Finite Group Databasebeta

← #41012000

Subgroups — #41012000

Order 27008.

212 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 81 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#27213203C211yesmaximal, characteristic
2C4×#16809946C211yesmaximal, characteristic
3#27212888C214yesmaximal, complemented
4C2×#125×C211C211yesmaximal, characteristic
5C2×#16809946—21no—
6C4⋊(C211⋊C8)—21no—
7C22⋊(C211⋊C8)—21no—
8#20269888C2212yes—
9#20269888C2212yes—
10C4×C211⋊C8—21no—
11#125×C211—21no—
12#125×C211—21no—
13#125×C211C412yes—
14#125×C211C2212yes—
15C2×C4⋊C4×C211—21no—
16C2×C4⋊C4×C211—21no—
17C2×C4⋊C4×C211C411yescharacteristic
18C2×C4⋊C4×C211C411yescharacteristic
19C2×C42×C211C2211yescharacteristic
20#16809946—24no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—22no—
23C22×C4×C211—21no—
24C22×C4×C211—21no—
25C22×C4×C211C2×C411yescharacteristic
26C22×C4×C211D411yescharacteristic
27C22×C4×C211D411yescharacteristic
28C4⋊C4×C211—22no—
29C4⋊C4×C211—22no—
30C4⋊C4×C211—21no—
31C4⋊C4×C211—21no—
32C4⋊C4×C211—22no—
33C4⋊C4×C211—22no—
34C4⋊C4×C211C2×C412yes—
35C4⋊C4×C211C2×C412yes—
36C42×C211C2×C411yescharacteristic
37C42×C211D411yescharacteristic
38C42×C211D411yescharacteristic
39C42×C211C2311yescharacteristic
40C211⋊C8—24no—
41C211⋊C8—42no—
42C23×C211C22⋊C411yescharacteristic
43C2×C4×C211—22no—
44C2×C4×C211—22no—
45C2×C4×C211—22no—
46C2×C4×C211—22no—
47C2×C4×C211—41no—
48C2×C4×C211—42no—
49C2×C4×C211—42no—
50C2×C4×C211C22⋊C411yescharacteristic
51C2×C4×C211C22⋊C411yescharacteristic
52C2×C4×C211C22⋊C411yescharacteristic
53C2×C4×C211C22×C411yescharacteristic
54C2×C4×C211C2×D411yescharacteristic
55C2×C4×C211C2×D411yescharacteristic
56C22×C211—21no—
57C22×C211—21no—
58C22×C211#10412yes—
59C22×C211C2×C22⋊C411yescharacteristic
60C4×C211—21no—
61C4×C211—21no—
62C4×C211—21no—
63C4×C211—21no—
64C4×C211—21no—
65C4×C211—42no—
66C4×C211—42no—
67C4×C211#10112yes—
68C2×C211—22no—
69C2×C211C2×#10111yescharacteristic
70C2×C211C2×#10411yescharacteristic
71C2×C211#42111yescharacteristic
72C211#152811yescharacteristic, complemented, Sylow
73#1528—2111nomaximal, Sylow
74C2×#125D21111yescharacteristic
75#432—2111no—
76C4×M16—2111no—
77#330—2114no—
78C2×M16—4221no—
79#125—21no—
80#125—21no—
81#125Dic21112yes—
82#125C2×D21112yes—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4Dic21111yescharacteristic
86C2×C4⋊C4Dic21111yescharacteristic
87C2×C42C2×D21111yescharacteristic
88C4⋊C8—4221no—
89C22⋊C8—4221no—
90#97—2112no—
91#97—2112no—
92C4×C8—4221no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4C2×Dic21111yescharacteristic
96C22×C4C22⋊D21111yescharacteristic
97C22×C4C22⋊D21111yescharacteristic
98M16—4224no—
99C2×C8—4222no—
100C2×C8—4222no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4C2×Dic21112yes—
108C4⋊C4C2×Dic21112yes—
109C42C2×Dic21111yescharacteristic
110C42C22⋊D21111yescharacteristic
111C42C22⋊D21111yescharacteristic
112C42C22×D21111yescharacteristic
113C23C22⋊Dic21111yescharacteristic
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4C22⋊Dic21111yescharacteristic
122C2×C4C22⋊Dic21111yescharacteristic
123C2×C4C22⋊Dic21111yescharacteristic
124C2×C4C22×Dic21111yescharacteristic
125C2×C4C2×C22⋊D21111yescharacteristic
126C2×C4C2×C22⋊D21111yescharacteristic
127C8—8442no—
128C8—4224no—
129C22—21no—
130C22—21no—
131C22#2026990412yes—
132C22C2×C22⋊Dic21111yescharacteristic
133C4—21no—
134C4—21no—
135C4—21no—
136C4—21no—
137C4—21no—
138C4—42no—
139C4—42no—
140C4#2026989612yes—
141C2—22no—
142C2C2×#2026989611yescharacteristic
143C2C2×#2026990411yescharacteristic
144C2#2721316311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
FittingC2×#125×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×C211C22⋊C4
cosocleC42×C211C23
hypercenterC2×#125D211
hypocenterC211#1528
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
3. #27212888C2semidirect
72. C211#1528semidirect

Maximal subgroups

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

subgroupindex
#272132032
C4×#168099462
#272128882
C2×#125×C2112
#1528211