Finite Group Databasebeta

← C2×#20252477 #27081227

Subgroups — C2×#20252477 #27081227

Order 12032.

2022 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
1#20252477C2116yesmaximal, direct product factor
2C2×#15689254C216yesmaximal, complemented
3C2×Q8×C47⋊C8C212yesmaximal, complemented
4C2×#15689180C213yesmaximal
5C2×#15689176C213yesmaximal
6C22×C4×Q8×C47C211yesmaximal, characteristic
7#15689254C22148yescomplemented
8Q8×C47⋊C8C22116yescomplemented
9#15689180C22124yes—
10#15689176C22124yes—
11C2×C4⋊(C47⋊C8)C2216yes—
12C2×C4⋊(C47⋊C8)C2216yes—
13C2×C22⋊(C47⋊C8)C2211yescharacteristic
14C2×C22⋊(C47⋊C8)C2213yes—
15C2×#13093935C2216yes—
16C2×C4×C47⋊C8C2216yes—
17C23×Q8×C47C411yescharacteristic
18C2×C4×Q8×C47C418yes—
19C2×C4×Q8×C47C413yes—
20C2×C4×Q8×C47C411yescharacteristic
21C2×C4×Q8×C47C2218yes—
22C2×C4×Q8×C47C2211yescharacteristic
23C2×C4×Q8×C47C2213yes—
24C22×C4⋊C4×C47C413yes—
25C22×C42×C47C2213yes—
26C22×C47⋊C8C2312yes—
27C22×C47⋊C8C2316yes—
28C4⋊(C47⋊C8)C23124yes—
29C4⋊(C47⋊C8)C23124yes—
30C22⋊(C47⋊C8)C2314yes—
31C22⋊(C47⋊C8)C23112yes—
32#13093935C23124yes—
33C4×C47⋊C8C23124yes—
34C22×Q8×C47—28no—
35C22×Q8×C47C2×C418yes—
36C22×Q8×C47C2×C413yes—
37C22×Q8×C47C2×C411yescharacteristic
38C23×C4×C47C2×C413yes—
39C23×C4×C47C2×C413yes—
40C23×C4×C47C2311yescharacteristic
41C4×Q8×C47C2×C4124yes—
42C4×Q8×C47C2×C4112yes—
43C4×Q8×C47C2×C418yes—
44C4×Q8×C47C2×C414yes—
45C4×Q8×C47C2314yes—
46C4×Q8×C47C23112yes—
47C2×C4⋊C4×C47C2×C4124yes—
48C2×C4⋊C4×C47C2×C416yes—
49C2×C4⋊C4×C47C2×C413yes—
50C2×C4⋊C4×C47C2×C413yes—
51C2×C42×C47C2×C413yes—
52C2×C42×C47C2×C4112yes—
53C2×C42×C47C2×C413yes—
54C2×C42×C47C2313yes—
55C2×C42×C47C2313yes—
56C2×C42×C47C23112yes—
57C24×C47C22×C411yescharacteristic
58C2×Q8×C47—232no—
59C2×Q8×C47—212no—
60C2×Q8×C47M1618yes—
61C2×Q8×C47C22×C4112yes—
62C2×Q8×C47C22×C414yes—
63C22×C4×C47—212no—
64C22×C4×C47—212no—
65C22×C4×C47—24no—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C22×C412yes—
68C22×C4×C47C22×C4112yes—
69C22×C4×C47C22×C413yes—
70C22×C4×C47C22×C413yes—
71C22×C4×C47C22×C413yes—
72C22×C4×C47C22×C4112yes—
73C22×C4×C47C22×C413yes—
74C22×C4×C47C2412yes—
75C22×C4×C47C2411yescharacteristic
76C4⋊C4×C47C22×C4124yes—
77C4⋊C4×C47C22×C4112yes—
78C4⋊C4×C47C22×C4112yes—
79C42×C47C22×C4112yes—
80C42×C47C22×C4112yes—
81C42×C47C22×C416yes—
82C42×C47C22×C416yes—
83C42×C47C2416yes—
84C42×C47C2416yes—
85C2×C47⋊C8—24no—
86C2×C47⋊C8—212no—
87C2×C47⋊C8C2414yes—
