Finite Group Databasebeta

← #41011404

Subgroups — #41011404

Order 27008.

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

normal subgroups only

This group has 124 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#398×C211C211yesmaximal, characteristic
4C2×#16809946—21no—
5C22⋊(C211⋊C8)—21no—
6C2×C4⋊C4×C211—21no—
7C2×C4⋊C4×C211C411yescharacteristic
8C2×C4⋊C4×C211C2211yescharacteristic
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—
22C4⋊C4×C211—41no—
23C4⋊C4×C211—41no—
24C211⋊C8—42no—
25C211⋊C8—81no—
26C23×C211C22⋊C411yescharacteristic
27C2×C4×C211—21no—
28C2×C4×C211—21no—
29C2×C4×C211—22no—
30C2×C4×C211—21no—
31C2×C4×C211—21no—
32C2×C4×C211—41no—
33C2×C4×C211—41no—
34C2×C4×C211—41no—
35C2×C4×C211—41no—
36C2×C4×C211—41no—
37C2×C4×C211—41no—
38C2×C4×C211—41no—
39C2×C4×C211D811yescharacteristic
40C2×C4×C211SD1611yescharacteristic
41C22×C211—21no—
42C22×C211—21no—
43C22×C211#9911yescharacteristic
44C22×C211#10211yescharacteristic
45C22×C211#10411yescharacteristic
46C4×C211—21no—
47C4×C211—21no—
48C4×C211—42no—
49C4×C211—41no—
50C4×C211—41no—
51C4×C211—81no—
52C2×C211—21no—
53C2×C211—21no—
54C2×C211#32711yescharacteristic
55C2×C211#35211yescharacteristic
56C2×C211#35611yescharacteristic
57C211#134611yescharacteristic, complemented, Sylow
58#1346—2111nomaximal, Sylow
59#398D21111yescharacteristic
60#343—2111no—
61#324—2111no—
62C2×M16—4221no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4Dic21111yescharacteristic
65C2×C4⋊C4C2×D21111yescharacteristic
66C22⋊C8—4221no—
67#95—21no—
68#95Dic21111yescharacteristic
69C22×C4—21no—
70C22×C4—21no—
71C22×C4C2×Dic21111yescharacteristic
72C22×C4C22⋊D21111yescharacteristic
73C22×C4C22⋊D21111yescharacteristic
74M16—8441no—
75M16—4222no—
76C2×C8—8441no—
77C2×C8—8441no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C23C22⋊Dic21111yescharacteristic
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4#1680994711yescharacteristic
96C2×C4#1680995011yescharacteristic
97C8—8442no—
98C8—16881no—
99C22—21no—
100C22—21no—
101C22#2026989211yescharacteristic
102C22#2026989811yescharacteristic
103C22#2026990411yescharacteristic
104C4—21no—
105C4—21no—
106C4—42no—
107C4—41no—
108C4—41no—
109C4—81no—
110C2—21no—
111C2—21no—
112C2#2721288011yescharacteristic
113C2#2721293611yescharacteristic
114C2#2721294511yescharacteristic
subgroupG/H
centerC22#20269892
commutatorC22×C4×C211C2×C4
FrattiniC2×C4⋊C4C2×D211
Fitting#398×C211C2
socleC22×C211#99
prime radicalC23×C211C22⋊C4
cosocleC2×C4⋊C4×C211C22
hypercenter#398D211
hypocenterC211#1346
LCSG ▷ C22×C4×C211 ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #398

Complements

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

NH ≅ G/Ntype
57. C211#1346semidirect

Maximal subgroups

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

subgroupindex
#272129152
#272128722
#398×C2112
#1346211