Finite Group Databasebeta

← C3×ΓL(1,𝔽8)×#104 #72244589

Subgroups — C3×ΓL(1,𝔽8)×#104 #72244589

Order 2016.

264 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C3×#41×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C3×M16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C3×C42×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4C3×C7×#104C311yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×#104C313yesmaximal, direct product factor
6C3×Q8×ΓL(1,𝔽8)C411yescharacteristic, complemented
7C3×D4×ΓL(1,𝔽8)—21no—
8C3×D4×ΓL(1,𝔽8)C411yescharacteristic, complemented
9C2×C3×C4×ΓL(1,𝔽8)—21no—
10C2×C3×C4×ΓL(1,𝔽8)—21no—
11C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C3×C8×ΓL(1,𝔽8)—21no—
13C3×C7×#41C2×C311yescharacteristic
14#41×ΓL(1,𝔽8)C2×C313yes—
15C3×C7×M16C2×C311yescharacteristic, complemented
16C3×C42×C7C2×C311yescharacteristic, complemented
17M16×ΓL(1,𝔽8)C2×C313yescomplemented
18C42×ΓL(1,𝔽8)C2×C313yescomplemented
19C32×#104—71nomaximal, Hall
20C22×C3×ΓL(1,𝔽8)—21no—
21C22×C3×ΓL(1,𝔽8)D411yescharacteristic
22C3×C4×ΓL(1,𝔽8)—22no—
23C3×C4×ΓL(1,𝔽8)—21no—
24C3×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
25C3×C4×ΓL(1,𝔽8)D411yescharacteristic
26C7×#104C3211yescharacteristic, complemented, Hall
27C3×C7×Q8C3×C411yescharacteristic, complemented
28C3×C7×D4—21no—
29C3×C7×D4C3×C411yescharacteristic, complemented
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C22×C311yescharacteristic
33Q8×ΓL(1,𝔽8)C3×C413yescomplemented
34D4×ΓL(1,𝔽8)—23no—
35D4×ΓL(1,𝔽8)C3×C413yescomplemented
36C2×C4×ΓL(1,𝔽8)—23no—
37C2×C4×ΓL(1,𝔽8)—23no—
38C2×C4×ΓL(1,𝔽8)C22×C313yes—
39C3×C7×C8—21no—
40C8×ΓL(1,𝔽8)—23no—
41C32×#41—71no—
42C32×M16—71no—
43C32×C42—71no—
44C2×C3×ΓL(1,𝔽8)—21no—
45C2×C3×ΓL(1,𝔽8)—41no—
46C2×C3×ΓL(1,𝔽8)C22⋊C411yescharacteristic
47C7×#41C2×C3211yescharacteristic
48C7×M16C2×C3211yescharacteristic, complemented
49C42×C7C2×C3211yescharacteristic, complemented
50C3×#104—73no—
51C3×#104ΓL(1,𝔽8)11yescharacteristic, direct product factor
52C22×C3×C7—21no—
53C22×C3×C7C3×D411yescharacteristic
54C22×ΓL(1,𝔽8)—23no—
55C22×ΓL(1,𝔽8)C3×D413yes—
56C3×C4×C7—22no—
57C3×C4×C7—21no—
58C3×C4×C7C2×C3×C411yescharacteristic
59C3×C4×C7C3×D411yescharacteristic
60C4×ΓL(1,𝔽8)—26no—
61C4×ΓL(1,𝔽8)—23no—
62C4×ΓL(1,𝔽8)C2×C3×C413yes—
63C4×ΓL(1,𝔽8)C3×D413yes—
64C32×Q8—71no—
65C32×D4—71no—
66C32×D4—141no—
67C2×C32×C4—71no—
68C2×C32×C4—141no—
69C2×C32×C4—141no—
70C32×C8—141no—
71C3×ΓL(1,𝔽8)#10411yescharacteristic, direct product factor, Hall
72C7×Q8C32×C411yescharacteristic, complemented
73C7×D4—21no—
74C7×D4C32×C411yescharacteristic, complemented
75C2×C4×C7—21no—
76C2×C4×C7—21no—
77C2×C4×C7C22×C3211yescharacteristic
78C7×C8—21no—
79C3×#41—73no—
80C3×#41C2×ΓL(1,𝔽8)11yescharacteristic
81C3×M16—73no—
82C3×M16C2×ΓL(1,𝔽8)11yescharacteristic, complemented
83C3×C42—73no—
84C3×C42C2×ΓL(1,𝔽8)11yescharacteristic, complemented
85C2×C3×C7—21no—
86C2×C3×C7—41no—
87C2×C3×C7C3×C22⋊C411yescharacteristic
88C2×ΓL(1,𝔽8)—23no—
89C2×ΓL(1,𝔽8)—43no—
90C2×ΓL(1,𝔽8)C3×C22⋊C413yes—
91C22×C32—71no—
92C22×C32—141no—
93C32×C4—71no—
94C32×C4—71no—
95C32×C4—141no—
96C32×C4—142no—
97#104C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
98C22×C7—21no—
99C22×C7C32×D411yescharacteristic
100C4×C7—22no—
