Finite Group Databasebeta

← Q8×C22⋊#157 #11804880

Subgroups — Q8×C22⋊#157 #11804880

Order 1152.

1670 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 1670 conjugacy classes of subgroups — too many to draw. Only its 228 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×Q8×#157C212yesmaximal, complemented
3#10593837C216yesmaximal, complemented
4#10593812C213yesmaximal
5C4×C22⋊#157C213yesmaximal
6C22×C32×Q8C411yescharacteristic, complemented
7C2×Q8×C3⋊D3—28no—
8C2×Q8×C3⋊D3C411yescharacteristic, complemented
9C2×Q8×C3⋊D3C413yescomplemented
10C2×Q8×C3⋊D3C2213yes—
11C2×Q8×C3⋊D3C2211yescharacteristic
12C22×C4×C3⋊D3C2213yes—
13C22×#634C413yescomplemented
14C2×C22⋊#157C2213yes—
15C2×C22⋊#157C2211yescharacteristic
16Q8×#157—28no—
17C2×C4⋊#157C2216yes—
18C2×C4×#157C2216yes—
19#63781C2216yes—
20#63781C2216yes—
21C22×D3×Q8—62no—
22C23×C3⋊D3C2311yescharacteristic
23C22×#157C2316yes—
24C22×#157C2312yes—
25C2×C32×Q8—24no—
26C2×C32×Q8C2×C413yes—
27C2×C32×Q8C2×C411yescharacteristic
28C22×C32×C4C2×C413yes—
29C22×C3⋊Dic3C2×C413yes—
30Q8×C3⋊D3—24no—
31Q8×C3⋊D3—212no—
32Q8×C3⋊D3—212no—
33Q8×C3⋊D3D418yes—
34C2×C4×C3⋊D3—212no—
35C2×C4×C3⋊D3C2×C413yes—
36C2×C4×C3⋊D3C2×C413yes—
37C2×C4×C3⋊D3C2313yes—
38C2×C4×C3⋊D3C2313yes—
39C2×#634—212no—
40C2×#634C2×C413yes—
41C2×#634C2×C416yes—
42C2×#634C2×C413yes—
43C22⋊#157—26no—
44C22⋊#157Q814yesdirect product factor
45C4⋊#157—212no—
46C4×#157—212no—
47Q8×C22⋊C4—91nomaximal, Sylow
48C22×C3×Q8—22no—
49C2×D3×Q8—66no—
50C2×D3×Q8—62no—
51C2×D3×Q8—62no—
52C2×D3×Q8—66no—
53C2×D3×Q8—632no—
54C22×C4×D3—66no—
55C22×Dic6—66no—
56C23×C32C22×C411yescharacteristic
57C22×C3⋊D3—22no—
58C22×C3⋊D3—22no—
59C22×C3⋊D3C22×C411yescharacteristic
60C22×C3⋊D3C2×Q812yes—
61C22×C3⋊D3#4112yes—
62C22×C3⋊D3C2411yescharacteristic
63C2×#157—22no—
64C2×#157—22no—
65C2×#157—26no—
66C2×#157—26no—
67C2×#157—26no—
68C2×#157C2×Q814yescomplemented
69C32×Q8—26no—
70C32×Q8C22⋊C414yescomplemented
71C2×C32×C4—26no—
72C2×C32×C4C22×C413yes—
73C2×C32×C4C22×C413yes—
74C2×C3⋊Dic3—26no—
75C2×C3⋊Dic3C22×C413yes—
76C2×C3⋊Dic3C22×C413yes—
77C4×C3⋊D3—26no—
78C4×C3⋊D3—26no—
79C4×C3⋊D3—26no—
80C4×C3⋊D3C2×D4112yes—
81#634—26no—
82#634—212no—
83#634C22⋊C4112yescomplemented
84C23×Q8—91no—
85C2×C4×Q8—92no—
86#391—96no—
87#380—93no—
88C4×C22⋊C4—93no—
89C23×D3—62no—
90C2×C3×Q8—216no—
91C2×C3×Q8—26no—
92C2×C3×Q8—22no—
93C22×C3×C4—26no—
94C22×Dic3—66no—
95D3×Q8—616no—
96D3×Q8—648no—
97D3×Q8—648no—
98D3×Q8—616no—
99C2×C4×D3—66no—
100C2×C4×D3—66no—
101C2×C4×D3—66no—
102C2×C4×D3—66no—
103C2×C4×D3—648no—
104C2×Dic6—66no—
105C2×Dic6—612no—
106C2×Dic6—66no—
107C2×Dic6—648no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32C4×Q812yescomplemented
111C22×C32C23×C411yescharacteristic
112C2×C3⋊D3—21no—
113C2×C3⋊D3—21no—
114C2×C3⋊D3—24no—
115C2×C3⋊D3—24no—
116C2×C3⋊D3C4×Q812yescomplemented
117C2×C3⋊D3C22×D412yes—
118C2×C3⋊D3C22×Q811yescharacteristic
119C2×C3⋊D3C2×#4111yescharacteristic
120C2×C3⋊D32−1+412yes—
121#157—24no—
122#157—46no—
123C32×C4—23no—
124C32×C4C2×C22⋊C416yes—
125C3⋊Dic3—23no—
126C3⋊Dic3C2×C22⋊C416yes—
127C22×Q8—91no—
128C22×Q8—93no—
129C22×Q8—93no—
130C22×Q8—91no—
131C22×Q8—93no—
132C22×Q8—188no—
133C22×Q8#15711yescharacteristic, complemented
134C23×C4—93no—
135C4×Q8—188no—
