Finite Group Databasebeta

← C3×ΓL(1,𝔽8)×#214 #72254173

Subgroups — C3×ΓL(1,𝔽8)×#214 #72254173

Order 3024.

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

normal subgroups only

This group has 368 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
2C3×ΓL(1,𝔽8)×Dic6C211yesmaximal, characteristic, complemented
3C3×ΓL(1,𝔽8)×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×C7×#214C311yesmaximal, characteristic, complemented
5C3×SD16×ΓL(1,𝔽8)—31nomaximal
6ΓL(1,𝔽8)×#214C313yesmaximal, direct product factor
7C22×C32×ΓL(1,𝔽8)—21no—
8C32×C4×ΓL(1,𝔽8)C2211yescharacteristic
9C3×Dic3×ΓL(1,𝔽8)—21no—
10C32×C7×D4C2×C311yescharacteristic
11C3×C7×Dic6C2×C311yescharacteristic, complemented
12C3×Q8×ΓL(1,𝔽8)—31no—
13C3×D4×ΓL(1,𝔽8)—21no—
14C3×D4×ΓL(1,𝔽8)—23no—
15C3×D4×ΓL(1,𝔽8)D311yescharacteristic
16C3×D4×ΓL(1,𝔽8)C2×C313yes—
17ΓL(1,𝔽8)×Dic6C2×C313yescomplemented
18C3×C7×C3⋊C8C2×C311yescharacteristic, complemented
19C3×C8×ΓL(1,𝔽8)—31no—
20ΓL(1,𝔽8)×C3⋊C8C2×C313yescomplemented
21C32×#214—71nomaximal, Hall
22C2×C32×ΓL(1,𝔽8)—41no—
23C2×C32×ΓL(1,𝔽8)D411yescharacteristic
24C3×C7×SD16—31no—
25C7×#214C3211yescharacteristic, complemented
26SD16×ΓL(1,𝔽8)—33no—
27C22×C32×C7—21no—
28C22×C3×ΓL(1,𝔽8)—22no—
29C22×C3×ΓL(1,𝔽8)—26no—
30C22×C3×ΓL(1,𝔽8)—21no—
31C22×C3×ΓL(1,𝔽8)—23no—
32C32×C4×C7C22×C311yescharacteristic
33C3×C7×Dic3—21no—
34C3×C4×ΓL(1,𝔽8)—21no—
35C3×C4×ΓL(1,𝔽8)—23no—
36C3×C4×ΓL(1,𝔽8)—61no—
37C3×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
38C3×C4×ΓL(1,𝔽8)C22×C313yes—
39Dic3×ΓL(1,𝔽8)—23no—
40C33×D4—71no—
41C32×Dic6—71no—
42C32×C3⋊C8—71no—
43C32×ΓL(1,𝔽8)SD1611yescharacteristic, complemented, Hall
44C3×C7×Q8—31no—
45C3×C7×D4—21no—
46C3×C7×D4C3×D311yescharacteristic
47C3×C7×D4C2×C3211yescharacteristic
48C7×Dic6C2×C3211yescharacteristic, complemented
49Q8×ΓL(1,𝔽8)—33no—
50D4×ΓL(1,𝔽8)—23no—
51D4×ΓL(1,𝔽8)C3×D313yes—
52C3×C7×C8—31no—
53C7×C3⋊C8C2×C3211yescharacteristic, complemented
54C8×ΓL(1,𝔽8)—33no—
55C32×SD16—211no—
56C3×#214—73no—
57C3×#214ΓL(1,𝔽8)11yescharacteristic, direct product factor
58C2×C32×C7—41no—
59C2×C32×C7C3×D411yescharacteristic
60C2×C3×ΓL(1,𝔽8)—21no—
61C2×C3×ΓL(1,𝔽8)—23no—
62C2×C3×ΓL(1,𝔽8)—42no—
63C2×C3×ΓL(1,𝔽8)—46no—
64C2×C3×ΓL(1,𝔽8)—41no—
65C2×C3×ΓL(1,𝔽8)—43no—
66C2×C3×ΓL(1,𝔽8)#6711yescharacteristic
