Finite Group Databasebeta

← C3×#1471 #72186

Subgroups — C3×#1471 #72186

Order 384.

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

normal subgroups only

This group has 322 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×#121C211yesmaximal, characteristic
2C3×#407C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
4C3×#323C214yesmaximal, complemented
5#1471C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×D4C411yescharacteristic
7C22×C3×D4C412yes—
8C23×C3×C4C2211yescharacteristic
9C2×C3×M16—21no—
10C22×C3×C8—21no—
11C3×#121—21no—
12C3×#121—21no—
13C3×#121—21no—
14C3×#121—21no—
15C2×C3×C4⋊C4C411yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C3×C22⋊C8—21no—
18C3×C22⋊C8—21no—
19C3×C22⋊C8C2212yes—
20C3×C22⋊C8C2212yes—
21C2×#121C2×C311yescharacteristic
22#407C2×C311yescharacteristic, complemented
23C2×C22⋊C8C2×C311yescharacteristic, complemented
24#323C2×C314yescomplemented
25C24×C3C2×C411yescharacteristic
26C24×C3C2×C412yes—
27C2×C3×D4—22no—
28C2×C3×D4—22no—
29C2×C3×D4—21no—
30C2×C3×D4—21no—
31C2×C3×D4—42no—
32C2×C3×D4—44no—
33C22×C3×C4—42no—
34C22×C3×C4C2×C411yescharacteristic
35C22×C3×C4C2×C412yes—
36C22×C3×C4D411yescharacteristic
37C22×C3×C4D411yescharacteristic
38C22×C3×C4D411yescharacteristic
39C22×C3×C4D411yescharacteristic
40C22×C3×C4C2311yescharacteristic
41C3×M16—42no—
42C2×C3×C8—22no—
43C2×C3×C8—22no—
44C2×C3×C8—42no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—21no—
47C3×C22⋊C4—22no—
48C3×C22⋊C4—22no—
49C22×D4C3×C411yescharacteristic
50C22×D4C3×C412yes—
51C23×C4C22×C311yescharacteristic
52C2×M16—21no—
53C22×C8—21no—
54#121—21no—
55#121—21no—
56#121—21no—
57#121—21no—
58C2×C4⋊C4C3×C411yescharacteristic
59C2×C22⋊C4C22×C312yes—
60C22⋊C8—21no—
61C22⋊C8—21no—
62C22⋊C8C22×C312yes—
63C22⋊C8C22×C312yes—
64C23×C3—22no—
65C23×C3—22no—
66C23×C3—22no—
67C23×C3—24no—
68C23×C3—41no—
69C23×C3—42no—
70C23×C3—42no—
71C23×C3C22⋊C411yescharacteristic
72C23×C3C22⋊C411yescharacteristic
73C23×C3C22×C411yescharacteristic
74C3×D4—42no—
75C3×D4—44no—
76C3×D4—44no—
77C3×D4—42no—
78C2×C3×C4—22no—
79C2×C3×C4—22no—
80C2×C3×C4—22no—
81C2×C3×C4—22no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—42no—
86C2×C3×C4C22⋊C411yescharacteristic
87C2×C3×C4C22⋊C411yescharacteristic
88C2×C3×C4C2×D411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C3×C8—42no—
91C3×C8—42no—
92C24C2×C3×C411yescharacteristic
93C24C2×C3×C412yes—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—42no—
99C2×D4—44no—
100C22×C4—42no—
101C22×C4C2×C3×C411yescharacteristic
102C22×C4C2×C3×C412yes—
103C22×C4C3×D411yescharacteristic
104C22×C4C3×D411yescharacteristic
105C22×C4C3×D411yescharacteristic
106C22×C4C3×D411yescharacteristic
107C22×C4C23×C311yescharacteristic
108M16—42no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—42no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C22⋊C4—22no—
115C22⋊C4—22no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—21no—
119C22×C3—21no—
120C22×C3—21no—
121C22×C3—21no—
122C22×C3—21no—
123C22×C3—44no—
124C22×C3—42no—
125C22×C3—42no—
126C22×C3—42no—
127C22×C3—84no—
128C22×C3#10012yes—
129C22×C3C2×C22⋊C411yescharacteristic
130C22×C3#13112yes—
131C3×C4—21no—
132C3×C4—21no—
133C3×C4—21no—
134C3×C4—42no—
135C3×C4#9912yes—
136C23—22no—
137C23—22no—
138C23—22no—
139C23—24no—
140C23—41no—
141C23—42no—
142C23—42no—
143C23C3×C22⋊C411yescharacteristic
144C23C3×C22⋊C411yescharacteristic
145C23C22×C3×C411yescharacteristic
146D4—42no—
147D4—44no—
148D4—44no—
149D4—42no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4C3×C22⋊C411yescharacteristic
159C2×C4C3×C22⋊C411yescharacteristic
160C2×C4C2×C3×D411yescharacteristic
161C2×C4C2×C3×D411yescharacteristic
162C8—42no—
163C8—42no—
164C2×C3—22no—
165C2×C3—22no—
166C2×C3—41no—
167C2×C3—82no—
168C2×C3#40811yescharacteristic
169C2×C3C2×#9911yescharacteristic
170C2×C3C2×#10011yescharacteristic
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—44no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—84no—
183C22C3×#10012yes—
184C22C2×C3×C22⋊C411yescharacteristic
185C22C3×#13112yes—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4C3×#9912yes—
191C3#147111yescharacteristic, direct product factor, Sylow
192C2—22no—
193C2—22no—
194C2—41no—
195C2—82no—
196C2C3×#40811yescharacteristic
197C2C2×C3×#9911yescharacteristic
198C2C2×C3×#10011yescharacteristic
subgroupG/H
center, socleC22×C3C2×C22⋊C4
commutatorC23C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×#121C2
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
2. C3×#407C2semidirect
3. C2×C3×C22⋊C8C2semidirect
4. C3×#323C2semidirect
5. #1471C3direct (N × H)
22. #407C2×C3semidirect
23. C2×C22⋊C8C2×C3semidirect
24. #323C2×C3semidirect
191. C3#1471direct (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
C3×#4072
C2×C3×C22⋊C82
C3×#3232
#14713