Finite Group Databasebeta

← D3×C11×ΓL(1,𝔽8)×D11 #44479115

Subgroups — D3×C11×ΓL(1,𝔽8)×D11 #44479115

Order 30492.

266 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C11×ΓL(1,𝔽8)×D33C211yesmaximal, characteristic, complemented
2D3×C112×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C3×C11×ΓL(1,𝔽8)×D11C211yesmaximal, characteristic, complemented
4D3×C7×C11×D11C311yesmaximal, characteristic, complemented
5C2×C11×ΓL(1,𝔽8)×D11—31nomaximal
6C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C7×C11×D33C2×C311yescharacteristic, complemented
8D3×C7×C112C2×C311yescharacteristic, complemented
9C3×C7×C11×D11C2×C311yescharacteristic, complemented
10C2×C112×ΓL(1,𝔽8)—31no—
11C11×ΓL(1,𝔽8)×D11—21no—
12C11×ΓL(1,𝔽8)×D11—31no—
13C11×ΓL(1,𝔽8)×D11D311yescharacteristic, direct product factor
14C3×D3×C11×D11—71nomaximal, Hall
15C2×C7×C11×D11—31noHall
16C2×D3×C11×ΓL(1,𝔽8)—111nomaximal
17D3×ΓL(1,𝔽8)×D11C1111yesmaximal, characteristic, direct product factor
18C112×ΓL(1,𝔽8)—21no—
19C112×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
20C3×C7×C112C22×C311yescharacteristic, complemented
21C3×C11×D33—71no—
22C3×D3×C112—71no—
23C32×C11×D11—71no—
24C2×C7×C112—31no—
25C7×C11×D11—31no—
26C7×C11×D11C3×D311yescharacteristic, complemented
27C2×C3×C11×D11—211no—
28D3×C11×D11ΓL(1,𝔽8)11yescharacteristic, direct product factor
29C2×C3×C11×ΓL(1,𝔽8)—111no—
30ΓL(1,𝔽8)×D33C2×C1111yescharacteristic, complemented
31D3×C11×ΓL(1,𝔽8)—25no—
32D3×C11×ΓL(1,𝔽8)—111no—
33D3×C11×ΓL(1,𝔽8)D1111yescharacteristic, direct product factor
34D3×C11×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
35C3×ΓL(1,𝔽8)×D11C2×C1111yescharacteristic, complemented
36C32×C112—71noHall
37C2×D3×C7×C11—111no—
38C22×C11×ΓL(1,𝔽8)—331no—
39D3×C7×D11C3×C1111yescharacteristic, complemented
40C2×ΓL(1,𝔽8)×D11—31no—
41C7×C112C2×C3×D311yescharacteristic, complemented, Hall
42C2×C3×C112—211no—
43C11×D33C2×ΓL(1,𝔽8)11yescharacteristic, complemented
44D3×C112C2×ΓL(1,𝔽8)11yescharacteristic, complemented
45C3×C11×D11—71no—
46C3×C11×D11—141no—
47C3×C11×D11—211no—
48C3×C11×D11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
49C3×C11×ΓL(1,𝔽8)—25no—
50C3×C11×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented
51C3×C11×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented
52C2×C11×D11—31noHall
53C2×C3×C7×C11—111no—
54C7×D33C2×C3×C1111yescharacteristic, complemented
55D3×C7×C11—25no—
56D3×C7×C11—111no—
57D3×C7×C11C3×D1111yescharacteristic, complemented
58D3×C7×C11C2×C3×C1111yescharacteristic, complemented
59C3×C7×D11C2×C3×C1111yescharacteristic, complemented
60C2×C11×ΓL(1,𝔽8)—31no—
61C2×C11×ΓL(1,𝔽8)—31no—
62C2×C11×ΓL(1,𝔽8)—65no—
63C2×C11×ΓL(1,𝔽8)—111no—
64C2×C11×ΓL(1,𝔽8)—221no—
65C2×C11×ΓL(1,𝔽8)—331no—
66ΓL(1,𝔽8)×D11—21no—
67ΓL(1,𝔽8)×D11—31no—
68ΓL(1,𝔽8)×D11D3×C1111yescharacteristic, direct product factor
69C2×C3×D3×C11—771no—
70C3×D3×D11—71no—
71C3×C112—71no—
72C3×C112—141no—
73C3×C112C22×ΓL(1,𝔽8)11yescharacteristic, complemented
74C22×C7×C11—331no—
75C2×C7×D11—31no—
76C2×D3×ΓL(1,𝔽8)—111noHall
77C2×C112—31no—
78C11×D11—31no—
79C11×D11D3×ΓL(1,𝔽8)11yescharacteristic, direct product factor
80C3×C7×C11—25no—
81C3×C7×C11C2×C3×D1111yescharacteristic, complemented
82C3×C7×C11C22×C3×C1111yescharacteristic, complemented
