Finite Group Databasebeta

← #13135784

Subgroups — #13135784

Order 1728.

352 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 61 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#11721864C211yesmaximal, characteristic
2H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
3#11720584C211yesmaximal, characteristic, complemented
4C3×#4477—34nomaximal
5C2×Q8×H(3,𝔽3)C2211yescharacteristic
6C2×C4⋊#272C411yescharacteristic
7M16×H(3,𝔽3)—21no—
8#88542—21no—
9#88542—21no—
10C4×#1084C411yescharacteristic
11#88538—21no—
12C3×#947—34no—
13C32×#101D314yescomplemented
14C3×#836—34no—
15C22×#272—21no—
16Q8×H(3,𝔽3)—41no—
17C2×C4×H(3,𝔽3)C2×C411yescharacteristic
18C2×C4×H(3,𝔽3)D411yescharacteristic
19C2×C4×H(3,𝔽3)D411yescharacteristic
20C2×#1084—21no—
21C4⋊#272—41no—
22C8×H(3,𝔽3)—41no—
23#6180—41no—
24C3×#355—91no—
25#4477—34no—
26C2×C32×Q8C2×D314yes—
27C2×C3×D12—34no—
28C32×M16—24no—
29C3×(C2×D3)⋊C4—64no—
30C3×(C2×D3)⋊C4—64no—
31C3×C4×Dic3—34no—
32C3×#208—64no—
33C22×H(3,𝔽3)C22⋊C411yescharacteristic
34C2×#272—42no—
35C2×#272—41no—
36C4×H(3,𝔽3)—21no—
37C4×H(3,𝔽3)—21no—
38C4×H(3,𝔽3)—21no—
39#1084—41no—
40C3×#124—91no—
41#947—34no—
42C3×#101—34no—
43C3×#101—91no—
44C3×#101C3⋊D311yescharacteristic
45#836—34no—
46C22×C3×D3—64no—
47C32×Q8—44no—
48C2×C32×C4C4×D314yes—
49C2×C32×C4D1214yes—
50C2×C32×C4#6714yes—
51C2×C3×Dic3—64no—
52C3×D12—124no—
53C32×C8—44no—
54C3×C3⋊C8—124no—
55#355—91noSylow
56C2×H(3,𝔽3)—21no—
57C2×H(3,𝔽3)#9911yescharacteristic
58#272—81no—
59C2×C3×Q8—34no—
60C2×C3×Q8C2×C3⋊D311yescharacteristic
61C2×C3×D4—91no—
62C2×D12—34no—
63C3×M16—21no—
64C3×M16—64no—
65C3×M16—181no—
66C3×C22⋊C4—181no—
67C3×C22⋊C4—181no—
68C3×C42—91no—
69(C2×D3)⋊C4—64no—
70(C2×D3)⋊C4—64no—
71C4×Dic3—34no—
72#208—64no—
73C22×C32(C2×D3)⋊C414yes—
74C2×C3×D3—128no—
75C2×C3×D3—124no—
76C32×C4—24no—
77C32×C4—24no—
78C32×C4—24no—
79C3×Dic3—124no—
80#124—91no—
81#101—91no—
82#101#27211yescharacteristic, complemented
83H(3,𝔽3)#35511yescharacteristic, complemented, Sylow
84C23×C3—181no—
85C22×D3—64no—
86C3×Q8—41no—
87C3×Q8—124no—
88C3×D4—361no—
89C2×C3×C4—34no—
90C2×C3×C4—34no—
91C2×C3×C4—34no—
92C2×C3×C4—181no—
93C2×C3×C4C4×C3⋊D311yescharacteristic
94C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
95C2×C3×C4#63811yescharacteristic
96C2×Dic3—64no—
97D12—124no—
98C3×C8—41no—
99C3×C8—124no—
100C3×C8—361no—
101C3⋊C8—124no—
102C2×C32—24no—
103C2×C32#80614yes—
104C3×D3—244no—
105C2×Q8C2×#27211yescharacteristic
106C2×D4—91no—
107M16—21no—
108M16—181no—
109C22⋊C4—181no—
110C22⋊C4—181no—
111C42—91no—
112C22×C3—34no—
113C22×C3—362no—
114C22×C3—361no—
115C22×C3#375211yescharacteristic
116C2×D3—124no—
117C2×D3—128no—
118C3×C4—21no—
119C3×C4—21no—
120C3×C4—21no—
121C3×C4—64no—
122C3×C4—64no—
123C3×C4—64no—
124C3×C4—361no—
125Dic3—124no—
126C32#447714yescomplemented
127C23—181no—
128Q8—41no—
129D4—361no—
130C2×C4—181no—
131C2×C4C4×#27211yescharacteristic
132C2×C4C4⋊#27211yescharacteristic
133C2×C4#623511yescharacteristic
134C8—41no—
135C8—361no—
136C2×C3—21no—
137C2×C3—34no—
138C2×C3—64no—
139C2×C3—721no—
140C2×C3#6363311yescharacteristic
141D3—244no—
142C22—362no—
143C22—361no—
144C22#8854211yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—361no—
149C3—34no—
150C3#1058934311yescharacteristic
151C2—21no—
152C2—721no—
153C2#1172052411yescharacteristic
subgroupG/H
center, socleC2×C3#63633
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×Q8C2×C3⋊D3
FittingH(3,𝔽3)×#101C2
prime radicalC2×C4⋊#272C4
cosocleC2×Q8×H(3,𝔽3)C22
hypercenterC3×#101C3⋊D3
hypocenterH(3,𝔽3)#355
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#101C2semidirect
3. #11720584C2semidirect
13. C32×#101D3semidirect
82. #101#272semidirect
83. H(3,𝔽3)#355semidirect
126. C32#4477semidirect

Maximal subgroups

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

subgroupindex
#117218642
H(3,𝔽3)×#1012
#117205842
C3×#44773