Finite Group Databasebeta

← #13135760

Subgroups — #13135760

Order 1728.

385 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721721C211yesmaximal, characteristic
2H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
3#11720520C211yesmaximal, characteristic, complemented
4C3×#4464—34nomaximal
5C2×C4⋊#272C411yescharacteristic
6C2×#6229C411yescharacteristic
7C2×C8×H(3,𝔽3)—21no—
8C42×H(3,𝔽3)C2211yescharacteristic
9#88542—21no—
10#88542—21no—
11C2×#6180—21no—
12C3×#875—34no—
13C32×#97D314yescomplemented
14C3×#803—34no—
15C22×#272—21no—
16C2×C4×H(3,𝔽3)C2×C411yescharacteristic
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×C4×H(3,𝔽3)D411yescharacteristic
19C2×#1084—21no—
20C4⋊#272—41no—
21#6229—41no—
22C8×H(3,𝔽3)—22no—
23#6180—22no—
24C3×#329—91no—
25#4464—34no—
26C2×C3×D12—34no—
27C2×C3×Dic6—34no—
28C2×C32×C8—24no—
29C32×C42C2×D314yes—
30C3×(C2×D3)⋊C4—64no—
31C3×(C2×D3)⋊C4—64no—
32C2×C3×C3⋊C8—64no—
33C22×H(3,𝔽3)C22⋊C411yescharacteristic
34C2×#272—41no—
35C2×#272—41no—
36C2×#272—41no—
37C4×H(3,𝔽3)—21no—
38C4×H(3,𝔽3)—21no—
39C4×H(3,𝔽3)—21no—
40#1084—41no—
41C3×#124—91no—
42#875—34no—
43C3×#97—34no—
44C3×#97—91no—
45C3×#97C3⋊D311yescharacteristic
46#803—34no—
47C22×C3×D3—64no—
48C2×C32×C4C4×D314yes—
49C2×C32×C4D1214yes—
50C2×C32×C4#6714yes—
51C2×C3×Dic3—64no—
52C3×D12—124no—
53C3×Dic6—124no—
54C32×C8—28no—
55C3×C3⋊C8—68no—
56#329—91noSylow
57C2×H(3,𝔽3)#10011yescharacteristic
58C2×H(3,𝔽3)#10411yescharacteristic
59C2×H(3,𝔽3)#10411yescharacteristic
60#272—81no—
61C2×C3×Q8—91no—
62C2×C3×D4—91no—
63C2×D12—34no—
64C2×Dic6—34no—
65C2×C3×C8—21no—
66C2×C3×C8—64no—
67C2×C3×C8—181no—
68C3×C22⋊C4—181no—
69C3×C22⋊C4—181no—
70C3×C42—34no—
71C3×C42C2×C3⋊D311yescharacteristic
72(C2×D3)⋊C4—64no—
73(C2×D3)⋊C4—64no—
74C2×C3⋊C8—64no—
75C22×C32(C2×D3)⋊C414yes—
76C2×C3×D3—124no—
77C2×C3×D3—124no—
78C2×C3×D3—124no—
79C32×C4—24no—
80C32×C4—24no—
81C32×C4—24no—
82C3×Dic3—124no—
83#124—91no—
84#97—91no—
85#97#27211yescharacteristic, complemented
86H(3,𝔽3)#32911yescharacteristic, complemented, Sylow
87C23×C3—181no—
88C22×D3—64no—
89C3×Q8—361no—
90C3×D4—361no—
91C2×C3×C4—34no—
92C2×C3×C4—34no—
93C2×C3×C4—34no—
94C2×C3×C4—181no—
95C2×C3×C4C4×C3⋊D311yescharacteristic
96C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
97C2×C3×C4#63811yescharacteristic
98C2×Dic3—64no—
99D12—124no—
100Dic6—124no—
101C3×C8—22no—
102C3×C8—68no—
103C3×C8—182no—
104C3⋊C8—68no—
105C2×C32#80514yes—
106C2×C32#82314yes—
107C2×C32#82514yes—
108C3×D3—244no—
109C2×Q8—91no—
110C2×D4—91no—
111C2×C8—21no—
112C2×C8—181no—
113C22⋊C4—181no—
114C22⋊C4—181no—
115C42C2×#27211yescharacteristic
116C22×C3—34no—
117C22×C3—361no—
118C22×C3—361no—
119C22×C3—361no—
120C22×C3#375211yescharacteristic
121C2×D3—124no—
122C2×D3—124no—
123C2×D3—124no—
124C3×C4—21no—
125C3×C4—21no—
126C3×C4—21no—
127C3×C4—64no—
128C3×C4—64no—
129C3×C4—64no—
130C3×C4—361no—
131Dic3—124no—
132C32#446414yescomplemented
133C23—181no—
134Q8—361no—
135D4—361no—
136C2×C4—181no—
137C2×C4C4×#27211yescharacteristic
138C2×C4C4⋊#27211yescharacteristic
139C2×C4#623511yescharacteristic
140C8—22no—
141C8—182no—
142C2×C3—34no—
143C2×C3—34no—
144C2×C3—34no—
145C2×C3—721no—
146C2×C3#6363211yescharacteristic
147C2×C3#6365011yescharacteristic
148C2×C3#6365211yescharacteristic
149D3—244no—
150C22—361no—
151C22—361no—
152C22—361no—
153C22#8854211yescharacteristic
154C4—21no—
155C4—21no—
156C4—21no—
157C4—361no—
158C3—34no—
159C3#1058933011yescharacteristic
160C2—721no—
161C2#1172052211yescharacteristic
162C2#1172056011yescharacteristic
163C2#1172056211yescharacteristic
subgroupG/H
center, socleC22×C3#3752
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC3×C42C2×C3⋊D3
FittingH(3,𝔽3)×#97C2
prime radicalC2×C4⋊#272C4
cosocleC42×H(3,𝔽3)C22
hypercenterC3×#97C3⋊D3
hypocenterH(3,𝔽3)#329
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#97C2semidirect
3. #11720520C2semidirect
13. C32×#97D3semidirect
85. #97#272semidirect
86. H(3,𝔽3)#329semidirect
132. C32#4464semidirect

Maximal subgroups

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

subgroupindex
#117217212
H(3,𝔽3)×#972
#117205202
C3×#44643