Finite Group Databasebeta

← #13137560

Subgroups — #13137560

Order 1728.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 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#11721746C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic
3C9×#807C211yesmaximal, characteristic
4C3×#63367C211yesmaximal, characteristic
5#11720279C211yesmaximal, characteristic
6#11720271C211yesmaximal, characteristic
7#11720264C211yesmaximal, characteristic
8#10590417—31nomaximal
9#10588558—31nomaximal
10C3×C9×C4⋊C4C2211yescharacteristic
11C4⋊(C3⋊Dic9)—21no—
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13#88534—21no—
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#64099—31no—
22#63872—31no—
23C3×#808—31no—
24C3×#807D311yescharacteristic
25#63575—31no—
26#63573—31no—
27#63553—31no—
28#63452—31no—
29#63452—31no—
30#63452—61no—
31C9×#107—31no—
32#63378—31no—
33C8×Dic9—31no—
34#63367—31no—
35#63367D311yescharacteristic
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C3⋊Dic9D411yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C9×C3⋊C8Q812yes—
44C3×C9⋊C8Q812yes—
45#4878—91no—
46#4858—91no—
47C32×C4⋊C4C2×D311yescharacteristic
48C4⋊(C3⋊Dic3)—31no—
49C4⋊(C3⋊Dic3)—61no—
50#3750—61no—
51C4×C3⋊Dic3—31no—
52C3×C4⋊Dic3—31no—
53C3×C4⋊Dic3C2×D311yescharacteristic
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8C2×D311yescharacteristic
56Dic3⋊Dic3—61no—
57Dic3⋊Dic3—61no—
58C2×C8×C9—31no—
59C9×C4⋊C4—21no—
60C9×C4⋊C4—31no—
61C9×C4⋊C4C2×D311yescharacteristic
62C4⋊Dic9—31no—
63C4⋊Dic9—61no—
64C4⋊Dic9—61no—
65C4⋊Dic9—61no—
66C4⋊Dic9—62no—
67C4⋊Dic9C2×D311yescharacteristic
68Dic9⋊C4—61no—
69Dic9⋊C4—61no—
70Dic9⋊C4—62no—
71C4×Dic9—31no—
72C4×Dic9—61no—
73C2×C9⋊C8C2×D311yescharacteristic
74C22×C3×C9C2×D411yescharacteristic
75C3×C4×C9—41no—
76C3×C4×C9C2×Q811yescharacteristic
77C3×C4×C9C2×Q811yescharacteristic
78C3⋊Dic9—21no—
79C3⋊Dic9—21no—
80C3⋊Dic9—41no—
81C9×Dic3—41no—
82C3×Dic9—41no—
83#890—91no—
84#890—91no—
85#890—91no—
86#890—91no—
87#890—181no—
88C3×#107—31no—
89C3×#106—91no—
90#818—91no—
91#817—91no—
92C8×Dic3—91no—
93C8×Dic3—91no—
94#808—31no—
95#808—91no—
96#807—91no—
97#807D911yescharacteristic
98C2×C32×C4—21no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—31no—
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—61no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—61no—
105C3×C3⋊C8—32no—
106C3×C3⋊C8Dic612yes—
107C2×C4×C9—22no—
108C2×C4×C9—21no—
109C2×C4×C9—21no—
110C2×C4×C9—61no—
111C2×C4×C9C22×D311yescharacteristic
112C2×Dic9—21no—
113C2×Dic9—31no—
114C2×Dic9—31no—
115C2×Dic9—61no—
116C2×Dic9—61no—
117C2×Dic9—62no—
118C2×Dic9—61no—
119C8×C9—32no—
120C9⋊C8Dic612yes—
121#499—271noSylow
122C2×C3×C9C4⋊Q811yescharacteristic
123C2×C3×C9#13511yescharacteristic
124C2×C3×C9#13511yescharacteristic
125C2×C3×C8—31no—
126C2×C3×C8—91no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—91no—
130C3×C4⋊C4C2×D911yescharacteristic
131C3×C4⋊C4D3211yescharacteristic
132C4⋊Dic3—31no—
133C4⋊Dic3—91no—
134C4⋊Dic3—91no—
135C4⋊Dic3—181no—
136C4⋊Dic3—181no—
137C4⋊Dic3—181no—
138C4⋊Dic3—181no—
139C4⋊Dic3—181no—
140C4⋊Dic3—182no—
141C4⋊Dic3C2×D911yescharacteristic
142Dic3⋊C4—181no—
143Dic3⋊C4—181no—
144Dic3⋊C4—181no—
145Dic3⋊C4—181no—
146Dic3⋊C4—182no—
147C4×Dic3—91no—
148C4×Dic3—91no—
149C4×Dic3—181no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8C2×D911yescharacteristic
