Finite Group Databasebeta

← #13156654

Subgroups — #13156654

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#11722080C211yesmaximal, characteristic
3#11722053C212yesmaximal
4Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
5C3×#63576C211yesmaximal, characteristic
6#10590413—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#63635—32no—
19C4×C3⋊(C3⋊C8)—31no—
20C3×#807—32no—
21#63576—31no—
22#63576D311yescharacteristic
23#63575—32no—
24#63574—32no—
25#63574—32no—
26#63574—62no—
27Dic3×C3⋊C8—32no—
28Dic3×C3⋊C8—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C32×Dic3—22no—
32C2×C32×Dic3D411yescharacteristic
33C2×C32×Dic3D411yescharacteristic
34C3⋊(C3⋊(C3⋊C8))Q812yes—
35C3×C3⋊(C3⋊C8)Q812yes—
36#5065—92no—
37#4878—91no—
38C32×C4⋊C4—32no—
39C32×C42—31no—
40C4⋊(C3⋊Dic3)C2×D312yes—
41C2×C3⋊(C3⋊C8)—32no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—61no—
44C2×C3⋊(C3⋊C8)—62no—
45C2×C3⋊(C3⋊C8)—121no—
46C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
47C3×C4⋊Dic3—22no—
48C3×C4⋊Dic3—32no—
49C3×C4⋊Dic3—32no—
50C3×C4⋊Dic3—62no—
51C3×C4⋊Dic3—62no—
52C3×C4⋊Dic3C2×D312yes—
53C3×C4⋊Dic3C2×D312yes—
54C3×Dic3⋊C4—62no—
55C3×Dic3⋊C4—62no—
56C3×Dic3⋊C4—62no—
57C3×C4×Dic3—21no—
58C3×C4×Dic3C2×D312yes—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—61no—
61#3724—22no—
62Dic3⋊Dic3—22no—
63C22×C33C2×D411yescharacteristic
64C33×C4C2×Q811yescharacteristic
65C33×C4C2×Q811yescharacteristic
66C3×C3⋊Dic3—42no—
67C32×Dic3—21no—
68C32×Dic3—21no—
69C32×Dic3—42no—
70C3×#125—92no—
71#890—32no—
72#870—32no—
73C3×#107—91no—
74#818—91no—
75C8×Dic3—91no—
76#808—92no—
77#808—92no—
78#808—92no—
79#808—182no—
80#807—32no—
81#807—92no—
82C4×C3⋊C8—92no—
83C4×C3⋊C8—181no—
84C2×C32×C4—21no—
85C2×C32×C4—22no—
86C2×C32×C4—31no—
87C2×C32×C4—31no—
88C2×C32×C4—41no—
89C2×C32×C4—62no—
90C2×C32×C4C22×D311yescharacteristic
91C2×C32×C4C22×D312yes—
92C2×C3⋊Dic3—22no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—21no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—42no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—122no—
101C2×C3×Dic3#6712yes—
102C2×C3×Dic3#6712yes—
103C3⋊(C3⋊C8)—32no—
104C3⋊(C3⋊C8)—34no—
105C3⋊(C3⋊C8)—62no—
106C3⋊(C3⋊C8)—64no—
107C3⋊(C3⋊C8)—122no—
108C3⋊(C3⋊C8)Dic612yes—
109C3×C3⋊C8—34no—
110C3×C3⋊C8—62no—
111#499—271noSylow
112C2×C33C4⋊Q811yescharacteristic
113C2×C33#13511yescharacteristic
114C2×C33#13511yescharacteristic
115C2×C3×C8—91no—
116C3×C4⋊C4—32no—
117C3×C4⋊C4—32no—
118C3×C4⋊C4—62no—
119C3×C4⋊C4—92no—
120C3×C4⋊C4—182no—
121C3×C4⋊C4—182no—
122C3×C42—32no—
123C3×C42—61no—
124C4⋊Dic3—32no—
125C4⋊Dic3—32no—
126C4⋊Dic3—62no—
127C4⋊Dic3—62no—
128C4⋊Dic3D3212yes—
129Dic3⋊C4—62no—
130Dic3⋊C4—62no—
131Dic3⋊C4—62no—
132C4×Dic3D3211yescharacteristic
133C2×C3⋊C8—32no—
134C2×C3⋊C8—61no—
135C2×C3⋊C8—92no—
136C2×C3⋊C8—91no—
137C2×C3⋊C8—181no—
138C2×C3⋊C8—182no—
139C2×C3⋊C8—361no—
140C22×C32—21no—
141C22×C32—22no—
142C22×C32—41no—
143C22×C32D3×D411yescharacteristic
144C22×C32C2×#6712yes—
145C32×C4—22no—
146C32×C4—21no—
147C32×C4—22no—
148C32×C4—21no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—61no—
152C32×C4—61no—
