Finite Group Databasebeta

← C22×#916 #87174

Subgroups — C22×#916 #87174

Order 384.

1504 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 525 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×#239C211yesmaximal, characteristic, complemented
2C22×#237C211yesmaximal, characteristic, complemented
3C22×C3×SD16C211yesmaximal, characteristic, complemented
4C22×#216C211yesmaximal, characteristic, complemented
5C22×#213C211yesmaximal, characteristic, complemented
6C2×#916C2124yesmaximal, direct product factor
7C22×#204C211yesmaximal, characteristic, complemented
8C22×D3×C8C211yesmaximal, characteristic, complemented
9C22×#135—31nomaximal, Sylow
10C22×C3×Q8C2211yescharacteristic
11C22×C3×D4C2211yescharacteristic
12C22×#67—21no—
13C23×Dic3—21no—
14C2×#239—26no—
15C2×#239C22112yescomplemented
16C2×#237—26no—
17C2×#237C22112yescomplemented
18C22×D12—21no—
19C22×D12C2211yescharacteristic
20C22×C4×D3—21no—
21C22×C4×D3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C2×C3×SD16C22112yescomplemented
24C22×C3×C8C2211yescharacteristic
25C2×#216C22112yescomplemented
26C2×#213C22112yescomplemented
27C22×C3⋊C8C2211yescharacteristic
28#916C22164yesdirect product factor
29C2×#204C22112yescomplemented
30C2×D3×C8C22112yescomplemented
31C22×#41—31no—
32C22×#41—31no—
33C2×#135—324no—
34C22×Q16—31no—
35C22×SD16—31no—
36C22×SD16D311yescharacteristic, complemented
37C22×D8—31no—
38C23×C8—31no—
39C24×C3—21no—
40C23×D3—21no—
41C23×D3D411yescharacteristic
42C2×C3×Q8—23no—
43C2×C3×Q8C2316yes—
44C2×C3×D4—23no—
45C2×C3×D4C2316yes—
46C22×C3×C4—21no—
47C22×C3×C4C2311yescharacteristic
48C2×#67—212no—
49C22×Dic3—21no—
50C22×Dic3—212no—
51C22×Dic3D411yescharacteristic
52#239—224no—
53#239C23116yescomplemented
54#237—224no—
55#237C23116yescomplemented
56C2×D12—212no—
57C2×D12—23no—
58C2×D12C2316yes—
59C2×C4×D3—212no—
60C2×C4×D3D416yes—
61C2×C4×D3C2316yes—
62C2×Dic6—23no—
63C2×Dic6C2316yes—
64C3×SD16C23116yescomplemented
65C2×C3×C8C2316yes—
66#216C23116yescomplemented
67#213C23116yescomplemented
68C2×C3⋊C8C2316yes—
69#204C23116yescomplemented
70D3×C8C23116yescomplemented
71C2×#41—312no—
72C2×#41—312no—
73C2×#41—66no—
74C2×#41—66no—
75C22×Q8—31no—
76C22×Q8C2×D311yescharacteristic
77C22×D4—31no—
78C22×D4—61no—
79C22×D4—61no—
80C22×D4C2×D311yescharacteristic
81C23×C4—31no—
82C23×C4—61no—
83C23×C4—61no—
84#135—364no—
85C2×Q16—312no—
86C2×SD16—312no—
87C2×SD16C2×D3112yescomplemented
88C2×D8—312no—
89C22×C8—31no—
90C22×C8—312no—
91C22×C8C2×D311yescharacteristic
92C23×C3—26no—
93C23×C3—44no—
94C23×C3C2×D411yescharacteristic
95C22×D3—24no—
96C22×D3—26no—
97C22×D3—44no—
98C22×D3C2×D416yes—
99C3×Q8—26no—
100C3×Q8C2414yes—
101C3×D4—26no—
102C3×D4C2414yes—
103C2×C3×C4—26no—
104C2×C3×C4C2×D413yes—
105C2×C3×C4C2413yes—
106#67—216no—
107C2×Dic3—26no—
108C2×Dic3—216no—
109C2×Dic3C2×D416yes—
110D12—216no—
111D12—26no—
112D12C2414yes—
113C4×D3—216no—
114C4×D3C2×D4112yes—