152C22×C32D3×D411yescharacteristic
153C32×C4—41no—
154C32×C4C2×Dic611yescharacteristic
155C32×C4D3×Q811yescharacteristic
156C3⋊Dic3—61no—
157C3⋊Dic3—61no—
158C3⋊Dic3—121no—
159C3×Dic3—41no—
160C3×Dic3—121no—
161C22×C9—21no—
162C22×C9D3×D411yescharacteristic
163C4×C9—21no—
164C4×C9—21no—
165C4×C9—42no—
166C4×C9—41no—
167C4×C9—121no—
168C4×C9C2×Dic611yescharacteristic
169C4×C9D3×Q811yescharacteristic
170Dic9—41no—
171Dic9—61no—
172Dic9—61no—
173Dic9—121no—
174Dic9—121no—
175Dic9—122no—
176Dic9—121no—
177#125—271no—
178#125—271no—
179#107—31no—
180#107—271no—
181#106—91no—
182#106—271no—
183C4×C8—271no—
184C3×C9#49911yescharacteristic, complemented, Sylow
185C2×C3×C4—22no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—61no—
190C2×C3×C4—181no—
191C2×C3×C4C22×D911yescharacteristic
192C2×C3×C4C2×D3211yescharacteristic
193C2×Dic3—21no—
194C2×Dic3—61no—
195C2×Dic3—91no—
196C2×Dic3—91no—
197C2×Dic3—91no—
198C2×Dic3—91no—
199C2×Dic3—181no—
200C2×Dic3—181no—
201C2×Dic3—182no—
202C2×Dic3—181no—
203C2×Dic3—181no—
204C3×C8—32no—
205C3×C8—92no—
206C3⋊C8—32no—
207C3⋊C8Dic1812yes—
208C2×C32#88811yescharacteristic
209C2×C32#91211yescharacteristic
210C2×C32#91911yescharacteristic
211C2×C9—21no—
212C2×C9—21no—
213C2×C9—21no—
214C2×C9#88811yescharacteristic
215C2×C9#91611yescharacteristic
216C2×C9#91611yescharacteristic
217C2×C8—31no—
218C2×C8—91no—
219C4⋊C4—31no—
220C4⋊C4—91no—
221C4⋊C4—271no—
222C4⋊C4—541no—
223C4⋊C4—541no—
224C4⋊C4—541no—
225C4⋊C4—541no—
226C4⋊C4D3×D911yescharacteristic
227C42—271no—
228C22×C3—21no—
229C22×C3D4×D911yescharacteristic
230C22×C3#381111yescharacteristic
231C3×C4—21no—
232C3×C4—21no—
233C3×C4—42no—
234C3×C4—41no—
235C3×C4—41no—
236C3×C4—121no—
237C3×C4—361no—
238C3×C4C2×Dic1811yescharacteristic
239C3×C4Q8×D911yescharacteristic
240C3×C4D3×Dic611yescharacteristic
241C3×C4D3×Dic611yescharacteristic
242Dic3—41no—
243Dic3—121no—
244Dic3—181no—
245Dic3—181no—
246Dic3—181no—
247Dic3—181no—
248Dic3—361no—
249Dic3—361no—
250Dic3—362no—
251Dic3—361no—
252Dic3—361no—
253C32#487811yescharacteristic
254C9—21no—
255C9#485811yescharacteristic, complemented
256C2×C4—21no—
257C2×C4—61no—
258C2×C4—181no—
259C2×C4—271no—
260C2×C4—271no—
261C2×C4—541no—
262C2×C4C2×D3×D911yescharacteristic
263C8—32no—
264C8—92no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3#6345011yescharacteristic
269C2×C3#6347411yescharacteristic
270C2×C3#6348111yescharacteristic
271C2×C3#6384011yescharacteristic
272C2×C3#6393611yescharacteristic
273C2×C3#6395011yescharacteristic
274C22#8866811yescharacteristic
275C4—41no—
276C4—121no—
277C4—361no—
278C4—541no—
279C4—541no—
280C4—1081no—
281C4D9×Dic611yescharacteristic
282C4D3×Dic1811yescharacteristic
283C3—21no—
284C3#1058855811yescharacteristic, complemented
285C3#1059041711yescharacteristic
286C2#1172090511yescharacteristic
287C2#1172117811yescharacteristic
288C2#1172122411yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socle, prime radicalC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#499
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
184. C3×C9#499semidirect
255. C9#4858semidirect
284. C3#10588558semidirect

Maximal subgroups

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

subgroupindex
#117217462
#117209372
C9×#8072
C3×#633672
#117202792
#117202712
#117202642
#105904173
#105885583