Finite Group Databasebeta

← #13139322

Subgroups — #13139322

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×C9×C4⋊Dic3C211yesmaximal, characteristic
3C3×#63389C211yesmaximal, characteristic
4#11720269C212yesmaximal, complemented
5#11720264C212yesmaximal, complemented
6#10591124—31nomaximal
7#10588647—31nomaximal
8C22×C3×C4×C9C2211yescharacteristic
9C2×#6226C2211yescharacteristic
10C22×C9×Dic3—21no—
11#88551—21no—
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13#88534—21no—
14C9×C4⋊Dic3—21no—
15C9×C4⋊Dic3C2212yes—
16C2×C3×C9⋊C8C2212yes—
17#64133—31no—
18C2×C3×C4⋊Dic3D311yescharacteristic
19C3×#830—31no—
20#63573—32no—
21#63567—32no—
22C2×C9×C4⋊C4—31no—
23#63486—31no—
24#63486—61no—
25#63389D311yescharacteristic
26#63368—32no—
27#63367—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×C9×Dic3—22no—
36C2×C9×Dic3—42no—
37C3×C9⋊C8—22no—
38#4971—91no—
39#4721—91no—
40C22×C32×C4C2×D311yescharacteristic
41C2×#634—31no—
42C22×C3×Dic3—21no—
43#3757—61no—
44C4⋊(C3⋊Dic3)—31no—
45#3750—61no—
46C3×C4⋊Dic3—21no—
47C3×C4⋊Dic3C2×D312yes—
48C2×C3×C3⋊C8—32no—
49C22×C4×C9—21no—
50C22×C4×C9—61no—
51C22×C4×C9C2×D311yescharacteristic
52C2×Dic18—31no—
53C2×Dic18—61no—
54C9×C4⋊C4—32no—
55C9×C4⋊C4—61no—
56#3676—62no—
57#3676—61no—
58C4⋊Dic9—31no—
59C4⋊Dic9—61no—
60Dic9⋊C4—62no—
61Dic9⋊C4—61no—
62C2×C9⋊C8C2×D312yes—
63C22×C3×C9—21no—
64C22×C3×C9—21no—
65C22×C3×C9SD1612yes—
66C22×C3×C9C2×D411yescharacteristic
67C3×C4×C9—21no—
68C3×C4×C9#4112yes—
69C3⋊Dic9—41no—
70C3⋊Dic9—41no—
71C9×Dic3—42no—
72C2×C3×C4⋊C4—31no—
73C2×C4⋊Dic3D911yescharacteristic
74#924—91no—
75#924—91no—
76#924—181no—
77C3×C22⋊C8—91no—
78#830—31no—
79#817—92no—
80#816—92no—
81#810—92no—
82#808—92no—
83C23×C32#6711yescharacteristic
84C2×C32×C4—22no—
85C2×C32×C4#6711yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C2×C3⋊Dic3—61no—
88C2×C3⋊Dic3—61no—
89#634—62no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—42no—
92C23×C9—21no—
93C23×C9D1211yescharacteristic
94C3×C3⋊C8—62no—
95C2×C4×C9—22no—
96C2×C4×C9—21no—
97C2×C4×C9—21no—
98C2×C4×C9—42no—
99C2×C4×C9—62no—
100C2×C4×C9—122no—
101C2×C4×C9D1211yescharacteristic
102C2×C4×C9C22×D311yescharacteristic
103C2×Dic9—62no—
104C2×Dic9—62no—
105C2×Dic9—61no—
106C2×Dic9—61no—
107Dic18—62no—
108Dic18—122no—
109C9⋊C8—22no—
110#483—271noSylow
111C2×C3×C9—22no—
112C2×C3×C9#12311yescharacteristic
113C2×C3×C9C2×SD1611yescharacteristic
114C2×C3×C9#13711yescharacteristic
115C22×C3×C4—21no—
116C22×C3×C4—61no—
117C22×C3×C4C2×D911yescharacteristic
118C22×C3×C4D3211yescharacteristic
119C22×Dic3—21no—
120C2×Dic6—91no—
121C2×Dic6—91no—
122C2×Dic6—181no—
123C2×C3×C8—92no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—61no—
126#217—182no—
127#217—181no—
128#217—181no—
129C4⋊Dic3—21no—
130C4⋊Dic3—91no—
131C4⋊Dic3—91no—
132C4⋊Dic3—181no—
133C4⋊Dic3C2×D912yes—
134Dic3⋊C4—182no—
135Dic3⋊C4—181no—
136Dic3⋊C4—181no—
137C2×C3⋊C8—32no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32#21412yes—
141C22×C32C2×#6711yescharacteristic
142C32×C4—21no—
143C32×C4#23512yes—
144C3⋊Dic3—121no—
145C3⋊Dic3—121no—
