Finite Group Databasebeta

← #13138258

Subgroups — #13138258

Order 1728.

324 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 91 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#11720969C211yesmaximal, characteristic
2Dic6×Dic9C211yesmaximal, characteristic
3#11720541C211yesmaximal, characteristic
4C9×#816C211yesmaximal, characteristic
5C3×Dic9⋊C8C211yesmaximal, characteristic
6#11720298C211yesmaximal, characteristic
7#11720262C211yesmaximal, characteristic
8#10590804—31nomaximal
9#10588473—31nomaximal
10C2×C9×Dic6C2211yescharacteristic
11C2×#6189—21no—
12C2×C3×C8×C9C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4×Dic9C2211yescharacteristic
17C2×C3×C9⋊C8C2211yescharacteristic
18Dic3⋊Dic9—21no—
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#63918—31no—
22Dic3×Dic6—31no—
23#63645—31no—
24C3×#816D311yescharacteristic
25C3×Dic3⋊C8—31no—
26#63575—31no—
27#63566—31no—
28Q8×Dic9—31no—
29#63450—31no—
30C9×#103—31no—
31#63395—31no—
32#63378—31no—
33#63378—61no—
34Dic9⋊C8D311yescharacteristic
35#63366—31no—
36C2×C3×C4×C9C2311yescharacteristic
37C2×C3⋊Dic9—21no—
38C2×C3⋊Dic9—21no—
39C2×C9×Dic3—21no—
40C2×C9×Dic3—21no—
41C2×C3×Dic9D411yescharacteristic
42C2×C3×Dic9D411yescharacteristic
43C9×Dic6—21no—
44C9×Dic6Q812yescomplemented
45#6189—42no—
46C3×C8×C9—21no—
47C3×C9⋊C8—21no—
48#4850—91no—
49#4793—91no—
50C2×C3×Dic6C2×D311yescharacteristic
51C2×#627—61no—
52C2×C32×C8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)—31no—
54C4⋊(C3⋊Dic3)—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C3×C4×Dic3—31no—
57C2×C3×C3⋊C8—31no—
58Dic3⋊Dic3—61no—
59Dic3⋊Dic3—61no—
60Dic32—61no—
61C2×Q8×C9—31no—
62C2×Dic18—61no—
63C2×C8×C9—21no—
64C2×C8×C9C2×D311yescharacteristic
65C9×C4⋊C4—31no—
66C4⋊Dic9—31no—
67C4⋊Dic9—31no—
68C4⋊Dic9—61no—
69C4⋊Dic9—61no—
70C4⋊Dic9—61no—
71Dic9⋊C4—61no—
72C4×Dic9—61no—
73C4×Dic9C2×D311yescharacteristic
74C2×C9⋊C8C2×D311yescharacteristic
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9C2×Q811yescharacteristic
77C3×C4×C9#4111yescharacteristic
78C3⋊Dic9—41no—
79C3⋊Dic9—41no—
80C9×Dic3—22no—
81C9×Dic3—41no—
82C3×Dic9—21no—
83C3×Dic9Q1612yes—
84Q8×Dic3—91no—
85#888—91no—
86#869—91no—
87C4×Dic6—91no—
88C3×C4⋊C8—91no—
89C3×#103—31no—
90#835—91no—
91#818—91no—
92#818—91no—
93#818—91no—
94#818—181no—
95#816—91no—
96#816D911yescharacteristic
97Dic3⋊C8—31no—
98#807—91no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—61no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—31no—
105C2×C3×Dic3—31no—
106C3×Dic6—21no—
107C3×Dic6Dic612yes—
108#627—122no—
109C32×C8—21no—
110C3×C3⋊C8—61no—
111Q8×C9—32no—
112Q8×C9—61no—
113C2×C4×C9—21no—
114C2×C4×C9—61no—
115C2×C4×C9—61no—
116C2×C4×C9C22×D311yescharacteristic
117C2×Dic9—61no—
118C2×Dic9—61no—
119C2×Dic9—121no—
120C2×Dic9—121no—
121C2×Dic9D1211yescharacteristic
122C2×Dic9D1211yescharacteristic
123Dic18—122no—
124C8×C9—22no—
125C8×C9—21no—
126C9⋊C8—21no—
127#477—271noSylow
128C2×C3×C9C22⋊Q811yescharacteristic
129C2×C3×C9C2×Q1611yescharacteristic
130C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
131C2×C3×Q8—31no—
132C2×Dic6—181no—
133C2×Dic6—181no—
134C2×Dic6C2×D911yescharacteristic
135C2×C3×C8—21no—
136C2×C3×C8—91no—
137C2×C3×C8C2×D911yescharacteristic
138C2×C3×C8D3211yescharacteristic
139C3×C4⋊C4—31no—
140C3×C42—91no—
141C4⋊Dic3—91no—
142C4⋊Dic3—91no—
143C4⋊Dic3—91no—
144C4⋊Dic3—91no—
145C4⋊Dic3—181no—
146C4⋊Dic3—181no—
147C4⋊Dic3—181no—
148C4⋊Dic3—181no—
