Finite Group Databasebeta

← #13154853

Subgroups — #13154853

Order 1728.

400 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 60 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×#88993C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic
3#11723595C211yesmaximal, characteristic
4#10590156—31nomaximal
5C22×C3×C3⋊Dic3C2211yescharacteristic
6#88993—21no—
7#88993C412yescomplemented
8C2×C3⋊#622—21no—
9C3×#3792—21no—
10C3×#3792—21no—
11C2×C3×#622C412yescomplemented
12#88864C411yescharacteristic
13C32×#217C411yescharacteristic
14#88807—21no—
15Dic3×C3⋊Dic3—21no—
16C2×#3792—31no—
17C2×#3792D311yescharacteristic
18#64086—31no—
19#63955—62no—
20C23×C33C2×C411yescharacteristic
21C2×C3⋊(C3⋊Dic3)—21no—
22C2×C3×C3⋊Dic3C2×C411yescharacteristic
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3D411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3Q811yescharacteristic
28C2×C32×Dic3—21no—
29C3⋊#622—22no—
30C3×#622—22no—
31#4556—91nomaximal
32C22×C3⋊Dic3C2×D311yescharacteristic
33C22×C3×Dic3—62no—
34#3792—21no—
35#3792—21no—
36#3792—32no—
37#3792—61no—
38C2×#622—61no—
39C2×#622Dic312yescomplemented
40C32×C22⋊C4—31no—
41#3757—31no—
42#3757—62no—
43#3757—122no—
44#3750—61no—
45C4×C3⋊Dic3—61no—
46C3×#217—22no—
47Dic3⋊Dic3—64no—
48Dic32—64no—
49C22×C33—41no—
50C22×C33C4211yescharacteristic
51C22×C33C22⋊C411yescharacteristic
52C22×C33C4⋊C411yescharacteristic
53C3⋊(C3⋊Dic3)—41no—
54C3×C3⋊Dic3C22⋊C411yescharacteristic
55C3×C3⋊Dic3C22⋊C411yescharacteristic
56C3×C3⋊Dic3C4⋊C411yescharacteristic
57C3×C3⋊Dic3C4⋊C411yescharacteristic
58C32×Dic3—41no—
59C2×C3×M16—91no—
60#926—91no—
61C2×#208—91no—
62#877—182no—
63C23×C32—22no—
64C23×C32—42no—
65C23×C32C4×D311yescharacteristic
66C2×C32×C4—61no—
67C2×C3⋊Dic3—61no—
68C2×C3⋊Dic3—64no—
69C2×C3⋊Dic3—124no—
70C2×C3⋊Dic3Dic611yescharacteristic
71C2×C3⋊Dic3C4×D311yescharacteristic
72C2×C3⋊Dic3D1211yescharacteristic
73C2×C3⋊Dic3C2×Dic311yescharacteristic
74C2×C3⋊Dic3#6711yescharacteristic
75C2×C3⋊Dic3#6711yescharacteristic
76C2×C3×Dic3—24no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83#622—22no—
84#622—62no—
85#344—271noSylow
86C2×C33—21no—
87C2×C33—21no—
88C2×C33—21no—
89C2×C33#9511yescharacteristic
90C22×C3×C4—91no—
91C22×Dic3—62no—
92C3×M16—181no—
93C3×M16—181no—
94C2×C3×C8—92no—
95C3×C22⋊C4—62no—
96#217—91no—
97#217—182no—
98#217—362no—
99#217#15711yescharacteristic
100C4⋊Dic3—181no—
101Dic3⋊C4—184no—
102C4×Dic3—181no—
103C4×Dic3—184no—
104#208—92no—
105#208—181no—
106C2×C3⋊C8—181no—
107C22×C32—22no—
108C22×C32—22no—
109C22×C32—22no—
110C22×C32—42no—
111C22×C32—42no—
112C22×C32—42no—
113C22×C32—41no—
114C22×C32—44no—
115C22×C32—84no—
116C22×C32C4×Dic311yescharacteristic
117C22×C32Dic3⋊C411yescharacteristic
118C22×C32(C2×D3)⋊C411yescharacteristic
119C32×C4—121no—
120C3⋊Dic3—121no—
