Finite Group Databasebeta

← #13156655

Subgroups — #13156655

Order 1728.

482 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C3×#63869C212yesmaximal
2#11722079C211yesmaximal, characteristic
3#11722056C212yesmaximal
4Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
5C3×#63574C211yesmaximal, characteristic
6#10590414—32nomaximal
7#10590314—31nomaximal
8C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
9C3×C4⋊(C3⋊Dic3)C2212yes—
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×C4⋊Dic3C2212yes—
12C32×C4×Dic3C2211yescharacteristic
13C3×#3724—22no—
14C3×Dic3⋊Dic3—22no—
15C3×#890—32no—
16C3×#870—32no—
17#63869D312yes—
18#63634—32no—
19C4×C3⋊(C3⋊C8)—31no—
20C3×#808—32no—
21#63576—32no—
22#63576—32no—
23#63576—62no—
24#63574—31no—
25#63574D311yescharacteristic
26#63573—32no—
27Dic3×C3⋊C8—32no—
28Dic3×C3⋊C8—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C32×Dic3—22no—
32C2×C32×Dic3D412yes—
33C3⋊(C3⋊(C3⋊C8))Q812yes—
34C3×C3⋊(C3⋊C8)Q812yes—
35#5065—92no—
36#4858—91no—
37C32×C4⋊C4—32no—
38C32×C42—31no—
39C4⋊(C3⋊Dic3)C2×D312yes—
40C2×C3⋊(C3⋊C8)—32no—
41C2×C3⋊(C3⋊C8)—31no—
42C2×C3⋊(C3⋊C8)—61no—
43C2×C3⋊(C3⋊C8)—62no—
44C2×C3⋊(C3⋊C8)—121no—
45C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
46C3×C4⋊Dic3—22no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—32no—
49C3×C4⋊Dic3—62no—
50C3×C4⋊Dic3—62no—
51C3×C4⋊Dic3C2×D312yes—
52C3×C4⋊Dic3C2×D312yes—
53C3×Dic3⋊C4—62no—
54C3×Dic3⋊C4—62no—
55C3×Dic3⋊C4—62no—
56C3×C4×Dic3—21no—
57C3×C4×Dic3C2×D312yes—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—61no—
60#3724—22no—
61Dic3⋊Dic3—22no—
62C22×C33C2×D411yescharacteristic
63C33×C4C2×Q811yescharacteristic
64C33×C4C2×Q811yescharacteristic
65C3×C3⋊Dic3—42no—
66C32×Dic3—22no—
67C32×Dic3—42no—
68C3×#125—92no—
69#890—32no—
70#870—32no—
71C3×#106—91no—
72#817—91no—
73C8×Dic3—91no—
74#808—32no—
75#808—92no—
76#807—92no—
77#807—92no—
78#807—92no—
79#807—182no—
80C4×C3⋊C8—92no—
81C4×C3⋊C8—181no—
82C2×C32×C4—21no—
83C2×C32×C4—22no—
84C2×C32×C4—32no—
85C2×C32×C4—41no—
86C2×C32×C4—62no—
87C2×C32×C4C22×D311yescharacteristic
88C2×C32×C4C22×D312yes—
89C2×C3⋊Dic3—22no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—42no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—122no—
97C2×C3×Dic3#6712yes—
98C2×C3×Dic3#6712yes—
99C3⋊(C3⋊C8)—32no—
100C3⋊(C3⋊C8)—34no—
101C3⋊(C3⋊C8)—62no—
102C3⋊(C3⋊C8)—64no—
103C3⋊(C3⋊C8)—122no—
104C3⋊(C3⋊C8)Dic612yes—
105C3×C3⋊C8—34no—
106C3×C3⋊C8—62no—
107#499—271noSylow
108C2×C33C4⋊Q811yescharacteristic
109C2×C33#13512yes—
110C2×C3×C8—91no—
111C3×C4⋊C4—32no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4—62no—
114C3×C4⋊C4—92no—
115C3×C4⋊C4—182no—
116C3×C4⋊C4—182no—
117C3×C42—32no—
118C3×C42—61no—
119C4⋊Dic3—32no—
120C4⋊Dic3—32no—
121C4⋊Dic3—62no—
122C4⋊Dic3—62no—
123C4⋊Dic3D3212yes—
124Dic3⋊C4—62no—
125Dic3⋊C4—62no—
126Dic3⋊C4—62no—
127C4×Dic3D3211yescharacteristic
128C2×C3⋊C8—32no—
129C2×C3⋊C8—61no—
130C2×C3⋊C8—92no—
131C2×C3⋊C8—91no—
132C2×C3⋊C8—181no—
133C2×C3⋊C8—182no—
134C2×C3⋊C8—361no—
135C22×C32—21no—
136C22×C32—22no—
