Finite Group Databasebeta

← #1010⋊C9 #11722435

Subgroups — #1010⋊C9 #11722435

Order 864.

173 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 173 conjugacy classes of subgroups — too many to draw. Only its 39 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×#41⋊C9C211yesmaximal, characteristic, complemented
2D3×Q8⋊C9C212yesmaximal, complemented
3C3×#1010C311yesmaximal, characteristic
42−1+4⋊C9—31nomaximal
5C9×D12—41nomaximal
6C3×Q8⋊C9C2211yescharacteristic
7C32×#41C2×C311yescharacteristic
8C3×D3×Q8C2×C312yes—
9C3×#237—32no—
10C3×#235—31no—
11C2×C3×Dic6—31no—
12#41⋊C9—21no—
13#41⋊C9D311yescharacteristic, complemented
14C2×Q8⋊C9—32no—
15C2×D3×C9—42no—
16C3×C4×C9—41no—
17C3×2−1+4—31no—
18#1010C911yescharacteristic, complemented
19C32×Q8C22×C311yescharacteristic
20C32×D4—31no—
21C2×C32×C4—31no—
22C3×#67—32no—
23C2×C3×Dic3—32no—
24C3×D12A411yescharacteristic
25C3×C4×D3—32no—
26C3×Dic6—31no—
27C3×Dic6—32no—
28D4×C9—121no—
29Q8⋊C9—21no—
30Q8⋊C9C2×D311yescharacteristic
31C2×C3×C9—41no—
32D3×C9—82no—
33C3×#41—21no—
34C3×#41—91no—
35C3×#41—92no—
36C3×#41C2×C911yescharacteristic, complemented
37C3×#41C3×D311yescharacteristic
38C2×C3×Q8—32no—
39C2×C3×Q8—91no—
40D3×Q8C2×C912yescomplemented
41#237—32no—
42#235—31no—
43C2×Dic6—31no—
44C22×C32—31no—
45C2×C3×D3C2×A412yes—
46C32×C4—31no—
47C32×C4C2×A411yescharacteristic
48C3×Dic3—32no—
49C22×C9—122no—
50C4×C9—41no—
51C4×C9—81no—
522−1+4—31noSylow
53C3×C9—41noSylow
54C3×Q8—21no—
55C3×Q8—92no—
56C3×Q8—91no—
57C3×Q8C22×C911yescharacteristic
58C3×Q8C2×C3×D311yescharacteristic
59C3×D4—31no—
60C3×D4—31no—
61C3×D4—31no—
62C3×D4—61no—
63C3×D4—92no—
64C2×C3×C4—31no—
65C2×C3×C4—31no—
66C2×C3×C4—61no—
67C2×C3×C4—92no—
68C2×C3×C4—92no—
69#67—32no—
70C2×Dic3—32no—
71D12C22⋊C911yescharacteristic
72C4×D3—32no—
73Dic6—31no—
74Dic6—32no—
75C2×C32—61no—
76C2×C32C22×A411yescharacteristic
77C3×D3—22no—
78C2×C9—41no—
79C2×C9—81no—
80C2×C9—242no—
81#41—91no—
82#41—92no—
83#41D3×C911yescharacteristic, complemented
84C2×Q8—32no—
85C2×Q8—91no—
86C22×C3—31no—
87C22×C3—32no—
88C22×C3—31no—
89C22×C3—61no—
90C2×D3C2×C22⋊C912yes—
91C3×C4—21no—
92C3×C4—31no—
93C3×C4—31no—
94C3×C4—61no—
95C3×C4—92no—
96C3×C4C2×C22⋊C911yescharacteristic
97C3×C4D3×A411yescharacteristic
98Dic3—32no—
99C322−1+4⋊C311yescharacteristic
100C9—41no—
101C9—81no—
102Q8—92no—
103Q8—91no—
104Q8C2×D3×C911yescharacteristic
105D4—31no—
106D4—31no—
107D4—92no—
108C2×C4—31no—
109C2×C4—92no—
110C2×C4—92no—
111C2×C3—21no—
112C2×C3—62no—
113C2×C3—61no—
114C2×C3—61no—
115C2×C3—121no—
116C2×C3C22×C22⋊C911yescharacteristic
117C2×C3C2×D3×A411yescharacteristic
118D3—22no—
119C22—31no—
120C22—32no—
121C4—31no—
122C4—92no—
123C4D3×C22⋊C911yescharacteristic
124C3—21no—
125C32−1+4⋊C911yescharacteristic, complemented
126C3#1010⋊C311yescharacteristic
127C2—62no—
128C2—61no—
129C2C2×D3×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC2×C3C2×D3×A4
commutator, hypocenterC3×Q8C22×C9
FittingC32×#41C2×C3
socleC2×C32C22×A4
prime radicalC3×#1010C3
cosocleC32×Q8C22×C3
hypercenterC3×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#41⋊C9C2semidirect
2. D3×Q8⋊C9C2semidirect
13. #41⋊C9D3semidirect
18. #1010C9semidirect
36. C3×#41C2×C9semidirect
40. D3×Q8C2×C9semidirect
83. #41D3×C9semidirect
125. C32−1+4⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×#41⋊C92
D3×Q8⋊C92
C3×#10103
2−1+4⋊C93
C9×D124