. Q8C2×#10625329semidirect
271. C2#20252408direct (N × H)

Maximal subgroups

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

subgroupindex
#202524082
C2×Q8×C47⋊C82
C2×#156892012
C2×#156891802
C2×C4×#106253292
C2×C4×#41×C472
C2×#297147