121C3⋊Dic3—124no—
122C3⋊Dic3—244no—
123C3⋊Dic3Dic3⋊C411yescharacteristic
124C3⋊Dic3C4⋊Dic311yescharacteristic
125C3⋊Dic3(C2×D3)⋊C411yescharacteristic
126C3⋊Dic3#21711yescharacteristic
127C3×Dic3—44no—
128C3×Dic3—62no—
129C3×Dic3—62no—
130C3×Dic3—62no—
131C3×Dic3—62no—
132C2×M16—91no—
133C2×M16—271no—
134#117—271no—
135C33#34411yescharacteristic, complemented, Sylow
136C23×C3—22no—
137C23×C3—42no—
138C23×C3C2×#15711yescharacteristic
139C2×C3×C4—64no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×Dic3—21no—
147C2×Dic3—62no—
148C2×Dic3—62no—
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—184no—
154C2×Dic3—181no—
155C2×Dic3—364no—
156C3×C8—182no—
157C3⋊C8—182no—
158C2×C32—21no—
159C2×C32—24no—
160C2×C32—22no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32—44no—
164C2×C32—42no—
165C2×C32—42no—
166C2×C32—42no—
167C2×C32—82no—
168C2×C32—82no—
169C2×C32C3⋊#9511yescharacteristic
170C22×C4—91no—
171M16—181no—
172M16—181no—
173M16—272no—
174M16—541no—
175C2×C8—92no—
176C2×C8—541no—
177C4⋊C4—541no—
178C22⋊C4—31no—
179C22⋊C4—271no—
180C42—541no—
181C22×C3—22no—
182C22×C3—22no—
183C22×C3—22no—
184C22×C3—41no—
185C22×C3—44no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—84no—
190C22×C3C4×#15711yescharacteristic
191C22×C3C4⋊#15711yescharacteristic
192C22×C3C22⋊#15711yescharacteristic
193C3×C4—91no—
194C3×C4—91no—
195C3×C4—91no—
196C3×C4—91no—
197C3×C4—124no—
198Dic3—41no—
199Dic3—62no—
200Dic3—62no—
201Dic3—62no—
202Dic3—62no—
203Dic3—361no—
204Dic3—364no—
205Dic3—724no—
206C32—22no—
207C32—42no—
208C32#455611yescharacteristic, complemented
209C23D3×#15711yescharacteristic
210C2×C4—61no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—541no—
218C8—182no—
219C8—542no—
220C2×C3—24no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—21no—
224C2×C3—22no—
225C2×C3—42no—
226C2×C3—42no—
227C2×C3—44no—
228C2×C3—42no—
229C2×C3—82no—
230C2×C3—82no—
231C2×C3#6378111yescharacteristic
232C22—41no—
233C22Dic3×#15711yescharacteristic
234C22(C2×D3)⋊#15711yescharacteristic
235C22Dic3⋊#15711yescharacteristic
236C4—91no—
237C4—91no—
238C4—91no—
239C4—91no—
240C4—121no—
241C4—1081no—
242C3—22no—
243C3—42no—
244C3#1059015611yescharacteristic, complemented
245C2—21no—
246C2—21no—
247C2—21no—
248C2#1172272211yescharacteristic
subgroupG/H
centerC2#11722722
commutatorC22×C33C42
Frattini, hypercenterC23D3×#157
Fitting, prime radicalC23×C33C2×C4
socleC2×C33#95
cosocleC22×C3×C3⋊Dic3C22
hypocenterC33#344
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
7. #88993C4semidirect
11. C2×C3×#622C4semidirect
39. C2×#622Dic3semidirect
135. C33#344semidirect
208. C32#4556semidirect
244. C3#10590156semidirect

Maximal subgroups

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

subgroupindex
C2×#889932
C2×C3×#37922
#117235952
#105901563
#45569