Finite Group Databasebeta

← C2×C3×#157⋊D3 #13049853

Subgroups — C2×C3×#157⋊D3 #13049853

Order 1296.

494 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 62 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×#1113C212yesmaximal, complemented
2C2×C32×#157C211yesmaximal, characteristic, complemented
3C3×#157⋊D3C214yesmaximal, direct product factor
4C2×#157⋊D3C311yesmaximal, characteristic, direct product factor
5C2×C3×GO+(4,𝔽2)—31nomaximal
6C2×C32×C3⋊D3—22no—
7C2×C32×C3⋊D3C2211yescharacteristic
8C3×#1113—22no—
9C3×#1113C2214yescomplemented
10C32×#157C2212yescomplemented
11C2×#1113C2×C312yescomplemented
12C2×C3×D32—32no—
13C2×C3×D32—62no—
14C2×C3×#157—21no—
15C2×C3×#157D311yescharacteristic, complemented
16C2×C3×#157C2×C311yescharacteristic, complemented
17#157⋊D3C2×C314yesdirect product factor
18C3×GO+(4,𝔽2)—34no—
19C2×C34D411yescharacteristic, complemented
20C32×C3⋊D3—24no—
21C32×C3⋊D3D411yescharacteristic, complemented
22C32×C3⋊D3C2311yescharacteristic
23C2×GO+(4,𝔽2)—31no—
24C2×C3×D12—91nomaximal
25C2×C3×C3⋊D3—22no—
26C2×C3×C3⋊D3—21no—
27C2×C3×C3⋊D3—22no—
28C2×C3×C3⋊D3—42no—
29C2×C3×C3⋊D3C2×D311yescharacteristic
30C2×C3×C3⋊D3C22×C311yescharacteristic
31C2×C32×D3—62no—
32C2×C32×D3—62no—
33C2×C32×D3—62no—
34C2×C32×D3—122no—
35#1113—22no—
36#1113C22×C314yescomplemented
37C3×D32—34no—
38C3×D32—64no—
39C3×D32—64no—
40C3×D32—62no—
41C3×#157—22no—
42C3×#157C2×D312yescomplemented
43C3×#157C22×C312yescomplemented
44C34C2×D411yescharacteristic, complemented, Sylow
45C22×C3×D3—92no—
46C22×C3×D3—182no—
47C2×D32—32no—
48C2×D32—62no—
49C2×#157C3×D311yescharacteristic, complemented
50GO+(4,𝔽2)—34no—
51C2×C32×C4—91no—
52C3×D12—94no—
53C2×C33—22no—
54C2×C33—21no—
55C2×C33—42no—
56C2×C33—42no—
57C2×C33—82no—
58C2×C33D1211yescharacteristic, complemented
59C2×C33C3×D411yescharacteristic, complemented
60C3×C3⋊D3—24no—
61C3×C3⋊D3—24no—
62C3×C3⋊D3—21no—
63C3×C3⋊D3—21no—
64C3×C3⋊D3—44no—
65C3×C3⋊D3D1211yescharacteristic, complemented
66C3×C3⋊D3C3×D411yescharacteristic, complemented
67C3×C3⋊D3C22×D311yescharacteristic
68C3×C3⋊D3C23×C311yescharacteristic
69C32×D3—62no—
70C32×D3—62no—
71C32×D3—64no—
72C32×D3—64no—
73C32×D3—124no—
74C2×C3×D4—271no—
75C2×D12—91no—
76C22×C32—91no—
77C22×C32—182no—
78C2×C3⋊D3—22no—
79C2×C3⋊D3C2×C3×D311yescharacteristic
80C2×C3×D3—62no—
81C2×C3×D3—62no—
82C2×C3×D3—62no—
83C2×C3×D3—62no—
84C2×C3×D3—62no—
85C2×C3×D3—62no—
86C2×C3×D3—94no—
87C2×C3×D3—122no—
88C2×C3×D3—122no—
89C2×C3×D3—122no—
90C2×C3×D3—122no—
91C2×C3×D3—122no—
92C2×C3×D3—124no—
93C2×C3×D3—184no—
94C2×C3×D3—182no—
95C2×C3×D3—184no—
96D32—34no—
97D32—64no—
98D32—64no—
99D32—62no—
100#157C2×C3×D312yescomplemented
101C32×C4—92no—
102C33—22no—
103C33—21no—
104C33—42no—
105C33—42no—
106C33—82no—
107C33C2×D1211yescharacteristic, complemented
108C33C2×C3×D411yescharacteristic, complemented
109C23×C3—272no—
110C22×D3—92no—
111C22×D3—182no—
112C3×D4—274no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—181no—
116D12—94no—
117C2×C32—22no—
118C2×C32—22no—
119C2×C32—42no—
120C2×C32—42no—
121C2×C32—42no—
122C2×C32—44no—
123C2×C32—42no—
124C2×C32—42no—
