Finite Group Databasebeta

← #1533×C211 #41009153

Subgroups — #1533×C211 #41009153

Order 27008.

248 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 88 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×#124×C211C211yesmaximal, characteristic
2#432×C211C212yesmaximal, complemented
3#329×C211C214yesmaximal, complemented
4C22×Q8×C211C411yescharacteristic
5C22×D4×C211C411yescharacteristic
6#124×C211—22no—
7#124×C211C412yes—
8#124×C211C2212yes—
9C2×C22⋊C4×C211—22no—
10C2×C42×C211C2211yescharacteristic
11C4⋊C8×C211—22no—
12C22⋊C8×C211—22no—
13#97×C211C2214yescomplemented
14C24×C211—21no—
15C2×Q8×C211—21no—
16C2×Q8×C211—42no—
17C2×Q8×C211C2×C412yes—
18C2×D4×C211—21no—
19C2×D4×C211—42no—
20C2×D4×C211C2×C412yes—
21C22×C4×C211—21no—
22C22×C4×C211C2×C411yescharacteristic
23C22×C4×C211D412yes—
24C2×C8×C211—24no—
25C22⋊C4×C211—24no—
26C22⋊C4×C211—24no—
27C42×C211C2×C411yescharacteristic
28C42×C211D412yes—
29C42×C211C2311yescharacteristic
30C23×C211—22no—
31C23×C211—44no—
32C23×C211—42no—
33C23×C211C22⋊C411yescharacteristic
34Q8×C211—42no—
35Q8×C211—42no—
36D4×C211—42no—
37D4×C211—42no—
38C2×C4×C211—22no—
39C2×C4×C211—22no—
40C2×C4×C211—42no—
41C2×C4×C211—42no—
42C2×C4×C211C22⋊C412yes—
43C2×C4×C211C22⋊C411yescharacteristic
44C2×C4×C211C22×C411yescharacteristic
45C2×C4×C211C2×D412yes—
46C8×C211—44no—
47C22×C211—22no—
48C22×C211—42no—
49C22×C211—44no—
50C22×C211—84no—
51C22×C211#10012yes—
52C22×C211C2×C22⋊C411yescharacteristic
53C4×C211—21no—
54C4×C211—21no—
55C4×C211—22no—
56C4×C211—22no—
57C4×C211—42no—
58C2×C211—22no—
59C2×C211—82no—
60C2×C211C2×#10011yescharacteristic
61C2×C211#42112yes—
62C211#153311yescharacteristic, direct product factor, Sylow
63#1533C21111yesmaximal, characteristic, direct product factor, Sylow
64C2×#124C2×C21111yescharacteristic
65#432C2×C21112yescomplemented
66#329C2×C21114yescomplemented
67C22×Q8C4×C21111yescharacteristic
68C22×D4C4×C21111yescharacteristic
69#124—22no—
70#124C4×C21112yes—
71#124C22×C21112yes—
72C2×C22⋊C4—22no—
73C2×C42C22×C21111yescharacteristic
74C4⋊C8—22no—
75C22⋊C8—22no—
76#97C22×C21114yescomplemented
77C24—21no—
78C2×Q8—21no—
79C2×Q8—42no—
80C2×Q8C2×C4×C21112yes—
81C2×D4—21no—
82C2×D4—42no—
83C2×D4C2×C4×C21112yes—
84C22×C4—21no—
85C22×C4C2×C4×C21111yescharacteristic
86C22×C4D4×C21112yes—
87C2×C8—24no—
88C22⋊C4—24no—
89C22⋊C4—24no—
90C42C2×C4×C21111yescharacteristic
91C42D4×C21112yes—
92C42C23×C21111yescharacteristic
93C23—22no—
94C23—44no—
95C23—42no—
96C23C22⋊C4×C21111yescharacteristic
97Q8—42no—
98Q8—42no—
99D4—42no—
100D4—42no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4C22⋊C4×C21112yes—
106C2×C4C22⋊C4×C21111yescharacteristic
107C2×C4C22×C4×C21111yescharacteristic
108C2×C4C2×D4×C21112yes—
109C8—44no—
110C22—22no—
111C22—42no—
112C22—44no—
113C22—84no—
114C22#100×C21112yes—
115C22C2×C22⋊C4×C21111yescharacteristic
116C4—21no—
117C4—21no—
118C4—22no—
119C4—22no—
120C4—42no—
121C2—22no—
122C2—82no—
123C2C2×#100×C21111yescharacteristic
124C2#421×C21112yes—
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC42C23×C211
prime radicalC22×D4×C211C4
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
2. #432×C211C2semidirect
3. #329×C211C2semidirect
13. #97×C211C22semidirect
62. C211#1533direct (N × H)
63. #1533C211direct (N × H)
65. #432C2×C211semidirect
66. #329C2×C211semidirect
76. #97C22×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#124×C2112
#432×C2112
#329×C2112
#1533211