Finite Group Databasebeta

← 3−1+2×#214 #13047866

Subgroups — 3−1+2×#214 #13047866

Order 1296.

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

normal subgroups only

This group has 248 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×D4×3−1+2C211yesmaximal, characteristic
2Dic6×3−1+2C211yesmaximal, characteristic, complemented
3C3⋊C8×3−1+2C211yesmaximal, characteristic, complemented
4C32×#214C311yesmaximal, characteristic
5SD16×3−1+2—31nomaximal
6C9×#214C313yesmaximal, complemented
7C22×C3×3−1+2—21no—
8C3×C4×3−1+2C2211yescharacteristic
9Dic3×3−1+2—21no—
10C33×D4C2×C311yescharacteristic
11C32×Dic6C2×C311yescharacteristic
12C32×C3⋊C8C2×C311yescharacteristic
13Q8×3−1+2—31no—
14D4×3−1+2—21no—
15D4×3−1+2—23no—
16D4×3−1+2D311yescharacteristic
17C3×D4×C9C2×C313yes—
18C9×Dic6C2×C313yescomplemented
19C8×3−1+2—31no—
20C9×C3⋊C8C2×C313yescomplemented
21C2×C3×3−1+2—41no—
22C2×C3×3−1+2D411yescharacteristic
23C32×SD16—31no—
24C3×#214—31no—
25C3×#214C3211yescharacteristic
26C9×SD16—33no—
27C22×C33—21no—
28C33×C4C22×C311yescharacteristic
29C32×Dic3—21no—
30C22×3−1+2—22no—
31C22×3−1+2—26no—
32C22×3−1+2—21no—
33C22×C3×C9—23no—
34C4×3−1+2—21no—
35C4×3−1+2—23no—
36C4×3−1+2—61no—
37C4×3−1+2C2×D311yescharacteristic
38C3×C4×C9C22×C313yes—
39C9×Dic3—23no—
40C3×3−1+2SD1611yescharacteristic, complemented, Sylow
41C32×Q8—31no—
42C32×D4—21no—
43C32×D4—31no—
44C32×D4—61no—
45C32×D4C3×D311yescharacteristic
46C32×D4C2×C3211yescharacteristic
47C3×Dic6—31no—
48C3×Dic6C2×C3211yescharacteristic
49C32×C8—31no—
50C3×C3⋊C8—31no—
51C3×C3⋊C8C2×C3211yescharacteristic
52Q8×C9—33no—
53D4×C9—23no—
54D4×C9C3×D313yes—
55C8×C9—33no—
56C2×C33—41no—
57C2×C33C3×D411yescharacteristic
58C2×3−1+2—21no—
59C2×3−1+2—23no—
60C2×3−1+2—42no—
61C2×3−1+2—46no—
62C2×3−1+2—41no—
63C2×3−1+2#6711yescharacteristic
64C2×C3×C9—43no—
65C2×C3×C9C3×D413yes—
66C3×SD16—31no—
67C3×SD16—91no—
68#2143−1+211yescharacteristic, direct product factor
69C22×C32—21no—
70C22×C32—22no—
71C22×C32—21no—
72C22×C32—61no—
73C22×C32—62no—
74C32×C4—21no—
75C32×C4—31no—
76C32×C4—61no—
77C32×C4—61no—
78C32×C4C2×C3×D311yescharacteristic
79C32×C4C22×C3211yescharacteristic
80C3×Dic3—21no—
81C3×Dic3—61no—
82C22×C9—26no—
83C22×C9—23no—
84C4×C9—23no—
85C4×C9—63no—
86C4×C9C2×C3×D313yes—
87C33C3×SD1611yescharacteristic
883−1+2—21no—
893−1+2—23no—
903−1+2#21411yescharacteristic, direct product factor
91C3×C9C3×SD1613yescomplemented
92C3×Q8—31no—
93C3×Q8—91no—
94C3×D4—21no—
95C3×D4—31no—
96C3×D4—61no—
97C3×D4C2×3−1+211yescharacteristic
98C3×D4C32×D311yescharacteristic
99Dic6C2×3−1+211yescharacteristic, complemented
100C3×C8—31no—
101C3×C8—91no—
102C3⋊C8C2×3−1+211yescharacteristic, complemented
103C2×C32—21no—
104C2×C32—31no—
105C2×C32—41no—
106C2×C32—42no—
107C2×C32—41no—
108C2×C32—61no—
109C2×C32—121no—
110C2×C32—122no—
111C2×C32C3×#6711yescharacteristic
112C2×C32C32×D411yescharacteristic
113C2×C9—23no—
114C2×C9—46no—
115C2×C9—43no—
116C2×C9C3×#6713yes—
117SD16—31noSylow
118C22×C3—22no—
119C22×C3—21no—
120C22×C3—21no—
121C22×C3—61no—
122C22×C3—62no—
123C3×C4—21no—
124C3×C4—31no—
125C3×C4—61no—
126C3×C4—61no—
127C3×C4—181no—
128C3×C4C22×3−1+211yescharacteristic
129C3×C4C2×C32×D311yescharacteristic
130Dic3—21no—
131C32—21no—
132C32—31no—
133C32—61no—
134C32C3×#21411yescharacteristic
135C32C32×SD1611yescharacteristic
136C9—23no—
137C9C3×#21413yescomplemented
138Q8—31no—
139D4D3×3−1+211yescharacteristic
140C8—31no—
141C2×C3—21no—
142C2×C3—31no—
143C2×C3—42no—
144C2×C3—41no—
145C2×C3—41no—
146C2×C3—61no—
147C2×C3—121no—
148C2×C3—122no—
149C2×C3D4×3−1+211yescharacteristic
150C2×C3C32×#6711yescharacteristic
151C22—21no—
152C4—61no—
153C4C2×D3×3−1+211yescharacteristic
154C3—21no—
155C3—31no—
156C3—61no—
157C3SD16×3−1+211yescharacteristic, complemented
158C3C32×#21411yescharacteristic
159C2—41no—
160C2#67×3−1+211yescharacteristic
subgroupG/H
centerC2×C3C32×#67
commutator, cosocleC32×C4C22×C32
FrattiniC3×C4C2×C32×D3
FittingC3×D4×3−1+2C2
socleC2×C32C32×D4
prime radicalC33×D4C2×C3
hypercenterD4×3−1+2D3
hypocenterC3SD16×3−1+2
LCSG ▷ C32×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C4×3−1+2 ◁ D4×3−1+2

Complements

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

NH ≅ G/Ntype
2. Dic6×3−1+2C2semidirect
3. C3⋊C8×3−1+2C2semidirect
6. C9×#214C3semidirect
18. C9×Dic6C2×C3semidirect
20. C9×C3⋊C8C2×C3semidirect
40. C3×3−1+2SD16semidirect
68. #2143−1+2direct (N × H)
90. 3−1+2#214direct (N × H)
91. C3×C9C3×SD16semidirect
99. Dic6C2×3−1+2semidirect
102. C3⋊C8C2×3−1+2semidirect
137. C9C3×#214semidirect
157. 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×D4×3−1+22
Dic6×3−1+22
C3⋊C8×3−1+22
C32×#2143
SD16×3−1+23
C9×#2143