Finite Group Databasebeta

← #1285×C211 #41008905

Subgroups — #1285×C211 #41008905

Order 27008.

164 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 68 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#432×C211C211yesmaximal, characteristic
2#432×C211C211yesmaximal, characteristic
3C4×C4⋊C4×C211C211yesmaximal, characteristic
4C2×C4⋊C4×C211—21no—
5C2×C42×C211C412yes—
6C2×C42×C211C2211yescharacteristic
7C4⋊C8×C211—21no—
8C4⋊C8×C211C412yescomplemented
9C22⋊C8×C211—21no—
10C22⋊C8×C211—21no—
11#97×C211—21no—
12#97×C211C412yescomplemented
13#95×C211—21no—
14C22×C4×C211—22no—
15C22×C4×C211C2×C411yescharacteristic
16C22×C4×C211D411yescharacteristic
17C22×C4×C211D411yescharacteristic
18C2×C8×C211—22no—
19C2×C8×C211—22no—
20C4⋊C4×C211—24no—
21C42×C211—42no—
22C42×C211C2×C411yescharacteristic
23C42×C211C2×C411yescharacteristic
24C42×C211D411yescharacteristic
25C42×C211Q811yescharacteristic
26C23×C211C22⋊C411yescharacteristic
27C2×C4×C211—21no—
28C2×C4×C211—21no—
29C2×C4×C211—22no—
30C2×C4×C211—24no—
31C2×C4×C211—41no—
32C2×C4×C211—42no—
33C2×C4×C211—42no—
34C2×C4×C211C4211yescharacteristic
35C2×C4×C211C22⋊C411yescharacteristic
36C2×C4×C211C22⋊C411yescharacteristic
37C2×C4×C211C4⋊C411yescharacteristic
38C2×C4×C211C4⋊C411yescharacteristic
39C2×C4×C211C4⋊C411yescharacteristic
40C8×C211—42no—
41C8×C211—42no—
42C22×C211—21no—
43C22×C211—21no—
44C22×C211#9511yescharacteristic
45C22×C211#10011yescharacteristic
46C22×C211#10111yescharacteristic
47C4×C211—21no—
48C4×C211—21no—
49C4×C211—21no—
50C4×C211—21no—
51C4×C211—21no—
52C4×C211—44no—
53C4×C211#10412yes—
54C2×C211—21no—
55C2×C211—21no—
56C2×C211H(3,ℤ/4)11yescharacteristic
57C2×C211#33911yescharacteristic
58C2×C211#34311yescharacteristic
59C211#128511yescharacteristic, direct product factor, Sylow
60#1285C21111yesmaximal, characteristic, direct product factor, Sylow
61#432C2×C21111yescharacteristic
62#432C2×C21111yescharacteristic
63C4×C4⋊C4C2×C21111yescharacteristic
64C2×C4⋊C4—21no—
65C2×C42C4×C21112yes—
66C2×C42C22×C21111yescharacteristic
67C4⋊C8—21no—
68C4⋊C8C4×C21112yescomplemented
69C22⋊C8—21no—
70C22⋊C8—21no—
71#97—21no—
72#97C4×C21112yescomplemented
73#95—21no—
74C22×C4—22no—
75C22×C4C2×C4×C21111yescharacteristic
76C22×C4D4×C21111yescharacteristic
77C22×C4D4×C21111yescharacteristic
78C2×C8—22no—
79C2×C8—22no—
80C4⋊C4—24no—
81C42—42no—
82C42C2×C4×C21111yescharacteristic
83C42C2×C4×C21111yescharacteristic
84C42D4×C21111yescharacteristic
85C42Q8×C21111yescharacteristic
86C23C22⋊C4×C21111yescharacteristic
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—24no—
91C2×C4—41no—
92C2×C4—42no—
93C2×C4—42no—
94C2×C4C42×C21111yescharacteristic
95C2×C4C22⋊C4×C21111yescharacteristic
96C2×C4C22⋊C4×C21111yescharacteristic
97C2×C4C4⋊C4×C21111yescharacteristic
98C2×C4C4⋊C4×C21111yescharacteristic
99C2×C4C4⋊C4×C21111yescharacteristic
100C8—42no—
101C8—42no—
102C22—21no—
103C22—21no—
104C22#95×C21111yescharacteristic
105C22#100×C21111yescharacteristic
106C22#101×C21111yescharacteristic
107C4—21no—
108C4—21no—
109C4—21no—
110C4—21no—
111C4—21no—
112C4—44no—
113C4#104×C21112yes—
114C2—21no—
115C2—21no—
116C2H(3,ℤ/4)×C21111yescharacteristic
117C2#339×C21111yescharacteristic
118C2#343×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#95
commutatorC2×C4C42×C211
Frattini, cosocleC2×C42C22×C211
prime radicalC23×C211C22⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C2×C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
8. C4⋊C8×C211C4semidirect
12. #97×C211C4semidirect
59. C211#1285direct (N × H)
60. #1285C211direct (N × H)
68. C4⋊C8C4×C211semidirect
72. #97C4×C211semidirect

Maximal subgroups

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

subgroupindex
#432×C2112
C4×C4⋊C4×C2112
#1285211