Finite Group Databasebeta

← #1528×C211 #41009148

Subgroups — #1528×C211 #41009148

Order 27008.

212 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 92 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×#125×C211C211yesmaximal, characteristic
2#432×C211C211yesmaximal, characteristic
3C4×M16×C211C211yesmaximal, characteristic
4#330×C211C214yesmaximal, complemented
5C2×M16×C211—21no—
6#125×C211—21no—
7#125×C211—21no—
8#125×C211C412yes—
9#125×C211C2212yes—
10C2×C4⋊C4×C211—21no—
11C2×C4⋊C4×C211—21no—
12C2×C4⋊C4×C211C411yescharacteristic
13C2×C4⋊C4×C211C411yescharacteristic
14C2×C42×C211C2211yescharacteristic
15C4⋊C8×C211—21no—
16C22⋊C8×C211—21no—
17#97×C211C2212yes—
18#97×C211C2212yes—
19C4×C8×C211—21no—
20C22×C4×C211—21no—
21C22×C4×C211—21no—
22C22×C4×C211C2×C411yescharacteristic
23C22×C4×C211D411yescharacteristic
24C22×C4×C211D411yescharacteristic
25M16×C211—24no—
26C2×C8×C211—22no—
27C2×C8×C211—22no—
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
40C23×C211C22⋊C411yescharacteristic
41C2×C4×C211—22no—
42C2×C4×C211—22no—
43C2×C4×C211—22no—
44C2×C4×C211—22no—
45C2×C4×C211—41no—
46C2×C4×C211—42no—
47C2×C4×C211—42no—
48C2×C4×C211C22⋊C411yescharacteristic
49C2×C4×C211C22⋊C411yescharacteristic
50C2×C4×C211C22⋊C411yescharacteristic
51C2×C4×C211C22×C411yescharacteristic
52C2×C4×C211C2×D411yescharacteristic
53C2×C4×C211C2×D411yescharacteristic
54C8×C211—24no—
55C8×C211—42no—
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, direct product factor, Sylow
73#1528C21111yesmaximal, characteristic, direct product factor, Sylow
74C2×#125C2×C21111yescharacteristic
75#432C2×C21111yescharacteristic
76C4×M16C2×C21111yescharacteristic
77#330C2×C21114yescomplemented
78C2×M16—21no—
79#125—21no—
80#125—21no—
81#125C4×C21112yes—
82#125C22×C21112yes—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C4×C21111yescharacteristic
86C2×C4⋊C4C4×C21111yescharacteristic
87C2×C42C22×C21111yescharacteristic
88C4⋊C8—21no—
89C22⋊C8—21no—
90#97C22×C21112yes—
91#97C22×C21112yes—
92C4×C8—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4C2×C4×C21111yescharacteristic
96C22×C4D4×C21111yescharacteristic
97C22×C4D4×C21111yescharacteristic
98M16—24no—
99C2×C8—22no—
100C2×C8—22no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4C2×C4×C21112yes—
108C4⋊C4C2×C4×C21112yes—
109C42C2×C4×C21111yescharacteristic
110C42D4×C21111yescharacteristic
111C42D4×C21111yescharacteristic
112C42C23×C21111yescharacteristic
113C23C22⋊C4×C21111yescharacteristic
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4C22⋊C4×C21111yescharacteristic
122C2×C4C22⋊C4×C21111yescharacteristic
123C2×C4C22⋊C4×C21111yescharacteristic
124C2×C4C22×C4×C21111yescharacteristic
125C2×C4C2×D4×C21111yescharacteristic
126C2×C4C2×D4×C21111yescharacteristic
127C8—24no—
128C8—42no—
129C22—21no—
130C22—21no—
131C22#104×C21112yes—
132C22C2×C22⋊C4×C21111yescharacteristic
133C4—21no—
134C4—21no—
135C4—21no—
136C4—21no—
137C4—21no—
138C4—42no—
139C4—42no—
140C4#101×C21112yes—
141C2—22no—
142C2C2×#101×C21111yescharacteristic
143C2C2×#104×C21111yescharacteristic
144C2#421×C21111yescharacteristic
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC42C23×C211
prime radicalC23×C211C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C2×C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
4. #330×C211C2semidirect
72. C211#1528direct (N × H)
73. #1528C211direct (N × H)
77. #330C2×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#125×C2112
#432×C2112
C4×M16×C2112
#330×C2112
#1528211