Finite Group Databasebeta

← C2×Q8×#10625329 #27080666

Subgroups — C2×Q8×#10625329 #27080666

Order 12032.

2154 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#10625329C2116yesmaximal, direct product factor
2C2×#15689254C216yesmaximal, complemented
3C2×Q8×C47⋊C8C212yesmaximal, complemented
4C2×#15689147C213yesmaximal
5C2×C4×#10625329C213yesmaximal
6C22×C4×Q8×C47C211yesmaximal, characteristic
7C22×#10625329C2213yes—
8C22×#10625329C2211yescharacteristic
9#15689254C22148yescomplemented
10Q8×C47⋊C8C22116yescomplemented
11#15689147C22124yes—
12C2×C4⋊(C47⋊C8)C2216yes—
13C2×C4⋊(C47⋊C8)C2216yes—
14C4×#10625329C22124yes—
15C2×#13093935C2216yes—
16C2×C4×C47⋊C8C2216yes—
17C23×Q8×C47C411yescharacteristic
18C2×C4×Q8×C47C413yes—
19C2×C4×Q8×C47C418yes—
20C2×C4×Q8×C47C411yescharacteristic
21C2×C4×Q8×C47C2211yescharacteristic
22C2×C4×Q8×C47C2213yes—
23C2×C4×Q8×C47C2218yes—
24C22×C4⋊C4×C47C413yes—
25C22×C42×C47C2213yes—
26C2×#10625329—212no—
27C2×#10625329Q818yesdirect product factor
28C2×#10625329C23112yes—
29C2×#10625329C2314yes—
30C22×C47⋊C8C2316yes—
31C22×C47⋊C8C2312yes—
32C4⋊(C47⋊C8)C23124yes—
33C4⋊(C47⋊C8)C23124yes—
34#13093935C23124yes—
35C4×C47⋊C8C23124yes—
36C22×Q8×C47—28no—
37C22×Q8×C47C2×C418yes—
38C22×Q8×C47C2×C413yes—
39C22×Q8×C47C2×C411yescharacteristic
40C23×C4×C47C2×C413yes—
41C23×C4×C47C2×C413yes—
42C23×C4×C47C2311yescharacteristic
43C4×Q8×C47C2×C4112yes—
44C4×Q8×C47C2×C4124yes—
45C4×Q8×C47C2×C414yes—
46C4×Q8×C47C2×C418yes—
47C4×Q8×C47C2314yes—
48C4×Q8×C47C23112yes—
49C2×C4⋊C4×C47C2×C416yes—
50C2×C4⋊C4×C47C2×C413yes—
51C2×C4⋊C4×C47C2×C413yes—
52C2×C4⋊C4×C47C2×C4124yes—
53C2×C42×C47C2×C413yes—
54C2×C42×C47C2×C413yes—
55C2×C42×C47C2×C4112yes—
56C2×C42×C47C2313yes—
57C2×C42×C47C2313yes—
58C2×C42×C47C23112yes—
59C24×C47C22×C411yescharacteristic
60C2×Q8×C47—232no—
61C2×Q8×C47—212no—
62C2×Q8×C47M1618yescomplemented
63C2×Q8×C47C22×C4112yes—
64C2×Q8×C47C22×C414yes—
65C22×C4×C47—212no—
66C22×C4×C47—46no—
67C22×C4×C47C22×C413yes—
68C22×C4×C47C22×C413yes—
69C22×C4×C47C22×C413yes—
70C22×C4×C47C22×C4112yes—
71C22×C4×C47C22×C4112yes—
72C22×C4×C47C22×C412yes—
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C22×C413yes—
75C22×C4×C47C2×Q814yes—
76C22×C4×C47#4114yes—
77C22×C4×C47C2412yes—
78C22×C4×C47C2411yescharacteristic
79C4⋊C4×C47C22×C4124yes—
80C4⋊C4×C47C22×C4112yes—
81C4⋊C4×C47C22×C4112yes—
82C42×C47C22×C416yes—
83C42×C47C22×C4112yes—
84C42×C47C22×C416yes—
85C42×C47C22×C4112yes—
86C42×C47C2416yes—
87C42×C47C2416yes—
88#10625329—224no—
89#10625329C2×Q8116yesdirect product factor
90C2×C47⋊C8—212no—
91C2×C47⋊C8C2×Q818yescomplemented
92C2×C47⋊C8C24112yes—
93C2×C47⋊C8C2414yes—
94C23×C47—22no—
