Finite Group Databasebeta

← ΓL(1,𝔽8)×D11×D33 #44479116

Subgroups — ΓL(1,𝔽8)×D11×D33 #44479116

Order 30492.

266 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 48 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
1ΓL(1,𝔽8)×C3⋊(C11⋊D11)C211yesmaximal, characteristic, complemented
2C11×ΓL(1,𝔽8)×D33C211yesmaximal, characteristic, complemented
3C3×C11×ΓL(1,𝔽8)×D11C211yesmaximal, characteristic, complemented
4C7×D11×D33C311yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×D112—31nomaximal
6C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C7×C3⋊(C11⋊D11)C2×C311yescharacteristic, complemented
8C7×C11×D33C2×C311yescharacteristic, complemented
9C3×C7×C11×D11C2×C311yescharacteristic, complemented
10ΓL(1,𝔽8)×C11⋊D11—31no—
11C11×ΓL(1,𝔽8)×D11—21no—
12C11×ΓL(1,𝔽8)×D11—31no—
13C11×ΓL(1,𝔽8)×D11D311yescharacteristic, complemented
14C3×D11×D33—71nomaximal, Hall
15C7×D112—31noHall
16C2×ΓL(1,𝔽8)×D33—111nomaximal
17D3×ΓL(1,𝔽8)×D11—111nomaximal
18C112×ΓL(1,𝔽8)—21no—
19C112×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
20C3×C7×C112C22×C311yescharacteristic, complemented
21C3×C3⋊(C11⋊D11)—71no—
22C3×C11×D33—71no—
23C32×C11×D11—71no—
24C7×C11⋊D11—31no—
25C7×C11×D11—31no—
26C7×C11×D11C3×D311yescharacteristic, complemented
27D11×D33ΓL(1,𝔽8)11yescharacteristic, direct product factor
28C3×D112—211no—
29C2×C3×C11×ΓL(1,𝔽8)—111no—
30ΓL(1,𝔽8)×D33—111no—
31ΓL(1,𝔽8)×D33—111no—
32ΓL(1,𝔽8)×D33—225no—
33ΓL(1,𝔽8)×D33D1111yescharacteristic, direct product factor
34D3×C11×ΓL(1,𝔽8)—111no—
35C3×ΓL(1,𝔽8)×D11D1111yescharacteristic, complemented
36C32×C112—71noHall
37C2×C7×D33—111no—
38D3×C7×D11—111no—
39C2×ΓL(1,𝔽8)×D11—331no—
40C2×ΓL(1,𝔽8)×D11—331no—
41C7×C112C2×C3×D311yescharacteristic, complemented, Hall
42C3⋊(C11⋊D11)C2×ΓL(1,𝔽8)11yescharacteristic, complemented
43C11×D33C2×ΓL(1,𝔽8)11yescharacteristic, complemented
44C3×C11⋊D11—211no—
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)C2×D1111yescharacteristic, complemented
52D112—31noHall
53C2×C3×C7×C11—111no—
54C7×D33—111no—
55C7×D33—111no—
56C7×D33—225no—
57C7×D33C3×D1111yescharacteristic, complemented
58D3×C7×C11—111no—
59C3×C7×D11C3×D1111yescharacteristic, complemented
60C2×C11×ΓL(1,𝔽8)—111no—
61C2×C11×ΓL(1,𝔽8)—221no—
62C2×C11×ΓL(1,𝔽8)—331no—
63ΓL(1,𝔽8)×D11—21no—
64ΓL(1,𝔽8)×D11—31no—
65ΓL(1,𝔽8)×D11—331no—
66ΓL(1,𝔽8)×D11—331no—
67ΓL(1,𝔽8)×D11—665no—
68ΓL(1,𝔽8)×D11D3311yescharacteristic, direct product factor
69C2×C3×D33—771no—
70C3×D3×D11—771no—
71C3×C112—71no—
72C3×C112—141no—
73C3×C112C22×ΓL(1,𝔽8)11yescharacteristic, complemented
74C2×C7×D11—331no—
75C2×C7×D11—331no—
76C2×D3×ΓL(1,𝔽8)—1211noHall
77C11⋊D11—31no—
78C11×D11—31no—
79C11×D11D3×ΓL(1,𝔽8)11yescharacteristic, complemented
80C3×C7×C11—25no—
81C3×C7×C11C2×C3×D1111yescharacteristic, complemented
82C3×C7×C11C2×C3×D1111yescharacteristic, 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, complemented
88C11×ΓL(1,𝔽8)C2×D3311yescharacteristic, complemented
89C2×C32×C11—771no—
90C3×D33—71no—
91C3×D33—771no—
92C3×D33—771no—
93C3×D33—1545no—
94C3×D3×C11—771no—
95C32×D11—71no—
96C2×C7×C11—111no—
97C2×C7×C11—331no—
98C7×D11—31no—
99C7×D11—331no—
100C7×D11—331no—
101C7×D11—665no—
102C7×D11C3×D3311yescharacteristic, complemented
103C2×D33—111no—
104C2×C3×D11—2311no—
105C2×C3×D11—2311no—
106D3×D11—111no—
107C2×C3×ΓL(1,𝔽8)—111no—
108D3×ΓL(1,𝔽8)—111no—
109D3×ΓL(1,𝔽8)—1211no—
110C112C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
111C32×C11—71no—
112C32×C11—71no—
113C32×C11—145no—
114C2×D3×C7—1211no—
115C22×ΓL(1,𝔽8)—3631no—