153C32×C4—122no—
154C32×C4C2×Dic611yescharacteristic
155C32×C4D3×Q812yes—
156C32×C4D3×Q811yescharacteristic
157C32×C4D3×Q812yes—
158C3⋊Dic3—42no—
159C3×Dic3—22no—
160C3×Dic3—22no—
161C3×Dic3—22no—
162C3×Dic3—22no—
163C3×Dic3—42no—
164C3×Dic3—42no—
165C3×Dic3—82no—
166C3×Dic3—122no—
167C3×Dic3—122no—
168C3×Dic3—242no—
169#125—92no—
170#107—91no—
171#107—271no—
172#106—272no—
173C4×C8—271no—
174C33#49911yescharacteristic, complemented, Sylow
175C2×C3×C4—21no—
176C2×C3×C4—22no—
177C2×C3×C4—32no—
178C2×C3×C4—32no—
179C2×C3×C4—41no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—122no—
185C2×C3×C4—182no—
186C2×C3×C4C2×D3212yes—
187C2×C3×C4C2×D3211yescharacteristic
188C2×Dic3—22no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—122no—
192C2×Dic3#62511yescharacteristic
193C2×Dic3#62511yescharacteristic
194C3×C8—92no—
195C3⋊C8—34no—
196C3⋊C8—62no—
197C3⋊C8—94no—
198C3⋊C8—92no—
199C3⋊C8—182no—
200C3⋊C8—184no—
201C3⋊C8—362no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—22no—
205C2×C32—21no—
206C2×C32—22no—
207C2×C32—22no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32#88811yescharacteristic
212C2×C32#91211yescharacteristic
213C2×C32#91911yescharacteristic
214C2×C32#94412yes—
215C2×C32#95012yes—
216C2×C32#95012yes—
217C2×C8—91no—
218C2×C8—271no—
219C4⋊C4—32no—
220C4⋊C4—92no—
221C4⋊C4—182no—
222C4⋊C4—182no—
223C42—31no—
224C22×C3—22no—
225C22×C3—21no—
226C22×C3—41no—
227C22×C3C2×#62511yescharacteristic
228C22×C3D3×#6712yes—
229C3×C4—21no—
230C3×C4—22no—
231C3×C4—21no—
232C3×C4—22no—
233C3×C4—41no—
234C3×C4—41no—
235C3×C4—62no—
236C3×C4—62no—
237C3×C4—62no—
238C3×C4—62no—
239C3×C4—122no—
240C3×C4—122no—
241C3×C4—242no—
242C3×C4—362no—
243C3×C4D3×Dic612yes—
244C3×C4#379711yescharacteristic
245C3×C4#379712yes—
246C3×C4#379711yescharacteristic
247Dic3—21no—
248Dic3—21no—
249Dic3—42no—
250Dic3—122no—
251Dic3—122no—
252Dic3—242no—
253C32—21no—
254C32—22no—
255C32—41no—
256C32#487811yescharacteristic, complemented
257C32#506512yescomplemented
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—62no—
261C2×C4—182no—
262C2×C4D3311yescharacteristic
263C8—92no—
264C8—272no—
265C2×C3—21no—
266C2×C3—22no—
267C2×C3—21no—
268C2×C3—22no—
269C2×C3—22no—
270C2×C3—21no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3#6382211yescharacteristic
275C2×C3#6382211yescharacteristic
276C2×C3#6383912yes—
277C2×C3#6387311yescharacteristic
278C2×C3#6393312yes—
279C2×C3#6394912yes—
280C22D3×#62511yescharacteristic
281C4—61no—
282C4—61no—
283C4—122no—
284C4—362no—
285C4#8895711yescharacteristic
286C4#8895811yescharacteristic
287C3—21no—
288C3—22no—
289C3—41no—
290C3#1059031411yescharacteristic, complemented
291C3#1059041312yescomplemented
292C2#1172285011yescharacteristic
293C2#1172285211yescharacteristic
294C2#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
174. C33#499semidirect
256. C32#4878semidirect
257. C32#5065semidirect
290. C3#10590314semidirect
291. C3#10590413semidirect

Maximal subgroups

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

subgroupindex
C3×#638692
#117220802
#117220532
Dic3×C3⋊(C3⋊C8)2
C3×#635762
#105904133
#105903143