Finite Group Databasebeta

← C2×#5711 #86117

Subgroups — C2×#5711 #86117

Order 384.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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
1C2×C3×#126C211yesmaximal, characteristic, complemented
2#5711C2116yesmaximal, direct product factor
3C2×#895C213yesmaximal, complemented
4C2×#894C213yesmaximal, complemented
5C2×#884C213yesmaximal, complemented
6C2×#882C213yesmaximal, complemented
7C2×#876C211yesmaximal, characteristic, complemented
8C2×#874C211yesmaximal, characteristic, complemented
9C2×#560—31nomaximal, Sylow
10C24×D3C2211yescharacteristic
11C22×#67C2213yes—
12C22×D12C2213yes—
13C22×D12C2213yes—
14C22×D12C2213yescomplemented
15C22×C4×D3C2213yescomplemented
16C3×#126C2218yescomplemented
17C2×C3×C4⋊C4C2213yescomplemented
18C2×C3×C22⋊C4C2213yes—
19C2×C3×C42C2211yescharacteristic, complemented
20C2×(C2×D3)⋊C4C2213yes—
21C2×(C2×D3)⋊C4C2213yes—
22C2×(C2×D3)⋊C4C2213yescomplemented
23C2×Dic3⋊C4C2213yescomplemented
24#895C22124yescomplemented
25#894C22124yescomplemented
26#884C22124yescomplemented
27#882C22124yescomplemented
28#876C2218yescomplemented
29#874C2218yescomplemented
30#560—316no—
31C2×#127—31no—
32C2×#126—31no—
33C2×#126D311yescharacteristic, complemented
34C2×#123—33no—
35C2×#121—33no—
36C2×#121—33no—
37C2×#120—33no—
38C24×C3C2311yescharacteristic
39C23×D3—46no—
40C23×D3C2311yescharacteristic
41C23×D3C2311yescharacteristic
42C23×D3C2313yes—
43C23×D3C2314yes—
44C22×C3×C4C2313yes—
45C22×C3×C4C2313yes—
46C2×#67—46no—
47C2×#67C23112yes—
48C22×Dic3C2313yes—
49C2×D12—46no—
50C2×D12—46no—
51C2×D12—46no—
52C2×D12C23112yes—
53C2×D12C23112yes—
54C2×D12C23112yescomplemented
55C2×C4×D3—46no—
56C2×C4×D3C23112yescomplemented
57C3×C4⋊C4C23112yescomplemented
58C3×C22⋊C4C23112yes—
59C3×C42C2314yescomplemented
60(C2×D3)⋊C4C23112yes—
61(C2×D3)⋊C4C23112yes—
62(C2×D3)⋊C4C23112yescomplemented
63Dic3⋊C4C23112yescomplemented
64C25—31no—
65C22×D4—33no—
66C22×D4—33no—
67C22×D4—33no—
68C22×D4—33no—
69C23×C4—33no—
70#127—38no—
71#126—38no—
72#126C2×D318yescomplemented
73#123—324no—
74#121—324no—
75#121—324no—
76#120—324no—
77C2×C4⋊C4—33no—
78C2×C4⋊C4C2×D313yescomplemented
79C2×C22⋊C4—33no—
80C2×C22⋊C4—33no—
81C2×C22⋊C4—33no—
82C2×C22⋊C4C2×D313yes—
83C2×C42C2×D311yescharacteristic, complemented
84C23×C3—26no—
85C23×C3C2412yes—
86C23×C3C2411yescharacteristic
87C22×D3—26no—
88C22×D3—26no—
89C22×D3—26no—
90C22×D3—26no—
91C22×D3—26no—
92C22×D3—412no—
93C22×D3—416no—
94C22×D3C2412yes—
95C22×D3C2412yes—
96C22×D3C2416yes—
97C2×C3×C4—26no—
98C2×C3×C4—26no—
99C2×C3×C4C2416yes—
100C2×C3×C4C2416yes—
101#67—412no—
102C2×Dic3—26no—
103C2×Dic3C2416yes—
104D12—412no—
105D12—412no—
106D12—412no—
107C4×D3—412no—
108C24—31no—
109C24—31no—
110C24—33no—
111C24—34no—
