Finite Group Databasebeta

← C2×C3×#6648 #13086488

Subgroups — C2×C3×#6648 #13086488

Order 1440.

326 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 136 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×D15C211yesmaximal, characteristic
2C2×C3×C5×C3⋊C8C211yesmaximal, characteristic, complemented
3C2×C32×#168C211yesmaximal, characteristic, complemented
4C3×#6648C214yesmaximal, direct product factor
5C2×C3×C8×D5—31nomaximal
6C2×#6648C311yesmaximal, characteristic, direct product factor
7C22×C3×D15C411yescharacteristic
8C2×C32×C4×C5C2211yescharacteristic
9C2×C3×Dic15C411yescharacteristic
10C3×C4×D15C412yes—
11C3×C4×D15C2212yes—
12C3×C5×C3⋊C8C2212yescomplemented
13C32×#168C2212yescomplemented
14C2×C3×D3×C8—51nomaximal, Hall
15C2×C4×D15C2×C311yescharacteristic
16C2×C3×C4×D5—31no—
17C2×C3×C5×C8—31no—
18C2×C5×C3⋊C8C2×C311yescharacteristic, complemented
19C2×C3×#168—21no—
20C2×C3×#168D311yescharacteristic, complemented
21C2×C3×#168C2×C311yescharacteristic, complemented
22C3×C8×D5—34no—
23#6648C2×C314yesdirect product factor
24C22×C32×C5C2×C411yescharacteristic
25C2×C3×D15C814yescomplemented
26C2×C3×D15C2×C412yes—
27C32×C4×C5C2×C411yescharacteristic
28C32×C4×C5C2311yescharacteristic
29C3×Dic15C2×C412yes—
30C2×C8×D5—31noHall
31C2×C3×C4×D3—51no—
32C2×C32×C8—51no—
33C2×C3×C3⋊C8D511yescharacteristic, complemented
34C3×D3×C8—54no—
35C22×D15C3×C411yescharacteristic
36C22×C3×D5—31no—
37C2×C3×C4×C5—21no—
38C2×C3×C4×C5C2×D311yescharacteristic
39C2×C3×C4×C5C22×C311yescharacteristic
40C2×Dic15C3×C411yescharacteristic
41C4×D15C3×C412yes—
42C4×D15C22×C312yes—
43C2×C3×Dic5—31no—
44C3×C4×D5—32no—
45C3×C4×D5—32no—
46C3×C5×C8—32no—
47C3×#168—22no—
48C3×#168C2×D312yescomplemented
49C3×#168C22×C312yescomplemented
50C5×C3⋊C8C22×C312yescomplemented
51C22×C3×C8—151no—
52C2×D3×C8—51no—
53C2×C32×C5C2×C812yescomplemented
54C2×C32×C5C22×C411yescharacteristic
55C3×D15C2×C814yescomplemented
56C2×C4×D5—31no—
57C2×C5×C8—31no—
58C2×#168C3×D311yescharacteristic, complemented
59C8×D5—34no—
60C22×C3×D3—51no—
61C2×C32×C4C2×D511yescharacteristic
62C2×C3×Dic3—51no—
63C3×C4×D3—52no—
64C3×C4×D3—52no—
65C32×C8—52no—
66C3×C3⋊C8C2×D512yescomplemented
67C22×C3×C5—21no—
68C22×C3×C5C4×D311yescharacteristic
69C22×C3×C5C2×C3×C411yescharacteristic
70C2×D15C3×C814yescomplemented
71C2×D15C2×C3×C412yes—
72C2×C3×D5—34no—
73C2×C3×D5—32no—
74C3×C4×C5—21no—
75C3×C4×C5—21no—
76C3×C4×C5C4×D311yescharacteristic
77C3×C4×C5C2×C3×C411yescharacteristic
78C3×C4×C5C22×D311yescharacteristic
79C3×C4×C5C23×C311yescharacteristic
80Dic15C2×C3×C412yes—
81C3×Dic5—32no—
82C22×C3×C4—151no—
83C2×C4×D3—51no—
84C2×C3×C8—31no—
85C2×C3×C8—51no—
86C2×C3×C8—51no—
87C2×C3×C8—101no—
88C2×C3×C8—154no—
89C2×C3⋊C8C3×D511yescharacteristic, complemented
90D3×C8—54no—
91C32×C5C22×C811yescharacteristic, complemented, Hall
92C22×D5—31no—
93C2×C4×C5C2×C3×D311yescharacteristic
94C2×Dic5—31no—
95C4×D5—32no—
96C4×D5—32no—
97C5×C8—32no—
98#168C2×C3×D312yescomplemented
99C22×C32C4×D511yescharacteristic
100C2×C3×D3—54no—
101C2×C3×D3—52no—
102C32×C4C4×D511yescharacteristic
103C32×C4C22×D511yescharacteristic
104C3×Dic3—52no—
105C22×C8—151noSylow
106C2×C3×C5—22no—
107C2×C3×C5—21no—
108C2×C3×C5D3×C812yescomplemented
109C2×C3×C5C2×C3×C812yescomplemented
110C2×C3×C5C2×C4×D311yescharacteristic
111C2×C3×C5C22×C3×C411yescharacteristic
112D15C2×C3×C814yescomplemented
113C3×D5—34no—
114C23×C3—151no—
115C22×D3—51no—
116C2×C3×C4—21no—
