Finite Group Databasebeta

← #13145555

Subgroups — #13145555

Order 1728.

452 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C2×C3×#3743C211yesmaximal, characteristic
2#11722182C211yesmaximal, characteristic, complemented
3C32×#812C211yesmaximal, characteristic, complemented
4#10589380—31nomaximal
5#10589109—34nomaximal
6C2×C3×C4×C3⋊D3C411yescharacteristic
7C2×C33×C8C2211yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
9C3×#3743—21no—
10C3×#3743—21no—
11C3⋊(C3⋊(C3⋊C16))—21no—
12C32×C3⋊C16—21no—
13C2×#3743D311yescharacteristic
14C2×C3×ΓL(1,𝔽25)—34no—
15C32×M32—31no—
16#63639—31no—
17#63639—34no—
18#63639—64no—
19C3×#812D314yescomplemented
20C22×C3×C3⋊D3C811yescharacteristic
21C2×C33×C4C2×C411yescharacteristic
22C2×C3×C3⋊Dic3C811yescharacteristic
23C3×C4×C3⋊D3—21no—
24C3×C4×C3⋊D3—21no—
25C33×C8D411yescharacteristic
26C33×C8D411yescharacteristic
27C3×C3⋊(C3⋊C8)—21no—
28#4553—91no—
29#4514—94no—
30C2×C4×C3⋊D3Dic311yescharacteristic
31C2×C3×C4×D3—34no—
32C2×C32×C8—24no—
33C2×C32×C8C2×D314yes—
34C2×C32×C8C2×D311yescharacteristic
35C2×C3⋊(C3⋊C8)Dic311yescharacteristic
36#3743—21no—
37#3743—21no—
38C2×C3×C3⋊C8—34no—
39C3×ΓL(1,𝔽25)—64no—
40C3×ΓL(1,𝔽25)—64no—
41C32×C16—61no—
42C3⋊(C3⋊C16)—68no—
43C3⋊(C3⋊C16)—64no—
44C3⋊(C3⋊C16)—61no—
45C3×C3⋊C16—24no—
46C22×C33C2×C811yescharacteristic
47C2×C3×C3⋊D3—21no—
48C2×C3×C3⋊D3—41no—
49C33×C4C22⋊C411yescharacteristic
50C33×C4M1611yescharacteristic
51C3×C3⋊Dic3—21no—
52C2×C3×M16—91no—
53C2×ΓL(1,𝔽25)—34no—
54C3×M32—34no—
55#812—94no—
56#812—91no—
57#812—184no—
58#812C3⋊D311yescharacteristic, complemented
59C22×C3⋊D3C3⋊C811yescharacteristic
60C22×C3×D3—34no—
61C2×C32×C4—24no—
62C2×C32×C4C4×D314yes—
63C2×C32×C4C2×Dic311yescharacteristic
64C2×C3⋊Dic3C3⋊C811yescharacteristic
65C4×C3⋊D3—21no—
66C4×C3⋊D3—21no—
67C2×C3×Dic3—34no—
68C3×C4×D3—64no—
69C3×C4×D3—64no—
70C32×C8—24no—
71C32×C8—24no—
72C32×C8D1214yes—
73C32×C8#6714yes—
74C32×C8#6711yescharacteristic
75C32×C8#6711yescharacteristic
76C3⋊(C3⋊C8)—21no—
77C3×C3⋊C8—64no—
78#349—271noSylow
79C2×C33—21no—
80C2×C33C22⋊C811yescharacteristic
81C3×C3⋊D3—41no—
82C22×C3×C4—91no—
83C2×C4×D3—34no—
84C3×M16—181no—
85C3×M16—181no—
86C2×C3×C8—24no—
87C2×C3×C8—91no—
88C2×C3×C8D3214yes—
89C2×C3×C8C2×C3⋊D311yescharacteristic
90C2×C3⋊C8—34no—
91ΓL(1,𝔽25)—64no—
92ΓL(1,𝔽25)—64no—
93C3×C16—64no—
94C3⋊C16—21no—
95C3⋊C16—188no—
96C3⋊C16—184no—
97C3⋊C16—181no—
98C22×C32—24no—
99C22×C32D3×C814yes—
100C22×C32C2×C3⋊C811yescharacteristic
101C2×C3⋊D3—21no—
102C2×C3⋊D3—41no—
103C2×C3×D3—64no—