137C22×C32—41no—
138C22×C32D3×D411yescharacteristic
139C22×C32C2×#6712yes—
140C32×C4—22no—
141C32×C4—21no—
142C32×C4—22no—
143C32×C4—21no—
144C32×C4—41no—
145C32×C4—41no—
146C32×C4—62no—
147C32×C4—122no—
148C32×C4C2×Dic611yescharacteristic
149C32×C4D3×Q812yes—
150C32×C4D3×Q811yescharacteristic
151C32×C4D3×Q812yes—
152C3⋊Dic3—42no—
153C3×Dic3—24no—
154C3×Dic3—22no—
155C3×Dic3—22no—
156C3×Dic3—42no—
157C3×Dic3—42no—
158C3×Dic3—82no—
159C3×Dic3—122no—
160C3×Dic3—122no—
161C3×Dic3—242no—
162#125—92no—
163#107—272no—
164#106—91no—
165#106—271no—
166C4×C8—271no—
167C33#49911yescharacteristic, complemented, Sylow
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—41no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—62no—
176C2×C3×C4—122no—
177C2×C3×C4—182no—
178C2×C3×C4C2×D3212yes—
179C2×C3×C4C2×D3211yescharacteristic
180C2×Dic3—22no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—122no—
184C2×Dic3#62512yes—
185C3×C8—92no—
186C3⋊C8—34no—
187C3⋊C8—62no—
188C3⋊C8—94no—
189C3⋊C8—92no—
190C3⋊C8—182no—
191C3⋊C8—184no—
192C3⋊C8—362no—
193C2×C32—22no—
194C2×C32—22no—
195C2×C32—21no—
196C2×C32—22no—
197C2×C32—22no—
198C2×C32—41no—
199C2×C32—42no—
200C2×C32#88811yescharacteristic
201C2×C32#91612yes—
202C2×C32#94412yes—
203C2×C32#95012yes—
204C2×C32#95012yes—
205C2×C8—91no—
206C2×C8—271no—
207C4⋊C4—32no—
208C4⋊C4—92no—
209C4⋊C4—182no—
210C4⋊C4—182no—
211C42—31no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—41no—
215C22×C3C2×#62511yescharacteristic
216C22×C3D3×#6712yes—
217C3×C4—21no—
218C3×C4—22no—
219C3×C4—21no—
220C3×C4—22no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—64no—
224C3×C4—62no—
225C3×C4—62no—
226C3×C4—122no—
227C3×C4—122no—
228C3×C4—242no—
229C3×C4—362no—
230C3×C4D3×Dic612yes—
231C3×C4#379711yescharacteristic
232C3×C4#379712yes—
233C3×C4#379711yescharacteristic
234Dic3—22no—
235Dic3—42no—
236Dic3—122no—
237Dic3—122no—
238Dic3—242no—
239C32—21no—
240C32—22no—
241C32—41no—
242C32#485811yescharacteristic, complemented
243C32#506512yescomplemented
244C2×C4—32no—
245C2×C4—62no—
246C2×C4—182no—
247C2×C4D3311yescharacteristic
248C8—92no—
249C8—272no—
250C2×C3—21no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—41no—
256C2×C3—42no—
257C2×C3#6382212yes—
258C2×C3#6383912yes—
259C2×C3#6387311yescharacteristic
260C2×C3#6393512yes—
261C2×C3#6394712yes—
262C22D3×#62511yescharacteristic
263C4—62no—
264C4—122no—
265C4—362no—
266C4#8895711yescharacteristic
267C4#8895811yescharacteristic
268C3—22no—
269C3—21no—
270C3—41no—
271C3#1059031411yescharacteristic, complemented
272C3#1059041412yescomplemented
273C2#1172285112yes—
274C2#1172303211yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#499
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
167. C33#499semidirect
242. C32#4858semidirect
243. C32#5065semidirect
271. C3#10590314semidirect
272. C3#10590414semidirect

Maximal subgroups

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

subgroupindex
C3×#638692
#117220792
#117220562
Dic3×C3⋊(C3⋊C8)2
C3×#635742
#105904143
#105903143