95C23×C47—22no—
96C23×C47C4×Q814yes—
97C23×C47C23×C412yes—
98C23×C47C23×C411yescharacteristic
99Q8×C47—224no—
100Q8×C47C2×M16116yescomplemented
101C2×C4×C47—26no—
102C2×C4×C47—26no—
103C2×C4×C47—26no—
104C2×C4×C47—224no—
105C2×C4×C47—412no—
106C2×C4×C47C4×Q818yes—
107C2×C4×C47C4×Q814yes—
108C2×C4×C47C2×M16112yes—
109C2×C4×C47C23×C416yes—
110C2×C4×C47C23×C416yes—
111C2×C4×C47C23×C416yes—
112C2×C4×C47C23×C412yes—
113C2×C4×C47C23×C411yescharacteristic
114C2×C4×C47C23×C416yes—
115C2×C4×C47C22×Q812yes—
116C2×C4×C47C22×Q814yes—
117C2×C4×C47C2×#4112yes—
118C2×C4×C47C2×#4114yes—
119C2×C4×C47C2511yescharacteristic
120C47⋊C8—212no—
121C47⋊C8C22×Q818yescomplemented
122C2×Q8×M16—471nomaximal, Sylow
123C22×C47—28no—
124C22×C47—22no—
125C22×C47—22no—
126C22×C47C2×C4×Q814yes—
127C22×C47C2×C4×Q812yes—
128C22×C47C22×M1612yes—
129C22×C47#56812yes—
130C22×C47C24×C411yescharacteristic
131C4×C47—26no—
132C4×C47—26no—
133C4×C47—26no—
134C4×C47C2×C4×Q812yes—
135C4×C47C2×C4×Q814yes—
136C4×C47C22×M16112yes—
137C4×C47C23×Q811yescharacteristic
138C4×C47C22×#4111yescharacteristic
139C22×C4×Q8D4711yescharacteristic
140Q8×M16—4716no—
141C2×#446—476no—
142C2×C8×Q8—472no—
143C2×#423—473no—
144C2×C4×M16—473no—
145C2×C47—24no—
146C2×C47Q8×M1614yescomplemented
147C2×C47C22×C4×Q811yescharacteristic
148C2×C47C23×M1611yescharacteristic
149C2×C47C2×#56811yescharacteristic
150C23×Q8Dic4711yescharacteristic
151C22×M16—471no—
152C22×M16—473no—
153C2×C4×Q8Dic4713yes—
154C2×C4×Q8Dic4718yes—
155C2×C4×Q8Dic4711yescharacteristic
156C2×C4×Q8C2×D4711yescharacteristic
157C2×C4×Q8C2×D4713yes—
158C2×C4×Q8C2×D4718yes—
159C22×C4⋊C4Dic4713yes—
160C22×C42C2×D4713yes—
161#446—4748no—
162C8×Q8—4716no—
163#423—4724no—
164C2×C4⋊C8—476no—
165C2×C4⋊C8—476no—
166C4×M16—4724no—
167C2×#97—476no—
168C2×C4×C8—476no—
169C47C2×Q8×M1611yescharacteristic, complemented, Sylow
170C22×Q8—28no—
171C22×Q8C2×Dic4718yes—
172C22×Q8C2×Dic4713yes—
173C22×Q8C2×Dic4711yescharacteristic
174C23×C4C2×Dic4713yes—
175C23×C4C2×Dic4713yes—
176C23×C4C22×D4711yescharacteristic
177C2×M16—478no—
178C2×M16—474no—
179C2×M16—4712no—
180C2×M16—9412no—
181C22×C8—472no—
182C22×C8—476no—
183C4×Q8C2×Dic47112yes—
184C4×Q8C2×Dic47124yes—
185C4×Q8C2×Dic4714yes—
186C4×Q8C2×Dic4718yes—
187C4×Q8C22×D4714yes—
188C4×Q8C22×D47112yes—
189C2×C4⋊C4C2×Dic4716yes—
190C2×C4⋊C4C2×Dic4713yes—
191C2×C4⋊C4C2×Dic4713yes—
192C2×C4⋊C4C2×Dic47124yes—
193C2×C42C2×Dic4713yes—
194C2×C42C2×Dic4713yes—
195C2×C42C2×Dic47112yes—
196C2×C42C22×D47112yes—
197C2×C42C22×D4713yes—
198C2×C42C22×D4713yes—
199C4⋊C8—4724no—
200C4⋊C8—4724no—
201#97—4724no—
202C4×C8—4724no—
203C24C22×Dic4711yescharacteristic
204C2×Q8—232no—
205C2×Q8—212no—