104C2×C3×D3—124no—
105C32×C4—24no—
106C32×C4—24no—
107C32×C4ΓL(1,𝔽25)14yes—
108C32×C4#20811yescharacteristic
109C32×C4(C2×D3)⋊C414yes—
110C32×C4#21711yescharacteristic
111C3⋊Dic3—21no—
112C3×Dic3—64no—
113C2×M16—91no—
114M32—31no—
115M32—271no—
116C33#34911yescharacteristic, complemented, Sylow
117C23×C3—91no—
118C22×D3—34no—
119C2×C3×C4—24no—
120C2×C3×C4—91no—
121C2×C3×C4—181no—
122C2×C3×C4—181no—
123C2×C3×C4D3×Dic314yes—
124C2×C3×C4C4×C3⋊D311yescharacteristic
125C2×Dic3—34no—
126C4×D3—64no—
127C4×D3—64no—
128C3×C8—24no—
129C3×C8—24no—
130C3×C8—181no—
131C3×C8#62514yes—
132C3×C8#62614yes—
133C3×C8C4⋊(C3⋊D3)11yescharacteristic
134C3×C8#63811yescharacteristic
135C3⋊C8—64no—
136C2×C32—21no—
137C2×C32—24no—
138C2×C32—24no—
139C2×C32—44no—
140C2×C32(C2×D3)⋊C814yes—
141C2×C32#83011yescharacteristic
142C3⋊D3—41no—
143C3×D3—124no—
144C22×C4—91no—
145M16—181no—
146M16—181no—
147C2×C8—91no—
148C2×C8D3×C3⋊D311yescharacteristic
149C16—61no—
150C16—541no—
151C22×C3—24no—
152C22×C3—181no—
153C22×C3—361no—
154C22×C3D3×C3⋊C814yes—
155C22×C3C8×C3⋊D311yescharacteristic
156C2×D3—64no—
157C2×D3—124no—
158C3×C4—24no—
159C3×C4—24no—
160C3×C4—181no—
161C3×C4#371114yes—
162C3×C4(C2×D3)⋊Dic314yes—
163C3×C4#374311yescharacteristic
164C3×C4#375211yescharacteristic
165Dic3—64no—
166C32—24no—
167C32#451414yescomplemented
168C32#455311yescharacteristic, complemented
169C23—91no—
170C2×C4—91no—
171C2×C4—181no—
172C2×C4—181no—
173C2×C4Dic3×C3⋊D311yescharacteristic
174C8—181no—
175C8#629011yescharacteristic
176C8#629211yescharacteristic
177C2×C3—24no—
178C2×C3—21no—
179C2×C3—24no—
180C2×C3—44no—
181C2×C3—361no—
182C2×C3(C2×D3)⋊(C3⋊C8)14yes—
183C2×C3#6364711yescharacteristic
184D3—124no—
185C22—181no—
186C22—361no—
187C22C3⋊D3×C3⋊C811yescharacteristic
188C4—181no—
189C4#8878711yescharacteristic
190C4#8880311yescharacteristic
191C3—24no—
192C3#1058910914yescomplemented
193C3#1058938011yescharacteristic, complemented
194C2—21no—
195C2—361no—
196C2#1172202311yescharacteristic
subgroupG/H
centerC4#88803
commutatorC22×C33C2×C8
Frattini, hypercenterC2×C8D3×C3⋊D3
Fitting, cosocleC2×C33×C8C22
socleC2×C33C22⋊C8
prime radicalC22×C3×C3⋊D3C8
hypocenterC33#349
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11722182C2semidirect
3. C32×#812C2semidirect
19. C3×#812D3semidirect
58. #812C3⋊D3semidirect
116. C33#349semidirect
167. C32#4514semidirect
168. C32#4553semidirect
192. C3#10589109semidirect
193. C3#10589380semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
#117221822
C32×#8122
#105893803
#105891093