Finite Group Databasebeta

← 3−1+2×#215 #13047877

Subgroups — 3−1+2×#215 #13047877

Order 1296.

226 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 77 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×Q8×3−1+2C211yesmaximal, characteristic, complemented
2D12×3−1+2C211yesmaximal, characteristic
3C3⋊C8×3−1+2C211yesmaximal, characteristic, complemented
4C32×#215C311yesmaximal, characteristic
5SD16×3−1+2—31nomaximal
6C9×#215C313yesmaximal, complemented
7C2×D3×3−1+2—21no—
8C3×C4×3−1+2—21no—
9C3×C4×3−1+2C2211yescharacteristic
10C33×Q8C2×C311yescharacteristic
11C32×D12C2×C311yescharacteristic
12C32×C3⋊C8C2×C311yescharacteristic
13Q8×3−1+2—21no—
14Q8×3−1+2—23no—
15Q8×3−1+2D311yescharacteristic, complemented
16C3×Q8×C9C2×C313yescomplemented
17D4×3−1+2—31no—
18C9×D12C2×C313yes—
19C8×3−1+2—31no—
20C9×C3⋊C8C2×C313yescomplemented
21C2×C3×3−1+2D411yescharacteristic
22D3×3−1+2—41no—
23C32×SD16—31no—
24C3×#215—31no—
25C3×#215C3211yescharacteristic
26C9×SD16—33no—
27C2×C32×D3—21no—
28C33×C4—21no—
29C33×C4C22×C311yescharacteristic
30C22×3−1+2—61no—
31C2×D3×C9—23no—
32C4×3−1+2—22no—
33C4×3−1+2—26no—
34C4×3−1+2—21no—
35C4×3−1+2—23no—
36C4×3−1+2—21no—
37C4×3−1+2C2×D311yescharacteristic
38C3×C4×C9—23no—
39C3×C4×C9C22×C313yes—
40C3×3−1+2SD1611yescharacteristic, complemented, Sylow
41C32×Q8—21no—
42C32×Q8—31no—
43C32×Q8—61no—
44C32×Q8C3×D311yescharacteristic
45C32×Q8C2×C3211yescharacteristic
46C32×D4—31no—
47C3×D12—31no—
48C3×D12C2×C3211yescharacteristic
49C32×C8—31no—
50C3×C3⋊C8—31no—
51C3×C3⋊C8C2×C3211yescharacteristic
52Q8×C9—23no—
53Q8×C9C3×D313yescomplemented
54D4×C9—33no—
55C8×C9—33no—
56C2×C33C3×D411yescharacteristic
57C32×D3—41no—
58C2×3−1+2—21no—
59C2×3−1+2—23no—
60C2×3−1+2—121no—
61C2×3−1+2#6711yescharacteristic
62C2×C3×C9C3×D413yes—
63D3×C9—43no—
64C3×SD16—31no—
65C3×SD16—91no—
66#2153−1+211yescharacteristic, direct product factor
67C22×C32—61no—
68C2×C3×D3—21no—
69C2×C3×D3—61no—
70C32×C4—21no—
71C32×C4—22no—
72C32×C4—21no—
73C32×C4—21no—
74C32×C4—31no—
75C32×C4—61no—
76C32×C4—61no—
77C32×C4—62no—
78C32×C4C2×C3×D311yescharacteristic
79C32×C4C22×C3211yescharacteristic
80C22×C9—63no—
81C4×C9—26no—
82C4×C9—23no—
83C4×C9—23no—
84C4×C9C2×C3×D313yes—
85C33C3×SD1611yescharacteristic
863−1+2—21no—
873−1+2—23no—
883−1+2#21511yescharacteristic, direct product factor
89C3×C9C3×SD1613yescomplemented
90C3×Q8—21no—
91C3×Q8—31no—
92C3×Q8—61no—
93C3×Q8C2×3−1+211yescharacteristic, complemented
94C3×Q8C32×D311yescharacteristic
95C3×D4—31no—
96C3×D4—91no—
97D12C2×3−1+211yescharacteristic
98C3×C8—31no—
99C3×C8—91no—
100C3⋊C8C2×3−1+211yescharacteristic, complemented
101C2×C32—21no—
102C2×C32—31no—
103C2×C32—61no—
104C2×C32—121no—
105C2×C32C3×#6711yescharacteristic
106C2×C32C32×D411yescharacteristic
107C3×D3—41no—
108C3×D3—121no—
109C2×C9—23no—
110C2×C9—123no—
111C2×C9C3×#6713yes—
112SD16—31noSylow
113C22×C3—61no—
114C22×C3—181no—
115C2×D3—21no—
116C3×C4—22no—
117C3×C4—21no—
118C3×C4—21no—
119C3×C4—21no—
120C3×C4—31no—
121C3×C4—61no—
122C3×C4—62no—
123C3×C4—61no—
124C3×C4C22×3−1+211yescharacteristic
125C3×C4C2×C32×D311yescharacteristic
126C32—21no—
127C32—31no—
128C32—61no—
129C32C3×#21511yescharacteristic
130C32C32×SD1611yescharacteristic
131C9—23no—
132C9C3×#21513yescomplemented
133Q8D3×3−1+211yescharacteristic, complemented
134D4—31no—
135C8—31no—
136C2×C3—21no—
137C2×C3—31no—
138C2×C3—61no—
139C2×C3—121no—
140C2×C3—361no—
141C2×C3D4×3−1+211yescharacteristic
142C2×C3C32×#6711yescharacteristic
143D3—41no—
144C22—61no—
145C4—21no—
146C4C2×D3×3−1+211yescharacteristic
147C3—21no—
148C3—31no—
149C3—61no—
150C3SD16×3−1+211yescharacteristic, complemented
151C3C32×#21511yescharacteristic
152C2—121no—
153C2#67×3−1+211yescharacteristic
subgroupG/H
centerC2×C3C32×#67
commutator, cosocleC32×C4C22×C32
FrattiniC3×C4C2×C32×D3
FittingC3×Q8×3−1+2C2
socleC2×C32C32×D4
prime radicalC32×D12C2×C3
hypercenterQ8×3−1+2D3
hypocenterC3SD16×3−1+2
LCSG ▷ C32×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C4×3−1+2 ◁ Q8×3−1+2

Complements

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

NH ≅ G/Ntype
1. C3×Q8×3−1+2C2semidirect
3. C3⋊C8×3−1+2C2semidirect
6. C9×#215C3semidirect
15. Q8×3−1+2D3semidirect
16. C3×Q8×C9C2×C3semidirect
20. C9×C3⋊C8C2×C3semidirect
40. C3×3−1+2SD16semidirect
53. Q8×C9C3×D3semidirect
66. #2153−1+2direct (N × H)
88. 3−1+2#215direct (N × H)
89. C3×C9C3×SD16semidirect
93. C3×Q8C2×3−1+2semidirect
100. C3⋊C8C2×3−1+2semidirect
132. C9C3×#215semidirect
133. Q8D3×3−1+2semidirect
150. C3SD16×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×3−1+22
D12×3−1+22
C3⋊C8×3−1+22
C32×#2153
SD16×3−1+23
C9×#2153