Finite Group Databasebeta

← Q8×C3⋊#157 #11724518

Subgroups — Q8×C3⋊#157 #11724518

Order 864.

240 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C4⋊(C3⋊#157)C213yesmaximal
3C4×C3⋊#157C213yesmaximal
4Q8×#157—31nomaximal
5C2×C3⋊#157C2213yes—
6C2×C3⋊#157C2211yescharacteristic
7C33×Q8C411yescharacteristic, complemented
8C3×C4×C3⋊D3C2213yes—
9C3×#634C413yescomplemented
10Q8×C3⋊D3D311yescharacteristic
11C3×D3×Q8—62no—
12C4⋊#157—33no—
13C4×#157—33no—
14C2×C3×C3⋊D3C2311yescharacteristic
15C3⋊#157—23no—
16C3⋊#157Q812yesdirect product factor
17C33×C4C2×C413yes—
18C3×C3⋊Dic3C2×C413yes—
19Q8×Dic3—91nomaximal
20C2×#157—31no—
21C2×#157—33no—
22C32×Q8—22no—
23C32×Q8—42no—
24C32×Q8Dic311yescharacteristic, complemented
25C4×C3⋊D3C2×D313yes—
26#634Dic313yescomplemented
27C3×C4×D3—66no—
28C3×Dic6—66no—
29C2×C33C22×C411yescharacteristic
30C3×C3⋊D3C2×Q811yescharacteristic
31C3×C3⋊D3#4111yescharacteristic
32C2×C3×Q8—91no—
33D3×Q8—62no—
34C4⋊Dic3—93no—
35C4×Dic3—93no—
36C2×C3⋊D3C22×D311yescharacteristic
37C2×C3×D3—62no—
38#157—32no—
39#157—63no—
40C32×C4—26no—
41C32×C4—46no—
42C32×C4C2×Dic313yes—
43C3⋊Dic3C2×Dic313yes—
44C3×Dic3—66no—
45C4×Q8—271noSylow
46C33C4×Q811yescharacteristic, complemented, Sylow
47C3×Q8—22no—
48C3×Q8—42no—
49C3×Q8—93no—
50C3×Q8#15711yescharacteristic, complemented
51C2×C3×C4—93no—
52C2×Dic3—91no—
53C2×Dic3—93no—
54C4×D3—66no—
55Dic6—66no—
56C2×C32—22no—
57C2×C32—42no—
58C2×C32C22×Dic311yescharacteristic
59C3⋊D3D3×Q811yescharacteristic
60C3⋊D3#23911yescharacteristic
61C3×D3—62no—
62C3×D3—62no—
63C2×Q8—91no—
64C4⋊C4—273no—
65C42—273no—
66C22×C3—91no—
67C2×D3—62no—
68C3×C4—26no—
69C3×C4—46no—
70C3×C4—93no—
71C3×C4C2×#15713yes—
72Dic3—66no—
73Dic3—92no—
74Dic3—183no—
75C32—22no—
76C32—42no—
77C32Q8×Dic311yescharacteristic, complemented
78Q8—93no—
79Q8C3⋊#15711yescharacteristic, direct product factor
80C2×C4—93no—
81C2×C4—271no—
82C2×C4—273no—
83C2×C3—22no—
84C2×C3—42no—
85C2×C3—91no—
86C2×C3—91no—
87C2×C3C22×#15711yescharacteristic
88D3—62no—
89D3—62no—
90C22—91no—
91C4—93no—
92C4—272no—
93C4—543no—
94C4C2×C3⋊#15713yes—
95C3—22no—
96C3—42no—
97C3Q8×#15711yescharacteristic, complemented
98C2—91no—
99C2—91no—
100C2C22×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×C3⋊#157
commutator, socleC2×C33C22×C4
FittingC33×Q8C4
prime radical, cosocleC2×C3×C3⋊D3C23
hypercenterQ8C3⋊#157
hypocenterC33C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
7. C33×Q8C4semidirect
9. C3×#634C4semidirect
16. C3⋊#157Q8direct (N × H)
24. C32×Q8Dic3semidirect
26. #634Dic3semidirect
46. C33C4×Q8semidirect
50. C3×Q8#157semidirect
77. C32Q8×Dic3semidirect
79. Q8C3⋊#157direct (N × H)
97. C3Q8×#157semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊D32
C4⋊(C3⋊#157)2
C4×C3⋊#1572
Q8×#1573
Q8×Dic39