Finite Group Databasebeta

← #13136662

Subgroups — #13136662

Order 1728.

174 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 23 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#11720843C211yesmaximal, characteristic
2#11720776C211yesmaximal, characteristic, complemented
3#11720251C211yesmaximal, characteristic, complemented
4#88649—21no—
5#88627—21no—
6C8×#272C2211yescharacteristic
7#3744⋊C3—21no—
8#88433C411yescharacteristic
9#88432C411yescharacteristic
10(C3⋊(C3⋊C16))⋊C3—21no—
11#63806—61no—
12D3×C3⋊C16—61no—
13C4×#272D411yescharacteristic
14C4⋊((C3⋊D3)⋊C3)—21no—
15#634⋊C3—21no—
16#6200—41no—
17((C3⋊D3)⋊C3)⋊C4—41no—
18#6188—41no—
19C8×H(3,𝔽3)D411yescharacteristic
20#6180C2×C411yescharacteristic
21#4497—91nomaximal
22#3799—61no—
23#3795—61no—
24#3744—21no—
25C3×#204—61no—
26C3×D3×C8—61no—
27C3×D3×C8—61no—
28#3719—61no—
29#3714—61no—
30C3⋊(C3⋊C16)—21no—
31C3×C3⋊C16—61no—
32C2×#272SD1611yescharacteristic
33C2×(C3⋊D3)⋊C3—41no—
34C4×H(3,𝔽3)C22⋊C411yescharacteristic
35#1084D811yescharacteristic
36(C3⋊Dic3)⋊C3—41no—
37#916—181no—
38#904—91no—
39C3×#108—91no—
40C2×C3⋊C16—91no—
41D3×C16—181no—
42C4⋊(C3⋊D3)—21no—
43#634—21no—
44C3×D12—61no—
45C3×C4×D3—61no—
46C3×C4×D3—61no—
47C3×Dic6—61no—
48#626—121no—
49#625—121no—
50#624—121no—
51D3×Dic3—121no—
52C32×C8—21no—
53C32×C8—21no—
54C3×C3⋊C8—61no—
55C3×C3⋊C8—61no—
56#359—271noSylow
57C2×H(3,𝔽3)#10211yescharacteristic
58#272—21no—
59(C3⋊D3)⋊C3—81no—
60#239—181no—
61#237—181no—
62#235—181no—
63C3×SD16—181no—
64C3×M16—181no—
65C2×C3×C8—91no—
66#216—181no—
67#213—181no—
68Dic12—91no—
69D24—91no—
70#204—61no—
71#204—181no—
72D3×C8—61no—
73D3×C8—61no—
74C3×C16—181no—
75C3⋊C16—61no—
76C3⋊C16—181no—
77C2×C3⋊D3—41no—
78C2×C3×D3—61no—
79C2×C3×D3—61no—
80C2×C3×D3—121no—
81C32×C4—21no—
82C32×C4—21no—
83C3⋊Dic3—41no—
84C3×Dic3—61no—
85C3×Dic3—61no—
86C3×Dic3—121no—
87#135—271no—
88C2×C16—271no—
89#108—91no—
90H(3,𝔽3)#35911yescharacteristic, complemented, Sylow
91C3×Q8—181no—
92C3×D4—181no—
93C2×C3×C4—91no—
94#67—361no—
95#67—361no—
96C2×Dic3—361no—
97D12—61no—
98D12—181no—
99D12—361no—
100C4×D3—61no—
101C4×D3—61no—
102C4×D3—361no—
103C4×D3—361no—
104Dic6—61no—
105Dic6—181no—
106C3×C8—61no—
107C3×C8—61no—
108C3×C8—91no—
109C3×C8—361no—
110C3×C8GO+(4,𝔽2)11yescharacteristic
111C3⋊C8—61no—
112C3⋊C8—61no—
113C2×C32—21no—
114C2×C32—21no—
115C3⋊D3—81no—
116C3×D3—62no—
117C3×D3—121no—
118C3×D3—241no—
119#41—541no—
120Q16—271no—
121SD16—181no—
122D8—271no—
123M16—181no—
124C2×C8—91no—
125C16—181no—
126C22×C3—91no—
127C22×C3—361no—
128C2×D3—61no—
129C2×D3—61no—
130C2×D3—121no—
131C2×D3—361no—
132C3×C4—61no—
133C3×C4—61no—
134C3×C4—91no—
135C3×C4—361no—
136C3×C4D32⋊C411yescharacteristic
137Dic3—61no—
138Dic3—61no—
139Dic3—121no—
140Dic3—361no—
141C32—21no—
142C32—21no—
143Q8—181no—
144D4—181no—
145D4—1081no—
146C2×C4—91no—
147C2×C4—1081no—
148C8—91no—
149C8—361no—
150C8#1088⋊C211yescharacteristic
151C2×C3—61no—
152C2×C3—61no—
153C2×C3—181no—
154C2×C3—721no—
155C2×C3#6372911yescharacteristic
156D3—62no—
157D3—121no—
158D3—241no—
159D3—721no—
160C22—91no—
161C22—361no—
162C4—91no—
163C4—361no—
164C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
165C3—61no—
166C3—61no—
167C3#1059001811yescharacteristic
168C2—181no—
169C2—721no—
170C2#1172069511yescharacteristic
subgroupG/H
centerC2#11720695
commutator#6180C2×C4
FrattiniC3×C8GO+(4,𝔽2)
FittingC8×H(3,𝔽3)D4
socleC2×C3#63729
prime radical#11720843C2
cosocleC8×#272C22
hypercenterC8#1088⋊C2
hypocenterH(3,𝔽3)#359
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. #11720776C2semidirect
3. #11720251C2semidirect
90. H(3,𝔽3)#359semidirect

Maximal subgroups

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

subgroupindex
#117208432
#117207762
#117202512
#44979