Finite Group Databasebeta

← #13154861

Subgroups — #13154861

Order 1728.

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

normal subgroups only

This group has 400 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#11723609C211yesmaximal, characteristic
2#11723111C211yesmaximal, characteristic
3C3×#63788C211yesmaximal, characteristic
4#10590154—31nomaximal
5C22×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
6C2×C3×#638C2211yescharacteristic
7#88994—21no—
8C3×#3792—21no—
9C32×#217C411yescharacteristic, complemented
10Dic3⋊(C3⋊Dic3)—21no—
11#88803—21no—
12#64095—31no—
13#63962—62no—
14#63788D311yescharacteristic
15#63786—31no—
16C23×C33C2×C411yescharacteristic
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3⋊#157—41no—
19C2×C3⋊(C3⋊Dic3)—21no—
20C2×C3⋊(C3⋊Dic3)—41no—
21C3×#638—41no—
22C2×C3×C3⋊Dic3D411yescharacteristic
23C2×C32×Dic3—21no—
24C3×#622—41no—
25#4476—91nomaximal
26C2×#638C2×D311yescharacteristic
27C22×C3⋊Dic3—31no—
28C22×C3⋊Dic3—62no—
29C22×C3⋊Dic3—122no—
30C2×C3×#67—62no—
31C22⋊#157—61no—
32#3792—21no—
33C32×C22⋊C4—31no—
34#3752—61no—
35C4⋊(C3⋊Dic3)—61no—
36C3×#217—22no—
37Dic3⋊Dic3—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#886—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—
59#638—41no—
60C2×C3⋊Dic3—61no—
61C2×C3⋊Dic3—64no—
62C2×C3⋊Dic3—124no—
63C2×C3⋊Dic3—124no—
64C2×C3⋊Dic3—128no—
65C2×C3⋊Dic3—121no—
66C2×C3⋊Dic3#6711yescharacteristic
67C3×#67—124no—
68C2×C3×Dic3—24no—
69C2×C3×Dic3—62no—
70#622—41no—
71#352—271noSylow
72C2×C33—21no—
73C2×C33—41no—
74C2×C33#9911yescharacteristic
75C3×C3⋊D3—41no—
76C2×C3×D4—91no—
77C2×#67—62no—
78C22×Dic3—91no—
79C22×Dic3—182no—
80C22×Dic3—362no—
81C3×M16—181no—
82C3×C22⋊C4—62no—
83#217—181no—
84#217—181no—
85#217#15711yescharacteristic, complemented
86(C2×D3)⋊C4—184no—
87C4⋊Dic3—184no—
88Dic3⋊C4—181no—
89C22×C32—21no—
90C22×C32—24no—
91C22×C32—22no—
92C22×C32—44no—
93C22×C32—42no—
94C22×C32—41no—
95C22×C32—44no—
96C22×C32—48no—
97C22×C32(C2×D3)⋊C411yescharacteristic
98C2×C3⋊D3—21no—
99C2×C3⋊D3—41no—
100C2×C3×D3—122no—
101C2×C3×D3—124no—
102#157—241no—
103C32×C4—121no—
104C3⋊Dic3—21no—
105C3⋊Dic3—124no—
106C3⋊Dic3—128no—
107C3⋊Dic3—121no—
108C3×Dic3—44no—
109C3×Dic3—122no—
110#123—271no—
111#100—91no—
112#99—271no—
113C33#35211yescharacteristic, complemented, Sylow
114C23×C3—22no—
115C23×C3—42no—
116C23×C3—91no—
117C23×C3C2×#15711yescharacteristic
118C22×D3—62no—
119C3×D4—361no—
120C2×C3×C4—64no—
121C2×C3×C4—91no—
122#67—124no—
123C2×Dic3—21no—
124C2×Dic3—62no—
125C2×Dic3—184no—
126C2×Dic3—181no—
127C2×Dic3—364no—
128C2×Dic3—364no—
129C2×Dic3—368no—
130C2×Dic3—361no—
131C2×Dic3—361no—
132C3×C8—361no—
133C2×C32—21no—
134C2×C32—24no—
135C2×C32—22no—
136C2×C32—44no—
137C2×C32—42no—
138C2×C32—44no—
139C2×C32—48no—
140C2×C32—41no—
141C2×C32#80611yescharacteristic
142C3⋊D3—41no—
143C3×D3—242no—
144C2×D4—91no—
145C22×C4—271no—
146M16—181no—
147C4⋊C4—541no—
148C22⋊C4—31no—
149C22⋊C4—541no—
150C22⋊C4—541no—
151C22×C3—24no—
152C22×C3—21no—
153C22×C3—22no—
154C22×C3—44no—
155C22×C3—41no—
156C22×C3—44no—
157C22×C3—48no—
158C22×C3—42no—
159C22×C3—181no—
160C22×C3—361no—
161C22×C3C22⋊#15711yescharacteristic
162C2×D3—122no—
163C2×D3—124no—
164C3×C4—124no—
165C3×C4—181no—
166Dic3—41no—
167Dic3—122no—
168Dic3—364no—
169Dic3—368no—
170Dic3—361no—
171Dic3—721no—
172C32—22no—
173C32—42no—
174C32#447611yescharacteristic, complemented
175C23—91no—
176C23D3×#15711yescharacteristic
177D4—361no—
178C2×C4—61no—
179C2×C4—91no—
180C2×C4—541no—
181C2×C4—1081no—
182C2×C4—1081no—
183C8—361no—
184C2×C3—24no—
185C2×C3—21no—
186C2×C3—22no—
187C2×C3—44no—
188C2×C3—44no—
189C2×C3—48no—
190C2×C3—41no—
191C2×C3—42no—
192C2×C3—361no—
193C2×C3#6377411yescharacteristic
194D3—242no—
195C22—21no—
196C22—41no—
197C22—181no—
198C22—361no—
199C22(C2×D3)⋊#15711yescharacteristic
200C4—121no—
201C4—181no—
202C4—1081no—
203C4—2161no—
204C3—22no—
205C3—42no—
206C3#1059015411yescharacteristic, complemented
207C2—21no—
208C2—41no—
209C2—361no—
210C2#1172272811yescharacteristic
subgroupG/H
centerC2#11722728
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
5. C22×C3⋊(C3⋊Dic3)C4semidirect
9. C32×#217C4semidirect
85. #217#157semidirect
113. C33#352semidirect
174. C32#4476semidirect
206. C3#10590154semidirect

Maximal subgroups

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

subgroupindex
#117236092
#117231112
C3×#637882
#105901543
#44769