112C24—126no—
113C24C22×D311yescharacteristic
114C2×D4—312no—
115C2×D4—312no—
116C2×D4—312no—
117C2×D4—312no—
118C2×D4—126no—
119C2×D4—126no—
120C2×D4—126no—
121C2×D4—126no—
122C22×C4—33no—
123C22×C4—312no—
124C22×C4—126no—
125C22×C4C22×D313yes—
126C22×C4C22×D313yes—
127C4⋊C4—312no—
128C4⋊C4C22×D3112yescomplemented
129C22⋊C4—312no—
130C22⋊C4—312no—
131C22⋊C4—312no—
132C22⋊C4C22×D3112yes—
133C42C22×D314yescomplemented
134C22×C3—26no—
135C22×C3—44no—
136C22×C3H(5,𝔽2)16yes—
137C22×C3C2511yescharacteristic
138C2×D3—26no—
139C2×D3—26no—
140C2×D3—26no—
141C2×D3—26no—
142C2×D3—26no—
143C2×D3—416no—
144C2×D3—412no—
145C2×D3—44no—
146C2×D3—44no—
147C3×C4—26no—
148C3×C4—26no—
149Dic3—26no—
150C23—26no—
151C23—32no—
152C23—32no—
153C23—36no—
154C23—66no—
155C23—66no—
156C23—66no—
157C23—66no—
158C23—66no—
159C23—1212no—
160C23—1216no—
161C23C23×D312yes—
162C23C23×D311yescharacteristic
163D4—1212no—
164D4—1212no—
165D4—1212no—
166D4—1212no—
167C2×C4—26no—
168C2×C4—26no—
169C2×C4—36no—
170C2×C4—66no—
171C2×C4—1212no—
172C2×C4C23×D316yes—
173C2×C4C23×D316yes—
174C2×C3—42no—
175C2×C3#56014yescomplemented
176C2×C3C2×H(5,𝔽2)13yes—
177D3—46no—
178D3—42no—
179D3—42no—
180C22—26no—
181C22—44no—
182C22—66no—
183C22—66no—
184C22—66no—
185C22—66no—
186C22—66no—
187C22—1216no—
188C22—124no—
189C22—124no—
190C22—1212no—
191C22#100916yes—
192C22C24×D311yescharacteristic
193C4—26no—
194C4—26no—
195C4—66no—
196C3C2×#56011yescharacteristic, complemented, Sylow
197C2—42no—
198C2—122no—
199C2—122no—
200C2—126no—
201C2#571114yesdirect product factor
202C2C2×#100913yes—
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#126C2
socleC23×C3C24
hypercenterC2×#126D3
hypocenterC3C2×#560
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126C2semidirect
2. #5711C2direct (N × H)
3. C2×#895C2semidirect
4. C2×#894C2semidirect
5. C2×#884C2semidirect
6. C2×#882C2semidirect
7. C2×#876C2semidirect
8. C2×#874C2semidirect
14. C22×D12C22semidirect
15. C22×C4×D3C22semidirect
16. C3×#126C22semidirect
17. C2×C3×C4⋊C4C22semidirect
19. C2×C3×C42C22semidirect
22. C2×(C2×D3)⋊C4C22semidirect
23. C2×Dic3⋊C4C22semidirect
24. #895C22semidirect
25. #894C22semidirect
26. #884C22semidirect
27. #882C22semidirect
28. #876C22semidirect
29. #874C22semidirect
33. C2×#126D3semidirect
54. C2×D12C23semidirect
56. C2×C4×D3C23semidirect
57. C3×C4⋊C4C23semidirect
59. C3×C42C23semidirect
62. (C2×D3)⋊C4C23semidirect
63. Dic3⋊C4C23semidirect
72. #126C2×D3semidirect
78. C2×C4⋊C4C2×D3semidirect
83. C2×C42C2×D3semidirect
128. C4⋊C4C22×D3semidirect
133. C42C22×D3semidirect
175. C2×C3#560semidirect
196. C3C2×#560semidirect
201. C2#5711direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1262
#57112
C2×#8952
C2×#8942
C2×#8842
C2×#8822
C2×#8762
C2×#8742
C2×#5603