Finite Group Databasebeta

← #41011401

Subgroups — #41011401

Order 27008.

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

normal subgroups only

This group has 168 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, complemented
2#27212872C211yesmaximal, characteristic, complemented
3#394×C211C211yesmaximal, characteristic
4C2×#16809946—21no—
5C22⋊(C211⋊C8)—21no—
6C22×D4×C211C411yescharacteristic
7C2×C4⋊C4×C211C2211yescharacteristic
8C2×C22⋊C4×C211—21no—
9C2×C22⋊C4×C211—21no—
10#95×C211C411yescharacteristic
11#16809946—22no—
12#16809946—41no—
13C2×C211⋊C8—41no—
14C2×C211⋊C8—41no—
15C24×C211—21no—
16C2×D4×C211—41no—
17C2×D4×C211—42no—
18C22×C4×C211—21no—
19C22×C4×C211C2×C411yescharacteristic
20C22×C4×C211D411yescharacteristic
21C22×C4×C211D411yescharacteristic
22C4⋊C4×C211—41no—
23C22⋊C4×C211—42no—
24C22⋊C4×C211—41no—
25C22⋊C4×C211—41no—
26C211⋊C8—42no—
27C211⋊C8—81no—
28C23×C211—22no—
29C23×C211—41no—
30C23×C211—41no—
31C23×C211—41no—
32C23×C211—41no—
33C23×C211—41no—
34C23×C211—41no—
35C23×C211C22⋊C411yescharacteristic
36D4×C211—42no—
37D4×C211—42no—
38C2×C4×C211—21no—
39C2×C4×C211—21no—
40C2×C4×C211—21no—
41C2×C4×C211—21no—
42C2×C4×C211—41no—
43C2×C4×C211—41no—
44C2×C4×C211—41no—
45C2×C4×C211—41no—
46C2×C4×C211—41no—
47C2×C4×C211D811yescharacteristic
48C2×C4×C211SD1611yescharacteristic
49C22×C211—21no—
50C22×C211—21no—
51C22×C211—42no—
52C22×C211—42no—
53C22×C211—42no—
54C22×C211—81no—
55C22×C211—81no—
56C22×C211—81no—
57C22×C211—81no—
58C22×C211#9911yescharacteristic
59C22×C211#10211yescharacteristic
60C22×C211#10411yescharacteristic
61C4×C211—21no—
62C4×C211—21no—
63C4×C211—41no—
64C4×C211—41no—
65C4×C211—81no—
66C2×C211—21no—
67C2×C211—21no—
68C2×C211—82no—
69C2×C211#32711yescharacteristic
70C2×C211AΣL(1,𝔽16)11yescharacteristic
71C2×C211#35511yescharacteristic
72C211#134511yescharacteristic, complemented, Sylow
73#1345—2111nomaximal, Sylow
74#394D21111yescharacteristic
75#343—2111no—
76#324—2111no—
77C22×D4Dic21111yescharacteristic
78C2×M16—4221no—
79C2×C4⋊C4C2×D21111yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C22⋊C8—4221no—
83#95Dic21111yescharacteristic
84C24—21no—
85C2×D4—41no—
86C2×D4—42no—
87C22×C4—21no—
88C22×C4C2×Dic21111yescharacteristic
89C22×C4C22⋊D21111yescharacteristic
90C22×C4C22⋊D21111yescharacteristic
91M16—8441no—
92M16—4222no—
93C2×C8—8441no—
94C2×C8—8441no—
95C4⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C23—22no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23C22⋊Dic21111yescharacteristic
107D4—42no—
108D4—42no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4#1680994711yescharacteristic
119C2×C4#1680995011yescharacteristic
120C8—8442no—
121C8—16881no—
122C22—21no—
123C22—21no—
124C22—42no—
125C22—42no—
126C22—42no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22—81no—
131C22#2026989211yescharacteristic
132C22#2026989811yescharacteristic
133C22#2026990411yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—81no—
139C2—21no—
140C2—21no—
141C2—82no—
142C2#2721288011yescharacteristic
143C2#2721293311yescharacteristic
144C2#2721294311yescharacteristic
subgroupG/H
centerC22#20269892
commutatorC22×C4×C211C2×C4
FrattiniC2×C4⋊C4C2×D211
Fitting#394×C211C2
socleC22×C211#99
prime radicalC22×D4×C211C4
cosocleC2×C4⋊C4×C211C22
hypercenter#394D211
hypocenterC211#1345
LCSG ▷ C22×C4×C211 ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #27212915C2semidirect
2. #27212872C2semidirect
72. C211#1345semidirect

Maximal subgroups

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

subgroupindex
#272129152
#272128722
#394×C2112
#1345211