Finite Group Databasebeta

← #11899696

Subgroups — #11899696

Order 1152.

638 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#10595444C212yesmaximal, complemented
2#10593838C211yesmaximal, characteristic, complemented
3#10593334C216yesmaximal, complemented
4#10593333C213yesmaximal, complemented
5C2×#63734C213yesmaximal, complemented
6C3×#1009—22no—
7#64157C2211yescharacteristic
8#63823—26no—
9#63822C2216yescomplemented
10C2×#3713C2216yescomplemented
11#63773C2213yescomplemented
12C2×#3708C2213yescomplemented
13#63736C2211yescharacteristic, complemented
14#63735C2216yescomplemented
15#63734C2213yescomplemented
16#63734C2216yescomplemented
17#5835—62no—
18C32×#41D411yescharacteristic
19C3×#239—42no—
20C3×D3×D4—46no—
21C3×#235—26no—
22C2×C3×D12—26no—
23#3748D413yes—
24C2×C3⋊(C3⋊C8)C2313yes—
25#3718C2312yes—
26#3715—26no—
27#3713—26no—
28#3713C2316yes—
29#3708C2311yescharacteristic
30#3708C2313yes—
31#3415—91nomaximal, Sylow
32C3×H(5,𝔽2)—62no—
33#1009—22no—
34#950—66no—
35#949—66no—
36#948—62no—
37#932—66no—
38C2×#213—66no—
39C22×C3×D3—46no—
40C32×Q8D811yescharacteristic
41C32×D4D813yes—
42C2×C32×C4C2×D413yes—
43C3×#67—46no—
44C3×D12—26no—
45C3×D12—46no—
46C3×C4×D3—46no—
47C3×Dic6—22no—
48C3⋊(C3⋊C8)C2×D413yes—
49C3⋊(C3⋊C8)C2411yescharacteristic
50#576—92no—
51#509—96no—
52#508—93no—
53C2×D16—93no—
54#504—91no—
55C3×#41—22no—
56C3×#41—66no—
57C3×#41—122no—
58C2×C3×D4—66no—
59C2×C3×D4—126no—
60#239—42no—
61D3×D4—46no—
62#235—26no—
63C2×D12—26no—
64#216—62no—
65#215—66no—
66#214—66no—
67#213—66no—
68#213—612no—
69#208—66no—
70C2×C3⋊C8—66no—
71C22×C32C2×D813yes—
72C2×C3×D3—46no—
73C2×C3×D3—166no—
74C32×C4C2×D813yes—
75C32×C4C22×D411yescharacteristic
76C3×Dic3—42no—
77H(5,𝔽2)—62no—
78AGL(1,ℤ/8)—186no—
79#135—96no—
80C2×D8—96no—
81#131—91no—
82Q32—91no—
83SD32—96no—
84D16—93no—
85D16—96no—
86M32—93no—
87C2×C16—93no—
88C23×C3—126no—
89C22×D3—46no—
90C3×Q8—22no—
91C3×Q8—62no—
92C3×D4—26no—
93C3×D4—66no—
94C3×D4—126no—
95C3×D4—126no—
96C2×C3×C4—26no—
97C2×C3×C4—126no—
98#67—46no—
99D12—26no—
100D12—46no—
101C4×D3—46no—
102Dic6—22no—
103C3⋊C8—66no—
104C3⋊C8—62no—
105C2×C32—23no—
106C2×C32C22×D811yescharacteristic
107C3×D3—86no—
108#41—66no—
109#41—122no—
110#41GO+(4,𝔽2)11yescharacteristic
111C2×D4—66no—
112C2×D4—126no—
113Q16—92no—
114SD16—186no—
115D8—96no—
116D8—186no—
117M16—93no—
118C2×C8—93no—
119C16—91no—
120C16—93no—
121C22×C3—26no—
122C22×C3—126no—
123C22×C3—486no—
124C2×D3—46no—
125C2×D3—166no—
126C3×C4—22no—
127C3×C4—26no—
128C3×C4—122no—
129Dic3—42no—
130C32#341511yescharacteristic, complemented, Sylow
131C23—126no—
132Q8—62no—
133Q8#377411yescharacteristic
134D4—66no—
135D4—126no—
136D4—126no—
137D4#377413yes—
138C2×C4—126no—
139C2×C4C2×GO+(4,𝔽2)13yes—
140C8—91no—
141C8—93no—
142C2×C3—22no—
143C2×C3—46no—
144C2×C3—246no—
145D3—86no—
146C22—126no—
147C22—486no—
148C22C2×#377413yes—
149C4—122no—
150C4C2×#377413yes—
151C4C22×GO+(4,𝔽2)11yescharacteristic
152C3—22no—
153C2—23no—
154C2—246no—
155C2C22×#377411yescharacteristic
subgroupG/H
centerC2C22×#3774
commutator, cosocleC3⋊(C3⋊C8)C24
FrattiniC4C22×GO+(4,𝔽2)
FittingC32×#41D4
socleC2×C32C22×D8
hypercenter#41GO+(4,𝔽2)
hypocenterC32#3415
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #10595444C2semidirect
2. #10593838C2semidirect
3. #10593334C2semidirect
4. #10593333C2semidirect
5. C2×#63734C2semidirect
9. #63822C22semidirect
10. C2×#3713C22semidirect
11. #63773C22semidirect
12. C2×#3708C22semidirect
13. #63736C22semidirect
14. #63735C22semidirect
15. #63734C22semidirect
16. #63734C22semidirect
130. C32#3415semidirect

Maximal subgroups

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

subgroupindex
#105954442
#105938382
#105933342
#105933332
C2×#637342
#34159