Finite Group Databasebeta

← C2×C3×#480 #84949

Subgroups — C2×C3×#480 #84949

Order 384.

512 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 216 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×#121C211yesmaximal, characteristic
2C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#480C218yesmaximal, direct product factor
4C2×C3×#107C212yesmaximal, complemented
5C2×C3×#102C212yesmaximal, complemented
6C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
7C2×#480C311yesmaximal, characteristic, direct product factor, Sylow
8C22×C3×D4—21no—
9C22×C3×D4C2211yescharacteristic
10C23×C3×C4—21no—
11C23×C3×C4C2211yescharacteristic
12C22×C3×C8C2212yes—
13C3×#121—22no—
14C3×#121C2214yes—
15C2×C3×C4⋊C4—22no—
16C2×C3×C4⋊C4—21no—
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2214yescomplemented
20C2×C3×C22⋊C4—21no—
21C3×#107C2218yescomplemented
22C3×#102C2218yescomplemented
23C3×C22⋊C8C2214yescomplemented
24C2×#121C2×C311yescharacteristic
25C22×C4⋊C4C2×C311yescharacteristic, complemented
26#480C2×C318yesdirect product factor
27C2×#107C2×C312yescomplemented
28C2×#102C2×C312yescomplemented
29C2×C22⋊C8C2×C311yescharacteristic, complemented
30C24×C3—21no—
31C24×C3D411yescharacteristic
32C2×C3×D4—24no—
33C2×C3×D4—21no—
34C2×C3×D4—24no—
35C2×C3×D4—44no—
36C2×C3×D4C2312yes—
37C22×C3×C4—24no—
38C22×C3×C4—24no—
39C22×C3×C4—21no—
40C22×C3×C4—22no—
41C22×C3×C4—44no—
42C22×C3×C4D412yes—
43C22×C3×C4D411yescharacteristic
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C2×C3×C8—24no—
47C2×C3×C8C2314yes—
48C3×C4⋊C4—22no—
49C3×C4⋊C4—24no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4C2314yes—
52C3×C4⋊C4C2312yes—
53C3×C22⋊C4—24no—
54C22×D4—21no—
55C22×D4C22×C311yescharacteristic
56C23×C4—21no—
57C23×C4C22×C311yescharacteristic
58C22×C8C22×C312yes—
59#121—22no—
60#121C22×C314yes—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C22×C312yes—
64C2×C4⋊C4C22×C311yescharacteristic
65C2×C4⋊C4C22×C314yescomplemented
66C2×C22⋊C4—21no—
67#107C22×C318yescomplemented
68#102C22×C318yescomplemented
69C22⋊C8C22×C314yescomplemented
70C23×C3—22no—
71C23×C3—22no—
72C23×C3—22no—
73C23×C3—42no—
74C23×C3—42no—
75C23×C3—42no—
76C23×C3D814yes—
77C23×C3C2×D412yes—
78C23×C3C2×D411yescharacteristic
79C3×D4—24no—
80C3×D4—48no—
81C3×D4—42no—
82C2×C3×C4—22no—
83C2×C3×C4—24no—
84C2×C3×C4—24no—
85C2×C3×C4—22no—
86C2×C3×C4—24no—
87C2×C3×C4—48no—
88C2×C3×C4—44no—
89C2×C3×C4—42no—
90C2×C3×C4C2×D412yes—
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4#4114yes—
93C2×C3×C4C2411yescharacteristic
94C3×C8—24no—
95C24—21no—
96C24C3×D411yescharacteristic
97C2×D4—24no—
98C2×D4—21no—
99C2×D4—24no—
100C2×D4—44no—
101C2×D4C23×C312yes—
102C22×C4—24no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—44no—
107C22×C4C3×D411yescharacteristic
108C22×C4C3×D412yes—
109C22×C4C23×C312yes—
110C22×C4C23×C311yescharacteristic
111C2×C8—24no—
112C2×C8C23×C314yes—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—21no—
116C4⋊C4C23×C314yes—
117C4⋊C4C23×C312yes—
118C22⋊C4—24no—
119C22×C3—22no—
120C22×C3—22no—
121C22×C3—28no—
122C22×C3—42no—
123C22×C3—42no—
124C22×C3—42no—
125C22×C3—84no—
126C22×C3#12312yes—
127C22×C3C2×D814yes—
128C22×C3C2×D812yes—
129C22×C3#13712yes—
130C22×C3C22×D411yescharacteristic
131C3×C4—22no—
132C3×C4—44no—
133C3×C4—42no—
134C3×C4#12312yes—
135C3×C4C2×#4112yes—
136C23—22no—
137C23—22no—
138C23—22no—
139C23—42no—
140C23—42no—
141C23—42no—
142C23C3×D814yes—
143C23C2×C3×D412yes—
144C23C2×C3×D411yescharacteristic
145D4—24no—
146D4—48no—
147D4—42no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—48no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4C2×C3×D412yes—
157C2×C4C2×C3×D411yescharacteristic
158C2×C4C3×#4114yes—
159C2×C4C24×C311yescharacteristic
160C8—24no—
161C2×C3—24no—
162C2×C3—82no—
163C2×C3#48014yesdirect product factor
164C2×C3C2×#12311yescharacteristic
165C2×C3C22×D811yescharacteristic
166C2×C3C2×#13711yescharacteristic
167C22—22no—
168C22—22no—
169C22—28no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—84no—
174C22C3×#12312yes—
175C22C2×C3×D814yes—
176C22C2×C3×D812yes—
177C22C3×#13712yes—
178C22C22×C3×D411yescharacteristic
179C4—22no—
180C4—44no—
181C4—42no—
182C4C3×#12312yes—
183C4C2×C3×#4112yes—
184C3C2×#48011yescharacteristic, direct product factor, Sylow
185C2—24no—
186C2—82no—
187C2C3×#48014yesdirect product factor
188C2C2×C3×#12311yescharacteristic
189C2C22×C3×D811yescharacteristic
190C2C2×C3×#13711yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC2×C3×#121C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4⋊C4C2semidirect
3. C3×#480C2direct (N × H)
4. C2×C3×#107C2semidirect
5. C2×C3×#102C2semidirect
6. C2×C3×C22⋊C8C2semidirect
7. C2×#480C3direct (N × H)
19. C2×C3×C4⋊C4C22semidirect
21. C3×#107C22semidirect
22. C3×#102C22semidirect
23. C3×C22⋊C8C22semidirect
25. C22×C4⋊C4C2×C3semidirect
26. #480C2×C3direct (N × H)
27. C2×#107C2×C3semidirect
28. C2×#102C2×C3semidirect
29. C2×C22⋊C8C2×C3semidirect
65. C2×C4⋊C4C22×C3semidirect
67. #107C22×C3semidirect
68. #102C22×C3semidirect
69. C22⋊C8C22×C3semidirect
163. C2×C3#480direct (N × H)
184. C3C2×#480direct (N × H)
187. C2C3×#480direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C22×C3×C4⋊C42
C3×#4802
C2×C3×#1072
C2×C3×#1022
C2×C3×C22⋊C82
C2×#4803