Finite Group Databasebeta

← #13139385

Subgroups — #13139385

Order 1728.

386 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 101 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#11721813C211yesmaximal, characteristic
2C2×C3×C4⋊Dic9C211yesmaximal, characteristic
3C9×#830C211yesmaximal, characteristic
4#11720279C212yesmaximal, complemented
5#11720277C212yesmaximal, complemented
6#10591135—31nomaximal
7#10588402—31nomaximal
8C22×C3×C4×C9C2211yescharacteristic
9C2×#6226C2211yescharacteristic
10C22×C3×Dic9—21no—
11#88551—21no—
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13#88534—21no—
14C3×C4⋊Dic9—21no—
15C3×C4⋊Dic9C2212yes—
16C2×C9×C3⋊C8C2212yes—
17#64133—31no—
18C2×C3×C4⋊Dic3—31no—
19C3×#830D311yescharacteristic
20#63575—32no—
21#63567—32no—
22C2×C4⋊Dic9D311yescharacteristic
23#63486—31no—
24#63486—61no—
25C9×C22⋊C8—31no—
26#63378—32no—
27#63376—32no—
28C23×C3×C9D411yescharacteristic
29C2×C3×C4×C9—22no—
30C2×C3×C4×C9D411yescharacteristic
31C2×C3×C4×C9C2311yescharacteristic
32C2×C3⋊Dic9—21no—
33C2×C3⋊Dic9—21no—
34#6226—22no—
35C2×C3×Dic9—22no—
36C2×C3×Dic9—42no—
37C9×C3⋊C8—22no—
38#4970—91no—
39#4722—91no—
40C22×C32×C4C2×D311yescharacteristic
41C2×#634—31no—
42C22×C3×Dic3—61no—
43#3757—61no—
44C4⋊(C3⋊Dic3)—31no—
45#3750—61no—
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—61no—
48C2×C3×C3⋊C8C2×D312yes—
49C22×C4×C9—21no—
50C22×C4×C9C2×D311yescharacteristic
51C22×Dic9—21no—
52C2×Dic18—31no—
53C2×Dic18—61no—
54C2×C8×C9—32no—
55#3676—62no—
56#3676—61no—
57C4⋊Dic9—21no—
58C4⋊Dic9—31no—
59C4⋊Dic9—61no—
60C4⋊Dic9C2×D312yes—
61Dic9⋊C4—62no—
62Dic9⋊C4—61no—
63C22×C3×C9—21no—
64C22×C3×C9—21no—
65C22×C3×C9Q1612yes—
66C22×C3×C9C2×D411yescharacteristic
67C3×C4×C9—21no—
68C3×C4×C9#4112yes—
69C3⋊Dic9—41no—
70C3⋊Dic9—41no—
71C3×Dic9—42no—
72C2×C3×C4⋊C4—91no—
73C2×C4⋊Dic3—31no—
74#924—91no—
75#924—91no—
76#924—181no—
77C3×C22⋊C8—31no—
78#830D911yescharacteristic
79#818—92no—
80#816—92no—
81#810—92no—
82#807—92no—
83C23×C32D1211yescharacteristic
84C2×C32×C4—22no—
85C2×C32×C4D1211yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C2×C3⋊Dic3—61no—
88C2×C3⋊Dic3—61no—
89#634—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—122no—
92C23×C9—21no—
93C23×C9#6711yescharacteristic
94C3×C3⋊C8—22no—
95C2×C4×C9—22no—
96C2×C4×C9—21no—
97C2×C4×C9—21no—
98C2×C4×C9—42no—
99C2×C4×C9#6711yescharacteristic
100C2×C4×C9C22×D311yescharacteristic
101C2×Dic9—22no—
102C2×Dic9—42no—
103C2×Dic9—62no—
104C2×Dic9—62no—
105C2×Dic9—61no—
106C2×Dic9—61no—
107Dic18—62no—
108Dic18—122no—
109C8×C9—62no—
110#484—271noSylow
111C2×C3×C9—22no—
112C2×C3×C9#12311yescharacteristic
113C2×C3×C9C2×Q1611yescharacteristic
114C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
115C22×C3×C4—21no—
116C22×C3×C4—181no—
117C22×C3×C4C2×D911yescharacteristic
118C22×C3×C4D3211yescharacteristic
119C22×Dic3—61no—
120C2×Dic6—91no—
121C2×Dic6—91no—
122C2×Dic6—181no—
123C2×C3×C8—32no—
124C3×C4⋊C4—92no—
125C3×C4⋊C4—181no—
126#217—182no—
127#217—181no—
128#217—181no—
129C4⋊Dic3—32no—
130C4⋊Dic3—61no—
131C4⋊Dic3—91no—
132C4⋊Dic3—91no—
133C4⋊Dic3—181no—
134Dic3⋊C4—182no—
135Dic3⋊C4—181no—
136Dic3⋊C4—181no—
137C2×C3⋊C8C2×D912yes—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32Dic1212yes—
141C22×C32C2×D1211yescharacteristic
142C32×C4—21no—
143C32×C4#23712yes—
