Finite Group Databasebeta

← #13047378

Subgroups — #13047378

Order 1296.

332 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 58 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
1Q8×AΣL(1,𝔽27)C211yesmaximal, characteristic, complemented
2C4⋊#4175C213yesmaximal, complemented
3C4×#4175C213yesmaximal, complemented
4#89080C311yesmaximal, characteristic, complemented
5#88722—31nomaximal
6C2×#4175C2213yes—
7C4×AΣL(1,𝔽27)C2213yes—
8#66322C2211yescharacteristic
9C33×Q8C2×C311yescharacteristic, complemented
10C4⋊(C3⋊(C3⋊D3))C2×C313yescomplemented
11C4×C3⋊(C3⋊D3)C2×C313yescomplemented
12Q8×3−1+2—21no—
13Q8×H(3,𝔽3)D311yescharacteristic, complemented
14C4⋊((C3⋊D3)⋊C3)—33no—
15C4×(C3⋊D3)⋊C3—33no—
16C2×AΣL(1,𝔽27)C2311yescharacteristic
17#4175—23no—
18#3832—31no—
19#3832—91no—
20#3832—93no—
21C3×#239—91no—
22C2×C3⋊(C3⋊D3)C22×C313yes—
23C33×C4C22×C313yes—
24C3⋊(C3⋊Dic3)C22×C311yescharacteristic
25C2×(C3⋊D3)⋊C3—33no—
26C4×3−1+2—23no—
27C4×H(3,𝔽3)C2×D313yes—
28(C3⋊Dic3)⋊C3—31no—
29AΣL(1,𝔽27)#4111yescharacteristic, complemented, Sylow
30C32×Q8—31no—
31C32×Q8—33no—
32C32×Q8—31no—
33C32×Q8C3×D311yescharacteristic
34C4⋊(C3⋊D3)—33no—
35C4⋊(C3⋊D3)—93no—
36C4⋊(C3⋊D3)—99no—
37C4×C3⋊D3—33no—
38C4×C3⋊D3—93no—
39C4×C3⋊D3—99no—
40C3×D12—93no—
41C3×C4×D3—93no—
42Q8×C9—61no—
43C2×C33C23×C311yescharacteristic
44C3⋊(C3⋊D3)—23no—
45C2×3−1+2—21no—
46C2×H(3,𝔽3)C22×D311yescharacteristic
47(C3⋊D3)⋊C3—63no—
48C3×#41—271no—
49#239—91no—
50#239—271no—
51#239—273no—
52C2×C3⋊D3—33no—
53C2×C3⋊D3—93no—
54C2×C3⋊D3—99no—
55C2×C3×D3—93no—
56C32×C4—33no—
57C32×C4—39no—
58C32×C4—33no—
59C32×C4C2×C3×D313yes—
60C3⋊Dic3—31no—
61C3⋊Dic3—91no—
62C3⋊Dic3—93no—
63C3×Dic3—91no—
64C4×C9—63no—
65C33C3×#4111yescharacteristic, complemented
663−1+2—21no—
67H(3,𝔽3)#23911yescharacteristic, complemented
68C3×Q8—31no—
69C3×Q8—33no—
70C3×Q8—91no—
71C3×Q8(C3⋊D3)⋊C311yescharacteristic
72C3×D4—273no—
73C2×C3×C4—273no—
74D12—93no—
75D12—273no—
76D12—279no—
77C4×D3—93no—
78C4×D3—273no—
79C4×D3—279no—
80C2×C32—31no—
81C2×C32—33no—
82C2×C32—31no—
83C2×C32C22×C3×D311yescharacteristic
84C3⋊D3—63no—
85C3⋊D3—183no—
86C3⋊D3—189no—
87C3×D3—183no—
88C2×C9—61no—
89#41—271noSylow
90C22×C3—273no—
91C2×D3—93no—
92C2×D3—273no—
93C2×D3—279no—
94C3×C4—39no—
95C3×C4—33no—
96C3×C4—93no—
97C3×C4—271no—
98C3×C4C2×(C3⋊D3)⋊C313yes—
99Dic3—91no—
100Dic3—271no—
101Dic3—273no—
102C32—33no—
103C32—31no—
104C32—31no—
105C32C3×#23911yescharacteristic
106C9—61no—
107Q8#417511yescharacteristic, complemented
108D4—273no—
109C2×C4—273no—
110C2×C3—33no—
111C2×C3—31no—
112C2×C3—91no—
113C2×C3—543no—
114C2×C3C22×(C3⋊D3)⋊C311yescharacteristic
115D3—183no—
116D3—543no—
117D3—549no—
118C22—273no—
119C4—271no—
120C4C2×#417513yes—
121C3—33no—
122C3—31no—
123C3—91no—
124C3#8872211yescharacteristic
125C2—543no—
126C2C22×#417511yescharacteristic
subgroupG/H
centerC2C22×#4175
commutator, cosocleC2×C33C23×C3
FrattiniC2×C32C22×C3×D3
FittingQ8×AΣL(1,𝔽27)C2
socleC2×C3C22×(C3⋊D3)⋊C3
hypercenterQ8#4175
hypocenterC33C3×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×AΣL(1,𝔽27)C2semidirect
2. C4⋊#4175C2semidirect
3. C4×#4175C2semidirect
4. #89080C3semidirect
9. C33×Q8C2×C3semidirect
10. C4⋊(C3⋊(C3⋊D3))C2×C3semidirect
11. C4×C3⋊(C3⋊D3)C2×C3semidirect
13. Q8×H(3,𝔽3)D3semidirect
29. AΣL(1,𝔽27)#41semidirect
65. C33C3×#41semidirect
67. H(3,𝔽3)#239semidirect
107. Q8#4175semidirect

Maximal subgroups

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

subgroupindex
Q8×AΣL(1,𝔽27)2
C4⋊#41752
C4×#41752
#890803
#887223