Finite Group Databasebeta

← #41011425

Subgroups — #41011425

Order 27008.

116 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 37 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#27212915C211yesmaximal, characteristic
2#27212872C211yesmaximal, characteristic
3#400×C211C211yesmaximal, characteristic
4C2×#16809946—21no—
5C22⋊(C211⋊C8)—21no—
6C2×C4⋊C4×C211C411yescharacteristic
7C2×C4⋊C4×C211C2211yescharacteristic
8#95×C211—21no—
9#95×C211—21no—
10#95×C211C411yescharacteristic
11#16809946—22no—
12#16809946—41no—
13C2×C211⋊C8—41no—
14C2×C211⋊C8—41no—
15C22×C4×C211—21no—
16C22×C4×C211—21no—
17C22×C4×C211C2×C411yescharacteristic
18C22×C4×C211D411yescharacteristic
19C22×C4×C211D411yescharacteristic
20C4⋊C4×C211—41no—
21C4⋊C4×C211—41no—
22C211⋊C8—42no—
23C211⋊C8—81no—
24C23×C211C22⋊C411yescharacteristic
25C2×C4×C211—21no—
26C2×C4×C211—21no—
27C2×C4×C211—21no—
28C2×C4×C211—21no—
29C2×C4×C211—41no—
30C2×C4×C211—41no—
31C2×C4×C211—41no—
32C2×C4×C211—41no—
33C2×C4×C211—42no—
34C2×C4×C211—41no—
35C2×C4×C211—41no—
36C2×C4×C211SD1611yescharacteristic
37C2×C4×C211Q1611yescharacteristic
38C22×C211—21no—
39C22×C211—21no—
40C22×C211#9911yescharacteristic
41C22×C211#10311yescharacteristic
42C22×C211#10411yescharacteristic
43C4×C211—21no—
44C4×C211—21no—
45C4×C211—41no—
46C4×C211—41no—
47C4×C211—81no—
48C4×C211—81no—
49C2×C211—21no—
50C2×C211—21no—
51C2×C211#32811yescharacteristic
52C2×C211#35211yescharacteristic
53C2×C211#35511yescharacteristic
54C211#135311yescharacteristic, complemented, Sylow
55#1353—2111nomaximal, Sylow
56#400D21111yescharacteristic
57#343—2111no—
58#324—2111no—
59C2×M16—4221no—
60C2×C4⋊C4Dic21111yescharacteristic
61C2×C4⋊C4C2×D21111yescharacteristic
62C22⋊C8—4221no—
63#95—21no—
64#95—21no—
65#95Dic21111yescharacteristic
66C22×C4—21no—
67C22×C4—21no—
68C22×C4C2×Dic21111yescharacteristic
69C22×C4C22⋊D21111yescharacteristic
70C22×C4C22⋊D21111yescharacteristic
71M16—8441no—
72M16—4222no—
73C2×C8—8441no—
74C2×C8—8441no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C23C22⋊Dic21111yescharacteristic
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—41no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—42no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4#1680994911yescharacteristic
90C2×C4#1680995211yescharacteristic
91C8—8442no—
92C8—16881no—
93C22—21no—
94C22—21no—
95C22#2026989211yescharacteristic
96C22#2026990111yescharacteristic
97C22#2026990411yescharacteristic
98C4—21no—
99C4—21no—
100C4—41no—
101C4—41no—
102C4—81no—
103C4—81no—
104C2—21no—
105C2—21no—
106C2#2721288311yescharacteristic
107C2#2721293611yescharacteristic
108C2#2721294311yescharacteristic
subgroupG/H
centerC22#20269892
commutatorC22×C4×C211C2×C4
FrattiniC2×C4⋊C4C2×D211
Fitting#400×C211C2
socleC22×C211#99
prime radicalC23×C211C22⋊C4
cosocleC2×C4⋊C4×C211C22
hypercenter#400D211
hypocenterC211#1353
LCSG ▷ C22×C4×C211 ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #400

Complements

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

NH ≅ G/Ntype
54. C211#1353semidirect

Maximal subgroups

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

subgroupindex
#272129152
#272128722
#400×C2112
#1353211