144C3⋊Dic3—121no—
145C3⋊Dic3—121no—
146C3×Dic3—122no—
147C22×C9—21no—
148C22×C9—21no—
149C22×C9—22no—
150C22×C9—21no—
151C22×C9—41no—
152C22×C9—41no—
153C22×C9#21612yes—
154C22×C9C2×#6711yescharacteristic
155C4×C9—21no—
156C4×C9—22no—
157C4×C9—41no—
158C4×C9#23512yes—
159Dic9—42no—
160Dic9—122no—
161Dic9—122no—
162Dic9—121no—
163Dic9—121no—
164C22⋊Q8—271no—
165C2×C4⋊C4—91no—
166#107—272no—
167#103—272no—
168C22⋊C8—31no—
169C3×C9#48411yescharacteristic, complemented, Sylow
170C23×C3—21no—
171C23×C3D3611yescharacteristic
172C23×C3#62611yescharacteristic
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—42no—
178C2×C3×C4—182no—
179C2×C3×C4—362no—
180C2×C3×C4D3611yescharacteristic
181C2×C3×C4C22×D911yescharacteristic
182C2×C3×C4#62611yescharacteristic
183C2×C3×C4C2×D3211yescharacteristic
184C2×Dic3—62no—
185C2×Dic3—122no—
186C2×Dic3—182no—
187C2×Dic3—182no—
188C2×Dic3—181no—
189C2×Dic3—181no—
190C2×Dic3—181no—
191C2×Dic3—181no—
192Dic6—182no—
193Dic6—182no—
194Dic6—362no—
195C3×C8—62no—
196C3⋊C8—22no—
197C2×C32—22no—
198C2×C32#88611yescharacteristic
199C2×C32C2×Dic1211yescharacteristic
200C2×C32#90811yescharacteristic
201C2×C9—22no—
202C2×C9—21no—
203C2×C9—21no—
204C2×C9—21no—
205C2×C9—42no—
206C2×C9#92911yescharacteristic
207C2×C9#93211yescharacteristic
208C2×C9C2×#21611yescharacteristic
209C2×Q8—271no—
210C22×C4—181no—
211C22×C4D3×D911yescharacteristic
212C2×C8—32no—
213C4⋊C4—92no—
214C4⋊C4—181no—
215C4⋊C4—271no—
216C4⋊C4—541no—
217C22⋊C4—541no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—22no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3Dic3612yes—
227C22×C3C2×D3611yescharacteristic
228C22×C3#371912yes—
229C22×C3C2×#62611yescharacteristic
230C3×C4—21no—
231C3×C4—21no—
232C3×C4—22no—
233C3×C4—41no—
234C3×C4—362no—
235C3×C4#369912yes—
236C3×C4#380412yes—
237Dic3—122no—
238Dic3—362no—
239Dic3—362no—
240Dic3—361no—
241Dic3—361no—
242Dic3—361no—
243Dic3—361no—
244C32#472211yescharacteristic
245C9—21no—
246C9#497011yescharacteristic, complemented
247C23#619211yescharacteristic
248Q8—542no—
249C2×C4—22no—
250C2×C4—182no—
251C2×C4—362no—
252C2×C4—541no—
253C2×C4—541no—
254C2×C4#619211yescharacteristic
255C2×C4C2×D3×D911yescharacteristic
256C8—62no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—21no—
261C2×C3—21no—
262C2×C3—42no—
263C2×C3#6344811yescharacteristic
264C2×C3C2×Dic3611yescharacteristic
265C2×C3#6347011yescharacteristic
266C2×C3#6383011yescharacteristic
267C2×C3C2×#371911yescharacteristic
268C2×C3#6396611yescharacteristic
269C22—21no—
270C22—21no—
271C22#8841812yes—
272C22C2×#619211yescharacteristic
273C4—21no—
274C4—362no—
275C4—1081no—
276C4—1081no—
277C4#8866212yes—
278C3—21no—
279C3#1058840211yescharacteristic, complemented
280C3#1059113511yescharacteristic
281C2—22no—
282C2#1172086811yescharacteristic
283C2C2×#8841811yescharacteristic
284C2#1172125511yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
prime radicalC23×C32D12
hypercenterC22×C4D3×D9
hypocenterC3×C9#484
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720279C2semidirect
5. #11720277C2semidirect
169. C3×C9#484semidirect
246. C9#4970semidirect
279. C3#10588402semidirect

Maximal subgroups

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

subgroupindex
#117218132
C2×C3×C4⋊Dic92
C9×#8302
#117202792
#117202772
#105911353
#105884023