Finite Group Databasebeta

← C2×C3×#509 #85276

Subgroups — C2×C3×#509 #85276

Order 384.

400 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 184 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×#135C211yesmaximal, characteristic, complemented
2C22×C3×D8C211yesmaximal, characteristic, complemented
3C3×#509C218yesmaximal, direct product factor
4C2×C3×SD32C212yesmaximal, complemented
5C2×C3×D16C212yesmaximal, complemented
6C2×C3×M32C211yesmaximal, characteristic, complemented
7C2×#509C311yesmaximal, characteristic, direct product factor, Sylow
8C2×C3×#41—21no—
9C22×C3×D4—21no—
10C3×#135—22no—
11C3×#135C2214yescomplemented
12C2×C3×Q16C2211yescharacteristic, complemented
13C2×C3×SD16—21no—
14C2×C3×D8—21no—
15C2×C3×D8—22no—
16C2×C3×D8C2212yes—
17C2×C3×D8C2214yescomplemented
18C2×C3×D8C2211yescharacteristic, complemented
19C22×C3×C8C2211yescharacteristic
20C3×SD32C2218yescomplemented
21C3×D16C2218yescomplemented
22C3×M32C2214yescomplemented
23C2×C3×C16C2212yescomplemented
24C2×#135C2×C311yescharacteristic, complemented
25C22×D8C2×C311yescharacteristic, complemented
26#509C2×C318yesdirect product factor
27C2×SD32C2×C312yescomplemented
28C2×D16C2×C312yescomplemented
29C2×M32C2×C311yescharacteristic, complemented
30C24×C3—41no—
31C3×#41—24no—
32C3×#41—42no—
33C2×C3×Q8—21no—
34C2×C3×D4—22no—
35C2×C3×D4—24no—
36C2×C3×D4—21no—
37C2×C3×D4—41no—
38C2×C3×D4—42no—
39C2×C3×D4—41no—
40C22×C3×C4—41no—
41C22×C3×C4D411yescharacteristic
42C3×Q16—21no—
43C3×Q16C2312yescomplemented
44C3×SD16—24no—
45C3×D8—22no—
46C3×D8—24no—
47C3×D8—21no—
48C3×D8C2314yes—
49C3×D8C2312yescomplemented
50C2×C3×C8D411yescharacteristic
51C2×C3×C8D412yes—
52C2×C3×C8C2311yescharacteristic
53C2×C3×C8C2312yes—
54C3×C16C2314yescomplemented
55C2×#41—21no—
56C22×D4—21no—
57#135—22no—
58#135C22×C314yescomplemented
59C2×Q16C22×C311yescharacteristic, complemented
60C2×SD16—21no—
61C2×D8—21no—
62C2×D8—22no—
63C2×D8C22×C312yes—
64C2×D8C22×C314yescomplemented
65C2×D8C22×C311yescharacteristic, complemented
66C22×C8C22×C311yescharacteristic
67SD32C22×C318yescomplemented
68D16C22×C318yescomplemented
69M32C22×C314yescomplemented
70C2×C16C22×C312yescomplemented
71C23×C3—44no—
72C23×C3—41no—
73C23×C3—42no—
74C23×C3—84no—
75C23×C3D811yescharacteristic
76C3×Q8—22no—
77C3×Q8—41no—
78C3×D4—24no—
79C3×D4—22no—
80C3×D4—44no—
81C3×D4—44no—
82C3×D4—42no—
83C3×D4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—44no—
86C2×C3×C4D812yes—
87C2×C3×C4D811yescharacteristic
88C2×C3×C4C2×D412yes—
89C2×C3×C4C2×D411yescharacteristic
90C3×C8C2×D411yescharacteristic
91C3×C8C2×D412yes—
92C3×C8C2411yescharacteristic
93C24—41no—
94#41—24no—
95#41—42no—
96C2×Q8—21no—
97C2×D4—22no—
98C2×D4—24no—
99C2×D4—21no—
100C2×D4—41no—
101C2×D4—41no—
102C2×D4—42no—