136C2×C4⋊C4—96no—
137C2×C22⋊C4—91no—
138C2×C22⋊C4—93no—
139C2×C42—96no—
140#95—96no—
141#95—96no—
142C23×C3—22no—
143C22×D3—62no—
144C22×D3—64no—
145C22×D3—64no—
146C22×D3—62no—
147C22×D3—68no—
148C22×D3—68no—
149C3×Q8—224no—
150C3×Q8—28no—
151C2×C3×C4—224no—
152C2×C3×C4—26no—
153C2×C3×C4—26no—
154C2×Dic3—66no—
155C2×Dic3—66no—
156C2×Dic3—624no—
157C4×D3—624no—
158C4×D3—624no—
159C4×D3—624no—
160C4×D3—624no—
161Dic6—624no—
162Dic6—648no—
163Dic6—624no—
164C2×C32—22no—
165C2×C32C22×C22⋊C411yescharacteristic
166C2×C32C2×C4×Q811yescharacteristic
167C2×C32#51911yescharacteristic
168C3⋊D3—22no—
169C3⋊D3#54812yes—
170C3⋊D3D4×Q812yes—
171C24—91no—
172C2×Q8—24no—
173C2×Q8—98no—
174C2×Q8—93no—
175C2×Q8—96no—
176C2×Q8—93no—
177C2×Q8—1812no—
178C2×Q8—1812no—
179C2×Q8—1812no—
180C2×Q8—184no—
181C2×Q8C2×#15713yes—
182C2×Q8C2×#15711yescharacteristic
183C22×C4—93no—
184C22×C4—92no—
185C22×C4—93no—
186C22×C4—93no—
187C22×C4—93no—
188C22×C4—93no—
189C22×C4—96no—
190C22×C4—1812no—
191C22×C4C2×#15713yes—
192C4⋊C4—1812no—
193C22⋊C4—94no—
194C22⋊C4—186no—
195C42—1812no—
196C22×C3—24no—
197C22×C3—24no—
198C22×C3—24no—
199C22×C3—22no—
200C2×D3—64no—
201C2×D3—64no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—64no—
205C2×D3—64no—
206C2×D3—616no—
207C2×D3—616no—
208C2×D3—64no—
209C3×C4—212no—
210C3×C4—212no—
211Dic3—612no—
212Dic3—612no—
213C32Q8×C22⋊C411yescharacteristic, complemented, Sylow
214C23—91no—
215C23—92no—
216C23—92no—
217C23—91no—
218C23—182no—
219C23—182no—
220C23C22×#15711yescharacteristic
221Q8—26no—
222Q8—912no—
223Q8—1812no—
224Q8—186no—
225Q8C22⋊#15714yesdirect product factor
226C2×C4—26no—
227C2×C4—93no—
228C2×C4—912no—
229C2×C4—94no—
230C2×C4—93no—
231C2×C4—186no—
232C2×C4—186no—
233C2×C4—186no—
234C2×C4—182no—
235C2×C4—182no—
236C2×C4—186no—
237C2×C4—186no—
238C2×C4—186no—
239C2×C4—186no—
240C2×C4C22×#15713yes—
241C2×C4C22×#15713yes—
242C2×C3—28no—
243C2×C3—22no—
244C2×C3—22no—
245C2×C3—22no—
246D3—64no—
247D3—64no—
248D3—68no—
249C22—21no—
250C22—21no—
251C22—91no—
252C22—91no—
253C22—92no—
254C22—92no—
255C22—92no—
256C22—184no—
257C22—184no—
258C22—181no—
259C22—181no—
260C22Q8×#15712yescomplemented
261C22C23×#15711yescharacteristic
262C4—23no—
263C4—96no—
264C4—184no—
265C4—183no—
266C4—366no—
267C4C2×C22⋊#15716yes—
268C3—22no—
269C2—22no—
270C2—92no—
271C2—92no—
272C2—182no—
273C2C2×Q8×#15711yescharacteristic
274C2#1059664611yescharacteristic
275C2C22×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C23×#157
commutator, socleC22×C32C23×C4
FittingC22×C32×Q8C4
prime radicalC23×C3⋊D3C23
cosocleC22×C3⋊D3C24
hypercenterC22×Q8#157
hypocenterC32Q8×C22⋊C4
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#157C2semidirect
3. #10593837C2semidirect
6. C22×C32×Q8C4semidirect
8. C2×Q8×C3⋊D3C4semidirect
9. C2×Q8×C3⋊D3C4semidirect
13. C22×#634C4semidirect
44. C22⋊#157Q8direct (N × H)
68. C2×#157C2×Q8semidirect
70. C32×Q8C22⋊C4semidirect
83. #634C22⋊C4semidirect
110. C22×C32C4×Q8semidirect
116. C2×C3⋊D3C4×Q8semidirect
133. C22×Q8#157semidirect
213. C32Q8×C22⋊C4semidirect
225. Q8C22⋊#157direct (N × H)
260. C22Q8×#157semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×C3⋊D32
C2×Q8×#1572
#105938372
#105938122
C4×C22⋊#1572
Q8×C22⋊C49