Finite Group Databasebeta

← #1662×C211 #41009282

Subgroups — #1662×C211 #41009282

Order 27008.

196 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 60 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#562×C211C211yesmaximal, characteristic
2#491×C211C211yesmaximal, characteristic, complemented
3#489×C211C212yesmaximal, complemented
4#330×C211C211yesmaximal, characteristic, complemented
5#329×C211C212yesmaximal, complemented
6C4⋊Q8×C211C2211yescharacteristic
7#125×C211C2211yescharacteristic
8#124×C211C2212yes—
9#123×C211—22no—
10C22⋊Q8×C211—22no—
11#121×C211—22no—
12#103×C211—22no—
13#103×C211—22no—
14#102×C211—22no—
15#97×C211C2211yescharacteristic
16#97×C211C2212yes—
17C2×Q8×C211—21no—
18C2×Q8×C211D412yes—
19C2×D4×C211—22no—
20C2×D4×C211D412yes—
21C22×C4×C211—22no—
22C2×C8×C211—22no—
23C2×C8×C211—21no—
24C4⋊C4×C211—22no—
25C4⋊C4×C211—22no—
26C4⋊C4×C211—22no—
27C4⋊C4×C211D412yes—
28C22⋊C4×C211—22no—
29C22⋊C4×C211—22no—
30C22⋊C4×C211—22no—
31C42×C211C2311yescharacteristic
32C23×C211—22no—
33Q8×C211—42no—
34Q8×C211—42no—
35D4×C211—42no—
36D4×C211—82no—
37C2×C4×C211—22no—
38C2×C4×C211—22no—
39C2×C4×C211—41no—
40C2×C4×C211—82no—
41C2×C4×C211C2×D411yescharacteristic
42C2×C4×C211C2×D412yes—
43C8×C211—41no—
44C8×C211—42no—
45C22×C211—42no—
46C22×C211—42no—
47C22×C211—42no—
48C22×C211#12011yescharacteristic
49C4×C211—21no—
50C4×C211—22no—
51C4×C211—42no—
52C4×C211—42no—
53C4×C211—81no—
54C2×C211—82no—
55C2×C211#45411yescharacteristic
56C2×C211#45512yes—
57C211#166211yescharacteristic, direct product factor, Sylow
58#1662C21111yesmaximal, characteristic, direct product factor, Sylow
59#562C2×C21111yescharacteristic
60#491C2×C21111yescharacteristic, complemented
61#489C2×C21112yescomplemented
62#330C2×C21111yescharacteristic, complemented
63#329C2×C21112yescomplemented
64C4⋊Q8C22×C21111yescharacteristic
65#125C22×C21111yescharacteristic
66#124C22×C21112yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—22no—
71#103—22no—
72#102—22no—
73#97C22×C21111yescharacteristic
74#97C22×C21112yes—
75C2×Q8—21no—
76C2×Q8D4×C21112yes—
77C2×D4—22no—
78C2×D4D4×C21112yes—
79C22×C4—22no—
80C2×C8—22no—
81C2×C8—21no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4D4×C21112yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C23×C21111yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—42no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—41no—
98C2×C4—82no—
99C2×C4C2×D4×C21111yescharacteristic
100C2×C4C2×D4×C21112yes—
101C8—41no—
102C8—42no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#120×C21111yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—81no—
112C2—82no—
113C2#454×C21111yescharacteristic
114C2#455×C21112yes—
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC42C23×C211
prime radical#562×C211C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #491×C211C2semidirect
3. #489×C211C2semidirect
4. #330×C211C2semidirect
5. #329×C211C2semidirect
57. C211#1662direct (N × H)
58. #1662C211direct (N × H)
60. #491C2×C211semidirect
61. #489C2×C211semidirect
62. #330C2×C211semidirect
63. #329C2×C211semidirect

Maximal subgroups

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

subgroupindex
#562×C2112
#491×C2112
#489×C2112
#330×C2112
#329×C2112
#1662211