146C3×Dic3—42no—
147C22×C9—21no—
148C22×C9—21no—
149C22×C9—22no—
150C22×C9—21no—
151C22×C9—41no—
152C22×C9—41no—
153C22×C9#20412yes—
154C22×C9C2×D1211yescharacteristic
155C4×C9—21no—
156C4×C9—22no—
157C4×C9—41no—
158C4×C9—122no—
159C4×C9#23712yes—
160Dic9—122no—
161Dic9—122no—
162Dic9—121no—
163Dic9—121no—
164C22⋊Q8—271no—
165C2×C4⋊C4—31no—
166#106—272no—
167#103—272no—
168C22⋊C8—91no—
169C3×C9#48311yescharacteristic, complemented, Sylow
170C23×C3—21no—
171C23×C3#61111yescharacteristic
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—62no—
179C2×C3×C4—122no—
180C2×C3×C4#61111yescharacteristic
181C2×C3×C4C22×D911yescharacteristic
182C2×C3×C4#62611yescharacteristic
183C2×C3×C4C2×D3211yescharacteristic
184C2×Dic3—22no—
185C2×Dic3—42no—
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—182no—
196C3⋊C8—62no—
197C2×C32—22no—
198C2×C32#92911yescharacteristic
199C2×C32C2×#21411yescharacteristic
200C2×C32#94211yescharacteristic
201C2×C9—22no—
202C2×C9—21no—
203C2×C9—21no—
204C2×C9—21no—
205C2×C9—42no—
206C2×C9#88611yescharacteristic
207C2×C9C2×#20411yescharacteristic
208C2×C9#90911yescharacteristic
209C2×Q8—271no—
210C22×C4—61no—
211C22×C4D3×D911yescharacteristic
212C2×C8—92no—
213C4⋊C4—32no—
214C4⋊C4—61no—
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×C3#367212yes—
227C22×C3C2×#61111yescharacteristic
228C22×C3#371612yes—
229C22×C3C2×#62611yescharacteristic
230C3×C4—21no—
231C3×C4—21no—
232C3×C4—22no—
233C3×C4—41no—
234C3×C4—122no—
235C3×C4#369712yes—
236C3×C4#380412yes—
237Dic3—42no—
238Dic3—362no—
239Dic3—362no—
240Dic3—361no—
241Dic3—361no—
242Dic3—361no—
243Dic3—361no—
244C32#497111yescharacteristic
245C9—21no—
246C9#472111yescharacteristic, complemented
247C23#619511yescharacteristic
248Q8—542no—
249C2×C4—22no—
250C2×C4—62no—
251C2×C4—122no—
252C2×C4—541no—
253C2×C4—541no—
254C2×C4#619511yescharacteristic
255C2×C4C2×D3×D911yescharacteristic
256C8—182no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—21no—
261C2×C3—21no—
262C2×C3—42no—
263C2×C3#6349111yescharacteristic
264C2×C3C2×#367211yescharacteristic
265C2×C3#6350511yescharacteristic
266C2×C3C2×#371611yescharacteristic
267C2×C3#6383311yescharacteristic
268C2×C3#6396611yescharacteristic
269C22—21no—
270C22—21no—
271C22#8842412yes—
272C22C2×#619511yescharacteristic
273C4—21no—
274C4—122no—
275C4—1081no—
276C4—1081no—
277C4#8865812yes—
278C3—21no—
279C3#1058864711yescharacteristic, complemented
280C3#1059112411yescharacteristic
281C2—22no—
282C2C2×#8842411yescharacteristic
283C2#1172089511yescharacteristic
284C2#1172125111yescharacteristic
subgroupG/H
centerC22C2×#6195
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×#67
prime radicalC23×C32#67
hypercenterC22×C4D3×D9
hypocenterC3×C9#483
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. #11720269C2semidirect
5. #11720264C2semidirect
169. C3×C9#483semidirect
246. C9#4721semidirect
279. C3#10588647semidirect

Maximal subgroups

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

subgroupindex
#117218132
C2×C9×C4⋊Dic32
C3×#633892
#117202692
#117202642
#105911243
#105886473