Finite Group Databasebeta

← #13156740

Subgroups — #13156740

Order 1728.

834 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 174 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×#88813C211yesmaximal, characteristic
2#11722061C213yesmaximal
3C3×#63579C213yesmaximal
4#10591248—33nomaximal
5C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
6C22×C3×C3⋊Dic3C2213yes—
7C22×C32×Dic3C2213yes—
8#88813C414yescomplemented
9C2×Dic32—33no—
10#63658—33no—
11C3×C3⋊#95—33no—
12C3×C3⋊#95—33no—
13#63579—33no—
14#63579—33no—
15#63579—63no—
16#63579D313yes—
17C23×C33C2311yescharacteristic
18C2×C3⋊(C3⋊Dic3)—23no—
19C2×C3×C3⋊Dic3—23no—
20C2×C3×C3⋊Dic3—23no—
21C2×C3×C3⋊Dic3C2×C416yes—
22C2×C32×Dic3—23no—
23C2×C32×Dic3—23no—
24C2×C32×Dic3—23no—
25#4651—93no—
26C22×C32×C4—33no—
27C22×C3⋊Dic3—33no—
28C22×C3⋊Dic3—63no—
29C22×C3⋊Dic3—121no—
30C22×C3⋊Dic3C2×D313yes—
31C22×C3×Dic3—23no—
32C22×C3×Dic3—33no—
33C22×C3×Dic3—33no—
34C22×C3×Dic3—63no—
35C22×C3×Dic3C2×D313yes—
36C22×C3×Dic3C2×D313yes—
37Dic32—312no—
38C22×C33C22×C411yescharacteristic
39C22×C33#4113yes—
40C22×C33#4113yes—
41C3⋊(C3⋊Dic3)—41no—
42C3×C3⋊Dic3—26no—
43C32×Dic3—43no—
44C2×C4×Dic3—93no—
45C3×#95—93no—
46C3⋊#95—33no—
47C3⋊#95—33no—
48C3⋊#95—93no—
49C3⋊#95—93no—
50C3⋊#95—93no—
51C3⋊#95—183no—
52C23×C32—23no—
53C23×C32—41no—
54C23×C32C22×D313yes—
55C2×C32×C4—63no—
56C2×C32×C4—63no—
57C2×C32×C4—63no—
58C2×C3⋊Dic3—23no—
59C2×C3⋊Dic3—23no—
60C2×C3⋊Dic3—63no—
61C2×C3⋊Dic3—66no—
62C2×C3⋊Dic3—63no—
63C2×C3⋊Dic3—63no—
64C2×C3⋊Dic3—123no—
65C2×C3⋊Dic3—123no—
66C2×C3⋊Dic3—126no—
67C2×C3⋊Dic3C4×D316yes—
68C2×C3×Dic3—26no—
69C2×C3×Dic3—23no—
70C2×C3×Dic3—23no—
71C2×C3×Dic3—23no—
72C2×C3×Dic3—23no—
73C2×C3×Dic3—23no—
74C2×C3×Dic3—23no—
75C2×C3×Dic3—36no—
76C2×C3×Dic3—36no—
77C2×C3×Dic3—43no—
78C2×C3×Dic3—43no—
79C2×C3×Dic3—66no—
80C2×C3×Dic3—63no—
81C2×C3×Dic3—63no—
82C2×C3×Dic3—63no—
83C2×C3×Dic3—63no—
84C2×C3×Dic3—123no—
85C2×C3×Dic3—123no—
86#383—271noSylow
87C2×C33#11713yes—
88C2×C33#12613yes—
89C2×C33#12611yescharacteristic
90C22×C3×C4—33no—
91C22×C3×C4—33no—
92C22×C3×C4—63no—
93C22×C3×C4—93no—
94C22×Dic3—33no—
95C22×Dic3—33no—
96C22×Dic3—63no—
97C22×Dic3—93no—
98C22×Dic3—183no—
99C22×Dic3—361no—
100C22×Dic3D3213yes—
101C4×Dic3—912no—
102C22×C32—23no—
103C22×C32—23no—
104C22×C32—23no—
105C22×C32—23no—
106C22×C32—23no—
107C22×C32—23no—
108C22×C32—23no—
109C22×C32—41no—
110C22×C32—43no—
111C22×C32—43no—
112C22×C32C2×C4×D313yes—
113C22×C32#23513yes—
114C22×C32#23513yes—
115C22×C32#23713yes—
116C22×C32#23713yes—
117C22×C32#23713yes—
118C22×C32#23913yes—
119C32×C4—123no—
120C3⋊Dic3—26no—
121C3⋊Dic3—126no—
122C3⋊Dic3—124no—
123C3⋊Dic3—123no—
124C3×Dic3—46no—
125C3×Dic3—43no—