67C2×C3×ΓL(1,𝔽8)C3×D413yes—
68C7×SD16—31noHall
69C22×C33—141no—
70C33×C4—71no—
71C32×Dic3—141no—
72C22×C3×C7—22no—
73C22×C3×C7—21no—
74C22×C3×C7—21no—
75C22×ΓL(1,𝔽8)—26no—
76C22×ΓL(1,𝔽8)—23no—
77C3×C4×C7—21no—
78C3×C4×C7—61no—
79C3×C4×C7C2×C3×D311yescharacteristic
80C3×C4×C7C22×C3211yescharacteristic
81C7×Dic3—21no—
82C4×ΓL(1,𝔽8)—23no—
83C4×ΓL(1,𝔽8)—63no—
84C4×ΓL(1,𝔽8)C2×C3×D313yes—
85C32×Q8—211no—
86C32×D4—71no—
87C32×D4—73no—
88C32×D4—143no—
89C32×D4—141no—
90C32×D4C2×ΓL(1,𝔽8)11yescharacteristic
91C3×Dic6—73no—
92C3×Dic6C2×ΓL(1,𝔽8)11yescharacteristic, complemented
93C32×C8—211no—
94C3×C3⋊C8—73no—
95C3×C3⋊C8C2×ΓL(1,𝔽8)11yescharacteristic, complemented
96C32×C7C3×SD1611yescharacteristic, complemented
97C3×ΓL(1,𝔽8)—21no—
98C3×ΓL(1,𝔽8)—23no—
99C3×ΓL(1,𝔽8)#21411yescharacteristic, direct product factor
100C3×ΓL(1,𝔽8)C3×SD1613yescomplemented
101C7×Q8—31no—
102C7×D4C32×D311yescharacteristic
103C7×C8—31no—
104C2×C33—71no—
105C2×C33—281no—
106C3×SD16—31no—
107C3×SD16—213no—
108#214C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor
109C2×C3×C7—21no—
110C2×C3×C7—42no—
111C2×C3×C7—41no—
112C2×C3×C7—41no—
113C2×C3×C7C3×#6711yescharacteristic
114C2×C3×C7C32×D411yescharacteristic
115C2×ΓL(1,𝔽8)—23no—
116C2×ΓL(1,𝔽8)—46no—
117C2×ΓL(1,𝔽8)—43no—
118C2×ΓL(1,𝔽8)C3×#6713yes—
119C22×C32—21no—
120C22×C32—143no—
121C22×C32—141no—
122C22×C32—146no—
123C22×C32—142no—
124C32×C4—71no—
125C32×C4—73no—
126C32×C4—143no—
127C32×C4—141no—
128C32×C4—421no—
129C32×C4C22×ΓL(1,𝔽8)11yescharacteristic
130C3×Dic3—21no—
131C3×Dic3—143no—
132C22×C7—21no—
133C4×C7—61no—
134C4×C7C2×C32×D311yescharacteristic
135C33—71noSylow
136C3×Q8—31no—
137C3×Q8—213no—
138C3×D4—21no—
139C3×D4—73no—
140C3×D4—143no—
141C3×D4D3×ΓL(1,𝔽8)11yescharacteristic
142C3×D4C2×C3×ΓL(1,𝔽8)11yescharacteristic
143Dic6C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
144C3×C8—31no—
145C3×C8—213no—
146C3⋊C8C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
147C3×C7—21no—
148C3×C7C3×#21411yescharacteristic, complemented
149C3×C7C32×SD1611yescharacteristic, complemented
150ΓL(1,𝔽8)—23no—
151ΓL(1,𝔽8)C3×#21413yesdirect product factor
152C2×C32—41no—
153C2×C32—71no—
154C2×C32—73no—
155C2×C32—143no—
156C2×C32—141no—
157C2×C32—283no—
158C2×C32—281no—
159C2×C32—286no—