117C2×C3×C4—151no—
118C2×C3×C4—152no—
119C2×C3×C4—152no—
120C2×C3×C4D3×D511yescharacteristic
121C2×C3×C4C2×C3×D511yescharacteristic
122C2×Dic3—51no—
123C4×D3—52no—
124C4×D3—52no—
125C3×C8—32no—
126C3×C8—52no—
127C3×C8—52no—
128C3×C8—102no—
129C3⋊C8C2×C3×D512yescomplemented
130C22×C5C3×C4×D311yescharacteristic
131C2×D5—34no—
132C2×D5—32no—
133C4×C5C3×C4×D311yescharacteristic
134C4×C5C22×C3×D311yescharacteristic
135Dic5—32no—
136C2×C32C8×D512yescomplemented
137C2×C32C2×C4×D511yescharacteristic
138C3×D3—54no—
139C22×C4—151no—
140C2×C8—31no—
141C2×C8—51no—
142C2×C8—154no—
143C3×C5—21no—
144C3×C5C2×D3×C811yescharacteristic, complemented
145C3×C5C22×C3×C811yescharacteristic, complemented
146C22×C3—21no—
147C22×C3—152no—
148C22×C3—154no—
149C22×C3#120011yescharacteristic
150C22×C3C3×C4×D511yescharacteristic
151C2×D3—54no—
152C2×D3—52no—
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—152no—
156C3×C4#120011yescharacteristic
157C3×C4C3×C4×D511yescharacteristic
158C3×C4C2×D3×D511yescharacteristic
159C3×C4C22×C3×D511yescharacteristic
160Dic3—52no—
161C2×C5C3×D3×C812yescomplemented
162C2×C5C2×C3×C4×D311yescharacteristic
163D5—34no—
164C32C2×C8×D511yescharacteristic, complemented, Sylow
165C23—151no—
166C2×C4—151no—
167C2×C4—152no—
168C2×C4—152no—
169C2×C4C3×D3×D511yescharacteristic
170C8—32no—
171C8—52no—
172C2×C3—21no—
173C2×C3—22no—
174C2×C3—154no—
175C2×C3#664812yesdirect product factor
176C2×C3C3×C8×D512yescomplemented
177C2×C3C2×#120011yescharacteristic
178C2×C3C2×C3×C4×D511yescharacteristic
179D3—54no—
180C5C2×C3×D3×C811yescharacteristic, complemented, Sylow
181C22—152no—
182C22—154no—
183C22C3×#120011yescharacteristic
184C4—152no—
185C4C3×#120011yescharacteristic
186C4C2×C3×D3×D511yescharacteristic
187C3—21no—
188C3C2×#664811yescharacteristic, direct product factor
189C3C2×C3×C8×D511yescharacteristic, complemented
190C2—154no—
191C2C3×#664812yesdirect product factor
192C2C2×C3×#120011yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×D5
commutator, hypocenterC3×C5C22×C3×C8
FrattiniC4C2×C3×D3×D5
FittingC2×C32×C4×C5C22
socleC22×C32×C5C2×C4
prime radicalC22×C3×D15C4
cosocleC3×C4×C5C23×C3
LCSG ▷ C3×C5
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×C3⋊C8C2semidirect
3. C2×C32×#168C2semidirect
4. C3×#6648C2direct (N × H)
6. C2×#6648C3direct (N × H)
12. C3×C5×C3⋊C8C22semidirect
13. C32×#168C22semidirect
18. C2×C5×C3⋊C8C2×C3semidirect
20. C2×C3×#168D3semidirect
21. C2×C3×#168C2×C3semidirect
23. #6648C2×C3direct (N × H)
25. C2×C3×D15C8semidirect
33. C2×C3×C3⋊C8D5semidirect
48. C3×#168C2×D3semidirect
49. C3×#168C22×C3semidirect
50. C5×C3⋊C8C22×C3semidirect
53. C2×C32×C5C2×C8semidirect
55. C3×D15C2×C8semidirect
58. C2×#168C3×D3semidirect
66. C3×C3⋊C8C2×D5semidirect
70. C2×D15C3×C8semidirect
89. C2×C3⋊C8C3×D5semidirect
91. C32×C5C22×C8semidirect
98. #168C2×C3×D3semidirect
108. C2×C3×C5D3×C8semidirect
109. C2×C3×C5C2×C3×C8semidirect
112. D15C2×C3×C8semidirect
129. C3⋊C8C2×C3×D5semidirect
136. C2×C32C8×D5semidirect
144. C3×C5C2×D3×C8semidirect
145. C3×C5C22×C3×C8semidirect
161. C2×C5C3×D3×C8semidirect
164. C32C2×C8×D5semidirect
175. C2×C3#6648direct (N × H)
176. C2×C3C3×C8×D5semidirect
180. C5C2×C3×D3×C8semidirect
188. C3C2×#6648direct (N × H)
189. C3C2×C3×C8×D5semidirect
191. C2C3×#6648direct (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×D152
C2×C3×C5×C3⋊C82
C2×C32×#1682
C3×#66482
C2×C3×C8×D53
C2×#66483
C2×C3×D3×C85