Finite Group Databasebeta

← #41011305

Subgroups — #41011305

Order 27008.

200 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 61 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#27213110C211yesmaximal, characteristic
2#27213006C211yesmaximal, characteristic
3#407×C211C211yesmaximal, characteristic
4C2×#16809946C412yescomplemented
5C2×C4⋊Dic211—21no—
6C2×C22⋊Dic211C412yes—
7C22⋊(C211⋊C8)—21no—
8C22⋊(C211⋊C8)—21no—
9C211⋊#95—21no—
10C23×C4×C211C2211yescharacteristic
11C2×M16×C211—21no—
12C22⋊C8×C211—21no—
13C22⋊C8×C211C412yescomplemented
14C22×Dic211—22no—
15#16809946—42no—
16C2×C211⋊C8—22no—
17C4⋊Dic211—42no—
18C22⋊Dic211—42no—
19C24×C211C2×C411yescharacteristic
20C22×C4×C211—42no—
21C22×C4×C211C2×C411yescharacteristic
22C22×C4×C211C2×C411yescharacteristic
23C22×C4×C211D411yescharacteristic
24C22×C4×C211D411yescharacteristic
25C22×C4×C211D411yescharacteristic
26C22×C4×C211Q811yescharacteristic
27M16×C211—42no—
28C2×C8×C211—22no—
29C2×Dic211—42no—
30C2×Dic211—42no—
31C2×Dic211—42no—
32C211⋊C8—42no—
33C23×C211—21no—
34C23×C211—21no—
35C23×C211—41no—
36C23×C211—41no—
37C23×C211C4211yescharacteristic
38C23×C211C22⋊C411yescharacteristic
39C23×C211C4⋊C411yescharacteristic
40C2×C4×C211—22no—
41C2×C4×C211—22no—
42C2×C4×C211—22no—
43C2×C4×C211—41no—
44C2×C4×C211—41no—
45C2×C4×C211—41no—
46C2×C4×C211C22⋊C411yescharacteristic
47C2×C4×C211C22⋊C411yescharacteristic
48C2×C4×C211C4⋊C411yescharacteristic
49C2×C4×C211C4⋊C411yescharacteristic
50C8×C211—42no—
51C22×C211—21no—
52C22×C211—21no—
53C22×C211—21no—
54C22×C211—21no—
55C22×C211—21no—
56C22×C211—21no—
57C22×C211—21no—
58C22×C211—21no—
59C22×C211—42no—
60C22×C211—42no—
61C22×C211#9511yescharacteristic
62C22×C211#10011yescharacteristic
63C22×C211#10111yescharacteristic
64C4×C211—21no—
65C4×C211—21no—
66C4×C211—21no—
67C4×C211#9912yes—
68Dic211—82no—
69C2×C211—21no—
70C2×C211—21no—
71C2×C211—41no—
72C2×C211—41no—
73C2×C211#34211yescharacteristic
74C2×C211#34311yescharacteristic
75C2×C211#34411yescharacteristic
76C211#130511yescharacteristic, complemented, Sylow
77#1305—2111nomaximal, Sylow
78#407—2111no—
79#407D21111yescharacteristic
80#380—2111no—
81C23×C4C2×D21111yescharacteristic
82C2×M16—21no—
83C2×M16—2112no—
84C2×C4⋊C4—4221no—
85C2×C22⋊C4—2112no—
86C22⋊C8—21no—
87C22⋊C8—4221no—
88C22⋊C8—4221no—
89C22⋊C8Dic21112yescomplemented
90#95—4221no—
91C24C4×D21111yescharacteristic
92C22×C4—42no—
93C22×C4—4222no—
94C22×C4C4×D21111yescharacteristic
95C22×C4C2×Dic21111yescharacteristic
96C22×C4C22⋊D21111yescharacteristic
97C22×C4C22⋊D21111yescharacteristic
98C22×C4D84411yescharacteristic
99C22×C4Dic42211yescharacteristic
100M16—42no—
101M16—8442no—
102C2×C8—22no—
103C2×C8—4222no—
104C4⋊C4—8442no—
105C22⋊C4—8442no—
106C23—21no—
107C23—21no—
108C23—41no—
109C23—41no—
110C23C4×Dic21111yescharacteristic
111C23(C2×D211)⋊C411yescharacteristic
112C23Dic211⋊C411yescharacteristic
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—8442no—
120C2×C4—8442no—
121C2×C4—8442no—
122C2×C4(C2×D211)⋊C411yescharacteristic
123C2×C4C22⋊Dic21111yescharacteristic
124C2×C4Dic211⋊C411yescharacteristic
125C2×C4C4⋊Dic21111yescharacteristic
126C8—42no—
127C8—8442no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—42no—
137C22—42no—
138C22C211⋊#9511yescharacteristic
139C22#2026989311yescharacteristic
140C22#2026989511yescharacteristic
141C4—21no—
142C4—21no—
143C4—21no—
144C4—16882no—
145C4#2026989112yes—
146C2—21no—
147C2—21no—
148C2—41no—
149C2—41no—
150C2#2721291211yescharacteristic
151C2#2721291411yescharacteristic
152C2#2721291711yescharacteristic
subgroupG/H
centerC22C211⋊#95
commutatorC23×C211C42
FrattiniC23×C4C2×D211
Fitting#407×C211C2
socleC22×C211#95
prime radicalC24×C211C2×C4
cosocleC23×C4×C211C22
hypercenter#407D211
hypocenterC211#1305
LCSG ▷ C23×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
4. C2×#16809946C4semidirect
13. C22⋊C8×C211C4semidirect
76. C211#1305semidirect
89. C22⋊C8Dic211semidirect

Maximal subgroups

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

subgroupindex
#272131102
#272130062
#407×C2112
#1305211