Finite Group Databasebeta

← #1282×C211 #41008902

Subgroups — #1282×C211 #41008902

Order 27008.

228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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
2C2×C22⋊C8×C211C211yesmaximal, characteristic
3C4×C22⋊C4×C211C211yesmaximal, characteristic
4C23×C4×C211C2211yescharacteristic
5C22×C8×C211—21no—
6C22×C8×C211C412yescomplemented
7C2×C22⋊C4×C211C412yes—
8C2×C42×C211—21no—
9C22⋊C8×C211—21no—
10C22⋊C8×C211—21no—
11C22⋊C8×C211—21no—
12C22⋊C8×C211C412yescomplemented
13#95×C211—21no—
14C24×C211C2×C411yescharacteristic
15C22×C4×C211—22no—
16C22×C4×C211—42no—
17C22×C4×C211C2×C411yescharacteristic
18C22×C4×C211C2×C411yescharacteristic
19C22×C4×C211D411yescharacteristic
20C22×C4×C211D411yescharacteristic
21C22×C4×C211D411yescharacteristic
22C22×C4×C211Q811yescharacteristic
23C2×C8×C211—22no—
24C2×C8×C211—22no—
25C2×C8×C211—42no—
26C2×C8×C211—42no—
27C22⋊C4×C211—22no—
28C22⋊C4×C211C814yescomplemented
29C42×C211—42no—
30C23×C211—41no—
31C23×C211—41no—
32C23×C211C4211yescharacteristic
33C23×C211C22⋊C411yescharacteristic
34C23×C211C4⋊C411yescharacteristic
35C23×C211C2×C812yes—
36C23×C211M1612yes—
37C2×C4×C211—22no—
38C2×C4×C211—22no—
39C2×C4×C211—22no—
40C2×C4×C211—24no—
41C2×C4×C211—41no—
42C2×C4×C211—41no—
43C2×C4×C211—41no—
44C2×C4×C211—42no—
45C2×C4×C211—42no—
46C2×C4×C211C22⋊C411yescharacteristic
47C2×C4×C211C22⋊C411yescharacteristic
48C2×C4×C211C4⋊C411yescharacteristic
49C2×C4×C211C4⋊C411yescharacteristic
50C8×C211—42no—
51C8×C211—42no—
52C22×C211—21no—
53C22×C211—21no—
54C22×C211—21no—
55C22×C211—21no—
56C22×C211—21no—
57C22×C211—21no—
58C22×C211—42no—
59C22×C211—42no—
60C22×C211#9511yescharacteristic
61C22×C211C4×C811yescharacteristic
62C22×C211#9711yescharacteristic
63C22×C211C22⋊C812yes—
64C22×C211C4⋊C812yes—
65C4×C211—21no—
66C4×C211—21no—
67C4×C211—21no—
68C4×C211—44no—
69C4×C211#9912yes—
70C2×C211—22no—
71C2×C211—21no—
72C2×C211—21no—
73C2×C211—22no—
74C2×C211#33611yescharacteristic
75C2×C211#34211yescharacteristic
76C2×C211#34411yescharacteristic
77C211#128211yescharacteristic, direct product factor, Sylow
78#1282C21111yesmaximal, characteristic, direct product factor, Sylow
79C2×C22⋊C8C2×C21111yescharacteristic
80C2×C22⋊C8C2×C21111yescharacteristic
81C4×C22⋊C4C2×C21111yescharacteristic
82C23×C4C22×C21111yescharacteristic
83C22×C8—21no—
84C22×C8C4×C21112yescomplemented
85C2×C22⋊C4C4×C21112yes—
86C2×C42—21no—
87C22⋊C8—21no—
88C22⋊C8—21no—
89C22⋊C8—21no—
90C22⋊C8C4×C21112yescomplemented
91#95—21no—
92C24C2×C4×C21111yescharacteristic
93C22×C4—22no—
94C22×C4—42no—
95C22×C4C2×C4×C21111yescharacteristic
96C22×C4C2×C4×C21111yescharacteristic
97C22×C4D4×C21111yescharacteristic
98C22×C4D4×C21111yescharacteristic
99C22×C4D4×C21111yescharacteristic
100C22×C4Q8×C21111yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—42no—
104C2×C8—42no—
105C22⋊C4—22no—
106C22⋊C4C8×C21114yescomplemented
107C42—42no—
108C23—41no—
109C23—41no—
110C23C42×C21111yescharacteristic
111C23C22⋊C4×C21111yescharacteristic
112C23C4⋊C4×C21111yescharacteristic
113C23C2×C8×C21112yes—
114C23M16×C21112yes—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4C22⋊C4×C21111yescharacteristic
125C2×C4C22⋊C4×C21111yescharacteristic
126C2×C4C4⋊C4×C21111yescharacteristic
127C2×C4C4⋊C4×C21111yescharacteristic
128C8—42no—
129C8—42no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—42no—
137C22—42no—
138C22#95×C21111yescharacteristic
139C22C4×C8×C21111yescharacteristic
140C22#97×C21111yescharacteristic
141C22C22⋊C8×C21112yes—
142C22C4⋊C8×C21112yes—
143C4—21no—
144C4—21no—
145C4—21no—
146C4—44no—
147C4#99×C21112yes—
148C2—22no—
149C2—21no—
150C2—21no—
151C2—22no—
152C2#336×C21111yescharacteristic
153C2#342×C21111yescharacteristic
154C2#344×C21111yescharacteristic
subgroupG/H
centerC2×C4×C211C22⋊C4
commutatorC22C4×C8×C211
Frattini, cosocleC23×C4C22×C211
socleC22×C211#95
prime radicalC24×C211C2×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
6. C22×C8×C211C4semidirect
12. C22⋊C8×C211C4semidirect
28. C22⋊C4×C211C8semidirect
77. C211#1282direct (N × H)
78. #1282C211direct (N × H)
84. C22×C8C4×C211semidirect
90. C22⋊C8C4×C211semidirect
106. C22⋊C4C8×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
C4×C22⋊C4×C2112
#1282211