101C4×C7—21no—
102C4×C7C2×C32×C411yescharacteristic
103C4×C7C32×D411yescharacteristic
104C3×Q8—73no—
105C3×Q8C4×ΓL(1,𝔽8)11yescharacteristic, complemented
106C3×D4—21no—
107C3×D4—73no—
108C3×D4—143no—
109C3×D4C4×ΓL(1,𝔽8)11yescharacteristic, complemented
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—73no—
113C2×C3×C4—143no—
114C2×C3×C4—143no—
115C2×C3×C4C22×ΓL(1,𝔽8)11yescharacteristic
116C3×C8—21no—
117C3×C8—143no—
118C3×C7C3×#10411yescharacteristic, complemented
119ΓL(1,𝔽8)C3×#10413yesdirect product factor
120C2×C32—71no—
121C2×C32—141no—
122C2×C32—281no—
123#41C2×C3×ΓL(1,𝔽8)11yescharacteristic
124M16C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
125C42C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
126C2×C7—21no—
127C2×C7—41no—
128C2×C7C32×C22⋊C411yescharacteristic
129C22×C3—21no—
130C22×C3—73no—
131C22×C3—143no—
132C22×C3D4×ΓL(1,𝔽8)11yescharacteristic
133C3×C4—22no—
134C3×C4—21no—
135C3×C4—73no—
136C3×C4—73no—
137C3×C4—143no—
138C3×C4—146no—
139C3×C4C2×C4×ΓL(1,𝔽8)11yescharacteristic
140C3×C4D4×ΓL(1,𝔽8)11yescharacteristic
141C32—71noSylow
142Q8C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
143D4—21no—
144D4C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
145C2×C4—21no—
146C2×C4—21no—
147C2×C4C22×C3×ΓL(1,𝔽8)11yescharacteristic
148C8—21no—
149C7C32×#10411yescharacteristic, complemented, Sylow
150C2×C3—21no—
151C2×C3—41no—
152C2×C3—73no—
153C2×C3—143no—
154C2×C3—283no—
155C2×C3C22⋊C4×ΓL(1,𝔽8)11yescharacteristic
156C22—21no—
157C22C3×D4×ΓL(1,𝔽8)11yescharacteristic
158C4—22no—
159C4—21no—
160C4C2×C3×C4×ΓL(1,𝔽8)11yescharacteristic
161C4C3×D4×ΓL(1,𝔽8)11yescharacteristic
162C3—73no—
163C3ΓL(1,𝔽8)×#10411yescharacteristic, direct product factor
164C2—21no—
165C2—41no—
166C2C3×C22⋊C4×ΓL(1,𝔽8)11yescharacteristic
subgroupG/H
centerC3×C4D4×ΓL(1,𝔽8)
commutatorC4×C7C2×C32×C4
FrattiniC2×C4C22×C3×ΓL(1,𝔽8)
FittingC3×C7×#104C3
socleC2×C3×C7C3×C22⋊C4
prime radicalC3×#41×ΓL(1,𝔽8)C2
cosocleC2×C4×C7C22×C32
hypercenterC3×#104ΓL(1,𝔽8)
hypocenterC7C32×#104
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
2. C3×M16×ΓL(1,𝔽8)C2semidirect
3. C3×C42×ΓL(1,𝔽8)C2semidirect
4. C3×C7×#104C3semidirect
5. ΓL(1,𝔽8)×#104C3direct (N × H)
6. C3×Q8×ΓL(1,𝔽8)C4semidirect
8. C3×D4×ΓL(1,𝔽8)C4semidirect
15. C3×C7×M16C2×C3semidirect
16. C3×C42×C7C2×C3semidirect
17. M16×ΓL(1,𝔽8)C2×C3semidirect
18. C42×ΓL(1,𝔽8)C2×C3semidirect
26. C7×#104C32semidirect
27. C3×C7×Q8C3×C4semidirect
29. C3×C7×D4C3×C4semidirect
33. Q8×ΓL(1,𝔽8)C3×C4semidirect
35. D4×ΓL(1,𝔽8)C3×C4semidirect
48. C7×M16C2×C32semidirect
49. C42×C7C2×C32semidirect
51. C3×#104ΓL(1,𝔽8)direct (N × H)
71. C3×ΓL(1,𝔽8)#104direct (N × H)
72. C7×Q8C32×C4semidirect
74. C7×D4C32×C4semidirect
82. C3×M16C2×ΓL(1,𝔽8)semidirect
84. C3×C42C2×ΓL(1,𝔽8)semidirect
97. #104C3×ΓL(1,𝔽8)direct (N × H)
105. C3×Q8C4×ΓL(1,𝔽8)semidirect
109. C3×D4C4×ΓL(1,𝔽8)semidirect
118. C3×C7C3×#104semidirect
119. ΓL(1,𝔽8)C3×#104direct (N × H)
124. M16C2×C3×ΓL(1,𝔽8)semidirect
125. C42C2×C3×ΓL(1,𝔽8)semidirect
142. Q8C3×C4×ΓL(1,𝔽8)semidirect
144. D4C3×C4×ΓL(1,𝔽8)semidirect
149. C7C32×#104semidirect
163. C3ΓL(1,𝔽8)×#104direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#41×ΓL(1,𝔽8)2
C3×M16×ΓL(1,𝔽8)2
C3×C42×ΓL(1,𝔽8)2
C3×C7×#1043
ΓL(1,𝔽8)×#1043
C32×#1047