Finite Group Databasebeta

← #41011427

Subgroups — #41011427

Order 27008.

140 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 35 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#27213066C211yesmaximal, characteristic
2#27212872C211yesmaximal, characteristic
3#343×C211C211yesmaximal, characteristic
4C22×Dic422C411yescharacteristic
5C2×Dic211⋊C4—21no—
6C22⋊(C211⋊C8)—21no—
7C211⋊#95—21no—
8C211⋊#95C411yescharacteristic
9C2×M16×C211—21no—
10C2×C4⋊C4×C211C2211yescharacteristic
11C2×Dic422—41no—
12C2×Dic422—42no—
13C22×Dic211—21no—
14C22×Dic211—21no—
15C2×C211⋊C8—41no—
16Dic211⋊C4—42no—
17C22×C4×C211C2×C411yescharacteristic
18C22×C4×C211D411yescharacteristic
19C22×C4×C211D411yescharacteristic
20M16×C211—22no—
21M16×C211—41no—
22C2×C8×C211—41no—
23C4⋊C4×C211—41no—
24Dic422—42no—
25Dic422—42no—
26C2×Dic211—22no—
27C2×Dic211—22no—
28C2×Dic211—41no—
29C2×Dic211—41no—
30C2×Dic211—41no—
31C2×Dic211—41no—
32C211⋊C8—81no—
33C23×C211C22⋊C411yescharacteristic
34C2×C4×C211—21no—
35C2×C4×C211—21no—
36C2×C4×C211—21no—
37C2×C4×C211—21no—
38C2×C4×C211—41no—
39C2×C4×C211—41no—
40C2×C4×C211SD1611yescharacteristic
41C2×C4×C211Q1611yescharacteristic
42C8×C211—42no—
43C22×C211—21no—
44C22×C211—21no—
45C22×C211#9911yescharacteristic
46C22×C211#10311yescharacteristic
47C22×C211#10411yescharacteristic
48C4×C211—21no—
49C4×C211—21no—
50C4×C211—41no—
51C4×C211—41no—
52Dic211—42no—
53Dic211—42no—
54C2×C211—21no—
55C2×C211—21no—
56C2×C211#32811yescharacteristic
57C2×C211AΣL(1,𝔽16)11yescharacteristic
58C2×C211#35611yescharacteristic
59C211#135411yescharacteristic, complemented, Sylow
60#1354—2111nomaximal, Sylow
61#395—2111no—
62#343D21111yescharacteristic
63#324—2111no—
64C22×Q8—2111no—
65C2×M16—21no—
66C2×C4⋊C4—4221no—
67C2×C4⋊C4C2×D21111yescharacteristic
68C22⋊C8—4221no—
69#95—4221no—
70#95—2111no—
71C2×Q8—8441no—
72C2×Q8—8442no—
73C22×C4—4221no—
74C22×C4—4221no—
75C22×C4C4×D21111yescharacteristic
76C22×C4C22⋊D21111yescharacteristic
77C22×C4D84411yescharacteristic
78M16—22no—
79M16—41no—
80C2×C8—41no—
81C2×C8—8441no—
82C4⋊C4—41no—
83C4⋊C4—8442no—
84C23(C2×D211)⋊C411yescharacteristic
85Q8—8442no—
86Q8—8442no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—8441no—
94C2×C4—8441no—
95C2×C4—8441no—
96C2×C4—8441no—
97C2×C4—4222no—
98C2×C4—4222no—
99C2×C4#1680995011yescharacteristic
100C2×C4#1680995211yescharacteristic
101C8—42no—
102C8—16881no—
103C22—21no—
104C22—21no—
105C22#2026989111yescharacteristic
106C22#2026990011yescharacteristic
107C22#2026990511yescharacteristic
108C4—21no—
109C4—21no—
110C4—41no—
111C4—41no—
112C4—8442no—
113C4—8442no—
114C2—21no—
115C2—21no—
116C2#2721288211yescharacteristic
117C2#2721293411yescharacteristic
118C2#2721294411yescharacteristic
subgroupG/H
centerC22#20269891
commutatorC22×C4×C211C2×C4
FrattiniC2×C4⋊C4C2×D211
Fitting#343×C211C2
socleC22×C211#99
prime radicalC23×C211C22⋊C4
cosocleC2×C4⋊C4×C211C22
hypercenter#343D211
hypocenterC211#1354
LCSG ▷ C22×C4×C211 ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #343

Complements

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

NH ≅ G/Ntype
59. C211#1354semidirect

Maximal subgroups

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

subgroupindex
#272130662
#272128722
#343×C2112
#1354211