116C7×C11—25no—
117C7×C11C3×D3×D1111yescharacteristic, complemented
118C7×C11C2×C3×D3311yescharacteristic, complemented
119C2×C3×C11—111no—
120C2×C3×C11—771no—
121C2×C3×C11—1541no—
122C2×C3×C11—2311no—
123D33—111no—
124D33—111no—
125D33—225no—
126D33ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
127C3×D11—71no—
128C3×D11—141no—
129C3×D11—211no—
130C3×D11—4625no—
131C3×D11—2311no—
132C3×D11—2311no—
133C3×D11ΓL(1,𝔽8)×D1111yescharacteristic, complemented
134D3×C11—111no—
135C3×ΓL(1,𝔽8)D11211yescharacteristic, complemented, Hall
136C2×D11—331no—
137C2×D11—331no—
138C2×C3×C7—111no—
139D3×C7—111no—
140D3×C7—1211no—
141C2×ΓL(1,𝔽8)—111no—
142C2×ΓL(1,𝔽8)—221no—
143C2×ΓL(1,𝔽8)—331no—
144C2×ΓL(1,𝔽8)—3631no—
145C2×C3×D3—8471noHall
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×C11C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
155C22×C7—3631noHall
156C2×C11—111no—
157C2×C11—331no—
158D11—31no—
159D11—331no—
160D11—331no—
161D11—665no—
162D11ΓL(1,𝔽8)×D3311yescharacteristic, direct product factor
163C3×C7C3×D11211yescharacteristic, complemented
164ΓL(1,𝔽8)—21no—
165ΓL(1,𝔽8)D11×D3311yescharacteristic, direct product factor
166C2×C32—771no—
167C3×D3—771no—
168C3×D3—8471no—
169C2×C7—111no—
170C2×C7—331no—
171C2×C7—3631no—
172C22×C3—25411no—
173C2×D3—1211no—
174C11—25no—
175C11D3×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
176C11C2×ΓL(1,𝔽8)×D3311yescharacteristic, complemented
177C32—71noSylow
178C7C3×D11×D3311yescharacteristic, complemented, Sylow
179C2×C3—111no—
180C2×C3—771no—
181C2×C3—1541no—
182C2×C3—2311no—
183C2×C3—25411no—
184D3—111no—
185D3—1211no—
186C22—3631noSylow
187C3—71no—
188C3—141no—
189C3ΓL(1,𝔽8)×D11211yescharacteristic, complemented
190C2—111no—
191C2—331no—
192C2—3631no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C112C22×C3
LCSG ▷ C3×C7×C112
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×C3⋊(C11⋊D11)C2semidirect
2. C11×ΓL(1,𝔽8)×D33C2semidirect
3. C3×C11×ΓL(1,𝔽8)×D11C2semidirect
4. C7×D11×D33C3semidirect
6. C3×C112×ΓL(1,𝔽8)C22semidirect
7. C7×C3⋊(C11⋊D11)C2×C3semidirect
8. C7×C11×D33C2×C3semidirect
9. C3×C7×C11×D11C2×C3semidirect
13. C11×ΓL(1,𝔽8)×D11D3semidirect
19. C112×ΓL(1,𝔽8)C2×D3semidirect
20. C3×C7×C112C22×C3semidirect
26. C7×C11×D11C3×D3semidirect
27. D11×D33ΓL(1,𝔽8)direct (N × H)
33. ΓL(1,𝔽8)×D33D11direct (N × H)
35. C3×ΓL(1,𝔽8)×D11D11semidirect
41. C7×C112C2×C3×D3semidirect
42. C3⋊(C11⋊D11)C2×ΓL(1,𝔽8)semidirect
43. C11×D33C2×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)C2×D11semidirect
57. C7×D33C3×D11semidirect
59. C3×C7×D11C3×D11semidirect
68. ΓL(1,𝔽8)×D11D33direct (N × H)
73. C3×C112C22×ΓL(1,𝔽8)semidirect
79. C11×D11D3×ΓL(1,𝔽8)semidirect
81. C3×C7×C11C2×C3×D11semidirect
82. C3×C7×C11C2×C3×D11semidirect
87. C11×ΓL(1,𝔽8)D3×D11semidirect
88. C11×ΓL(1,𝔽8)C2×D33semidirect
102. C7×D11C3×D33semidirect
110. C112C2×D3×ΓL(1,𝔽8)semidirect
117. C7×C11C3×D3×D11semidirect
118. C7×C11C2×C3×D33semidirect
126. D33ΓL(1,𝔽8)×D11direct (N × H)
133. C3×D11ΓL(1,𝔽8)×D11semidirect
135. C3×ΓL(1,𝔽8)D112semidirect
153. C3×C11C2×ΓL(1,𝔽8)×D11semidirect
154. C3×C11C2×ΓL(1,𝔽8)×D11semidirect
162. D11ΓL(1,𝔽8)×D33direct (N × H)
163. C3×C7C3×D112semidirect
165. ΓL(1,𝔽8)D11×D33direct (N × H)
175. C11D3×ΓL(1,𝔽8)×D11semidirect
176. C11C2×ΓL(1,𝔽8)×D33semidirect
178. C7C3×D11×D33semidirect
189. C3ΓL(1,𝔽8)×D112semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×C3⋊(C11⋊D11)2
C11×ΓL(1,𝔽8)×D332
C3×C11×ΓL(1,𝔽8)×D112
C7×D11×D333
ΓL(1,𝔽8)×D1123
C3×D11×D337
C2×ΓL(1,𝔽8)×D3311
D3×ΓL(1,𝔽8)×D1111