83C11×ΓL(1,𝔽8)—21no—
84C11×ΓL(1,𝔽8)—21no—
85C11×ΓL(1,𝔽8)—25no—
86C11×ΓL(1,𝔽8)—45no—
87C11×ΓL(1,𝔽8)D3×D1111yescharacteristic, direct product factor
88C11×ΓL(1,𝔽8)C2×D3×C1111yescharacteristic, complemented
89C2×C32×C11—771no—
90C3×D33—71no—
91C3×D3×C11—71no—
92C3×D3×C11—71no—
93C3×D3×C11—145no—
94C3×D3×C11—771no—
95C32×D11—71no—
96C2×C7×C11—31no—
97C2×C7×C11—31no—
98C2×C7×C11—65no—
99C2×C7×C11—111no—
100C2×C7×C11—331no—
101C7×D11—31no—
102C7×D11C3×D3×C1111yescharacteristic, complemented
103C22×C3×C11—2311no—
104C2×D3×C11—111no—
105C2×C3×D11—211no—
106D3×D11C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
107C2×C3×ΓL(1,𝔽8)—111no—
108D3×ΓL(1,𝔽8)—111no—
109D3×ΓL(1,𝔽8)C11×D1111yescharacteristic, direct product factor
110C112C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
111C32×C11—71no—
112C32×C11—71no—
113C32×C11—145no—
114C2×D3×C7—111no—
115C22×ΓL(1,𝔽8)—331no—
116C7×C11—25no—
117C7×C11C3×D3×D1111yescharacteristic, complemented
118C7×C11C2×C3×D3×C1111yescharacteristic, complemented
119C2×C3×C11—111no—
120C2×C3×C11—211no—
121C2×C3×C11—211no—
122C2×C3×C11—425no—
123C2×C3×C11—771no—
124C2×C3×C11—1541no—
125C2×C3×C11—2311no—
126D33C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
127C3×D11—71no—
128C3×D11—141no—
129C3×D11—211no—
130C3×D11C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
131D3×C11—25no—
132D3×C11—111no—
133D3×C11ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
134D3×C11C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
135C3×ΓL(1,𝔽8)C2×C11×D1111yescharacteristic, complemented, Hall
136C22×C11—331no—
137C2×D11—31no—
138C2×C3×C7—111no—
139D3×C7—111no—
140D3×C7C3×C11×D1111yescharacteristic, complemented
141C2×ΓL(1,𝔽8)—31no—
142C2×ΓL(1,𝔽8)—111no—
143C2×ΓL(1,𝔽8)—221no—
144C2×ΓL(1,𝔽8)—331no—
145C2×C3×D3—771noHall
146C3×C11—25no—
147C3×C11—71no—
148C3×C11—71no—
149C3×C11—145no—
150C3×C11—141no—
151C3×C11—141no—
152C3×C11—285no—
153C3×C11C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
154C3×C11C22×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
155C22×C7—331noHall
156C2×C11—31no—
157C2×C11—31no—
158C2×C11—65no—
159C2×C11—111no—
160C2×C11—331no—
161D11—31no—
162D11D3×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
163C3×C7C2×C3×C11×D1111yescharacteristic, complemented
164ΓL(1,𝔽8)—21no—
165ΓL(1,𝔽8)D3×C11×D1111yescharacteristic, direct product factor
166C2×C32—771no—
167C3×D3—71no—
168C3×D3—771no—
169C2×C7—31no—
170C2×C7—111no—
171C2×C7—331no—
172C22×C3—2311no—
173C2×D3—111no—
174C11—25no—
175C11D3×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
176C11C2×D3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
177C32—71noSylow
178C7C3×D3×C11×D1111yescharacteristic, complemented, Sylow
179C2×C3—111no—
180C2×C3—211no—
181C2×C3—771no—
182C2×C3—1541no—
183C2×C3—2311no—
184D3—111no—
185D3C11×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
186C22—331noSylow
187C3—71no—
188C3—141no—
189C3C2×C11×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
190C2—31no—
191C2—111no—
192C2—331no—
subgroupG/H
center, hypercenterC11D3×ΓL(1,𝔽8)×D11
commutator, cosocle, hypocenterC3×C7×C11C22×C3×C11
FrattiniC1G
Fitting, socleC3×C7×C112C22×C3
LCSG ▷ C3×C7×C11
UCSC1 ◁ C11