126C3×Dic3—43no—
127C3×Dic3—66no—
128C3×Dic3—66no—
129C3×Dic3—126no—
130C2×C42—271no—
131#95—93no—
132#95—273no—
133C33#38311yescharacteristic, complemented, Sylow
134C23×C3—23no—
135C23×C3—41no—
136C23×C3C2×D3213yes—
137C2×C3×C4—66no—
138C2×C3×C4—63no—
139C2×C3×C4—63no—
140C2×C3×C4—63no—
141C2×C3×C4—63no—
142C2×C3×C4—63no—
143C2×C3×C4—63no—
144C2×C3×C4—96no—
145C2×C3×C4—123no—
146C2×C3×C4—123no—
147C2×C3×C4—183no—
148C2×C3×C4—183no—
149C2×Dic3—23no—
150C2×Dic3—23no—
151C2×Dic3—23no—
152C2×Dic3—36no—
153C2×Dic3—36no—
154C2×Dic3—66no—
155C2×Dic3—63no—
156C2×Dic3—63no—
157C2×Dic3—63no—
158C2×Dic3—63no—
159C2×Dic3—123no—
160C2×Dic3—123no—
161C2×Dic3—186no—
162C2×Dic3—183no—
163C2×Dic3—183no—
164C2×Dic3—183no—
165C2×Dic3—363no—
166C2×Dic3—363no—
167C2×Dic3—366no—
168C2×C32—23no—
169C2×C32—23no—
170C2×C32—23no—
171C2×C32—23no—
172C2×C32—23no—
173C2×C32—23no—
174C2×C32—23no—
175C2×C32—43no—
176C2×C32—41no—
177C2×C32—43no—
178C2×C32#87213yes—
179C2×C32#87713yes—
180C2×C32#87913yes—
181C2×C32#87913yes—
182C2×C32#89213yes—
183C2×C32#89813yes—
184C2×C32#89813yes—
185C22×C4—33no—
186C22×C4—93no—
187C22×C4—271no—
188C42—274no—
189C22×C3—23no—
190C22×C3—23no—
191C22×C3—23no—
192C22×C3—23no—
193C22×C3—23no—
194C22×C3—23no—
195C22×C3—23no—
196C22×C3—41no—
197C22×C3—43no—
198C22×C3—43no—
199C22×C3#379513yes—
200C22×C3#379613yes—
201C22×C3#379913yes—
202C22×C3C4×D3213yes—
203C22×C3#380413yes—
204C22×C3#380413yes—
205C22×C3#380513yes—
206C3×C4—126no—
207C3×C4—123no—
208C3×C4—123no—
209C3×C4—186no—
210Dic3—43no—
211Dic3—66no—
212Dic3—66no—
213Dic3—126no—
214Dic3—366no—
215Dic3—364no—
216Dic3—363no—
217C32—23no—
218C32—41no—
219C32#465113yescomplemented
220C23D3311yescharacteristic
221C2×C4—63no—
222C2×C4—63no—
223C2×C4—63no—
224C2×C4—96no—
225C2×C4—183no—
226C2×C4—183no—
227C2×C4—543no—
228C2×C3—23no—
229C2×C3—23no—
230C2×C3—23no—
231C2×C3—23no—
232C2×C3—23no—
233C2×C3—23no—
234C2×C3—23no—
235C2×C3—43no—
236C2×C3—41no—
237C2×C3—43no—
238C2×C3#6383613yes—
239C2×C3#6384813yes—
240C2×C3#6385813yes—
241C2×C3#6386113yes—
242C2×C3#6387713yes—
243C2×C3#6387813yes—
244C2×C3#6395613yes—
245C22#8894911yescharacteristic
246C22#8896113yes—
247C22#8896213yes—
248C4—123no—
249C4—186no—
250C4—1081no—
251C3—23no—
252C3—41no—
253C3#1059124813yescomplemented
254C2#1172292213yes—
255C2#1172302613yes—
256C2#1172302811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#383
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
8. #88813C4semidirect
133. C33#383semidirect
219. C32#4651semidirect
253. C3#10591248semidirect

Maximal subgroups

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

subgroupindex
C2×#888132
#117220612
C3×#635792
#105912483