125C2×C32—82no—
126C2×C32—82no—
127C2×C32—84no—
128C2×C32—91no—
129C2×C32—91no—
130C2×C32—184no—
131C2×C32C3×D1211yescharacteristic, complemented
132C2×C32GO+(4,𝔽2)11yescharacteristic, complemented
133C3⋊D3—24no—
134C3⋊D3C3×D1211yescharacteristic, complemented
135C3⋊D3C22×C3×D311yescharacteristic
136C3×D3—62no—
137C3×D3—62no—
138C3×D3—62no—
139C3×D3—62no—
140C3×D3—64no—
141C3×D3—64no—
142C3×D3—64no—
143C3×D3—64no—
144C3×D3—122no—
145C3×D3—122no—
146C3×D3—124no—
147C3×D3—124no—
148C3×D3—124no—
149C3×D3—124no—
150C3×D3—128no—
151C2×D4—271noSylow
152C22×C3—91no—
153C22×C3—91no—
154C22×C3—181no—
155C22×C3—182no—
156C22×C3—182no—
157C22×C3—274no—
158C22×C3—362no—
159C22×C3—542no—
160C2×D3—62no—
161C2×D3—62no—
162C2×D3—62no—
163C2×D3—94no—
164C2×D3—122no—
165C2×D3—184no—
166C2×D3—182no—
167C2×D3—184no—
168C3×C4—92no—
169C3×C4—92no—
170C3×C4—182no—
171C32—22no—
172C32—22no—
173C32—42no—
174C32—42no—
175C32—42no—
176C32—44no—
177C32—42no—
178C32—42no—
179C32—82no—
180C32—82no—
181C32—84no—
182C32C2×C3×D1211yescharacteristic, complemented
183C32C2×GO+(4,𝔽2)11yescharacteristic, complemented
184C23—272no—
185D4—274no—
186C2×C4—91no—
187C2×C3—22no—
188C2×C3—21no—
189C2×C3—42no—
190C2×C3—42no—
191C2×C3—82no—
192C2×C3—91no—
193C2×C3—91no—
194C2×C3—91no—
195C2×C3—91no—
196C2×C3—181no—
197C2×C3—181no—
198C2×C3—184no—
199C2×C3—184no—
200C2×C3—364no—
201C2×C3C3×GO+(4,𝔽2)11yescharacteristic, complemented
202C2×C3#157⋊D311yescharacteristic, direct product factor
203D3—62no—
204D3—62no—
205D3—64no—
206D3—64no—
207D3—124no—
208C22—91no—
209C22—182no—
210C22—274no—
211C22—542no—
212C4—92no—
213C3—22no—
214C3—21no—
215C3—42no—
216C3—42no—
217C3—82no—
218C3C2×C3×GO+(4,𝔽2)11yescharacteristic, complemented
219C3C2×#157⋊D311yescharacteristic, direct product factor
220C2—91no—
221C2—91no—
222C2—184no—
223C2C3×#157⋊D311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#157⋊D3
commutator, cosocleC3×C3⋊D3C23×C3
FrattiniC1G
Fitting, socleC2×C34D4
hypocenterC33C2×C3×D4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1113C2semidirect
2. C2×C32×#157C2semidirect
3. C3×#157⋊D3C2direct (N × H)
4. C2×#157⋊D3C3direct (N × H)
9. C3×#1113C22semidirect
10. C32×#157C22semidirect
11. C2×#1113C2×C3semidirect
15. C2×C3×#157D3semidirect
16. C2×C3×#157C2×C3semidirect
17. #157⋊D3C2×C3direct (N × H)
19. C2×C34D4semidirect
21. C32×C3⋊D3D4semidirect
36. #1113C22×C3semidirect
42. C3×#157C2×D3semidirect
43. C3×#157C22×C3semidirect
44. C34C2×D4semidirect
49. C2×#157C3×D3semidirect
58. C2×C33D12semidirect
59. C2×C33C3×D4semidirect
65. C3×C3⋊D3D12semidirect
66. C3×C3⋊D3C3×D4semidirect
100. #157C2×C3×D3semidirect
107. C33C2×D12semidirect
108. C33C2×C3×D4semidirect
131. C2×C32C3×D12semidirect
132. C2×C32GO+(4,𝔽2)semidirect
134. C3⋊D3C3×D12semidirect
182. C32C2×C3×D12semidirect
183. C32C2×GO+(4,𝔽2)semidirect
201. C2×C3C3×GO+(4,𝔽2)semidirect
202. C2×C3#157⋊D3direct (N × H)
218. C3C2×C3×GO+(4,𝔽2)semidirect
219. C3C2×#157⋊D3direct (N × H)
223. C2C3×#157⋊D3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#11132
C2×C32×#1572
C3×#157⋊D32
C2×#157⋊D33
C2×C3×GO+(4,𝔽2)3
C2×C3×D129