Finite Group Databasebeta

← C3×#1472 #72187

Subgroups — C3×#1472 #72187

Order 384.

270 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 96 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×C22⋊Q8C211yesmaximal, characteristic
2C3×#407C211yesmaximal, characteristic
3C2×C3×C22⋊C8C211yesmaximal, characteristic
4C3×#324C214yesmaximal, complemented
5#1472C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×Q8C411yescharacteristic
7C23×C3×C4C2211yescharacteristic
8C2×C3×M16—21no—
9C22×C3×C8—21no—
10C3×C22⋊Q8—21no—
11C3×C22⋊Q8—21no—
12C3×C22⋊Q8—21no—
13C3×C22⋊Q8—21no—
14C2×C3×C4⋊C4C411yescharacteristic
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C22⋊C4C412yes—
17C3×C22⋊C8—21no—
18C3×C22⋊C8—21no—
19C3×C22⋊C8C2212yes—
20C3×C22⋊C8C2212yes—
21C2×C22⋊Q8C2×C311yescharacteristic
22#407C2×C311yescharacteristic
23C2×C22⋊C8C2×C311yescharacteristic
24#324C2×C314yescomplemented
25C24×C3C2×C411yescharacteristic
26C2×C3×Q8—21no—
27C2×C3×Q8—21no—
28C2×C3×Q8—42no—
29C22×C3×C4—42no—
30C22×C3×C4C2×C412yes—
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4C2×C412yes—
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4C2311yescharacteristic
38C3×M16—42no—
39C2×C3×C8—22no—
40C2×C3×C8—22no—
41C2×C3×C8—42no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4—22no—
45C3×C4⋊C4—22no—
46C3×C22⋊C4—22no—
47C3×C22⋊C4—22no—
48C22×Q8C3×C411yescharacteristic
49C23×C4C22×C311yescharacteristic
50C2×M16—21no—
51C22×C8—21no—
52C22⋊Q8—21no—
53C22⋊Q8—21no—
54C22⋊Q8—21no—
55C22⋊Q8—21no—
56C2×C4⋊C4C3×C411yescharacteristic
57C2×C4⋊C4C22×C312yes—
58C2×C22⋊C4C3×C412yes—
59C22⋊C8—21no—
60C22⋊C8—21no—
61C22⋊C8C22×C312yes—
62C22⋊C8C22×C312yes—
63C23×C3—22no—
64C23×C3—22no—
65C23×C3—41no—
66C23×C3C22⋊C411yescharacteristic
67C23×C3C22⋊C411yescharacteristic
68C23×C3C22×C411yescharacteristic
69C3×Q8—42no—
70C3×Q8—42no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—22no—
75C2×C3×C4—22no—
76C2×C3×C4—24no—
77C2×C3×C4—41no—
78C2×C3×C4—41no—
79C2×C3×C4—41no—
80C2×C3×C4—42no—
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×C3×C4C2×D411yescharacteristic
85C3×C8—42no—
86C3×C8—42no—
87C24C2×C3×C411yescharacteristic
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—42no—
91C22×C4—42no—
92C22×C4C2×C3×C412yes—
93C22×C4C2×C3×C411yescharacteristic
94C22×C4C2×C3×C412yes—
95C22×C4C3×D411yescharacteristic
96C22×C4C3×D411yescharacteristic
97C22×C4C3×D411yescharacteristic
98C22×C4C3×D411yescharacteristic
99C22×C4C23×C311yescharacteristic
100M16—42no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—42no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—22no—
110C22×C3—21no—
111C22×C3—21no—
112C22×C3—21no—
113C22×C3—21no—
114C22×C3—21no—
115C22×C3—21no—
116C22×C3—21no—
117C22×C3—44no—
118C22×C3#10112yes—
119C22×C3C2×C22⋊C411yescharacteristic
120C22×C3#13112yes—
121C3×C4—21no—
122C3×C4—21no—
123C3×C4—21no—
124C3×C4—42no—
125C3×C4—42no—
126C3×C4#9912yes—
127C23—22no—
128C23—22no—
129C23—41no—
130C23C3×C22⋊C411yescharacteristic
131C23C3×C22⋊C411yescharacteristic
132C23C22×C3×C411yescharacteristic
133Q8—42no—
134Q8—42no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4C3×C22⋊C411yescharacteristic
146C2×C4C3×C22⋊C411yescharacteristic
147C2×C4C2×C3×D411yescharacteristic
148C2×C4C2×C3×D411yescharacteristic
149C8—42no—
150C8—42no—
151C2×C3—22no—
152C2×C3—22no—
153C2×C3—41no—
154C2×C3#40811yescharacteristic
155C2×C3C2×#9911yescharacteristic
156C2×C3C2×#10111yescharacteristic
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—44no—
165C22C3×#10112yes—
166C22C2×C3×C22⋊C411yescharacteristic
167C22C3×#13112yes—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—42no—
172C4—42no—
173C4C3×#9912yes—
174C3#147211yescharacteristic, direct product factor, Sylow
175C2—22no—
176C2—22no—
177C2—41no—
178C2C3×#40811yescharacteristic
179C2C2×C3×#9911yescharacteristic
180C2C2×C3×#10111yescharacteristic
subgroupG/H
center, socleC22×C3C2×C22⋊C4
commutatorC23C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC24×C3C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#324C2semidirect
5. #1472C3direct (N × H)
24. #324C2×C3semidirect
174. C3#1472direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C3×#4072
C2×C3×C22⋊C82
C3×#3242
#14723