103C22×C4—41no—
104C22×C4C3×D411yescharacteristic
105Q16—21no—
106Q16C23×C312yescomplemented
107SD16—24no—
108D8—22no—
109D8—24no—
110D8—21no—
111D8C23×C314yes—
112D8C23×C312yescomplemented
113C2×C8C3×D411yescharacteristic
114C2×C8C3×D412yes—
115C2×C8C23×C311yescharacteristic
116C2×C8C23×C312yes—
117C16C23×C314yescomplemented
118C22×C3—22no—
119C22×C3—42no—
120C22×C3—44no—
121C22×C3—88no—
122C22×C3—82no—
123C22×C3—84no—
124C22×C3C2×D812yes—
125C22×C3C2×D811yescharacteristic
126C3×C4—42no—
127C3×C4C2×D812yes—
128C3×C4C2×D811yescharacteristic
129C3×C4C22×D411yescharacteristic
130C23—41no—
131C23—44no—
132C23—42no—
133C23—84no—
134C23C3×D811yescharacteristic
135Q8—22no—
136Q8—41no—
137D4—24no—
138D4—22no—
139D4—44no—
140D4—44no—
141D4—42no—
142D4—41no—
143C2×C4—41no—
144C2×C4—44no—
145C2×C4C3×D811yescharacteristic
146C2×C4C3×D812yes—
147C2×C4C2×C3×D411yescharacteristic
148C2×C4C2×C3×D412yes—
149C8C2×C3×D411yescharacteristic
150C8C2×C3×D412yes—
151C8C24×C311yescharacteristic
152C2×C3—22no—
153C2×C3—82no—
154C2×C3—84no—
155C2×C3#50912yesdirect product factor
156C2×C3C22×D811yescharacteristic
157C22—22no—
158C22—42no—
159C22—44no—
160C22—88no—
161C22—84no—
162C22—82no—
163C22C2×C3×D812yes—
164C22C2×C3×D811yescharacteristic
165C4—42no—
166C4C2×C3×D812yes—
167C4C2×C3×D811yescharacteristic
168C4C22×C3×D411yescharacteristic
169C3C2×#50911yescharacteristic, direct product factor, Sylow
170C2—22no—
171C2—84no—
172C2—82no—
173C2C3×#50912yesdirect product factor
174C2C22×C3×D811yescharacteristic
subgroupG/H
center, socleC22×C3C2×D8
commutator, Frattini, cosocleC8C24×C3
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#135C2semidirect
2. C22×C3×D8C2semidirect
3. C3×#509C2direct (N × H)
4. C2×C3×SD32C2semidirect
5. C2×C3×D16C2semidirect
6. C2×C3×M32C2semidirect
7. C2×#509C3direct (N × H)
11. C3×#135C22semidirect
12. C2×C3×Q16C22semidirect
17. C2×C3×D8C22semidirect
18. C2×C3×D8C22semidirect
20. C3×SD32C22semidirect
21. C3×D16C22semidirect
22. C3×M32C22semidirect
23. C2×C3×C16C22semidirect
24. C2×#135C2×C3semidirect
25. C22×D8C2×C3semidirect
26. #509C2×C3direct (N × H)
27. C2×SD32C2×C3semidirect
28. C2×D16C2×C3semidirect
29. C2×M32C2×C3semidirect
43. C3×Q16C23semidirect
49. C3×D8C23semidirect
54. C3×C16C23semidirect
58. #135C22×C3semidirect
59. C2×Q16C22×C3semidirect
64. C2×D8C22×C3semidirect
65. C2×D8C22×C3semidirect
67. SD32C22×C3semidirect
68. D16C22×C3semidirect
69. M32C22×C3semidirect
70. C2×C16C22×C3semidirect
106. Q16C23×C3semidirect
112. D8C23×C3semidirect
117. C16C23×C3semidirect
155. C2×C3#509direct (N × H)
169. C3C2×#509direct (N × H)
173. C2C3×#509direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1352
C22×C3×D82
C3×#5092
C2×C3×SD322
C2×C3×D162
C2×C3×M322
C2×#5093