149C4⋊Dic3C2×D911yescharacteristic
150Dic3⋊C4—181no—
151Dic3⋊C4—181no—
152C4×Dic3—31no—
153C4×Dic3—181no—
154C4×Dic3—181no—
155C2×C3⋊C8—31no—
156C22×C32D3×D411yescharacteristic
157C32×C4C2×Dic611yescharacteristic
158C32×C4#23711yescharacteristic
159C3⋊Dic3—121no—
160C3⋊Dic3—121no—
161C3×Dic3—22no—
162C3×Dic3—32no—
163C3×Dic3—41no—
164C3×Dic3—61no—
165C22×C9—21no—
166C22×C9C2×D1211yescharacteristic
167C4×C9—21no—
168C4×C9—21no—
169C4×C9—62no—
170C4×C9—121no—
171C4×C9#23711yescharacteristic
172C4×C9D3×Q811yescharacteristic
173Dic9—21no—
174Dic9—121no—
175Dic9—121no—
176Dic9—241no—
177Dic9—241no—
178Dic9Dic1212yes—
179C4⋊Q8—271no—
180C4×Q8—271no—
181#107—271no—
182#107—271no—
183C4⋊C8—91no—
184#103—31no—
185#103—271no—
186C3×C9#47711yescharacteristic, complemented, Sylow
187C3×Q8—32no—
188C3×Q8—61no—
189C2×C3×C4—21no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—91no—
193C2×C3×C4—91no—
194C2×C3×C4C22×D911yescharacteristic
195C2×C3×C4C2×D3211yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—181no—
203C2×Dic3—181no—
204C2×Dic3—361no—
205C2×Dic3—361no—
206Dic6—21no—
207Dic6—362no—
208Dic6—362no—
209Dic6Dic1812yescomplemented
210C3×C8—22no—
211C3×C8—21no—
212C3×C8—21no—
213C3×C8—181no—
214C3⋊C8—61no—
215C2×C32#87811yescharacteristic
216C2×C32#91411yescharacteristic
217C2×C32D3×Q1611yescharacteristic
218C2×C9—21no—
219C2×C9—21no—
220C2×C9—21no—
221C2×C9#89711yescharacteristic
222C2×C9C2×Dic1211yescharacteristic
223C2×C9#90811yescharacteristic
224C2×Q8—31no—
225C2×Q8—541no—
226C2×C8—91no—
227C2×C8D3×D911yescharacteristic
228C4⋊C4—31no—
229C4⋊C4—271no—
230C4⋊C4—271no—
231C4⋊C4—541no—
232C4⋊C4—541no—
233C42—91no—
234C42—541no—
235C22×C3—21no—
236C22×C3D4×D911yescharacteristic
237C22×C3D3×D1211yescharacteristic
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—62no—
241C3×C4—92no—
242C3×C4—121no—
243C3×C4—181no—
244C3×C4C2×Dic1811yescharacteristic
245C3×C4#369911yescharacteristic
246C3×C4D3×Dic611yescharacteristic
247C3×C4#380511yescharacteristic
248Dic3—22no—
249Dic3—32no—
250Dic3—41no—
251Dic3—61no—
252Dic3—361no—
253Dic3—361no—
254Dic3—361no—
255Dic3—361no—
256Dic3—721no—
257Dic3—721no—
258C32#479311yescharacteristic
259C9—21no—
260C9#485011yescharacteristic, complemented
261Q8—32no—
262Q8—61no—
263Q8—1082no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—541no—
269C2×C4—541no—
270C2×C4C2×D3×D911yescharacteristic
271C8—21no—
272C8—181no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3#6344011yescharacteristic
277C2×C3#6347611yescharacteristic
278C2×C3Q16×D911yescharacteristic
279C2×C3#6379411yescharacteristic
280C2×C3D3×Dic1211yescharacteristic
281C2×C3#6389311yescharacteristic
282C22D9×D1211yescharacteristic
283C4—62no—
284C4—92no—
285C4—121no—
286C4—181no—
287C4—1081no—
288C4—1081no—
289C4D3×Dic1811yescharacteristic
290C4#8866311yescharacteristic
291C3—21no—
292C3#1058847311yescharacteristic, complemented
293C3#1059080411yescharacteristic
294C2#1172081511yescharacteristic
295C2D9×Dic1211yescharacteristic
296C2#1172102311yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#477
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
44. C9×Dic6Q8semidirect
186. C3×C9#477semidirect
209. Dic6Dic18semidirect
260. C9#4850semidirect
292. C3#10588473semidirect

Maximal subgroups

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

subgroupindex
#117209692
Dic6×Dic92
#117205412
C9×#8162
C3×Dic9⋊C82
#117202982
#117202622
#105908043
#105884733