Finite Group Databasebeta

← C3×#1459 #72174

Subgroups — C3×#1459 #72174

Order 384.

332 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 124 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
2C2×C3×C22⋊C8C212yesmaximal, complemented
3C3×#324C212yesmaximal, complemented
4C3×#323C212yesmaximal, complemented
5#1459C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×D4C411yescharacteristic
7C23×C3×C4C411yescharacteristic
8C23×C3×C4C2211yescharacteristic
9C22×C3×C8—22no—
10C3×C4×D4—22no—
11C3×C4×D4—22no—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C22⋊C4C411yescharacteristic
14C2×C3×C22⋊C4C2211yescharacteristic
15C2×C3×C42C411yescharacteristic
16C3×C22⋊C8—22no—
17C3×C22⋊C8C2214yes—
18C2×C4×D4C2×C311yescharacteristic
19C2×C22⋊C8C2×C312yescomplemented
20#324C2×C312yescomplemented
21#323C2×C312yescomplemented
22C24×C3C2×C411yescharacteristic
23C24×C3C2×C411yescharacteristic
24C2×C3×D4—22no—
25C2×C3×D4—22no—
26C2×C3×D4C814yescomplemented
27C22×C3×C4—21no—
28C22×C3×C4—21no—
29C22×C3×C4—42no—
30C22×C3×C4—42no—
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4C2×C411yescharacteristic
34C22×C3×C4C2×C411yescharacteristic
35C22×C3×C4D412yes—
36C22×C3×C4D412yes—
37C22×C3×C4C2311yescharacteristic
38C2×C3×C8—24no—
39C2×C3×C8—44no—
40C3×C4⋊C4—22no—
41C3×C22⋊C4—22no—
42C3×C22⋊C4—22no—
43C3×C42—21no—
44C3×C42—21no—
45C22×D4C3×C411yescharacteristic
46C23×C4C3×C411yescharacteristic
47C23×C4C22×C311yescharacteristic
48C22×C8—22no—
49C4×D4—22no—
50C4×D4—22no—
51C2×C4⋊C4C22×C311yescharacteristic
52C2×C22⋊C4C3×C411yescharacteristic
53C2×C22⋊C4C22×C311yescharacteristic
54C2×C42C3×C411yescharacteristic
55C22⋊C8—22no—
56C22⋊C8C22×C314yes—
57C23×C3—21no—
58C23×C3—21no—
59C23×C3—41no—
60C23×C3—41no—
61C23×C3—41no—
62C23×C3—41no—
63C23×C3C22⋊C412yes—
64C23×C3C2×C812yes—
65C23×C3C2×C812yescomplemented
66C23×C3M1612yes—
67C23×C3C22×C411yescharacteristic
68C3×D4—28no—
69C2×C3×C4—21no—
70C2×C3×C4—21no—
71C2×C3×C4—24no—
72C2×C3×C4—22no—
73C2×C3×C4—21no—
74C2×C3×C4—21no—
75C2×C3×C4—42no—
76C2×C3×C4—41no—
77C2×C3×C4—42no—
78C2×C3×C4—42no—
79C2×C3×C4—41no—
80C2×C3×C4—41no—
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4C2×C812yescomplemented
84C2×C3×C4C2×D412yes—
85C3×C8—44no—
86C24C2×C3×C411yescharacteristic
87C24C2×C3×C411yescharacteristic
88C2×D4—22no—
89C2×D4—22no—
90C2×D4C3×C814yescomplemented
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4C2×C3×C411yescharacteristic
96C22×C4C2×C3×C411yescharacteristic
97C22×C4C2×C3×C411yescharacteristic
98C22×C4C2×C3×C411yescharacteristic
99C22×C4C3×D412yes—
100C22×C4C3×D412yes—
101C22×C4C23×C311yescharacteristic
102C2×C8—24no—
103C2×C8—44no—
104C4⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C42—21no—
108C42—21no—
109C22×C3—21no—
110C22×C3—22no—
111C22×C3—21no—
112C22×C3—22no—
113C22×C3—22no—
114C22×C3—42no—
115C22×C3—42no—
116C22×C3—42no—
117C22×C3—42no—
118C22×C3—41no—
119C22×C3—41no—
120C22×C3C22⋊C814yes—
121C22×C3C2×C22⋊C411yescharacteristic
122C22×C3C22×C811yescharacteristic
123C22×C3C2×M1611yescharacteristic
124C3×C4—22no—
125C3×C4—22no—
126C3×C4—21no—
127C3×C4—41no—
128C3×C4—41no—
129C3×C4#9912yes—
130C23—21no—
131C23—21no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23C3×C22⋊C412yes—
137C23C2×C3×C812yes—
138C23C2×C3×C812yescomplemented
139C23C3×M1612yes—
140C23C22×C3×C411yescharacteristic
141D4—28no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4C3×C22⋊C411yescharacteristic
155C2×C4C3×C22⋊C411yescharacteristic
156C2×C4C2×C3×C812yescomplemented
157C2×C4C2×C3×D412yes—
158C8—44no—
159C2×C3—21no—
160C2×C3—21no—
161C2×C3—24no—
162C2×C3—42no—
163C2×C3C2×C22⋊C811yescharacteristic
164C2×C3C2×#9911yescharacteristic
165C2×C3#41311yescharacteristic
166C22—21no—
167C22—22no—
168C22—21no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22C3×C22⋊C814yes—
178C22C2×C3×C22⋊C411yescharacteristic
179C22C22×C3×C811yescharacteristic
180C22C2×C3×M1611yescharacteristic
181C4—22no—
182C4—22no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4C3×#9912yes—
187C3#145911yescharacteristic, direct product factor, Sylow
188C2—21no—
189C2—21no—
190C2—24no—
191C2—42no—
192C2C2×C3×C22⋊C811yescharacteristic
193C2C2×C3×#9911yescharacteristic
194C2C3×#41311yescharacteristic
subgroupG/H
centerC2×C3×C4C22⋊C4
commutatorC22C22×C3×C8
Frattini, cosocleC22×C4C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC22×C3×D4C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊C8C2semidirect
3. C3×#324C2semidirect
4. C3×#323C2semidirect
5. #1459C3direct (N × H)
19. C2×C22⋊C8C2×C3semidirect
20. #324C2×C3semidirect
21. #323C2×C3semidirect
26. C2×C3×D4C8semidirect
65. C23×C3C2×C8semidirect
83. C2×C3×C4C2×C8semidirect
90. C2×D4C3×C8semidirect
138. C23C2×C3×C8semidirect
156. C2×C4C2×C3×C8semidirect
187. C3#1459direct (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
C2×C3×C22⋊C82
C3×#3242
C3×#3232
#14593