Finite Group Databasebeta

← #13136657

Subgroups — #13136657

Order 1728.

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

normal subgroups only

This group has 175 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#11720774C211yesmaximal, characteristic, complemented
3#11720252C211yesmaximal, characteristic, complemented
4#88649—21no—
5C4⋊#1088—21no—
6C8×#272C2211yescharacteristic
7#3744⋊C3—21no—
8#88433C411yescharacteristic, complemented
9#88432C411yescharacteristic, complemented
10(C3⋊(C3⋊C16))⋊C3—21no—
11#63806—61no—
12#63543—61no—
13C2×#1088—41no—
14C4×#272D411yescharacteristic
15C4⋊((C3⋊D3)⋊C3)—21no—
16#634⋊C3—21no—
17#6200—41no—
18((C3⋊D3)⋊C3)⋊C4—41no—
19C8×H(3,𝔽3)D411yescharacteristic
20#6180C2×C411yescharacteristic
21#4488—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×#272D811yescharacteristic
33C2×(C3⋊D3)⋊C3—41no—
34#1088—81no—
35C4×H(3,𝔽3)C22⋊C411yescharacteristic
36#1084SD1611yescharacteristic
37(C3⋊Dic3)⋊C3—41no—
38#916—181no—
39#904—91no—
40C3×#106—91no—
41#812—91no—
42#798—181no—
43C4⋊(C3⋊D3)—21no—
44#634—21no—
45C3×D12—61no—
46C3×C4×D3—61no—
47C3×C4×D3—61no—
48C3×Dic6—61no—
49#626—121no—
50#625—121no—
51#624—121no—
52D3×Dic3—121no—
53C32×C8—21no—
54C32×C8—21no—
55C3×C3⋊C8—61no—
56C3×C3⋊C8—61no—
57#360—271noSylow
58C2×H(3,𝔽3)#10211yescharacteristic
59#272—21no—
60(C3⋊D3)⋊C3—81no—
61#239—181no—
62#237—181no—
63#235—181no—
64C3×SD16—181no—
65C2×C3×C8—91no—
66C3×C4⋊C4—181no—
67#216—181no—
68#213—181no—
69Dic12—91no—
70D24—91no—
71#204—61no—
72#204—181no—
73D3×C8—61no—
74D3×C8—61no—
75C3×C16—181no—
76C3⋊C16—61no—
77C3⋊C16—181no—
78C2×C3⋊D3—41no—
79C2×C3×D3—61no—
80C2×C3×D3—61no—
81C2×C3×D3—121no—
82C32×C4—21no—
83C32×C4—21no—
84C3⋊Dic3—41no—
85C3×Dic3—61no—
86C3×Dic3—61no—
87C3×Dic3—121no—
88#135—271no—
89M32—271no—
90#106—91no—
91H(3,𝔽3)#36011yescharacteristic, complemented, Sylow
92C3×Q8—181no—
93C3×D4—181no—
94C2×C3×C4—91no—
95C2×C3×C4—361no—
96#67—361no—
97#67—361no—
98C2×Dic3—361no—
99D12—61no—
100D12—181no—
101D12—361no—
102C4×D3—61no—
103C4×D3—61no—
104C4×D3—361no—
105C4×D3—361no—
106Dic6—61no—
107Dic6—181no—
108C3×C8—61no—
109C3×C8—61no—
110C3×C8—91no—
111C3×C8GO+(4,𝔽2)11yescharacteristic
112C3⋊C8—61no—
113C3⋊C8—61no—
114C2×C32—21no—
115C2×C32—21no—
116C3⋊D3—81no—
117C3×D3—121no—
118C3×D3—121no—
119C3×D3—241no—
120#41—541no—
121Q16—271no—
122SD16—181no—
123D8—271no—
124C2×C8—91no—
125C4⋊C4—181no—
126C16—181no—
127C22×C3—91no—
128C22×C3—361no—
129C2×D3—61no—
130C2×D3—61no—
131C2×D3—121no—
132C2×D3—361no—
133C3×C4—61no—
134C3×C4—61no—
135C3×C4—91no—
136C3×C4—361no—
137C3×C4—721no—
138C3×C4D32⋊C411yescharacteristic
139Dic3—61no—
140Dic3—61no—
141Dic3—121no—
142Dic3—361no—
143C32—21no—
144C32—21no—
145Q8—181no—
146D4—181no—
147D4—1081no—
148C2×C4—91no—
149C2×C4—361no—
150C2×C4—1081no—
151C8—91no—
152C8#1088⋊C211yescharacteristic
153C2×C3—61no—
154C2×C3—61no—
155C2×C3—181no—
156C2×C3—721no—
157C2×C3#6372911yescharacteristic
158D3—121no—
159D3—121no—
160D3—241no—
161D3—721no—
162C22—91no—
163C22—361no—
164C4—91no—
165C4—361no—
166C4—721no—
167C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
168C3—61no—
169C3—61no—
170C3#1059001911yescharacteristic
171C2—181no—
172C2—721no—
173C2#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)#360
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. #11720774C2semidirect
3. #11720252C2semidirect
8. #88433C4semidirect
9. #88432C4semidirect
91. H(3,𝔽3)#360semidirect

Maximal subgroups

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

subgroupindex
#117208432
#117207742
#117202522
#44889