Finite Group Databasebeta

← #1821×C211 #41009441

Subgroups — #1821×C211 #41009441

Order 27008.

316 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C4⋊C4×C211C211yesmaximal, characteristic
2#407×C211C212yesmaximal
3#343×C211C214yesmaximal, complemented
4C23×C4×C211C412yes—
5C23×C4×C211C2211yescharacteristic
6C2×M16×C211C2214yes—
7C2×C4⋊C4×C211—22no—
8C2×C4⋊C4×C211—22no—
9C2×C4⋊C4×C211C412yes—
10C2×C4⋊C4×C211C2212yes—
11C22⋊C8×C211—22no—
12C22⋊C8×C211—22no—
13C24×C211C2×C411yescharacteristic
14C22×C4×C211—24no—
15C22×C4×C211—22no—
16C22×C4×C211—22no—
17C22×C4×C211—44no—
18C22×C4×C211C2×C411yescharacteristic
19C22×C4×C211C2×C414yes—
20C22×C4×C211D412yes—
21C22×C4×C211D412yes—
22C22×C4×C211C2311yescharacteristic
23M16×C211—28no—
24C2×C8×C211—24no—
25C4⋊C4×C211—24no—
26C4⋊C4×C211—24no—
27C23×C211—22no—
28C23×C211—22no—
29C23×C211—22no—
30C23×C211C22⋊C411yescharacteristic
31C23×C211C22⋊C411yescharacteristic
32C23×C211C22×C411yescharacteristic
33C2×C4×C211—22no—
34C2×C4×C211—24no—
35C2×C4×C211—24no—
36C2×C4×C211—24no—
37C2×C4×C211—44no—
38C2×C4×C211—44no—
39C2×C4×C211—44no—
40C2×C4×C211C22⋊C412yes—
41C2×C4×C211C2×D412yes—
42C2×C4×C211#4114yes—
43C8×C211—44no—
44C22×C211—28no—
45C22×C211—21no—
46C22×C211—21no—
47C22×C211—21no—
48C22×C211—21no—
49C22×C211—21no—
50C22×C211—21no—
51C22×C211#10012yes—
52C22×C211#10112yes—
53C22×C211C2×C22⋊C411yescharacteristic
54C22×C211#11712yes—
55C4×C211—22no—
56C4×C211—44no—
57C4×C211#12314yes—
58C2×C211—22no—
59C2×C211—22no—
60C2×C211—22no—
61C2×C211#38111yescharacteristic
62C2×C211C2×#10011yescharacteristic
63C2×C211C2×#10111yescharacteristic
64C211#182111yescharacteristic, direct product factor, Sylow
65#1821C21111yesmaximal, characteristic, direct product factor, Sylow
66C22×C4⋊C4C2×C21111yescharacteristic
67#407C2×C21112yes—
68#343C2×C21114yescomplemented
69C23×C4C4×C21112yes—
70C23×C4C22×C21111yescharacteristic
71C2×M16C22×C21114yes—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4C4×C21112yes—
75C2×C4⋊C4C22×C21112yes—
76C22⋊C8—22no—
77C22⋊C8—22no—
78C24C2×C4×C21111yescharacteristic
79C22×C4—24no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—44no—
83C22×C4C2×C4×C21111yescharacteristic
84C22×C4C2×C4×C21114yes—
85C22×C4D4×C21112yes—
86C22×C4D4×C21112yes—
87C22×C4C23×C21111yescharacteristic
88M16—28no—
89C2×C8—24no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C23—22no—
93C23—22no—
94C23—22no—
95C23C22⋊C4×C21111yescharacteristic
96C23C22⋊C4×C21111yescharacteristic
97C23C22×C4×C21111yescharacteristic
98C2×C4—22no—
99C2×C4—24no—
100C2×C4—24no—
101C2×C4—24no—
102C2×C4—44no—
103C2×C4—44no—
104C2×C4—44no—
105C2×C4C22⋊C4×C21112yes—
106C2×C4C2×D4×C21112yes—
107C2×C4#41×C21114yes—
108C8—44no—
109C22—28no—
110C22—21no—
111C22—21no—
112C22—21no—
113C22—21no—
114C22—21no—
115C22—21no—
116C22#100×C21112yes—
117C22#101×C21112yes—
118C22C2×C22⋊C4×C21111yescharacteristic
119C22#117×C21112yes—
120C4—22no—
121C4—44no—
122C4#123×C21114yes—
123C2—22no—
124C2—22no—
125C2—22no—
126C2#381×C21111yescharacteristic
127C2C2×#100×C21111yescharacteristic
128C2C2×#101×C21111yescharacteristic
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC23C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC24×C211C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
3. #343×C211C2semidirect
64. C211#1821direct (N × H)
65. #1821C211direct (N × H)
68. #343C2×C211semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×C2112
#407×C2112
#343×C2112
#1821211