Finite Group Databasebeta

← C3×#1486 #72201

Subgroups — C3×#1486 #72201

Order 384.

312 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 116 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×C4×D4C211yesmaximal, characteristic
2C3×#431C211yesmaximal, characteristic, complemented
3C3×#423C211yesmaximal, characteristic, complemented
4C3×D4⋊C8C214yesmaximal, complemented
5#1486C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×D4C411yescharacteristic
7C23×C3×C4—21no—
8C2×C3×M16—21no—
9C3×C4×D4—21no—
10C3×C4×D4—21no—
11C3×C4×D4C412yes—
12C3×C4×D4C2212yes—
13C2×C3×C4⋊C4C411yescharacteristic
14C2×C3×C22⋊C4—21no—
15C2×C3×C42C2211yescharacteristic
16C3×C4⋊C8—21no—
17C3×C4⋊C8—21no—
18C3×C4⋊C8C2212yescomplemented
19C3×C22⋊C8—21no—
20C3×C4×C8C2212yescomplemented
21C2×C4×D4C2×C311yescharacteristic
22#431C2×C311yescharacteristic, complemented
23#423C2×C311yescharacteristic, complemented
24D4⋊C8C2×C314yescomplemented
25C24×C3—21no—
26C2×C3×D4—24no—
27C2×C3×D4—21no—
28C2×C3×D4C2×C412yes—
29C22×C3×C4—21no—
30C22×C3×C4—22no—
31C22×C3×C4—22no—
32C22×C3×C4—42no—
33C22×C3×C4—42no—
34C22×C3×C4C2×C411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C3×M16—42no—
38C2×C3×C8—22no—
39C2×C3×C8—22no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4C2×C412yes—
42C3×C22⋊C4—22no—
43C3×C22⋊C4—22no—
44C3×C42C2×C411yescharacteristic
45C3×C42D411yescharacteristic
46C3×C42D411yescharacteristic
47C3×C42C2311yescharacteristic
48C22×D4C3×C411yescharacteristic
49C23×C4—21no—
50C2×M16—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4C3×C412yes—
54C4×D4C22×C312yes—
55C2×C4⋊C4C3×C411yescharacteristic
56C2×C22⋊C4—21no—
57C2×C42C22×C311yescharacteristic
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8C22×C312yescomplemented
61C22⋊C8—21no—
62C4×C8C22×C312yescomplemented
63C23×C3—24no—
64C23×C3—22no—
65C23×C3—42no—
66C23×C3—42no—
67C23×C3C22⋊C411yescharacteristic
68C3×D4—24no—
69C3×D4—22no—
70C3×D4M1614yescomplemented
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—41no—
74C2×C3×C4—42no—
75C2×C3×C4—44no—
76C2×C3×C4—42no—
77C2×C3×C4—42no—
78C2×C3×C4C22⋊C411yescharacteristic
79C2×C3×C4C22⋊C411yescharacteristic
80C2×C3×C4C22⋊C411yescharacteristic
81C2×C3×C4D812yes—
82C2×C3×C4SD1612yes—
83C2×C3×C4C22×C411yescharacteristic
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86C3×C8—42no—
87C3×C8—42no—
88C24—21no—
89C2×D4—24no—
90C2×D4—21no—
91C2×D4C2×C3×C412yes—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4C2×C3×C411yescharacteristic
98C22×C4C3×D411yescharacteristic
99C22×C4C3×D411yescharacteristic
100M16—42no—
101C2×C8—22no—
102C2×C8—22no—
103C4⋊C4—21no—
104C4⋊C4C2×C3×C412yes—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C42C2×C3×C411yescharacteristic
108C42C3×D411yescharacteristic
109C42C3×D411yescharacteristic
110C42C23×C311yescharacteristic
111C22×C3—21no—
112C22×C3—21no—
113C22×C3—24no—
114C22×C3—48no—
115C22×C3—42no—
116C22×C3—42no—
117C22×C3#10212yes—
118C22×C3C2×C22⋊C411yescharacteristic
119C3×C4—21no—
120C3×C4—21no—
121C3×C4—21no—
122C3×C4—42no—
123C3×C4#10212yes—
124C3×C4C2×M1612yes—
125C3×C4C2×D811yescharacteristic
126C3×C4C2×SD1611yescharacteristic
127C23—24no—
128C23—22no—
129C23—42no—
130C23—42no—
131C23C3×C22⋊C411yescharacteristic
132D4—24no—
133D4—22no—
134D4C3×M1614yescomplemented
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4C3×C22⋊C411yescharacteristic
143C2×C4C3×C22⋊C411yescharacteristic
144C2×C4C3×C22⋊C411yescharacteristic
145C2×C4C3×D812yes—
146C2×C4C3×SD1612yes—
147C2×C4C22×C3×C411yescharacteristic
148C2×C4C2×C3×D411yescharacteristic
149C2×C4C2×C3×D411yescharacteristic
150C8—42no—
151C8—42no—
152C2×C3—22no—
153C2×C3—44no—
154C2×C3#40711yescharacteristic
155C2×C3C2×#10211yescharacteristic
156C2×C3#42111yescharacteristic
157C22—21no—
158C22—21no—
159C22—24no—
160C22—48no—
161C22—42no—
162C22—42no—
163C22C3×#10212yes—
164C22C2×C3×C22⋊C411yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—42no—
169C4C3×#10212yes—
170C4C2×C3×M1612yes—
171C4C2×C3×D811yescharacteristic
172C4C2×C3×SD1611yescharacteristic
173C3#148611yescharacteristic, direct product factor, Sylow
174C2—22no—
175C2—44no—
176C2C3×#40711yescharacteristic
177C2C2×C3×#10211yescharacteristic
178C2C3×#42111yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC42C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC22×C3×D4C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#431C2semidirect
3. C3×#423C2semidirect
4. C3×D4⋊C8C2semidirect
5. #1486C3direct (N × H)
18. C3×C4⋊C8C22semidirect
20. C3×C4×C8C22semidirect
22. #431C2×C3semidirect
23. #423C2×C3semidirect
24. D4⋊C8C2×C3semidirect
60. C4⋊C8C22×C3semidirect
62. C4×C8C22×C3semidirect
70. C3×D4M16semidirect
134. D4C3×M16semidirect
173. C3#1486direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C3×#4312
C3×#4232
C3×D4⋊C82
#14863