Finite Group Databasebeta

← C3×ΓL(1,𝔽8)×#213 #72254213

Subgroups — C3×ΓL(1,𝔽8)×#213 #72254213

Order 3024.

392 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C32×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C3×ΓL(1,𝔽8)×D12C211yesmaximal, characteristic, complemented
3C3×ΓL(1,𝔽8)×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×C7×#213C311yesmaximal, characteristic, complemented
5C3×D8×ΓL(1,𝔽8)—31nomaximal
6ΓL(1,𝔽8)×#213C313yesmaximal, direct product factor
7C22×C32×ΓL(1,𝔽8)—21no—
8C2×C3×D3×ΓL(1,𝔽8)—21no—
9C32×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C32×C7×D4C2×C311yescharacteristic, complemented
11C3×C7×D12C2×C311yescharacteristic, complemented
12C3×D4×ΓL(1,𝔽8)—21no—
13C3×D4×ΓL(1,𝔽8)—23no—
14C3×D4×ΓL(1,𝔽8)—31no—
15C3×D4×ΓL(1,𝔽8)D311yescharacteristic, complemented
16C3×D4×ΓL(1,𝔽8)C2×C313yescomplemented
17ΓL(1,𝔽8)×D12C2×C313yescomplemented
18C3×C7×C3⋊C8C2×C311yescharacteristic, complemented
19C3×C8×ΓL(1,𝔽8)—31no—
20ΓL(1,𝔽8)×C3⋊C8C2×C313yescomplemented
21C32×#213—71nomaximal, Hall
22C2×C32×ΓL(1,𝔽8)—41no—
23C2×C32×ΓL(1,𝔽8)D411yescharacteristic
24C3×D3×ΓL(1,𝔽8)—41no—
25C3×C7×D8—31no—
26C7×#213C3211yescharacteristic, complemented
27D8×ΓL(1,𝔽8)—33no—
28C22×C32×C7—21no—
29C2×C3×D3×C7—21no—
30C22×C3×ΓL(1,𝔽8)—22no—
31C22×C3×ΓL(1,𝔽8)—26no—
32C22×C3×ΓL(1,𝔽8)—21no—
33C22×C3×ΓL(1,𝔽8)—23no—
34C22×C3×ΓL(1,𝔽8)—61no—
35C2×D3×ΓL(1,𝔽8)—23no—
36C32×C4×C7C22×C311yescharacteristic
37C3×C4×ΓL(1,𝔽8)—21no—
38C3×C4×ΓL(1,𝔽8)—23no—
39C3×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
40C3×C4×ΓL(1,𝔽8)C22×C313yes—
41C33×D4—71no—
42C32×D12—71no—
43C32×C3⋊C8—71no—
44C32×ΓL(1,𝔽8)D811yescharacteristic, complemented, Hall
45C3×C7×D4—21no—
46C3×C7×D4—31no—
47C3×C7×D4C3×D311yescharacteristic, complemented
48C3×C7×D4C2×C3211yescharacteristic, complemented
49C7×D12C2×C3211yescharacteristic, complemented
50D4×ΓL(1,𝔽8)—23no—
51D4×ΓL(1,𝔽8)—33no—
52D4×ΓL(1,𝔽8)C3×D313yescomplemented
53C3×C7×C8—31no—
54C7×C3⋊C8C2×C3211yescharacteristic, complemented
55C8×ΓL(1,𝔽8)—33no—
56C32×D8—211no—
57C3×#213—73no—
58C3×#213ΓL(1,𝔽8)11yescharacteristic, direct product factor
59C2×C32×C7—41no—
60C2×C32×C7C3×D411yescharacteristic
61C3×D3×C7—41no—
62C2×C3×ΓL(1,𝔽8)—21no—
63C2×C3×ΓL(1,𝔽8)—23no—
64C2×C3×ΓL(1,𝔽8)—42no—
65C2×C3×ΓL(1,𝔽8)—46no—
66C2×C3×ΓL(1,𝔽8)—41no—
67C2×C3×ΓL(1,𝔽8)—43no—
68C2×C3×ΓL(1,𝔽8)—121no—
69C2×C3×ΓL(1,𝔽8)#6711yescharacteristic
70C2×C3×ΓL(1,𝔽8)C3×D413yes—
71D3×ΓL(1,𝔽8)—43no—
72C7×D8—31noHall
73C22×C33—141no—
74C2×C32×D3—141no—
75C33×C4—71no—
76C22×C3×C7—22no—
77C22×C3×C7—21no—
78C22×C3×C7—21no—
79C22×C3×C7—61no—