Complements

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

NH ≅ G/Ntype
1. C11×ΓL(1,𝔽8)×D33C2semidirect
2. D3×C112×ΓL(1,𝔽8)C2semidirect
3. C3×C11×ΓL(1,𝔽8)×D11C2semidirect
4. D3×C7×C11×D11C3semidirect
6. C3×C112×ΓL(1,𝔽8)C22semidirect
7. C7×C11×D33C2×C3semidirect
8. D3×C7×C112C2×C3semidirect
9. C3×C7×C11×D11C2×C3semidirect
13. C11×ΓL(1,𝔽8)×D11D3direct (N × H)
17. D3×ΓL(1,𝔽8)×D11C11direct (N × H)
19. C112×ΓL(1,𝔽8)C2×D3semidirect
20. C3×C7×C112C22×C3semidirect
26. C7×C11×D11C3×D3semidirect
28. D3×C11×D11ΓL(1,𝔽8)direct (N × H)
30. ΓL(1,𝔽8)×D33C2×C11semidirect
33. D3×C11×ΓL(1,𝔽8)D11direct (N × H)
34. D3×C11×ΓL(1,𝔽8)C2×C11semidirect
35. C3×ΓL(1,𝔽8)×D11C2×C11semidirect
39. D3×C7×D11C3×C11semidirect
41. C7×C112C2×C3×D3semidirect
43. C11×D33C2×ΓL(1,𝔽8)semidirect
44. D3×C112C2×ΓL(1,𝔽8)semidirect
48. C3×C11×D11C2×ΓL(1,𝔽8)semidirect
50. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
51. C3×C11×ΓL(1,𝔽8)C22×C11semidirect
54. C7×D33C2×C3×C11semidirect
57. D3×C7×C11C3×D11semidirect
58. D3×C7×C11C2×C3×C11semidirect
59. C3×C7×D11C2×C3×C11semidirect
68. ΓL(1,𝔽8)×D11D3×C11direct (N × H)
73. C3×C112C22×ΓL(1,𝔽8)semidirect
79. C11×D11D3×ΓL(1,𝔽8)direct (N × H)
81. C3×C7×C11C2×C3×D11semidirect
82. C3×C7×C11C22×C3×C11semidirect
87. C11×ΓL(1,𝔽8)D3×D11direct (N × H)
88. C11×ΓL(1,𝔽8)C2×D3×C11semidirect
102. C7×D11C3×D3×C11semidirect
106. D3×D11C11×ΓL(1,𝔽8)direct (N × H)
109. D3×ΓL(1,𝔽8)C11×D11direct (N × H)
110. C112C2×D3×ΓL(1,𝔽8)semidirect
117. C7×C11C3×D3×D11semidirect
118. C7×C11C2×C3×D3×C11semidirect
126. D33C2×C11×ΓL(1,𝔽8)semidirect
130. C3×D11C2×C11×ΓL(1,𝔽8)semidirect
133. D3×C11ΓL(1,𝔽8)×D11direct (N × H)
134. D3×C11C2×C11×ΓL(1,𝔽8)semidirect
135. C3×ΓL(1,𝔽8)C2×C11×D11semidirect
140. D3×C7C3×C11×D11semidirect
153. C3×C11C2×ΓL(1,𝔽8)×D11semidirect
154. C3×C11C22×C11×ΓL(1,𝔽8)semidirect
162. D11D3×C11×ΓL(1,𝔽8)direct (N × H)
163. C3×C7C2×C3×C11×D11semidirect
165. ΓL(1,𝔽8)D3×C11×D11direct (N × H)
175. C11D3×ΓL(1,𝔽8)×D11direct (N × H)
176. C11C2×D3×C11×ΓL(1,𝔽8)semidirect
178. C7C3×D3×C11×D11semidirect
185. D3C11×ΓL(1,𝔽8)×D11direct (N × H)
189. C3C2×C11×ΓL(1,𝔽8)×D11semidirect

Maximal subgroups

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

subgroupindex
C11×ΓL(1,𝔽8)×D332
D3×C112×ΓL(1,𝔽8)2
C3×C11×ΓL(1,𝔽8)×D112
D3×C7×C11×D113
C2×C11×ΓL(1,𝔽8)×D113
C3×D3×C11×D117
C2×D3×C11×ΓL(1,𝔽8)11
D3×ΓL(1,𝔽8)×D1111