Finite Group Databasebeta

← #1289×C211 #41008909

Subgroups — #1289×C211 #41008909

Order 27008.

228 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 84 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⋊C8×C211C211yesmaximal, characteristic
2#380×C211C212yesmaximal
3C23×C4×C211C2211yescharacteristic
4C22×C8×C211C412yescomplemented
5C2×C4⋊C4×C211—22no—
6C2×C22⋊C4×C211—22no—
7C22⋊C8×C211—22no—
8#95×C211C414yescomplemented
9C24×C211D411yescharacteristic
10C22×C4×C211—24no—
11C22×C4×C211—42no—
12C22×C4×C211C2×C411yescharacteristic
13C22×C4×C211C2×C412yes—
14C22×C4×C211D412yes—
15C22×C4×C211Q811yescharacteristic
16C2×C8×C211—22no—
17C2×C8×C211—42no—
18C4⋊C4×C211—44no—
19C22⋊C4×C211—42no—
20C22⋊C4×C211—42no—
21C23×C211—41no—
22C23×C211—41no—
23C23×C211C22⋊C412yes—
24C23×C211C4⋊C411yescharacteristic
25C23×C211D811yescharacteristic
26C23×C211SD1612yes—
27C23×C211Q1611yescharacteristic
28C2×C4×C211—24no—
29C2×C4×C211—22no—
30C2×C4×C211—42no—
31C2×C4×C211—41no—
32C2×C4×C211—44no—
33C2×C4×C211—44no—
34C2×C4×C211—44no—
35C2×C4×C211C4211yescharacteristic
36C2×C4×C211C22⋊C411yescharacteristic
37C2×C4×C211C4⋊C412yes—
38C8×C211—42no—
39C22×C211—22no—
40C22×C211—21no—
41C22×C211—22no—
42C22×C211—21no—
43C22×C211—41no—
44C22×C211—42no—
45C22×C211—41no—
46C22×C211#9511yescharacteristic
47C22×C211#10212yes—
48C22×C211#10312yes—
49C22×C211#10611yescharacteristic
50C22×C211#10711yescharacteristic
51C4×C211—21no—
52C4×C211—22no—
53C4×C211—84no—
54C4×C211#9912yes—
55C2×C211—22no—
56C2×C211—22no—
57C2×C211—21no—
58C2×C211—21no—
59C2×C211H(3,ℤ/4)11yescharacteristic
60C2×C211#34011yescharacteristic
61C2×C211#34211yescharacteristic
62C211#128911yescharacteristic, direct product factor, Sylow
63#1289C21111yesmaximal, characteristic, direct product factor, Sylow
64C2×C22⋊C8C2×C21111yescharacteristic
65#380C2×C21112yes—
66C23×C4C22×C21111yescharacteristic
67C22×C8C4×C21112yescomplemented
68C2×C4⋊C4—22no—
69C2×C22⋊C4—22no—
70C22⋊C8—22no—
71#95C4×C21114yescomplemented
72C24D4×C21111yescharacteristic
73C22×C4—24no—
74C22×C4—42no—
75C22×C4C2×C4×C21111yescharacteristic
76C22×C4C2×C4×C21112yes—
77C22×C4D4×C21112yes—
78C22×C4Q8×C21111yescharacteristic
79C2×C8—22no—
80C2×C8—42no—
81C4⋊C4—44no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C23—41no—
85C23—41no—
86C23C22⋊C4×C21112yes—
87C23C4⋊C4×C21111yescharacteristic
88C23D8×C21111yescharacteristic
89C23SD16×C21112yes—
90C23Q16×C21111yescharacteristic
91C2×C4—24no—
92C2×C4—22no—
93C2×C4—42no—
94C2×C4—41no—
95C2×C4—44no—
96C2×C4—44no—
97C2×C4—44no—
98C2×C4C42×C21111yescharacteristic
99C2×C4C22⋊C4×C21111yescharacteristic
100C2×C4C4⋊C4×C21112yes—
101C8—42no—
102C22—22no—
103C22—21no—
104C22—22no—
105C22—21no—
106C22—41no—
107C22—42no—
108C22—41no—
109C22#95×C21111yescharacteristic
110C22#102×C21112yes—
111C22#103×C21112yes—
112C22#106×C21111yescharacteristic
113C22#107×C21111yescharacteristic
114C4—21no—
115C4—22no—
116C4—84no—
117C4#99×C21112yes—
118C2—22no—
119C2—22no—
120C2—21no—
121C2—21no—
122C2H(3,ℤ/4)×C21111yescharacteristic
123C2#340×C21111yescharacteristic
124C2#342×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#95
commutatorC2×C4C42×C211
Frattini, cosocleC23×C4C22×C211
prime radicalC24×C211D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
4. C22×C8×C211C4semidirect
8. #95×C211C4semidirect
62. C211#1289direct (N × H)
63. #1289C211direct (N × H)
67. C22×C8C4×C211semidirect
71. #95C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×C2112
#380×C2112
#1289211