115C4×D3C2414yes—
116Dic6—26no—
117Dic6C2414yes—
118C3×C8C2414yes—
119C3⋊C8C2414yes—
120C24—21no—
121C24—31no—
122C24—61no—
123#41—316no—
124#41—316no—
125#41—624no—
126#41—624no—
127C2×Q8—23no—
128C2×Q8—36no—
129C2×Q8—63no—
130C2×Q8C22×D316yes—
131C2×D4—23no—
132C2×D4—36no—
133C2×D4—612no—
134C2×D4—612no—
135C2×D4—63no—
136C2×D4C22×D316yes—
137C22×C4—21no—
138C22×C4—31no—
139C22×C4—36no—
140C22×C4—36no—
141C22×C4—612no—
142C22×C4—612no—
143C22×C4—61no—
144C22×C4C22×D311yescharacteristic
145Q16—316no—
146SD16—316no—
147SD16C22×D3116yescomplemented
148D8—316no—
149C2×C8—36no—
150C2×C8—316no—
151C2×C8C22×D316yes—
152C22×C3—24no—
153C22×C3—412no—
154C22×C3#13514yescomplemented
155C22×C3C22×D413yes—
156C2×D3—212no—
157C2×D3—24no—
158C2×D3—412no—
159C2×D3C22×D414yes—
160C3×C4—24no—
161C3×C4C22×D413yes—
162C3×C4C2511yescharacteristic
163Dic3—24no—
164Dic3C22×D414yes—
165C23—26no—
166C23—36no—
167C23—44no—
168C23—64no—
169C23—66no—
170C23—124no—
171C23D3×D411yescharacteristic
172Q8—26no—
173Q8—34no—
174Q8—66no—
175Q8C23×D314yes—
176D4—26no—
177D4—34no—
178D4—616no—
179D4—616no—
180D4—66no—
181D4C23×D314yes—
182C2×C4—26no—
183C2×C4—36no—
184C2×C4—312no—
185C2×C4—34no—
186C2×C4—616no—
187C2×C4—616no—
188C2×C4—66no—
189C2×C4D3×D413yes—
190C2×C4C23×D313yes—
191C8—34no—
192C8C23×D314yes—
193C2×C3—44no—
194C2×C3C2×#13516yescomplemented
195C2×C3C23×D411yescharacteristic
196D3—24no—
197D3—44no—
198C22—24no—
199C22—34no—
200C22—412no—
201C22—612no—
202C22—64no—
203C22—1212no—
204C22#91614yesdirect product factor
205C22C2×D3×D413yes—
206C4—24no—
207C4—34no—
208C4—64no—
209C4C2×D3×D413yes—
210C4C24×D311yescharacteristic
211C3C22×#13511yescharacteristic, complemented, Sylow
212C2—44no—
213C2—64no—
214C2—124no—
215C2C2×#91616yesdirect product factor
216C2C22×D3×D411yescharacteristic
subgroupG/H
centerC23D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC22×C3×SD16C2
socleC23×C3C2×D4
hypercenterC22×SD16D3
hypocenterC3C22×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#239C2semidirect
2. C22×#237C2semidirect
3. C22×C3×SD16C2semidirect
4. C22×#216C2semidirect
5. C22×#213C2semidirect
6. C2×#916C2direct (N × H)
7. C22×#204C2semidirect
8. C22×D3×C8C2semidirect
15. C2×#239C22semidirect
17. C2×#237C22semidirect
23. C2×C3×SD16C22semidirect
25. C2×#216C22semidirect
26. C2×#213C22semidirect
28. #916C22direct (N × H)
29. C2×#204C22semidirect
30. C2×D3×C8C22semidirect
36. C22×SD16D3semidirect
53. #239C23semidirect
55. #237C23semidirect
64. C3×SD16C23semidirect
66. #216C23semidirect
67. #213C23semidirect
69. #204C23semidirect
70. D3×C8C23semidirect
87. C2×SD16C2×D3semidirect
147. SD16C22×D3semidirect
154. C22×C3#135semidirect
194. C2×C3C2×#135semidirect
204. C22#916direct (N × H)
211. C3C22×#135semidirect
215. C2C2×#916direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#2392
C22×#2372
C22×C3×SD162
C22×#2162
C22×#2132
C2×#9162
C22×#2042
C22×D3×C82
C22×#1353