Finite Group Databasebeta

← #1326×C211 #41008946

Subgroups — #1326×C211 #41008946

Order 27008.

158 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 44 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#385×C211C211yesmaximal, characteristic
2#323×C211C211yesmaximal, characteristic
3#323×C211C211yesmaximal, characteristic
4C23×C4×C211C411yescharacteristic
5C2×C4⋊C4×C211—21no—
6C2×C22⋊C4×C211C411yescharacteristic
7C2×C22⋊C4×C211C2211yescharacteristic
8C22⋊C8×C211—21no—
9C22⋊C8×C211—21no—
10#95×C211—21no—
11C24×C211C2×C411yescharacteristic
12C22×C4×C211—21no—
13C22×C4×C211—21no—
14C22×C4×C211—42no—
15C22×C4×C211—41no—
16C22×C4×C211D411yescharacteristic
17C22×C4×C211D411yescharacteristic
18C2×C8×C211—41no—
19C2×C8×C211—41no—
20C4⋊C4×C211—42no—
21C22⋊C4×C211—21no—
22C22⋊C4×C211—41no—
23C22⋊C4×C211C812yescomplemented
24C23×C211—21no—
25C23×C211—21no—
26C23×C211—41no—
27C23×C211—41no—
28C23×C211C22⋊C411yescharacteristic
29C23×C211C2×C811yescharacteristic
30C23×C211M1611yescharacteristic
31C2×C4×C211—22no—
32C2×C4×C211—22no—
33C2×C4×C211—21no—
34C2×C4×C211—21no—
35C2×C4×C211—41no—
36C2×C4×C211—41no—
37C2×C4×C211—41no—
38C2×C4×C211—42no—
39C2×C4×C211—41no—
40C2×C4×C211—41no—
41C2×C4×C211—41no—
42C2×C4×C211—42no—
43C8×C211—81no—
44C8×C211—81no—
45C22×C211—21no—
46C22×C211—21no—
47C22×C211—21no—
48C22×C211—21no—
49C22×C211—41no—
50C22×C211—41no—
51C22×C211—42no—
52C22×C211—41no—
53C22×C211—41no—
54C22×C211C22⋊C811yescharacteristic
55C22×C211#9911yescharacteristic
56C22×C211#10111yescharacteristic
57C4×C211—42no—
58C4×C211—41no—
59C4×C211—42no—
60C4×C211—41no—
61C2×C211—21no—
62C2×C211—21no—
63C2×C211—41no—
64C2×C211—41no—
65C2×C211#32411yescharacteristic
66C2×C211AΣL(1,𝔽16)11yescharacteristic
67C2×C211#35211yescharacteristic
68C211#132611yescharacteristic, direct product factor, Sylow
69#1326C21111yesmaximal, characteristic, direct product factor, Sylow
70#385C2×C21111yescharacteristic
71#323C2×C21111yescharacteristic
72#323C2×C21111yescharacteristic
73C23×C4C4×C21111yescharacteristic
74C2×C4⋊C4—21no—
75C2×C22⋊C4C4×C21111yescharacteristic
76C2×C22⋊C4C22×C21111yescharacteristic
77C22⋊C8—21no—
78C22⋊C8—21no—
79#95—21no—
80C24C2×C4×C21111yescharacteristic
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—42no—
84C22×C4—41no—
85C22×C4D4×C21111yescharacteristic
86C22×C4D4×C21111yescharacteristic
87C2×C8—41no—
88C2×C8—41no—
89C4⋊C4—42no—
90C22⋊C4—21no—
91C22⋊C4—41no—
92C22⋊C4C8×C21112yescomplemented
93C23—21no—
94C23—21no—
95C23—41no—
96C23—41no—
97C23C22⋊C4×C21111yescharacteristic
98C23C2×C8×C21111yescharacteristic
99C23M16×C21111yescharacteristic
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—42no—
112C8—81no—
113C8—81no—
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—42no—
121C22—41no—
122C22—41no—
123C22C22⋊C8×C21111yescharacteristic
124C22#99×C21111yescharacteristic
125C22#101×C21111yescharacteristic
126C4—42no—
127C4—41no—
128C4—42no—
129C4—41no—
130C2—21no—
131C2—21no—
132C2—41no—
133C2—41no—
134C2#324×C21111yescharacteristic
135C2AΣL(1,𝔽16)×C21111yescharacteristic
136C2#352×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#99
commutatorC23C2×C8×C211
Frattini, cosocleC2×C22⋊C4C22×C211
prime radicalC24×C211C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C211 ◁ C2×C22⋊C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
23. C22⋊C4×C211C8semidirect
68. C211#1326direct (N × H)
69. #1326C211direct (N × H)
92. C22⋊C4C8×C211semidirect

Maximal subgroups

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

subgroupindex
#385×C2112
#323×C2112
#1326211