Finite Group Databasebeta

← #1339×C211 #41008959

Subgroups — #1339×C211 #41008959

Order 27008.

114 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 36 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
1#469×C211C211yesmaximal, characteristic
2#349×C211C211yesmaximal, characteristic, complemented
3H(3,ℤ/4)×C211C211yesmaximal, characteristic, complemented
4C2×D8×C211C411yescharacteristic, complemented
5C2×M16×C211C2211yescharacteristic
6#121×C211—21no—
7#117×C211—21no—
8M32×C211—21no—
9#106×C211C411yescharacteristic, complemented
10#102×C211—21no—
11C2×D4×C211—21no—
12C2×D4×C211—41no—
13C22×C4×C211D411yescharacteristic
14D8×C211—41no—
15M16×C211—41no—
16C2×C8×C211C2×C411yescharacteristic
17C2×C8×C211D411yescharacteristic
18C4⋊C4×C211—21no—
19C4⋊C4×C211—41no—
20C22⋊C4×C211—41no—
21C22⋊C4×C211—41no—
22C42×C211—22no—
23C16×C211—41no—
24C23×C211—41no—
25C23×C211D811yescharacteristic
26D4×C211—41no—
27D4×C211—41no—
28D4×C211—82no—
29C2×C4×C211—21no—
30C2×C4×C211—21no—
31C2×C4×C211—42no—
32C2×C4×C211—41no—
33C2×C4×C211C22⋊C411yescharacteristic
34C2×C4×C211SD1611yescharacteristic
35C8×C211—21no—
36C8×C211—21no—
37C22×C211—21no—
38C22×C211—41no—
39C22×C211—81no—
40C22×C211—81no—
41C22×C211—81no—
42C22×C211#10211yescharacteristic
43C4×C211—21no—
44C4×C211—81no—
45C4×C211—82no—
46C4×C211#9911yescharacteristic
47C4×C211#10411yescharacteristic
48C2×C211—21no—
49C2×C211—41no—
50C2×C211—161no—
51C2×C211#32711yescharacteristic
52C211#133911yescharacteristic, direct product factor, Sylow
53#1339C21111yesmaximal, characteristic, direct product factor, Sylow
54#469C2×C21111yescharacteristic
55#349C2×C21111yescharacteristic, complemented
56H(3,ℤ/4)C2×C21111yescharacteristic, complemented
57C2×D8C4×C21111yescharacteristic, complemented
58C2×M16C22×C21111yescharacteristic
59#121—21no—
60#117—21no—
61M32—21no—
62#106C4×C21111yescharacteristic, complemented
63#102—21no—
64C2×D4—21no—
65C2×D4—41no—
66C22×C4D4×C21111yescharacteristic
67D8—41no—
68M16—41no—
69C2×C8C2×C4×C21111yescharacteristic
70C2×C8D4×C21111yescharacteristic
71C4⋊C4—21no—
72C4⋊C4—41no—
73C22⋊C4—41no—
74C22⋊C4—41no—
75C42—22no—
76C16—41no—
77C23—41no—
78C23D8×C21111yescharacteristic
79D4—41no—
80D4—41no—
81D4—82no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—42no—
85C2×C4—41no—
86C2×C4C22⋊C4×C21111yescharacteristic
87C2×C4SD16×C21111yescharacteristic
88C8—21no—
89C8—21no—
90C22—21no—
91C22—41no—
92C22—81no—
93C22—81no—
94C22—81no—
95C22#102×C21111yescharacteristic
96C4—21no—
97C4—81no—
98C4—82no—
99C4#99×C21111yescharacteristic
100C4#104×C21111yescharacteristic
101C2—21no—
102C2—41no—
103C2—161no—
104C2#327×C21111yescharacteristic
subgroupG/H
center, socleC2×C211#327
commutatorC2×C8C2×C4×C211
Frattini, cosocleC2×M16C22×C211
prime radical#469×C211C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C2×C4×C211 ◁ C2×M16×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #349×C211C2semidirect
3. H(3,ℤ/4)×C211C2semidirect
4. C2×D8×C211C4semidirect
9. #106×C211C4semidirect
52. C211#1339direct (N × H)
53. #1339C211direct (N × H)
55. #349C2×C211semidirect
56. H(3,ℤ/4)C2×C211semidirect
57. C2×D8C4×C211semidirect
62. #106C4×C211semidirect

Maximal subgroups

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

subgroupindex
#469×C2112
#349×C2112
H(3,ℤ/4)×C2112
#1339211