Finite Group Databasebeta

← #1819×C211 #41009439

Subgroups — #1819×C211 #41009439

Order 27008.

504 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C22⋊C4×C211C211yesmaximal, characteristic
2#379×C211C212yesmaximal
3#342×C211C214yesmaximal, complemented
4C25×C211C2211yescharacteristic
5C23×C4×C211C412yescomplemented
6C2×C22⋊C4×C211—24no—
7C2×C22⋊C4×C211—24no—
8C2×C22⋊C4×C211C412yescomplemented
9C2×C22⋊C4×C211C2214yes—
10C2×C22⋊C4×C211C2212yes—
11C24×C211—24no—
12C24×C211—44no—
13C24×C211C2×C411yescharacteristic
14C24×C211C2×C411yescharacteristic
15C24×C211D414yes—
16C24×C211C2311yescharacteristic
17C22×C4×C211—22no—
18C22×C4×C211—22no—
19C22×C4×C211—24no—
20C22×C4×C211—44no—
21C22×C4×C211C2×C414yes—
22C22⋊C4×C211—24no—
23C22⋊C4×C211—24no—
24C22⋊C4×C211—28no—
25C22⋊C4×C211—48no—
26C23×C211—24no—
27C23×C211—22no—
28C23×C211—28no—
29C23×C211—22no—
30C23×C211—28no—
31C23×C211—42no—
32C23×C211—44no—
33C23×C211—42no—
34C23×C211—416no—
35C23×C211C22⋊C412yes—
36C23×C211C22⋊C412yes—
37C23×C211C22×C411yescharacteristic
38C23×C211C2×D412yes—
39C23×C211#4114yes—
40C2×C4×C211—24no—
41C2×C4×C211—44no—
42C2×C4×C211—44no—
43C2×C4×C211—44no—
44C2×C4×C211—48no—
45C2×C4×C211—44no—
46C22×C211—28no—
47C22×C211—22no—
48C22×C211—22no—
49C22×C211—22no—
50C22×C211—24no—
51C22×C211—22no—
52C22×C211—24no—
53C22×C211—44no—
54C22×C211—416no—
55C22×C211—44no—
56C22×C211#9914yes—
57C22×C211C2×C22⋊C411yescharacteristic
58C22×C211#11712yes—
59C22×C211#12314yes—
60C4×C211—44no—
61C4×C211—84no—
62C2×C211—24no—
63C2×C211—24no—
64C2×C211—22no—
65C2×C211—42no—
66C2×C211#38111yescharacteristic
67C2×C211C2×#9912yes—
68C211#181911yescharacteristic, direct product factor, Sylow
69#1819C21111yesmaximal, characteristic, direct product factor, Sylow
70C22×C22⋊C4C2×C21111yescharacteristic
71#379C2×C21112yes—
72#342C2×C21114yescomplemented
73C25C22×C21111yescharacteristic
74C23×C4C4×C21112yescomplemented
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4C4×C21112yescomplemented
78C2×C22⋊C4C22×C21114yes—
79C2×C22⋊C4C22×C21112yes—
80C24—24no—
81C24—44no—
82C24C2×C4×C21111yescharacteristic
83C24C2×C4×C21111yescharacteristic
84C24D4×C21114yes—
85C24C23×C21111yescharacteristic
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—24no—
89C22×C4—44no—
90C22×C4C2×C4×C21114yes—
91C22⋊C4—24no—
92C22⋊C4—24no—
93C22⋊C4—28no—
94C22⋊C4—48no—
95C23—24no—
96C23—22no—
97C23—28no—
98C23—22no—
99C23—28no—
100C23—42no—
101C23—44no—
102C23—42no—
103C23—416no—
104C23C22⋊C4×C21112yes—
105C23C22⋊C4×C21112yes—
106C23C22×C4×C21111yescharacteristic
107C23C2×D4×C21112yes—
108C23#41×C21114yes—
109C2×C4—24no—
110C2×C4—44no—
111C2×C4—44no—
112C2×C4—44no—
113C2×C4—48no—
114C2×C4—44no—
115C22—28no—
116C22—22no—
117C22—22no—
118C22—22no—
119C22—24no—
120C22—22no—
121C22—24no—
122C22—44no—
123C22—416no—
124C22—44no—
125C22#99×C21114yes—
126C22C2×C22⋊C4×C21111yescharacteristic
127C22#117×C21112yes—
128C22#123×C21114yes—
129C4—44no—
130C4—84no—
131C2—24no—
132C2—24no—
133C2—22no—
134C2—42no—
135C2#381×C21111yescharacteristic
136C2C2×#99×C21112yes—
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC23C22×C4×C211
Frattini, cosocleC24C23×C211
prime radicalC25×C211C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C25×C211 ◁ G

Complements

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

NH ≅ G/Ntype
3. #342×C211C2semidirect
5. C23×C4×C211C4semidirect
8. C2×C22⋊C4×C211C4semidirect
68. C211#1819direct (N × H)
69. #1819C211direct (N × H)
72. #342C2×C211semidirect
74. C23×C4C4×C211semidirect
77. C2×C22⋊C4C4×C211semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C4×C2112
#379×C2112
#342×C2112
#1819211