88C2×C47⋊C8C24112yes—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47#13114yes—
92C23×C47C23×C412yes—
93C23×C47C23×C411yescharacteristic
94Q8×C47—224no—
95Q8×C47C2×M16116yes—
96C2×C4×C47—22no—
97C2×C4×C47—26no—
98C2×C4×C47—224no—
99C2×C4×C47—26no—
100C2×C4×C47—26no—
101C2×C4×C47—24no—
102C2×C4×C47—224no—
103C2×C4×C47—24no—
104C2×C4×C47C2×M16112yes—
105C2×C4×C47C23×C412yes—
106C2×C4×C47C23×C416yes—
107C2×C4×C47C23×C416yes—
108C2×C4×C47C23×C416yes—
109C2×C4×C47C23×C411yescharacteristic
110C2×C4×C47C23×C416yes—
111C2×C4×C472−1+414yes—
112C2×C4×C47C2511yescharacteristic
113C47⋊C8—24no—
114C47⋊C8—212no—
115C2×#2984—471nomaximal, Sylow
116C22×C47—22no—
117C22×C47—28no—
118C22×C47—22no—
119C22×C47#51912yes—
120C22×C47C22×M1612yes—
121C22×C47C2×#13114yes—
122C22×C47C2×#13112yes—
123C22×C47C24×C411yescharacteristic
124C4×C47—22no—
125C4×C47—26no—
126C4×C47—26no—
127C4×C47—26no—
128C4×C47#51912yes—
129C4×C47C22×M16112yes—
130C4×C47C2×2−1+412yes—
131C22×C4×Q8D4711yescharacteristic
132#2984—4716no—
133C2×#446—476no—
134C2×C8×Q8—472no—
135C2×#432—473no—
136C2×#431—473no—
137C2×C47—24no—
138C2×C47#298414yescomplemented
139C2×C47C2×#51911yescharacteristic
140C2×C47C23×M1611yescharacteristic
141C2×C47C22×#13111yescharacteristic
142C23×Q8Dic4711yescharacteristic
143C2×C4×Q8Dic4718yes—
144C2×C4×Q8Dic4713yes—
145C2×C4×Q8Dic4711yescharacteristic
146C2×C4×Q8C2×D4711yescharacteristic
147C2×C4×Q8C2×D4713yes—
148C2×C4×Q8C2×D4718yes—
149C22×C4⋊C4Dic4713yes—
150C22×C42C2×D4713yes—
151#446—4748no—
152C8×Q8—4716no—
153#432—4724no—
154#431—4724no—
155C2×C4⋊C8—476no—
156C2×C4⋊C8—476no—
157C2×C22⋊C8—471no—
158C2×C22⋊C8—473no—
159C2×#97—476no—
160C2×C4×C8—476no—
161C47C2×#298411yescharacteristic, complemented, Sylow
162C22×Q8—28no—
163C22×Q8C2×Dic4718yes—
164C22×Q8C2×Dic4713yes—
165C22×Q8C2×Dic4711yescharacteristic
166C23×C4C2×Dic4713yes—
167C23×C4C2×Dic4713yes—
168C23×C4C22×D4711yescharacteristic
169C22×C8—472no—
170C22×C8—476no—
171C4×Q8C2×Dic47112yes—
172C4×Q8C2×Dic47124yes—
173C4×Q8C2×Dic4714yes—
174C4×Q8C2×Dic4718yes—
175C4×Q8C22×D4714yes—
176C4×Q8C22×D47112yes—
177C2×C4⋊C4C2×Dic4716yes—
178C2×C4⋊C4C2×Dic47124yes—
179C2×C4⋊C4C2×Dic4713yes—
180C2×C4⋊C4C2×Dic4713yes—
181C2×C42C2×Dic4713yes—
182C2×C42C2×Dic47112yes—
183C2×C42C2×Dic4713yes—
184C2×C42C22×D47112yes—
185C2×C42C22×D4713yes—
186C2×C42C22×D4713yes—
187C4⋊C8—4724no—
188C4⋊C8—4724no—
189C22⋊C8—474no—
190C22⋊C8—4712no—
