Finite Group Databasebeta

← C2×D3×#126 #86111

Subgroups — C2×D3×#126 #86111

Order 384.

1320 conjugacy classes of subgroups; only the 517 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 517 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
2D3×#126C2116yesmaximal, direct product factor
3C2×#898C213yesmaximal, complemented
4C2×D3×C4⋊C4C213yesmaximal, complemented
5C2×D3×C22⋊C4C213yesmaximal
6C2×#879C213yesmaximal, complemented
7C2×#876C211yesmaximal, characteristic, complemented
8C2×C42×D3C211yesmaximal, characteristic, complemented
9C22×#126—31nomaximal, Sylow
10C24×D3C2211yescharacteristic
11C22×C4×D3C2213yes—
12C22×C4×D3C2213yes—
13C3×#126C2218yescomplemented
14C2×C3×C4⋊C4C2213yescomplemented
15C2×C3×C22⋊C4C2213yes—
16C2×C3×C42C2211yescharacteristic, complemented
17C2×#217C2213yes—
18C2×(C2×D3)⋊C4C2213yes—
19C2×(C2×D3)⋊C4C2213yes—
20C2×C4⋊Dic3C2213yescomplemented
21C2×Dic3⋊C4C2213yescomplemented
22C2×Dic3⋊C4C2213yescomplemented
23C2×C4×Dic3C2213yescomplemented
24#898C22124yescomplemented
25D3×C4⋊C4C22124yescomplemented
26D3×C22⋊C4C22124yes—
27#879C22124yescomplemented
28#876C2218yescomplemented
29C42×D3C2218yescomplemented
30C2×#126—316no—
31C2×#126—33no—
32C2×#126—33no—
33C2×#126—31no—
34C2×#126D311yescharacteristic, direct product factor
35C22×C4⋊C4—33no—
36C22×C22⋊C4—33no—
37C22×C42—31no—
38C24×C3C2311yescharacteristic
39C23×D3—212no—
40C23×D3C2314yes—
41C23×D3C2311yescharacteristic
42C23×D3C2311yescharacteristic
43C22×C3×C4C2313yes—
44C22×C3×C4C2313yes—
45C22×Dic3C2313yes—
46C22×Dic3C2313yes—
47C2×C4×D3—212no—
48C2×C4×D3—212no—
49C2×C4×D3C23112yes—
50C2×C4×D3C23112yes—
51C3×C4⋊C4C23112yescomplemented
52C3×C22⋊C4C23112yes—
53C3×C42C2314yescomplemented
54#217C23112yes—
55(C2×D3)⋊C4C23112yes—
56(C2×D3)⋊C4C23112yes—
57C4⋊Dic3C23112yescomplemented
58Dic3⋊C4C23112yescomplemented
59Dic3⋊C4C23112yescomplemented
60C4×Dic3C23112yescomplemented
61C25—31no—
62C23×C4—33no—
63C23×C4—33no—
64#126—324no—
65#126—324no—
66#126—38no—
67#126C2×D318yesdirect product factor
68C2×C4⋊C4—33no—
69C2×C4⋊C4—33no—
70C2×C4⋊C4—33no—
71C2×C4⋊C4—324no—
72C2×C4⋊C4C2×D313yescomplemented
73C2×C22⋊C4—33no—
74C2×C22⋊C4—33no—
75C2×C22⋊C4—33no—
76C2×C22⋊C4—324no—
77C2×C22⋊C4C2×D313yes—
78C2×C42—33no—
79C2×C42—38no—
80C2×C42C2×D311yescharacteristic, complemented
81C23×C3—26no—
82C23×C3C2412yes—
83C23×C3C2411yescharacteristic
84C22×D3—26no—
85C22×D3—224no—
86C22×D3—416no—
87C22×D3#41112yes—
88C22×D3C2412yes—
89C22×D3C2412yes—
90C2×C3×C4—26no—
91C2×C3×C4—26no—
92C2×C3×C4C2416yes—
93C2×C3×C4C2416yes—
94C2×Dic3—26no—
95C2×Dic3—26no—
96C2×Dic3C2416yes—
97C2×Dic3C2416yes—
98C4×D3—224no—
99C4×D3—224no—
100C24—31no—
101C24—34no—
102C24—31no—
