Finite Group Databasebeta

← #41013478

Subgroups — #41013478

Order 27008.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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#27213110C212yesmaximal
2#27212915C214yesmaximal, complemented
3C22×C4⋊C4×C211C211yesmaximal, characteristic
4C2×#16809946C2214yes—
5C22⋊(C211⋊C8)—22no—
6C22⋊(C211⋊C8)—22no—
7C23×C4×C211C412yes—
8C23×C4×C211C2211yescharacteristic
9C2×C4⋊C4×C211—22no—
10C2×C4⋊C4×C211—22no—
11C2×C4⋊C4×C211C412yes—
12C2×C4⋊C4×C211C2212yes—
13#16809946—28no—
14C2×C211⋊C8—24no—
15C24×C211C2×C411yescharacteristic
16C22×C4×C211—24no—
17C22×C4×C211—22no—
18C22×C4×C211—22no—
19C22×C4×C211—44no—
20C22×C4×C211C2×C411yescharacteristic
21C22×C4×C211C2×C414yes—
22C22×C4×C211D412yes—
23C22×C4×C211D412yes—
24C22×C4×C211C2311yescharacteristic
25C4⋊C4×C211—24no—
26C4⋊C4×C211—24no—
27C211⋊C8—44no—
28C23×C211—22no—
29C23×C211—22no—
30C23×C211—22no—
31C23×C211C22⋊C411yescharacteristic
32C23×C211C22⋊C411yescharacteristic
33C23×C211C22×C411yescharacteristic
34C2×C4×C211—22no—
35C2×C4×C211—24no—
36C2×C4×C211—24no—
37C2×C4×C211—24no—
38C2×C4×C211—44no—
39C2×C4×C211—44no—
40C2×C4×C211—44no—
41C2×C4×C211C22⋊C412yes—
42C2×C4×C211C2×D412yes—
43C2×C4×C211#4114yes—
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, complemented, Sylow
65#1821—2111nomaximal, Sylow
66C22×C4⋊C4D21111yescharacteristic
67#407—2112no—
68#343—2114no—
69C23×C4Dic21112yes—
70C23×C4C2×D21111yescharacteristic
71C2×M16—2114no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4Dic21112yes—
75C2×C4⋊C4C2×D21112yes—
76C22⋊C8—4222no—
77C22⋊C8—4222no—
78C24C2×Dic21111yescharacteristic
79C22×C4—24no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—44no—
83C22×C4C2×Dic21111yescharacteristic
84C22×C4C2×Dic21114yes—
85C22×C4C22⋊D21112yes—
86C22×C4C22⋊D21112yes—
87C22×C4C22×D21111yescharacteristic
88M16—4228no—
89C2×C8—4224no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C23—22no—
93C23—22no—
94C23—22no—
95C23C22⋊Dic21111yescharacteristic
96C23C22⋊Dic21111yescharacteristic
97C23C22×Dic21111yescharacteristic
98C2×C4—22no—
99C2×C4—24no—
100C2×C4—24no—
101C2×C4—24no—
102C2×C4—44no—
103C2×C4—44no—
104C2×C4—44no—
105C2×C4C22⋊Dic21112yes—
106C2×C4C2×C22⋊D21112yes—
107C2×C4#1680996214yes—
108C8—8444no—
109C22—28no—
110C22—21no—
111C22—21no—
112C22—21no—
113C22—21no—
114C22—21no—
115C22—21no—
116C22#2026989412yes—
117C22#2026989612yes—
118C22C2×C22⋊Dic21111yescharacteristic
119C22#2026993612yes—
120C4—22no—
121C4—44no—
122C4#2026996014yes—
123C2—22no—
124C2—22no—
125C2—22no—
126C2#2721300911yescharacteristic
127C2C2×#2026989411yescharacteristic
128C2C2×#2026989611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
FittingC22×C4⋊C4×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211C2×C4
cosocleC22×C4×C211C23
hypercenterC22×C4⋊C4D211
hypocenterC211#1821
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #27212915C2semidirect
64. C211#1821semidirect

Maximal subgroups

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

subgroupindex
#272131102
#272129152
C22×C4⋊C4×C2112
#1821211