Finite Group Databasebeta

← #13157204

Subgroups — #13157204

Order 1728.

276 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C3×#64116C211yesmaximal, characteristic
2#11723098C211yesmaximal, characteristic, complemented
3C2×C3⋊#3708C211yesmaximal, characteristic, complemented
4#10590187—31nomaximal
5C3×#3828C411yescharacteristic
6C33×M16C411yescharacteristic
7C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
8C3×#3743—21no—
9C3×C8×C3⋊D3—21no—
10C3⋊#3708—21no—
11C3⋊#3708—21no—
12#64116D311yescharacteristic
13C3×#907—62no—
14#63773—31no—
15C2×#3708—31no—
16C2×C33×C4C2×C411yescharacteristic
17C3×#638—21no—
18C3×C4⋊(C3⋊D3)C811yescharacteristic
19C3×C4×C3⋊D3—21no—
20C3×#634C811yescharacteristic
21C33×C8—21no—
22C3×C3⋊(C3⋊C8)D411yescharacteristic
23C3×C3⋊(C3⋊C8)D411yescharacteristic
24#4561—91nomaximal
25#3828Dic311yescharacteristic
26C3×#235—62no—
27C32×M16—22no—
28C32×M16—42no—
29C32×M16Dic311yescharacteristic
30C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
31#3743—21no—
32C8×C3⋊D3—21no—
33C2×C3×C3⋊C8—62no—
34C3×ΓL(1,𝔽25)—64no—
35C3×D3×C8—64no—
36#3708—61no—
37#3708—61no—
38C22×C33M1611yescharacteristic
39C2×C3×C3⋊D3—21no—
40C33×C4C22⋊C411yescharacteristic
41C33×C4C2×C811yescharacteristic
42C3×C3⋊Dic3—21no—
43C3×#131—91no—
44#907—62no—
45#812—91no—
46C2×C3⋊C16—91no—
47C2×C32×C4—22no—
48C2×C32×C4—42no—
49C2×C32×C4C2×Dic311yescharacteristic
50#638—21no—
51C4⋊(C3⋊D3)C3⋊C811yescharacteristic
52C4×C3⋊D3—21no—
53#634C3⋊C811yescharacteristic
54C3×#67—64no—
55C3×D12—62no—
56C3×C4×D3—64no—
57C3×Dic6—62no—
58C32×C8—21no—
59C32×C8—24no—
60C32×C8—44no—
61C3⋊(C3⋊C8)#6711yescharacteristic
62C3⋊(C3⋊C8)#6711yescharacteristic
63C3×C3⋊C8—62no—
64C3×C3⋊C8—62no—
65#350—271noSylow
66C2×C33—21no—
67C2×C33C22⋊C811yescharacteristic
68C3×C3⋊D3—41no—
69C3×#41—91no—
70#235—62no—
71C3×M16—22no—
72C3×M16—42no—
73C3×M16—181no—
74C3×M16#15711yescharacteristic
75C2×C3×C8—91no—
76C2×C3×C8—181no—
77C2×C3⋊C8—62no—
78ΓL(1,𝔽25)—64no—
79D3×C8—64no—
80C3⋊C16—181no—
81C3⋊C16—181no—
82C22×C32—22no—
83C22×C32—42no—
84C22×C32#20811yescharacteristic
85C2×C3⋊D3—21no—
86C2×C3×D3—64no—
87C32×C4—22no—
88C32×C4—22no—
89C32×C4—42no—
90C32×C4—42no—
91C32×C4C2×C3⋊C811yescharacteristic
92C32×C4#21711yescharacteristic
93C3⋊Dic3—21no—
94C3×Dic3—64no—
95#131—91no—
96M32—271no—
97C2×C16—271no—
98C33#35011yescharacteristic, complemented, Sylow
99C3×Q8—91no—
100C3×D4—91no—
101C3×D4—181no—
102C2×C3×C4—22no—
103C2×C3×C4—42no—
104C2×C3×C4—181no—
105C2×C3×C4C2×#15711yescharacteristic
106#67—64no—
107D12—62no—
108C4×D3—64no—
109Dic6—62no—
110C3×C8—24no—
111C3×C8—21no—
112C3×C8—44no—
113C3×C8—91no—
114C3×C8—91no—
115C3⋊C8—62no—
116C3⋊C8—62no—
117C2×C32—22no—
118C2×C32—21no—
119C2×C32—42no—
120C2×C32—42no—
121C2×C32—82no—
122C2×C32#83011yescharacteristic
123C3⋊D3—41no—
124C3×D3—124no—
125#41—91no—
126M16—181no—
127M16C3⋊#15711yescharacteristic
128C2×C8—91no—
129C2×C8—181no—
130C16—541no—
131C16—541no—
132C22×C3—22no—
133C22×C3—42no—
134C22×C3—181no—
135C22×C3#378811yescharacteristic
136C2×D3—64no—
137C3×C4—22no—
138C3×C4—22no—
139C3×C4—42no—
140C3×C4—42no—
141C3×C4—181no—
142C3×C4#378711yescharacteristic
143C3×C4C22⋊#15711yescharacteristic
144Dic3—64no—
145C32—22no—
146C32—42no—
147C32#456111yescharacteristic, complemented
148Q8—91no—
149D4—91no—
150D4—181no—
151C2×C4—181no—
152C2×C4C2×C3⋊#15711yescharacteristic
153C8—21no—
154C8—91no—
155C8—91no—
156C2×C3—21no—
157C2×C3—22no—
158C2×C3—42no—
159C2×C3—42no—
160C2×C3—82no—
161C2×C3—361no—
162C2×C3#6377811yescharacteristic
163D3—124no—
164C22—181no—
165C22#8898911yescharacteristic
166C4—181no—
167C4#8898811yescharacteristic
168C4#8899411yescharacteristic
169C3—22no—
170C3—42no—
171C3#1059018711yescharacteristic, complemented
172C2—21no—
173C2—361no—
174C2#1172310311yescharacteristic
subgroupG/H
centerC4#88994
commutatorC33×C4C2×C8
FrattiniC2×C4C2×C3⋊#157
FittingC33×M16C4
socleC2×C33C22⋊C8
prime radicalC3×#3828C4
cosocleC2×C3×C3⋊(C3⋊C8)C22
hypercenterM16C3⋊#157
hypocenterC33#350
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. #11723098C2semidirect
3. C2×C3⋊#3708C2semidirect
98. C33#350semidirect
147. C32#4561semidirect
171. C3#10590187semidirect

Maximal subgroups

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

subgroupindex
C3×#641162
#117230982
C2×C3⋊#37082
#105901873
#45619