80C2×D3×C7—21no—
81C22×ΓL(1,𝔽8)—26no—
82C22×ΓL(1,𝔽8)—23no—
83C22×ΓL(1,𝔽8)—63no—
84C3×C4×C7—21no—
85C3×C4×C7C2×C3×D311yescharacteristic
86C3×C4×C7C22×C3211yescharacteristic
87C4×ΓL(1,𝔽8)—23no—
88C4×ΓL(1,𝔽8)C2×C3×D313yes—
89C32×D4—71no—
90C32×D4—73no—
91C32×D4—143no—
92C32×D4—141no—
93C32×D4—211no—
94C32×D4C2×ΓL(1,𝔽8)11yescharacteristic, complemented
95C3×D12—73no—
96C3×D12C2×ΓL(1,𝔽8)11yescharacteristic, complemented
97C32×C8—211no—
98C3×C3⋊C8—73no—
99C3×C3⋊C8C2×ΓL(1,𝔽8)11yescharacteristic, complemented
100C32×C7C3×D811yescharacteristic, complemented
101C3×ΓL(1,𝔽8)—21no—
102C3×ΓL(1,𝔽8)—23no—
103C3×ΓL(1,𝔽8)#21311yescharacteristic, direct product factor
104C3×ΓL(1,𝔽8)C3×D813yescomplemented
105C7×D4—31no—
106C7×D4C32×D311yescharacteristic, complemented
107C7×C8—31no—
108C2×C33—71no—
109C2×C33—281no—
110C32×D3—281no—
111C3×D8—31no—
112C3×D8—213no—
113#213C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor
114C2×C3×C7—21no—
115C2×C3×C7—42no—
116C2×C3×C7—41no—
117C2×C3×C7—41no—
118C2×C3×C7—121no—
119C2×C3×C7C3×#6711yescharacteristic
120C2×C3×C7C32×D411yescharacteristic
121D3×C7—41no—
122C2×ΓL(1,𝔽8)—23no—
123C2×ΓL(1,𝔽8)—46no—
124C2×ΓL(1,𝔽8)—43no—
125C2×ΓL(1,𝔽8)—123no—
126C2×ΓL(1,𝔽8)C3×#6713yes—
127C22×C32—21no—
128C22×C32—143no—
129C22×C32—141no—
130C22×C32—146no—
131C22×C32—142no—
132C22×C32—421no—
133C2×C3×D3—21no—
134C2×C3×D3—143no—
135C32×C4—71no—
136C32×C4—73no—
137C32×C4—143no—
138C32×C4—141no—
139C32×C4C22×ΓL(1,𝔽8)11yescharacteristic
140C22×C7—21no—
141C22×C7—61no—
142C4×C7C2×C32×D311yescharacteristic
143C33—71noSylow
144C3×D4—21no—
145C3×D4—31no—
146C3×D4—73no—
147C3×D4—143no—
148C3×D4—213no—
149C3×D4D3×ΓL(1,𝔽8)11yescharacteristic, complemented
150C3×D4C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
151D12C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
152C3×C8—31no—
153C3×C8—213no—
154C3⋊C8C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
155C3×C7—21no—
156C3×C7C3×#21311yescharacteristic, complemented
157C3×C7C32×D811yescharacteristic, complemented
158ΓL(1,𝔽8)—23no—
159ΓL(1,𝔽8)C3×#21313yesdirect product factor
160C2×C32—41no—
161C2×C32—71no—
162C2×C32—73no—
163C2×C32—143no—
164C2×C32—141no—
165C2×C32—283no—
166C2×C32—281no—
167C2×C32—286no—
168C2×C32—282no—
169C2×C32—841no—
170C2×C32D4×ΓL(1,𝔽8)11yescharacteristic
171C3×D3—41no—
172C3×D3—283no—
173D8—31noSylow
174C2×C7—41no—
175C2×C7—121no—
176C2×C7C32×#6711yescharacteristic
177C22×C3—22no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—61no—
181C22×C3—143no—
182C22×C3—146no—
183C22×C3—423no—
184C2×D3—21no—
185C3×C4—21no—
186C3×C4—73no—
