Finite Group Databasebeta

← #41013484

Subgroups — #41013484

Order 27008.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
2#27213006C211yesmaximal, characteristic
3#27212908C214yesmaximal, complemented
4C22×C4⋊C4×C211C211yesmaximal, characteristic
5C22×C211⋊C8C2212yes—
6C2×C4⋊Dic211C2212yes—
7C2×C22⋊Dic211—21no—
8C22⋊(C211⋊C8)—22no—
9C211⋊#95—21no—
10C23×C4×C211—21no—
11C23×C4×C211C2211yescharacteristic
12C2×C4⋊C4×C211—21no—
13C2×C4⋊C4×C211—22no—
14C2×C4⋊C4×C211C414yescomplemented
15C2×C4⋊C4×C211C2212yes—
16C22×Dic211—22no—
17C2×C211⋊C8—22no—
18C2×C211⋊C8—24no—
19C4⋊Dic211—24no—
20C22⋊Dic211—42no—
21C24×C211D411yescharacteristic
22C22×C4×C211—24no—
23C22×C4×C211—24no—
24C22×C4×C211—22no—
25C22×C4×C211—42no—
26C22×C4×C211—42no—
27C22×C4×C211C2×C412yes—
28C22×C4×C211D411yescharacteristic
29C22×C4×C211D412yes—
30C22×C4×C211C2311yescharacteristic
31C4⋊C4×C211—24no—
32C4⋊C4×C211—22no—
33C4⋊C4×C211C2×C414yes—
34C2×Dic211—42no—
35C2×Dic211—42no—
36C2×Dic211—42no—
37C211⋊C8—42no—
38C23×C211—22no—
39C23×C211—22no—
40C23×C211C22⋊C412yes—
41C23×C211D812yes—
42C23×C211SD1612yes—
43C23×C211C2×D411yescharacteristic
44C2×C4×C211—22no—
45C2×C4×C211—24no—
46C2×C4×C211—24no—
47C2×C4×C211—24no—
48C2×C4×C211—48no—
49C2×C4×C211—42no—
50C2×C4×C211—42no—
51C2×C4×C211C22⋊C412yes—
52C2×C4×C211C22×C411yescharacteristic
53C2×C4×C211C2×D411yescharacteristic
54C2×C4×C211#4114yes—
55C22×C211—22no—
56C22×C211—22no—
57C22×C211—24no—
58C22×C211—24no—
59C22×C211#10214yes—
60C22×C211C2×C22⋊C411yescharacteristic
61C22×C211#12312yes—
62C22×C211C2×D811yescharacteristic
63C22×C211C2×SD1611yescharacteristic
64C22×C211#13712yes—
65C4×C211—22no—
66C4×C211—44no—
67C4×C211#11712yes—
68C4×C211#12312yes—
69Dic211—82no—
70C2×C211—24no—
71C2×C211#38111yescharacteristic
72C2×C211C2×#10211yescharacteristic
73C2×C211#41911yescharacteristic
74C2×C211#48012yes—
75C2×C211#48312yes—
76C211#182311yescharacteristic, complemented, Sylow
77#1823—2111nomaximal, Sylow
78C22×C4⋊C4D21111yescharacteristic
79C2×C22⋊C8—2111no—
80#380—2111no—
81#340—2114no—
82C23×C4—21no—
83C23×C4C2×D21111yescharacteristic
84C22×C8—2112no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—2112no—
88C2×C4⋊C4Dic21114yescomplemented
89C2×C4⋊C4C2×D21112yes—
90C2×C22⋊C4—4221no—
91C22⋊C8—4222no—
92#95—4221no—
93C24C22⋊D21111yescharacteristic
94C22×C4—24no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—4222no—
100C22×C4C2×Dic21112yes—
101C22×C4C22⋊D21111yescharacteristic
102C22×C4C22⋊D21112yes—
103C22×C4C22×D21111yescharacteristic
104C2×C8—4222no—
105C2×C8—4224no—
106C4⋊C4—24no—
107C4⋊C4—22no—
108C4⋊C4—4224no—
109C4⋊C4C2×Dic21114yes—
110C22⋊C4—8442no—
111C23—22no—
112C23—22no—
113C23C22⋊Dic21112yes—
114C23#1680994712yes—
115C23#1680995012yes—
116C23C2×C22⋊D21111yescharacteristic
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—24no—
120C2×C4—24no—
121C2×C4—48no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—8442no—
125C2×C4—8442no—
126C2×C4—8442no—
127C2×C4C22⋊Dic21112yes—
128C2×C4C22×Dic21111yescharacteristic
129C2×C4C2×C22⋊D21111yescharacteristic
130C2×C4#1680996214yes—
131C8—8442no—
132C22—22no—
133C22—24no—
134C22—24no—
135C22—22no—
136C22#2026989814yes—
137C22C2×C22⋊Dic21111yescharacteristic
138C22#2026996012yes—
139C22C2×#1680994711yescharacteristic
140C22C2×#1680995011yescharacteristic
141C22#2027000812yes—
142C4—22no—
143C4—44no—
144C4—16882no—
145C4#2026993612yes—
146C4#2026996012yes—
147C2—24no—
148C2#2721300911yescharacteristic
149C2C2×#2026989811yescharacteristic
150C2#2721315511yescharacteristic
151C2#2721345912yes—
152C2#2721347612yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC22×C4⋊C4×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC22×C4⋊C4D211
hypocenterC211#1823
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #27212908C2semidirect
14. C2×C4⋊C4×C211C4semidirect
76. C211#1823semidirect
88. C2×C4⋊C4Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C211⋊C8)2
#272130062
#272129082
C22×C4⋊C4×C2112
#1823211