103C24—612no—
104C24C22×D311yescharacteristic
105C22×C4—33no—
106C22×C4—33no—
107C22×C4—312no—
108C22×C4—312no—
109C22×C4—612no—
110C22×C4—612no—
111C22×C4C22×D313yes—
112C22×C4C22×D313yes—
113C4⋊C4—312no—
114C4⋊C4—312no—
115C4⋊C4—312no—
116C4⋊C4C22×D3112yescomplemented
117C22⋊C4—312no—
118C22⋊C4—312no—
119C22⋊C4—312no—
120C22⋊C4C22×D3112yes—
121C42—312no—
122C42C22×D314yescomplemented
123C22×C3—26no—
124C22×C3—44no—
125C22×C3C2×#4116yes—
126C22×C3C2511yescharacteristic
127C2×D3—26no—
128C2×D3—44no—
129C2×D3—416no—
130C2×D3#126116yesdirect product factor
131C2×D3C2×#41112yes—
132C3×C4—26no—
133C3×C4—26no—
134Dic3—26no—
135Dic3—26no—
136C23—26no—
137C23—312no—
138C23—32no—
139C23—32no—
140C23—624no—
141C23—66no—
142C23—1216no—
143C23C23×D312yes—
144C23C23×D311yescharacteristic
145C2×C4—26no—
146C2×C4—26no—
147C2×C4—36no—
148C2×C4—36no—
149C2×C4—624no—
150C2×C4—624no—
151C2×C4—66no—
152C2×C4—66no—
153C2×C4C23×D316yes—
154C2×C4C23×D316yes—
155C2×C3—42no—
156C2×C3C2×#12614yescomplemented
157C2×C3C22×#4113yes—
158D3—42no—
159D3C2×#12618yesdirect product factor
160C22—26no—
161C22—316no—
162C22—312no—
163C22—44no—
164C22—66no—
165C22—1216no—
166C22—124no—
167C22D3×#4116yes—
168C22C24×D311yescharacteristic
169C4—26no—
170C4—26no—
171C4—66no—
172C4—66no—
173C3C22×#12611yescharacteristic, complemented, Sylow
174C2—38no—
175C2—42no—
176C2—122no—
177C2D3×#12614yesdirect product factor
178C2C2×D3×#4113yes—
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#126C2
socleC23×C3C24
prime radicalC24×D3C22
hypercenterC2×#126D3
hypocenterC3C22×#126
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. D3×#126C2direct (N × H)
3. C2×#898C2semidirect
4. C2×D3×C4⋊C4C2semidirect
6. C2×#879C2semidirect
7. C2×#876C2semidirect
8. C2×C42×D3C2semidirect
13. C3×#126C22semidirect
14. C2×C3×C4⋊C4C22semidirect
16. C2×C3×C42C22semidirect
20. C2×C4⋊Dic3C22semidirect
21. C2×Dic3⋊C4C22semidirect
22. C2×Dic3⋊C4C22semidirect
23. C2×C4×Dic3C22semidirect
24. #898C22semidirect
25. D3×C4⋊C4C22semidirect
27. #879C22semidirect
28. #876C22semidirect
29. C42×D3C22semidirect
34. C2×#126D3direct (N × H)
51. C3×C4⋊C4C23semidirect
53. C3×C42C23semidirect
57. C4⋊Dic3C23semidirect
58. Dic3⋊C4C23semidirect
59. Dic3⋊C4C23semidirect
60. C4×Dic3C23semidirect
67. #126C2×D3direct (N × H)
72. C2×C4⋊C4C2×D3semidirect
80. C2×C42C2×D3semidirect
116. C4⋊C4C22×D3semidirect
122. C42C22×D3semidirect
130. C2×D3#126direct (N × H)
156. C2×C3C2×#126semidirect
159. D3C2×#126direct (N × H)
173. C3C22×#126semidirect
177. C2D3×#126direct (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
D3×#1262
C2×#8982
C2×D3×C4⋊C42
C2×D3×C22⋊C42
C2×#8792
C2×#8762
C2×C42×D32
C22×#1263