Finite Group Databasebeta

← #1663×C211 #41009283

Subgroups — #1663×C211 #41009283

Order 27008.

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

normal subgroups only

This group has 172 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#564×C211C211yesmaximal, characteristic
2#501×C211C212yesmaximal
3#491×C211C211yesmaximal, characteristic
4#330×C211C212yesmaximal
5#330×C211C211yesmaximal, characteristic
6C4⋊Q8×C211C2211yescharacteristic
7#125×C211—22no—
8#125×C211—22no—
9#125×C211—22no—
10#125×C211C2211yescharacteristic
11#125×C211C2212yes—
12#107×C211—22no—
13#106×C211—22no—
14#103×C211—22no—
15#97×C211C2211yescharacteristic
16#97×C211C2212yes—
17C2×Q8×C211—21no—
18C2×C8×C211—22no—
19C2×C8×C211—21no—
20C4⋊C4×C211—22no—
21C4⋊C4×C211—22no—
22C4⋊C4×C211—22no—
23C4⋊C4×C211—22no—
24C4⋊C4×C211—22no—
25C4⋊C4×C211—22no—
26C4⋊C4×C211—22no—
27C4⋊C4×C211D412yes—
28C4⋊C4×C211D412yes—
29C4⋊C4×C211D412yes—
30C42×C211—22no—
31C42×C211C2311yescharacteristic
32Q8×C211—42no—
33C2×C4×C211—22no—
34C2×C4×C211—22no—
35C2×C4×C211—22no—
36C2×C4×C211—41no—
37C2×C4×C211C2×D412yes—
38C2×C4×C211C2×D411yescharacteristic
39C8×C211—24no—
40C8×C211—41no—
41C22×C211#12011yescharacteristic
42C4×C211—21no—
43C4×C211—22no—
44C4×C211—42no—
45C4×C211—42no—
46C4×C211—42no—
47C4×C211—81no—
48C2×C211#45411yescharacteristic
49C2×C211#45612yes—
50C211#166311yescharacteristic, direct product factor, Sylow
51#1663C21111yesmaximal, characteristic, direct product factor, Sylow
52#564C2×C21111yescharacteristic
53#501C2×C21112yes—
54#491C2×C21111yescharacteristic
55#330C2×C21112yes—
56#330C2×C21111yescharacteristic
57C4⋊Q8C22×C21111yescharacteristic
58#125—22no—
59#125—22no—
60#125—22no—
61#125C22×C21111yescharacteristic
62#125C22×C21112yes—
63#107—22no—
64#106—22no—
65#103—22no—
66#97C22×C21111yescharacteristic
67#97C22×C21112yes—
68C2×Q8—21no—
69C2×C8—22no—
70C2×C8—21no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—22no—
78C4⋊C4D4×C21112yes—
79C4⋊C4D4×C21112yes—
80C4⋊C4D4×C21112yes—
81C42—22no—
82C42C23×C21111yescharacteristic
83Q8—42no—
84C2×C4—22no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—41no—
88C2×C4C2×D4×C21112yes—
89C2×C4C2×D4×C21111yescharacteristic
90C8—24no—
91C8—41no—
92C22#120×C21111yescharacteristic
93C4—21no—
94C4—22no—
95C4—42no—
96C4—42no—
97C4—42no—
98C4—81no—
99C2#454×C21111yescharacteristic
100C2#456×C21112yes—
subgroupG/H
center, socle, prime radicalC22×C211#120
commutator, Frattini, cosocleC42C23×C211
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
50. C211#1663direct (N × H)
51. #1663C211direct (N × H)

Maximal subgroups

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

subgroupindex
#564×C2112
#501×C2112
#491×C2112
#330×C2112
#1663211