Finite Group Databasebeta

← #13154859

Subgroups — #13154859

Order 1728.

360 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 30 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#11723605C211yesmaximal, characteristic
2#11723111C211yesmaximal, characteristic
3C3×#63788C211yesmaximal, characteristic
4#10590205—31nomaximal
5C2×C3×#638C2211yescharacteristic
6C22×C32×Dic3C411yescharacteristic, complemented
7#88994—21no—
8C3×#3792—21no—
9#88864C411yescharacteristic, complemented
10#88806—21no—
11#88803—21no—
12#64138—31no—
13#63961—62no—
14#63788D311yescharacteristic
15#63786—31no—
16C23×C33C2×C411yescharacteristic
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3⋊#157—41no—
19C2×C3⋊(C3⋊Dic3)—21no—
20C3×#638—41no—
21C2×C3×C3⋊Dic3D411yescharacteristic
22C2×C32×Dic3—21no—
23C2×C32×Dic3—41no—
24C3×#622—41no—
25#4476—91nomaximal
26C22×C32×C4—31no—
27C2×#638C2×D311yescharacteristic
28C2×C3×#67—62no—
29C22×C3×Dic3—22no—
30C22⋊#157—61no—
31#3792—21no—
32#3757—31no—
33#3757—62no—
34#3757—122no—
35#3752—61no—
36#3750—61no—
37#3724—64no—
38(C2×D3)⋊Dic3—64no—
39C22×C33—21no—
40C22×C33—41no—
41C22×C33C22⋊C411yescharacteristic
42C2×C3×C3⋊D3—21no—
43C2×C3×C3⋊D3—41no—
44C3⋊#157—81no—
45C3⋊(C3⋊Dic3)—41no—
46C3×C3⋊Dic3—21no—
47C32×Dic3—41no—
48#935—91no—
49#929—182no—
50C3×#100—91no—
51#834—91no—
52C23×C32—22no—
53C23×C32—42no—
54C23×C32C4×D311yescharacteristic
55C22×C3⋊D3D1211yescharacteristic
56C22×C3×D3—62no—
57C2×#157—121no—
58C2×C32×C4—61no—
59C2×C32×C4—121no—
60#638—41no—
61C2×C3⋊Dic3—61no—
62C2×C3⋊Dic3—64no—
63C2×C3⋊Dic3—124no—
64C2×C3⋊Dic3#6711yescharacteristic
65C3×#67—124no—
66C2×C3×Dic3—24no—
67C2×C3×Dic3—44no—
68C2×C3×Dic3—62no—
69#622—41no—
70#352—271noSylow
71C2×C33—21no—
72C2×C33—41no—
73C2×C33#9911yescharacteristic
74C3×C3⋊D3—41no—
75C2×C3×D4—91no—
76C22×C3×C4—62no—
77C2×#67—62no—
78C22×Dic3#15711yescharacteristic, complemented
79C3×M16—181no—
80#217—91no—
81#217—182no—
82#217—181no—
83#217—181no—
84#217—362no—
85(C2×D3)⋊C4—184no—
86Dic3⋊C4—181no—
87Dic3⋊C4—184no—
88C22×C32—21no—
89C22×C32—24no—
90C22×C32—22no—
91C22×C32—44no—
92C22×C32—42no—
93C22×C32—41no—
94C22×C32—44no—
95C22×C32—84no—
96C22×C32(C2×D3)⋊C411yescharacteristic
97C2×C3⋊D3—21no—
98C2×C3⋊D3—41no—
99C2×C3×D3—122no—
100C2×C3×D3—124no—
101#157—241no—
102C32×C4—121no—
103C3⋊Dic3—21no—
104C3⋊Dic3—121no—
105C3⋊Dic3—124no—
106C3⋊Dic3—244no—
107C3×Dic3—44no—
108C3×Dic3—122no—
109#123—271no—
110#100—91no—
111#99—271no—
112C33#35211yescharacteristic, complemented, Sylow
113C23×C3—22no—
114C23×C3—42no—
115C23×C3—91no—
116C23×C3C2×#15711yescharacteristic
117C22×D3—62no—
118C3×D4—361no—
119C2×C3×C4—64no—
120C2×C3×C4—91no—
121C2×C3×C4—124no—
122#67—124no—
123C2×Dic3—21no—
124C2×Dic3—41no—
125C2×Dic3—62no—
126C2×Dic3—184no—
127C2×Dic3—181no—
128C2×Dic3—364no—
129C2×Dic3—361no—
130C3×C8—361no—
131C2×C32—21no—
132C2×C32—24no—
133C2×C32—22no—
134C2×C32—44no—
135C2×C32—42no—
136C2×C32—41no—
137C2×C32—44no—
138C2×C32—84no—
139C2×C32#80611yescharacteristic
140C3⋊D3—41no—
141C3×D3—242no—
142C2×D4—91no—
143C22×C4—31no—
144M16—181no—
145C4⋊C4—541no—
146C22⋊C4—271no—
147C22⋊C4—541no—
148C22⋊C4—541no—
149C22×C3—24no—
150C22×C3—21no—
151C22×C3—22no—
152C22×C3—44no—
153C22×C3—41no—
154C22×C3—44no—
155C22×C3—42no—
156C22×C3—84no—
157C22×C3—181no—
158C22×C3—361no—
159C22×C3C22⋊#15711yescharacteristic
160C2×D3—122no—
161C2×D3—124no—
162C3×C4—124no—
163C3×C4—181no—
164Dic3—41no—
165Dic3—122no—
166Dic3—361no—
167Dic3—364no—
168Dic3—724no—
169Dic3—721no—
170C32—22no—
171C32—42no—
172C32#447611yescharacteristic, complemented
173C23—91no—
174C23D3×#15711yescharacteristic
175D4—361no—
176C2×C4—61no—
177C2×C4—91no—
178C2×C4—121no—
179C2×C4—541no—
180C2×C4—1081no—
181C8—361no—
182C2×C3—24no—
183C2×C3—21no—
184C2×C3—22no—
185C2×C3—44no—
186C2×C3—44no—
187C2×C3—41no—
188C2×C3—42no—
189C2×C3—84no—
190C2×C3—361no—
191C2×C3#6378611yescharacteristic
192D3—242no—
193C22—21no—
194C22—41no—
195C22—181no—
196C22—361no—
197C22(C2×D3)⋊#15711yescharacteristic
198C4—121no—
199C4—181no—
200C4—1081no—
201C4—2161no—
202C3—22no—
203C3—42no—
204C3#1059020511yescharacteristic, complemented
205C2—21no—
206C2—41no—
207C2—361no—
208C2#1172272711yescharacteristic
subgroupG/H
centerC2#11722727
commutator, FittingC23×C33C2×C4
Frattini, hypercenterC23D3×#157
socleC2×C33#99
prime radical, cosocleC2×C3×#638C22
hypocenterC33#352
LCSG ▷ C23×C33 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
6. C22×C32×Dic3C4semidirect
9. #88864C4semidirect
78. C22×Dic3#157semidirect
112. C33#352semidirect
172. C32#4476semidirect
204. C3#10590205semidirect

Maximal subgroups

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

subgroupindex
#117236052
#117231112
C3×#637882
#105902053
#44769