Finite Group Databasebeta

← C3×GO+(4,𝔽2)⋊C4 #11724095

Subgroups — C3×GO+(4,𝔽2)⋊C4 #11724095

Order 864.

142 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 32 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×C3×GO+(4,𝔽2)C211yesmaximal, characteristic
2C2×C3×AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
3C3×#3777C211yesmaximal, characteristic, complemented
4GO+(4,𝔽2)⋊C4C311yesmaximal, characteristic, direct product factor
5C2×C3×D32—21no—
6C2×C3×#157—21no—
7C2×C3×#157C2211yescharacteristic
8C3×GO+(4,𝔽2)—21no—
9C3×GO+(4,𝔽2)C412yescomplemented
10C3×AGL(1,𝔽9)—21no—
11C2×GO+(4,𝔽2)C2×C311yescharacteristic
12C2×AGL(1,𝔽9)C2×C311yescharacteristic, complemented
13#3777C2×C311yescharacteristic, complemented
14C2×C3×C3⋊D3D411yescharacteristic
15C2×C32×D3—41no—
16C3×D32—22no—
17C3×D32—41no—
18C3×#157—41no—
19C3×#157C2×C411yescharacteristic
20C3×#157D411yescharacteristic
21C3×#102—91nomaximal
22C22×C3×D3—121no—
23C2×D32—21no—
24C2×#157—21no—
25C2×#157C22×C311yescharacteristic
26GO+(4,𝔽2)—21no—
27GO+(4,𝔽2)C3×C412yescomplemented
28AGL(1,𝔽9)—21no—
29C2×C33SD1611yescharacteristic
30C3×C3⋊D3C22⋊C411yescharacteristic
31C3×C3⋊D3D811yescharacteristic
32C32×D3—42no—
33C2×C3×D4—91no—
34C2×C3×C8—91no—
35C3×C4⋊C4—91no—
36C22×C32—121no—
37C2×C3⋊D3C3×D411yescharacteristic
38C2×C3×D3—41no—
39C2×C3×D3—41no—
40C2×C3×D3—81no—
41C2×C3×D3—121no—
42C2×C3×D3—122no—
43C2×C3×D3—122no—
44D32—22no—
45D32—41no—
46#157—41no—
47#157C2×C3×C411yescharacteristic
48#157C3×D411yescharacteristic
49#102—91noSylow
50C33#10211yescharacteristic, complemented, Sylow
51C23×C3—181no—
52C22×D3—121no—
53C3×D4—92no—
54C3×D4—181no—
55C2×C3×C4—91no—
56C2×C3×C4—181no—
57C3×C8—181no—
58C2×C32—41no—
59C2×C32—81no—
60C2×C32—122no—
61C2×C32C3×SD1611yescharacteristic
62C3⋊D3C3×C22⋊C411yescharacteristic
63C3⋊D3C3×D811yescharacteristic
64C3×D3—42no—
65C3×D3—42no—
66C3×D3—82no—
67C3×D3—121no—
68C3×D3—121no—
69C2×D4—91no—
70C2×C8—91no—
71C4⋊C4—91no—
72C22×C3—91no—
73C22×C3—121no—
74C22×C3—121no—
75C22×C3—182no—
76C22×C3—241no—
77C22×C3—361no—
78C2×D3—41no—
79C2×D3—121no—
80C2×D3—122no—
81C2×D3—122no—
82C3×C4—91no—
83C3×C4—91no—
84C3×C4—361no—
85C32—41no—
86C32—81no—
87C32C3×#10211yescharacteristic, complemented
88C23—181no—
89D4—92no—
90D4—181no—
91C2×C4—91no—
92C2×C4—181no—
93C8—181no—
94C2×C3—41no—
95C2×C3—81no—
96C2×C3—91no—
97C2×C3—91no—
98C2×C3—122no—
99C2×C3—122no—
100C2×C3—242no—
101C2×C3AΓL(1,𝔽9)11yescharacteristic
102D3—42no—
103D3—121no—
104D3—121no—
105C22—91no—
106C22—121no—
107C22—182no—
108C22—361no—
109C4—91no—
110C4—91no—
111C4—361no—
112C3—41no—
113C3—81no—
114C3GO+(4,𝔽2)⋊C411yescharacteristic, direct product factor
115C2—91no—
116C2—91no—
117C2—122no—
118C2C3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3AΓL(1,𝔽9)
commutator#157C2×C3×C4
FrattiniC2C3×AΓL(1,𝔽9)
Fitting, socleC2×C33SD16
prime radicalC2×C3×GO+(4,𝔽2)C2
cosocleC2×#157C22×C3
hypocenterC32C3×#102
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×AGL(1,𝔽9)C2semidirect
3. C3×#3777C2semidirect
4. GO+(4,𝔽2)⋊C4C3direct (N × H)
9. C3×GO+(4,𝔽2)C4semidirect
12. C2×AGL(1,𝔽9)C2×C3semidirect
13. #3777C2×C3semidirect
27. GO+(4,𝔽2)C3×C4semidirect
50. C33#102semidirect
87. C32C3×#102semidirect
114. C3GO+(4,𝔽2)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×GO+(4,𝔽2)2
C2×C3×AGL(1,𝔽9)2
C3×#37772
GO+(4,𝔽2)⋊C43
C3×#1029