Finite Group Databasebeta

← #41011402

Subgroups — #41011402

Order 27008.

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

normal subgroups only

This group has 168 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#27213063C211yesmaximal, characteristic
2#27212915C211yesmaximal, characteristic, complemented
3#324×C211C211yesmaximal, characteristic, complemented
4C22×D844C411yescharacteristic
5C2×#16809946—21no—
6C2×(C2×D211)⋊C4—21no—
7C2×(C2×D211)⋊C4—21no—
8C211⋊#95C411yescharacteristic
9C2×C4⋊C4×C211C2211yescharacteristic
10C22⋊C8×C211—21no—
11C23×D211—21no—
12C2×D844—41no—
13C2×D844—42no—
14C22×Dic211—21no—
15#16809946—22no—
16#16809946—41no—
17C2×C211⋊C8—41no—
18(C2×D211)⋊C4—42no—
19(C2×D211)⋊C4—41no—
20(C2×D211)⋊C4—41no—
21C22×C4×C211C2×C411yescharacteristic
22C22×C4×C211D411yescharacteristic
23C22×C4×C211D411yescharacteristic
24C2×C8×C211—41no—
25C4⋊C4×C211—41no—
26C22×D211—22no—
27C22×D211—41no—
28C22×D211—41no—
29C22×D211—41no—
30C22×D211—41no—
31C22×D211—41no—
32C22×D211—41no—
33D844—42no—
34D844—42no—
35C2×Dic211—41no—
36C2×Dic211—41no—
37C2×Dic211—41no—
38C211⋊C8—42no—
39C23×C211C22⋊C411yescharacteristic
40C2×C4×C211—21no—
41C2×C4×C211—21no—
42C2×C4×C211—21no—
43C2×C4×C211—21no—
44C2×C4×C211—41no—
45C2×C4×C211—41no—
46C2×C4×C211D811yescharacteristic
47C2×C4×C211SD1611yescharacteristic
48C8×C211—81no—
49C22×C211—21no—
50C22×C211—21no—
51C22×C211#9911yescharacteristic
52C22×C211#10211yescharacteristic
53C22×C211#10411yescharacteristic
54C2×D211—42no—
55C2×D211—42no—
56C2×D211—42no—
57C2×D211—81no—
58C2×D211—81no—
59C2×D211—81no—
60C2×D211—81no—
61C4×C211—21no—
62C4×C211—21no—
63C4×C211—41no—
64C4×C211—41no—
65Dic211—81no—
66C2×C211—21no—
67C2×C211—21no—
68C2×C211#32711yescharacteristic
69C2×C211AΣL(1,𝔽16)11yescharacteristic
70C2×C211#35511yescharacteristic
71D211—82no—
72C211#134511yescharacteristic, complemented, Sylow
73#1345—2111nomaximal, Sylow
74#394—2111no—
75#343—2111no—
76#324D21111yescharacteristic, complemented
77C22×D4—2111no—
78C2×M16—4221no—
79C2×C4⋊C4C2×D21111yescharacteristic
80C2×C22⋊C4—4221no—
81C2×C22⋊C4—4221no—
82C22⋊C8—21no—
83#95—2111no—
84C24—4221no—
85C2×D4—8441no—
86C2×D4—8442no—
87C22×C4—4221no—
88C22×C4C4×D21111yescharacteristic
89C22×C4C22⋊D21111yescharacteristic
90C22×C4D84411yescharacteristic
91M16—8441no—
92M16—4222no—
93C2×C8—41no—
94C2×C8—8441no—
95C4⋊C4—41no—
96C22⋊C4—8442no—
97C22⋊C4—8441no—
98C22⋊C4—8441no—
99C23—8441no—
100C23—8441no—
101C23—8441no—
102C23—8441no—
103C23—8441no—
104C23—8441no—
105C23—4222no—
106C23(C2×D211)⋊C411yescharacteristic
107D4—8442no—
108D4—8442no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—8441no—
116C2×C4—8441no—
117C2×C4—8441no—
118C2×C4D168811yescharacteristic
119C2×C4#1680995111yescharacteristic
120C8—81no—
121C8—8442no—
122C22—21no—
123C22—21no—
124C22—8442no—
125C22—8442no—
126C22—8442no—
127C22—16881no—
128C22—16881no—
129C22—16881no—
130C22—16881no—
131C22#2026989111yescharacteristic
132C22#2026989911yescharacteristic
133C22#2026990311yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—16881no—
139C2—21no—
140C2—21no—
141C2—16882no—
142C2#2721288111yescharacteristic
143C2#2721293211yescharacteristic
144C2#2721294211yescharacteristic
subgroupG/H
centerC22#20269891
commutatorC22×C4×C211C2×C4
FrattiniC2×C4⋊C4C2×D211
Fitting#324×C211C2
socleC22×C211#99
prime radicalC22×D844C4
cosocleC2×C4⋊C4×C211C22
hypercenter#324D211
hypocenterC211#1345
LCSG ▷ C22×C4×C211 ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27212915C2semidirect
3. #324×C211C2semidirect
72. C211#1345semidirect
76. #324D211semidirect

Maximal subgroups

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

subgroupindex
#272130632
#272129152
#324×C2112
#1345211