Finite Group Databasebeta

← C22×#930 #87221

Subgroups — C22×#930 #87221

Order 384.

2528 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 725 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
1C24×C3×C4C211yesmaximal, characteristic, complemented
2C23×#67C212yesmaximal, complemented
3C23×D12C211yesmaximal, characteristic, complemented
4C2×#930C2124yesmaximal, direct product factor
5C22×(C2×D3)⋊C4C212yesmaximal, complemented
6C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
7C22×#121—31nomaximal, Sylow
8C25×C3C2211yescharacteristic
9C24×D3C2212yes—
10C23×C3×C4—28no—
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×C3×C4C22112yescomplemented
14C22×#67—216no—
15C22×#67C22124yescomplemented
16C23×Dic3C2212yes—
17C22×D12—28no—
18C22×D12C22112yescomplemented
19#930C22164yesdirect product factor
20C2×(C2×D3)⋊C4C22124yescomplemented
21C2×C4⋊Dic3C22112yescomplemented
22C23×D4—32no—
23C23×D4—31no—
24C24×C4D311yescharacteristic, complemented
25C2×#121—324no—
26C22×C4⋊C4—31no—
27C22×C22⋊C4—32no—
28C24×C3—24no—
29C24×C3—24no—
30C24×C3D418yescomplemented
31C24×C3C2316yes—
32C24×C3C2311yescharacteristic
33C23×D3—28no—
34C23×D3—28no—
35C23×D3—28no—
36C23×D3C23112yes—
37C22×C3×C4—24no—
38C22×C3×C4—248no—
39C22×C3×C4D418yescomplemented
40C22×C3×C4C2316yes—
41C22×C3×C4C2316yes—
42C22×C3×C4C23116yescomplemented
43C2×#67—296no—
44C2×#67C23132yescomplemented
45C22×Dic3—28no—
46C22×Dic3C23112yes—
47C2×D12—248no—
48C2×D12C23116yescomplemented
49(C2×D3)⋊C4C23132yescomplemented
50C4⋊Dic3C23116yescomplemented
51C25—32no—
52C25C2×D311yescharacteristic
53C22×D4—324no—
54C22×D4—312no—
55C22×D4—616no—
56C22×D4—68no—
57C23×C4—28no—
58C23×C4—32no—
59C23×C4C2×D311yescharacteristic
60C23×C4C2×D311yescharacteristic
61C23×C4C2×D3112yescomplemented
62#121—364no—
63C2×C4⋊C4—312no—
64C2×C22⋊C4—324no—
65C23×C3—232no—
66C23×C3—212no—
67C23×C3—212no—
68C23×C3C2×D414yes—
69C23×C3C2×D414yes—
70C23×C3C2×D4124yescomplemented
71C23×C3#4114yes—
72C23×C3C2414yes—
73C23×C3C2413yes—
74C22×D3—224no—
75C22×D3—224no—
76C22×D3—224no—
77C22×D3—432no—
78C22×D3C2418yes—
79C2×C3×C4—212no—
80C2×C3×C4—232no—
81C2×C3×C4C2×D4124yescomplemented
82C2×C3×C4C2414yes—
83C2×C3×C4C2414yes—
84#67—264no—
85C2×Dic3—224no—
86C2×Dic3C2418yes—
87D12—232no—
88C24—24no—
89C24—24no—
90C24—312no—
91C24—68no—
92C24—68no—
93C24—68no—
94C24D1218yescomplemented
95C24C22×D316yes—
96C24C22×D311yescharacteristic
97C2×D4—332no—
98C2×D4—316no—
99C2×D4—696no—
100C2×D4—648no—
101C22×C4—24no—
102C22×C4—248no—
103C22×C4—312no—
104C22×C4—68no—
105C22×C4#6718yescomplemented
106C22×C4C22×D316yes—
107C22×C4C22×D316yes—
108C22×C4C22×D3116yescomplemented
109C4⋊C4—316no—
110C22⋊C4—332no—
111C22×C3—248no—