160C2×C32—282no—
161C2×C32D4×ΓL(1,𝔽8)11yescharacteristic
162SD16—31noSylow
163C2×C7—41no—
164C2×C7C32×#6711yescharacteristic
165C22×C3—22no—
166C22×C3—21no—
167C22×C3—21no—
168C22×C3—143no—
169C22×C3—146no—
170C3×C4—21no—
171C3×C4—61no—
172C3×C4—73no—
173C3×C4—143no—
174C3×C4—423no—
175C3×C4C2×D3×ΓL(1,𝔽8)11yescharacteristic
176C3×C4C22×C3×ΓL(1,𝔽8)11yescharacteristic
177Dic3—21no—
178C32—71no—
179C32—73no—
180C32—143no—
181C32—141no—
182C32SD16×ΓL(1,𝔽8)11yescharacteristic, complemented
183Q8—31no—
184D4C3×D3×ΓL(1,𝔽8)11yescharacteristic
185C8—31no—
186C7C32×#21411yescharacteristic, complemented, Sylow
187C2×C3—21no—
188C2×C3—42no—
189C2×C3—41no—
190C2×C3—41no—
191C2×C3—73no—
192C2×C3—143no—
193C2×C3—283no—
194C2×C3—286no—
195C2×C3ΓL(1,𝔽8)×#6711yescharacteristic
196C2×C3C3×D4×ΓL(1,𝔽8)11yescharacteristic
197C22—21no—
198C4—61no—
199C4C2×C3×D3×ΓL(1,𝔽8)11yescharacteristic
200C3—21no—
201C3—73no—
202C3—143no—
203C3ΓL(1,𝔽8)×#21411yescharacteristic, direct product factor
204C3C3×SD16×ΓL(1,𝔽8)11yescharacteristic, complemented
205C2—41no—
206C2C3×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
prime radicalC32×D4×ΓL(1,𝔽8)C2
hypercenterC3×D4D3×ΓL(1,𝔽8)
hypocenterC3×C7C32×SD16
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
2. C3×ΓL(1,𝔽8)×Dic6C2semidirect
3. C3×ΓL(1,𝔽8)×C3⋊C8C2semidirect
4. C3×C7×#214C3semidirect
6. ΓL(1,𝔽8)×#214C3direct (N × H)
11. C3×C7×Dic6C2×C3semidirect
17. ΓL(1,𝔽8)×Dic6C2×C3semidirect
18. C3×C7×C3⋊C8C2×C3semidirect
20. ΓL(1,𝔽8)×C3⋊C8C2×C3semidirect
25. C7×#214C32semidirect
43. C32×ΓL(1,𝔽8)SD16semidirect
48. C7×Dic6C2×C32semidirect
53. C7×C3⋊C8C2×C32semidirect
57. C3×#214ΓL(1,𝔽8)direct (N × H)
92. C3×Dic6C2×ΓL(1,𝔽8)semidirect
95. C3×C3⋊C8C2×ΓL(1,𝔽8)semidirect
96. C32×C7C3×SD16semidirect
99. C3×ΓL(1,𝔽8)#214direct (N × H)
100. C3×ΓL(1,𝔽8)C3×SD16semidirect
108. #214C3×ΓL(1,𝔽8)direct (N × H)
143. Dic6C2×C3×ΓL(1,𝔽8)semidirect
146. C3⋊C8C2×C3×ΓL(1,𝔽8)semidirect
148. C3×C7C3×#214semidirect
149. C3×C7C32×SD16semidirect
151. ΓL(1,𝔽8)C3×#214direct (N × H)
182. C32SD16×ΓL(1,𝔽8)semidirect
186. C7C32×#214semidirect
203. C3ΓL(1,𝔽8)×#214direct (N × H)
204. C3C3×SD16×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)×Dic62
C3×ΓL(1,𝔽8)×C3⋊C82
C3×C7×#2143
C3×SD16×ΓL(1,𝔽8)3
ΓL(1,𝔽8)×#2143
C32×#2147