Finite Group Databasebeta

← C22×#938 #87235

Subgroups — C22×#938 #87235

Order 384.

2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 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
1C23×C3×D4C211yesmaximal, characteristic, complemented
2C23×#67C212yesmaximal, complemented
3C23×C4×D3C211yesmaximal, characteristic, complemented
4C22×#217C212yesmaximal, complemented
5C2×#938C2124yesmaximal, direct product factor
6C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
7C22×#121—31nomaximal, Sylow
8C25×C3C2212yes—
9C24×D3C2211yescharacteristic
10C22×C3×D4—28no—
11C22×C3×D4C22112yescomplemented
12C23×C3×C4C2211yescharacteristic
13C22×#67—216no—
14C22×#67C22124yescomplemented
15C23×Dic3C2211yescharacteristic
16C23×Dic3C2212yes—
17C22×C4×D3—28no—
18C22×C4×D3C22112yescomplemented
19C2×#217C22124yescomplemented
20#938C22164yesdirect product factor
21C2×C4⋊Dic3C22112yescomplemented
22C23×D4—32no—
23C23×D4D311yescharacteristic, complemented
24C24×C4—31no—
25C2×#121—324no—
26C22×C4⋊C4—31no—
27C22×C22⋊C4—32no—
28C24×C3—28no—
29C24×C3—28no—
30C24×C3—28no—
31C24×C3C23112yes—
32C24×C3C2311yescharacteristic
33C23×D3—24no—
34C23×D3—24no—
35C23×D3D418yescomplemented
36C23×D3C2316yes—
37C2×C3×D4—248no—
38C2×C3×D4C23116yescomplemented
39C22×C3×C4D418yescomplemented
40C22×C3×C4C2316yes—
41C2×#67—296no—
42C2×#67C23132yescomplemented
43C22×Dic3—24no—
44C22×Dic3—28no—
45C22×Dic3C23112yes—
46C22×Dic3C2316yes—
47C2×C4×D3—248no—
48C2×C4×D3C23116yescomplemented
49#217C23132yescomplemented
50C4⋊Dic3C23116yescomplemented
51C25—31no—
52C25C2×D312yes—
53C22×D4—28no—
54C22×D4—324no—
55C22×D4—616no—
56C22×D4C2×D3112yescomplemented
57C23×C4—31no—
58C23×C4—32no—
59C23×C4—312no—
60C23×C4—68no—
61C23×C4C2×D311yescharacteristic
62#121—364no—
63C2×C4⋊C4—312no—
64C2×C22⋊C4—324no—
65C23×C3—224no—
66C23×C3—224no—
67C23×C3—224no—
68C23×C3—432no—
69C23×C3C2×D414yes—
70C23×C3C2×D414yes—
71C23×C3#4114yes—
72C23×C3C2418yes—
73C23×C3C2413yes—
74C22×D3—232no—
75C22×D3—212no—
76C22×D3—212no—
77C22×D3C2×D4124yescomplemented
78C22×D3C2414yes—
79C3×D4—232no—
80C2×C3×C4C2×D4124yescomplemented
81C2×C3×C4C2414yes—
82#67—264no—
83C2×Dic3—212no—
84C2×Dic3—224no—
85C2×Dic3C2418yes—
86C2×Dic3C2414yes—
87C4×D3—232no—
88C24—28no—
89C24—28no—
90C24—28no—
91C24—38no—
92C24—36no—
93C24—64no—
94C24—64no—
95C24C22×D3112yes—
96C24C22×D311yescharacteristic
97C2×D4—248no—
98C2×D4—332no—
99C2×D4—696no—
100C2×D4C22×D3116yescomplemented
101C22×C4—36no—
102C22×C4—312no—
103C22×C4—316no—
104C22×C4—68no—
105C22×C4—64no—
106C22×C4—648no—
107C22×C4#6718yescomplemented
108C22×C4C22×D316yes—
109C4⋊C4—316no—
110C22⋊C4—332no—
111C22×C3—28no—
112C22×C3—28no—
113C22×C3—28no—
114C22×C3—448no—
115C22×C3#121116yescomplemented
116C22×C3C22×D416yes—
117C22×C3C22×D416yes—
118C22×C3C2×#4116yes—
119C22×C3C2511yescharacteristic
120C2×D3—248no—
121C2×D3—24no—
122C2×D3—24no—
123C2×D3C22×D418yescomplemented
124C3×C4C22×D418yescomplemented
125Dic3—24no—
126Dic3—28no—
127C23—224no—
128C23—224no—
129C23—224no—
130C23—324no—
131C23—34no—
132C23—432no—
133C23—632no—
134C23—612no—
135C23—612no—
136C23D3×D414yes—
137C23#23714yes—
138C23C2×#6714yes—
139C23C23×D318yes—
140C23C23×D313yes—
141D4—232no—
142D4—664no—
143C2×C4—34no—
144C2×C4—38no—
145C2×C4—624no—
146C2×C4—612no—
147C2×C4—632no—
148C2×C4C2×#67124yescomplemented
149C2×C4C23×D314yes—
150C2×C3—48no—
151C2×C3C2×#121112yescomplemented
152C2×C3C23×D411yescharacteristic
153C2×C3C23×D411yescharacteristic
154C2×C3C22×#4111yescharacteristic
155D3—28no—
156C22—28no—
157C22—28no—
158C22—28no—
159C22—38no—
160C22—448no—
161C22—64no—
162C22—64no—
163C22—648no—
164C22#938116yesdirect product factor
165C22C2×D3×D416yes—
166C22C2×#23716yes—
167C22C22×#6716yes—
168C22C24×D311yescharacteristic
169C4—68no—
170C4—64no—
171C4C22×#6718yescomplemented
172C3C22×#12111yescharacteristic, complemented, Sylow
173C2—48no—
174C2—68no—
175C2C2×#938112yesdirect product factor
176C2C22×D3×D411yescharacteristic
177C2C22×#23711yescharacteristic
178C2C23×#6711yescharacteristic
subgroupG/H
centerC24C22×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC23×C3×D4C2
socleC24×C3C23
hypercenterC23×D4D3
hypocenterC3C22×#121
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×D4C2semidirect
2. C23×#67C2semidirect
3. C23×C4×D3C2semidirect
4. C22×#217C2semidirect
5. C2×#938C2direct (N × H)
6. C22×C4⋊Dic3C2semidirect
11. C22×C3×D4C22semidirect
14. C22×#67C22semidirect
18. C22×C4×D3C22semidirect
19. C2×#217C22semidirect
20. #938C22direct (N × H)
21. C2×C4⋊Dic3C22semidirect
23. C23×D4D3semidirect
35. C23×D3D4semidirect
38. C2×C3×D4C23semidirect
39. C22×C3×C4D4semidirect
42. C2×#67C23semidirect
48. C2×C4×D3C23semidirect
49. #217C23semidirect
50. C4⋊Dic3C23semidirect
56. C22×D4C2×D3semidirect
77. C22×D3C2×D4semidirect
80. C2×C3×C4C2×D4semidirect
100. C2×D4C22×D3semidirect
107. C22×C4#67semidirect
115. C22×C3#121semidirect
123. C2×D3C22×D4semidirect
124. C3×C4C22×D4semidirect
148. C2×C4C2×#67semidirect
151. C2×C3C2×#121semidirect
164. C22#938direct (N × H)
171. C4C22×#67semidirect
172. C3C22×#121semidirect
175. C2C2×#938direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D42
C23×#672
C23×C4×D32
C22×#2172
C2×#9382
C22×C4⋊Dic32
C22×#1213