187C3×C4—143no—
188C3×C4C2×D3×ΓL(1,𝔽8)11yescharacteristic
189C3×C4C22×C3×ΓL(1,𝔽8)11yescharacteristic
190C32—71no—
191C32—73no—
192C32—143no—
193C32—141no—
194C32D8×ΓL(1,𝔽8)11yescharacteristic, complemented
195D4—31no—
196D4C3×D3×ΓL(1,𝔽8)11yescharacteristic, complemented
197C8—31no—
198C7C32×#21311yescharacteristic, complemented, Sylow
199C2×C3—21no—
200C2×C3—42no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—73no—
204C2×C3—121no—
205C2×C3—143no—
206C2×C3—283no—
207C2×C3—286no—
208C2×C3—843no—
209C2×C3ΓL(1,𝔽8)×#6711yescharacteristic
210C2×C3C3×D4×ΓL(1,𝔽8)11yescharacteristic
211D3—41no—
212C22—21no—
213C22—61no—
214C4C2×C3×D3×ΓL(1,𝔽8)11yescharacteristic
215C3—21no—
216C3—73no—
217C3—143no—
218C3ΓL(1,𝔽8)×#21311yescharacteristic, direct product factor
219C3C3×D8×ΓL(1,𝔽8)11yescharacteristic, complemented
220C2—41no—
221C2—121no—
222C2C3×ΓL(1,𝔽8)×#6711yescharacteristic
subgroupG/H
centerC2×C3ΓL(1,𝔽8)×#67
commutator, cosocleC3×C4×C7C22×C32
FrattiniC4C2×C3×D3×ΓL(1,𝔽8)
FittingC32×C7×D4C2×C3
socleC2×C32×C7C3×D4
hypercenterC3×D4D3×ΓL(1,𝔽8)
hypocenterC3×C7C32×D8
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C32×D4×ΓL(1,𝔽8)C2semidirect
2. C3×ΓL(1,𝔽8)×D12C2semidirect
3. C3×ΓL(1,𝔽8)×C3⋊C8C2semidirect
4. C3×C7×#213C3semidirect
6. ΓL(1,𝔽8)×#213C3direct (N × H)
10. C32×C7×D4C2×C3semidirect
11. C3×C7×D12C2×C3semidirect
15. C3×D4×ΓL(1,𝔽8)D3semidirect
16. C3×D4×ΓL(1,𝔽8)C2×C3semidirect
17. ΓL(1,𝔽8)×D12C2×C3semidirect
18. C3×C7×C3⋊C8C2×C3semidirect
20. ΓL(1,𝔽8)×C3⋊C8C2×C3semidirect
26. C7×#213C32semidirect
44. C32×ΓL(1,𝔽8)D8semidirect
47. C3×C7×D4C3×D3semidirect
48. C3×C7×D4C2×C32semidirect
49. C7×D12C2×C32semidirect
52. D4×ΓL(1,𝔽8)C3×D3semidirect
54. C7×C3⋊C8C2×C32semidirect
58. C3×#213ΓL(1,𝔽8)direct (N × H)
94. C32×D4C2×ΓL(1,𝔽8)semidirect
96. C3×D12C2×ΓL(1,𝔽8)semidirect
99. C3×C3⋊C8C2×ΓL(1,𝔽8)semidirect
100. C32×C7C3×D8semidirect
103. C3×ΓL(1,𝔽8)#213direct (N × H)
104. C3×ΓL(1,𝔽8)C3×D8semidirect
106. C7×D4C32×D3semidirect
113. #213C3×ΓL(1,𝔽8)direct (N × H)
149. C3×D4D3×ΓL(1,𝔽8)semidirect
150. C3×D4C2×C3×ΓL(1,𝔽8)semidirect
151. D12C2×C3×ΓL(1,𝔽8)semidirect
154. C3⋊C8C2×C3×ΓL(1,𝔽8)semidirect
156. C3×C7C3×#213semidirect
157. C3×C7C32×D8semidirect
159. ΓL(1,𝔽8)C3×#213direct (N × H)
194. C32D8×ΓL(1,𝔽8)semidirect
196. D4C3×D3×ΓL(1,𝔽8)semidirect
198. C7C32×#213semidirect
218. C3ΓL(1,𝔽8)×#213direct (N × H)
219. C3C3×D8×ΓL(1,𝔽8)semidirect

Maximal subgroups

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

subgroupindex
C32×D4×ΓL(1,𝔽8)2
C3×ΓL(1,𝔽8)×D122
C3×ΓL(1,𝔽8)×C3⋊C82
C3×C7×#2133
C3×D8×ΓL(1,𝔽8)3
ΓL(1,𝔽8)×#2133
C32×#2137