Finite Group Databasebeta

← #1314×C211 #41008934

Subgroups — #1314×C211 #41008934

Order 27008.

140 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C2×C22⋊C8×C211C211yesmaximal, characteristic
2C22⋊C16×C211C212yesmaximal, complemented
3C23×C4×C211C411yescharacteristic
4C22×C8×C211C411yescharacteristic
5C22×C8×C211C2211yescharacteristic
6C2×C16×C211—22no—
7C22⋊C8×C211—22no—
8C24×C211C811yescharacteristic
9C22×C4×C211—21no—
10C22×C4×C211—21no—
11C22×C4×C211—41no—
12C22×C4×C211—41no—
13C22×C4×C211C811yescharacteristic
14C22×C4×C211C2×C411yescharacteristic
15C2×C8×C211—21no—
16C2×C8×C211—41no—
17C2×C8×C211—41no—
18C2×C8×C211D412yes—
19C16×C211—42no—
20C23×C211—21no—
21C23×C211—21no—
22C23×C211—41no—
23C23×C211—41no—
24C23×C211C1612yescomplemented
25C23×C211C2×C811yescharacteristic
26C2×C4×C211—21no—
27C2×C4×C211—21no—
28C2×C4×C211—21no—
29C2×C4×C211—42no—
30C2×C4×C211—41no—
31C2×C4×C211—41no—
32C2×C4×C211—41no—
33C2×C4×C211C22⋊C411yescharacteristic
34C2×C4×C211M1611yescharacteristic
35C8×C211—22no—
36C8×C211—41no—
37C22×C211—21no—
38C22×C211—21no—
39C22×C211—22no—
40C22×C211—42no—
41C22×C211—42no—
42C22×C211—41no—
43C22×C211—41no—
44C22×C211C22⋊C811yescharacteristic
45C22×C211C2×C1611yescharacteristic
46C22×C211M3211yescharacteristic
47C4×C211—21no—
48C4×C211—41no—
49C4×C211#9911yescharacteristic
50C4×C211#10011yescharacteristic
51C2×C211—21no—
52C2×C211—21no—
53C2×C211—42no—
54C2×C211#32311yescharacteristic
55C2×C211C22⋊C1611yescharacteristic
56C2×C211#34911yescharacteristic
57C211#131411yescharacteristic, direct product factor, Sylow
58#1314C21111yesmaximal, characteristic, direct product factor, Sylow
59C2×C22⋊C8C2×C21111yescharacteristic
60C22⋊C16C2×C21112yescomplemented
61C23×C4C4×C21111yescharacteristic
62C22×C8C4×C21111yescharacteristic
63C22×C8C22×C21111yescharacteristic
64C2×C16—22no—
65C22⋊C8—22no—
66C24C8×C21111yescharacteristic
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4C8×C21111yescharacteristic
72C22×C4C2×C4×C21111yescharacteristic
73C2×C8—21no—
74C2×C8—41no—
75C2×C8—41no—
76C2×C8D4×C21112yes—
77C16—42no—
78C23—21no—
79C23—21no—
80C23—41no—
81C23—41no—
82C23C16×C21112yescomplemented
83C23C2×C8×C21111yescharacteristic
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—42no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4C22⋊C4×C21111yescharacteristic
92C2×C4M16×C21111yescharacteristic
93C8—22no—
94C8—41no—
95C22—21no—
96C22—21no—
97C22—22no—
98C22—42no—
99C22—42no—
100C22—41no—
101C22—41no—
102C22C22⋊C8×C21111yescharacteristic
103C22C2×C16×C21111yescharacteristic
104C22M32×C21111yescharacteristic
105C4—21no—
106C4—41no—
107C4#99×C21111yescharacteristic
108C4#100×C21111yescharacteristic
109C2—21no—
110C2—21no—
111C2—42no—
112C2#323×C21111yescharacteristic
113C2C22⋊C16×C21111yescharacteristic
114C2#349×C21111yescharacteristic
subgroupG/H
centerC2×C4×C211C22⋊C4
commutatorC22C2×C16×C211
Frattini, cosocleC22×C8C22×C211
socleC22×C211C22⋊C8
prime radicalC24×C211C8
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C211 ◁ C22×C8×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×C211C2semidirect
24. C23×C211C16semidirect
57. C211#1314direct (N × H)
58. #1314C211direct (N × H)
60. C22⋊C16C2×C211semidirect
82. C23C16×C211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×C2112
C22⋊C16×C2112
#1314211