191#97—4724no—
192C4×C8—4724no—
193C24C22×Dic4711yescharacteristic
194C2×Q8—232no—
195C2×Q8—212no—
196C2×Q8#1062532918yes—
197C2×Q8C22×Dic47112yes—
198C2×Q8C22×Dic4714yes—
199C22×C4—212no—
200C22×C4—212no—
201C22×C4—24no—
202C22×C4C22×Dic4711yescharacteristic
203C22×C4C22×Dic4712yes—
204C22×C4C22×Dic47112yes—
205C22×C4C22×Dic4713yes—
206C22×C4C22×Dic4713yes—
207C22×C4C22×Dic4713yes—
208C22×C4C22×Dic47112yes—
209C22×C4C22×Dic4713yes—
210C22×C4C23×D4712yes—
211C22×C4C23×D4711yescharacteristic
212C2×C8—474no—
213C2×C8—4712no—
214C2×C8—944no—
215C2×C8—9412no—
216C4⋊C4C22×Dic47124yes—
217C4⋊C4C22×Dic47112yes—
218C4⋊C4C22×Dic47112yes—
219C42C22×Dic47112yes—
220C42C22×Dic47112yes—
221C42C22×Dic4716yes—
222C42C22×Dic4716yes—
223C42C23×D4716yes—
224C42C23×D4716yes—
225C23—22no—
226C23—22no—
227C23#1309403114yes—
228C23C23×Dic4712yes—
229C23C23×Dic4711yescharacteristic
230Q8—224no—
231Q8C2×#10625329116yes—
232C2×C4—22no—
233C2×C4—26no—
234C2×C4—224no—
235C2×C4—26no—
236C2×C4—26no—
237C2×C4—24no—
238C2×C4—224no—
239C2×C4—24no—
240C2×C4C2×#10625329112yes—
241C2×C4C23×Dic4712yes—
242C2×C4C23×Dic4716yes—
243C2×C4C23×Dic4716yes—
244C2×C4C23×Dic4716yes—
245C2×C4C23×Dic4711yescharacteristic
246C2×C4C23×Dic4716yes—
247C2×C4#1309407214yes—
248C2×C4C24×D4711yescharacteristic
249C8—944no—
250C8—9412no—
251C22—22no—
252C22—28no—
253C22—22no—
254C22#1568960112yes—
255C22C22×#1062532912yes—
256C22C2×#1309403114yes—
257C22C2×#1309403112yes—
258C22C24×Dic4711yescharacteristic
259C4—26no—
260C4—26no—
261C4—26no—
262C4—22no—
263C4#1568960112yes—
264C4C22×#10625329112yes—
265C4C2×#1309407212yes—
266C2—24no—
267C2#2025247714yesdirect product factor
268C2C2×#1568960111yescharacteristic
269C2C23×#1062532911yescharacteristic
270C2C22×#1309403111yescharacteristic
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC2×C4C24×D47
FittingC22×C4×Q8×C47C2
socleC23×C47C23×C4
prime radicalC24×C47C22×C4
cosocleC2×C4×C47C25
hypercenterC22×C4×Q8D47
hypocenterC47C2×#2984
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20252477C2direct (N × H)
2. C2×#15689254C2semidirect
3. C2×Q8×C47⋊C8C2semidirect
7. #15689254C22semidirect
8. Q8×C47⋊C8C22semidirect
138. C2×C47#2984semidirect
161. C47C2×#2984semidirect
267. C2#20252477direct (N × H)

Maximal subgroups

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

subgroupindex
#202524772
C2×#156892542
C2×Q8×C47⋊C82
C2×#156891802
C2×#156891762
C22×C4×Q8×C472
C2×#298447