206C2×Q8#1062532918yesdirect product factor
207C2×Q8C22×Dic47112yes—
208C2×Q8C22×Dic4714yes—
209C22×C4—212no—
210C22×C4—46no—
211C22×C4Q8×D4714yes—
212C22×C4#1062535414yes—
213C22×C4C22×Dic4713yes—
214C22×C4C22×Dic4713yes—
215C22×C4C22×Dic4713yes—
216C22×C4C22×Dic47112yes—
217C22×C4C22×Dic47112yes—
218C22×C4C22×Dic4711yescharacteristic
219C22×C4C22×Dic4712yes—
220C22×C4C22×Dic4713yes—
221C22×C4C23×D4712yes—
222C22×C4C23×D4711yescharacteristic
223M16—4716no—
224M16—9424no—
225C2×C8—478no—
226C2×C8—474no—
227C2×C8—4712no—
228C2×C8—9412no—
229C4⋊C4C22×Dic47124yes—
230C4⋊C4C22×Dic47112yes—
231C4⋊C4C22×Dic47112yes—
232C42C22×Dic47112yes—
233C42C22×Dic4716yes—
234C42C22×Dic47112yes—
235C42C22×Dic4716yes—
236C42C23×D4716yes—
237C42C23×D4716yes—
238C23—22no—
239C23—22no—
240C23Q8×Dic4714yes—
241C23C23×Dic4712yes—
242C23C23×Dic4711yescharacteristic
243Q8—224no—
244Q8C2×#10625329116yesdirect product factor
245C2×C4—26no—
246C2×C4—26no—
247C2×C4—26no—
248C2×C4—224no—
249C2×C4—412no—
250C2×C4Q8×Dic4718yes—
251C2×C4Q8×Dic4714yes—
252C2×C4C2×#10625329112yes—
253C2×C4C23×Dic4716yes—
254C2×C4C23×Dic4716yes—
255C2×C4C23×Dic4716yes—
256C2×C4C23×Dic4712yes—
257C2×C4C23×Dic4711yescharacteristic
258C2×C4C23×Dic4716yes—
259C2×C4C2×Q8×D4712yes—
260C2×C4C2×Q8×D4714yes—
261C2×C4C2×#1062535412yes—
262C2×C4C2×#1062535414yes—
263C2×C4C24×D4711yescharacteristic
264C8—478no—
265C8—9412no—
266C22—22no—
267C22—22no—
268C22—28no—
269C22C2×Q8×Dic4714yes—
270C22C2×Q8×Dic4712yes—
271C22C22×#1062532912yes—
272C22#1568989012yes—
273C22C24×Dic4711yescharacteristic
274C4—26no—
275C4—26no—
276C4—26no—
277C4C2×Q8×Dic4714yes—
278C4C2×Q8×Dic4712yes—
279C4C22×#10625329112yes—
280C4C22×Q8×D4711yescharacteristic
281C4C22×#1062535411yescharacteristic
282C2—24no—
283C2Q8×#1062532914yesdirect product factor
284C2C22×Q8×Dic4711yescharacteristic
285C2C23×#1062532911yescharacteristic
286C2C2×#1568989011yescharacteristic
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×Q8×M16
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. Q8×#10625329C2direct (N × H)
2. C2×#15689254C2semidirect
3. C2×Q8×C47⋊C8C2semidirect
9. #15689254C22semidirect
10. Q8×C47⋊C8C22semidirect
27. C2×#10625329Q8direct (N × H)
62. C2×Q8×C47M16semidirect
89. #10625329C2×Q8direct (N × H)
91. C2×C47⋊C8C2×Q8semidirect
100. Q8×C47C2×M16semidirect
121. C47⋊C8C22×Q8semidirect
146. C2×C47Q8×M16semidirect
169. C47C2×Q8×M16semidirect
206. C2×Q8#10625329direct (N × H)
244. Q8C2×#10625329direct (N × H)
283. C2Q8×#10625329direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#106253292
C2×#156892542
C2×Q8×C47⋊C82
C2×#156891472
C2×C4×#106253292
C22×C4×Q8×C472
C2×Q8×M1647