Finite Group Databasebeta

← #13136658

Subgroups — #13136658

Order 1728.

203 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 25 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#11720838C211yesmaximal, characteristic
2#11720775C211yesmaximal, characteristic, complemented
3#11720251C211yesmaximal, characteristic, complemented
4#88655—21no—
5C4⋊#1088—21no—
6C8×#272C2211yescharacteristic
7#3745⋊C3—21no—
8#88432C412yescomplemented
9(C3⋊(C3⋊C16))⋊C3—21no—
10D3×D24—61no—
11D3×C3⋊C16—61no—
12C2×((C3⋊D3)⋊C3)⋊C2—41no—
13C2×#1088—41no—
14C4×#272D411yescharacteristic
15C4⋊((C3⋊D3)⋊C3)—22no—
16#6198—41no—
17C8×H(3,𝔽3)D411yescharacteristic
18#6180C2×C411yescharacteristic
19#4491—91nomaximal
20D3×D12—62no—
21#3745—21no—
22C3×D24—61no—
23C3×D3×C8—61no—
24C3×D3×C8—61no—
25#3714—62no—
26C3⋊(C3⋊C16)—21no—
27C3×C3⋊C16—61no—
28C2×#272D811yescharacteristic
29C2×(C3⋊D3)⋊C3—42no—
30((C3⋊D3)⋊C3)⋊C2—81no—
31((C3⋊D3)⋊C3)⋊C2—81no—
32#1088—81no—
33C4×H(3,𝔽3)C22⋊C411yescharacteristic
34#1084SD1611yescharacteristic
35D3×D8—181no—
36C2×D24—91no—
37C3×#107—91no—
38C2×C3⋊C16—91no—
39D3×C16—181no—
40C2×D32—122no—
41C4⋊(C3⋊D3)—22no—
42C3×D12—62no—
43C3×C4×D3—61no—
44C3×C4×D3—61no—
45#626—122no—
46C32×C8—21no—
47C32×C8—21no—
48C3×C3⋊C8—61no—
49C3×C3⋊C8—61no—
50#357—271noSylow
51C2×H(3,𝔽3)#10211yescharacteristic
52#272D1611yescharacteristic
53#272SD3211yescharacteristic
54(C3⋊D3)⋊C3—82no—
55D3×D4—182no—
56C2×D12—181no—
57C3×D8—181no—
58C2×C3×C8—91no—
59C3×C4⋊C4—181no—
60#213—182no—
61D24—61no—
62D24—92no—
63D24—181no—
64D3×C8—61no—
65D3×C8—61no—
66C3×C16—181no—
67C3⋊C16—61no—
68C3⋊C16—181no—
69C2×C3⋊D3—42no—
70C2×C3×D3—61no—
71C2×C3×D3—61no—
72C2×C3×D3—122no—
73D32—242no—
74D32—242no—
75C32×C4—21no—
76C32×C4—21no—
77C3×Dic3—61no—
78C3×Dic3—61no—
79C2×D8—271no—
80C2×C16—271no—
81#107—91no—
82H(3,𝔽3)#35711yescharacteristic, complemented, Sylow
83C22×D3—362no—
84C22×D3—361no—
85C3×D4—182no—
86C2×C3×C4—91no—
87C2×C3×C4—361no—
88#67—362no—
89D12—62no—
90D12—182no—
91D12—361no—
92C4×D3—61no—
93C4×D3—61no—
94C3×C8—61no—
95C3×C8—61no—
96C3×C8—91no—
97C3×C8GO+(4,𝔽2)11yescharacteristic
98C3⋊C8—61no—
99C3⋊C8—61no—
100C2×C32—21no—
101C2×C32—21no—
102C3⋊D3—82no—
103C3×D3—61no—
104C3×D3—61no—
105C3×D3—61no—
106C3×D3—61no—
107C3×D3—242no—
108C2×D4—541no—
109D8—181no—
110D8—272no—
111C2×C8—91no—
112C4⋊C4—181no—
113C16—181no—
114C22×C3—91no—
115C22×C3—362no—
116C2×D3—61no—
117C2×D3—61no—
118C2×D3—122no—
119C2×D3—362no—
120C2×D3—722no—
121C2×D3—722no—
122C2×D3—721no—
123C2×D3—721no—
124C3×C4—61no—
125C3×C4—61no—
126C3×C4—91no—
127C3×C4—721no—
128C3×C4D32⋊C411yescharacteristic
129Dic3—61no—
130Dic3—61no—
131C32—21no—
132C32—21no—
133C23—1081no—
134D4—182no—
135D4—1081no—
136C2×C4—91no—
137C2×C4—361no—
138C8—91no—
139C8#1088⋊C211yescharacteristic
140C2×C3—61no—
141C2×C3—61no—
142C2×C3—91no—
143C2×C3—91no—
144C2×C3—722no—
145C2×C3#6372911yescharacteristic
146D3—61no—
147D3—61no—
148D3—61no—
149D3—61no—
150D3—242no—
151D3—722no—
152C22—91no—
153C22—362no—
154C22—2161no—
155C22—2161no—
156C4—91no—
157C4—721no—
158C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
159C3—61no—
160C3—61no—
161C3#1059001611yescharacteristic
162C2—91no—
163C2—91no—
164C2—722no—
165C2#1172069511yescharacteristic
subgroupG/H
centerC2#11720695
commutator#6180C2×C4
FrattiniC3×C8GO+(4,𝔽2)
FittingC8×H(3,𝔽3)D4
socleC2×C3#63729
prime radical#11720838C2
cosocleC8×#272C22
hypercenterC8#1088⋊C2
hypocenterH(3,𝔽3)#357
LCSG ▷ #6180 ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11720775C2semidirect
3. #11720251C2semidirect
8. #88432C4semidirect
82. H(3,𝔽3)#357semidirect

Maximal subgroups

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

subgroupindex
#117208382
#117207752
#117202512
#44919