112C22×C3—24no—
113C22×C3—24no—
114C22×C3#121116yescomplemented
115C22×C3C22×D416yes—
116C22×C3C22×D416yes—
117C22×C3C22×D418yescomplemented
118C22×C3C2×#4116yes—
119C22×C3C2511yescharacteristic
120C2×D3—28no—
121C2×D3—28no—
122C2×D3—28no—
123C2×D3—448no—
124C3×C4—24no—
125C3×C4C22×D418yescomplemented
126Dic3—28no—
127C23—232no—
128C23—212no—
129C23—212no—
130C23—38no—
131C23—624no—
132C23—624no—
133C23—624no—
134C23—1232no—
135C23C2×D1214yes—
136C23C2×D12124yescomplemented
137C23#23514yes—
138C23C2×#6714yes—
139C23C23×D314yes—
140C23C23×D313yes—
141D4—664no—
142D4—632no—
143C2×C4—212no—
144C2×C4—232no—
145C2×C4—38no—
146C2×C4—624no—
147C2×C4C2×#67124yescomplemented
148C2×C4C23×D314yes—
149C2×C4C23×D314yes—
150C2×C3—28no—
151C2×C3C2×#121112yescomplemented
152C2×C3C23×D411yescharacteristic
153C2×C3C23×D411yescharacteristic
154C2×C3C22×#4111yescharacteristic
155D3—48no—
156C22—24no—
157C22—24no—
158C22—248no—
159C22—68no—
160C22—68no—
161C22—68no—
162C22—1248no—
163C22#930116yesdirect product factor
164C22C22×D1216yes—
165C22C22×D1218yescomplemented
166C22C2×#23516yes—
167C22C22×#6716yes—
168C22C24×D311yescharacteristic
169C4—24no—
170C4—68no—
171C4C22×#6718yescomplemented
172C3C22×#12111yescharacteristic, complemented, Sylow
173C2—28no—
174C2—128no—
175C2C2×#930112yesdirect product factor
176C2C23×D1211yescharacteristic
177C2C22×#23511yescharacteristic
178C2C23×#6711yescharacteristic
subgroupG/H
centerC24C22×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC24×C3×C4C2
socleC24×C3C23
hypercenterC24×C4D3
hypocenterC3C22×#121
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. C24×C3×C4C2semidirect
2. C23×#67C2semidirect
3. C23×D12C2semidirect
4. C2×#930C2direct (N × H)
5. C22×(C2×D3)⋊C4C2semidirect
6. C22×C4⋊Dic3C2semidirect
13. C23×C3×C4C22semidirect
15. C22×#67C22semidirect
18. C22×D12C22semidirect
19. #930C22direct (N × H)
20. C2×(C2×D3)⋊C4C22semidirect
21. C2×C4⋊Dic3C22semidirect
24. C24×C4D3semidirect
30. C24×C3D4semidirect
39. C22×C3×C4D4semidirect
42. C22×C3×C4C23semidirect
44. C2×#67C23semidirect
48. C2×D12C23semidirect
49. (C2×D3)⋊C4C23semidirect
50. C4⋊Dic3C23semidirect
61. C23×C4C2×D3semidirect
70. C23×C3C2×D4semidirect
81. C2×C3×C4C2×D4semidirect
94. C24D12semidirect
105. C22×C4#67semidirect
108. C22×C4C22×D3semidirect
114. C22×C3#121semidirect
117. C22×C3C22×D4semidirect
125. C3×C4C22×D4semidirect
136. C23C2×D12semidirect
147. C2×C4C2×#67semidirect
151. C2×C3C2×#121semidirect
163. C22#930direct (N × H)
165. C22C22×D12semidirect
171. C4C22×#67semidirect
172. C3C22×#121semidirect
175. C2C2×#930direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C42
C23×#672
C23×D122
C2×#9302
C22×(C2×D3)⋊C42
C22×C4⋊Dic32
C22×#1213