Finite Group Databasebeta

← #13137535

Subgroups — #13137535

Order 1728.

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

normal subgroups only

This group has 324 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
1C3×#63449C211yesmaximal, characteristic
2#11720969C211yesmaximal, characteristic
3#11720526C211yesmaximal, characteristic
4C9×#808C211yesmaximal, characteristic
5#11720276C211yesmaximal, characteristic
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720268C211yesmaximal, characteristic
8#10590556—31nomaximal
9#10588534—31nomaximal
10C2×C3×Dic18C2211yescharacteristic
11C2×#6189—21no—
12C3×C9×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C3×C4×Dic9—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic9⋊Dic3—21no—
21C3×#887—31no—
22#63918—31no—
23#63635—31no—
24C3×#808D311yescharacteristic
25Dic3⋊(C3⋊C8)—31no—
26#63566—31no—
27#63565—31no—
28#63450—31no—
29#63449D311yescharacteristic
30C9×#106—31no—
31#63376—31no—
32Dic9⋊C8—31no—
33#63368—31no—
34#63367—31no—
35#63367—61no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C9×Dic3—21no—
40C2×C3×Dic9—21no—
41C2×C3×Dic9D411yescharacteristic
42C2×C3×Dic9D411yescharacteristic
43C3×Dic18—21no—
44C3×Dic18Q812yescomplemented
45#6189—42no—
46C3⋊(C9⋊C8)—21no—
47C9×C3⋊C8—21no—
48#5020—91no—
49#4854—91no—
50C2×C3×Dic6—31no—
51C2×#627—61no—
52C32×C4⋊C4C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)—31no—
54C2×C3⋊(C3⋊C8)—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C3×Dic3⋊C4—61no—
57C3×C4×Dic3—31no—
58C3×C4×Dic3—61no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60Dic3⋊Dic3—61no—
61C2×Dic18—61no—
62C2×Dic18C2×D311yescharacteristic
63C2×C8×C9—31no—
64C9×C4⋊C4—21no—
65C9×C4⋊C4—31no—
66C9×C4⋊C4C2×D311yescharacteristic
67C4⋊Dic9—31no—
68C4⋊Dic9—61no—
69Dic9⋊C4—21no—
70Dic9⋊C4—61no—
71C4×Dic9—21no—
72C4×Dic9C2×D311yescharacteristic
73C2×C9⋊C8—31no—
74C2×C9⋊C8—61no—
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9—41no—
77C3×C4×C9C2×Q811yescharacteristic
78C3×C4×C9#4111yescharacteristic
79C3⋊Dic9—41no—
80C9×Dic3—41no—
81C3×Dic9—22no—
82C3×Dic9—21no—
83C3×Dic9SD1612yes—
84C3×C4×Q8—91no—
85#888—91no—
86#887—31no—
87#869—91no—
88C3×#106—31no—
89#835—91no—
90#835—91no—
91#816—91no—
92Dic3⋊C8—91no—
93#810—91no—
94#808—91no—
95#808—91no—
96#808—181no—
97#808D911yescharacteristic
98C4⋊(C3⋊C8)—91no—
99C2×C32×C4—21no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—61no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—31no—
104C2×C3×Dic3—31no—
105C2×C3×Dic3—61no—
106C3×Dic6—32no—
107C3×Dic6—61no—
108#627—122no—
109C3⋊(C3⋊C8)—61no—
110C3×C3⋊C8—21no—
111C2×C4×C9—21no—
112C2×C4×C9—21no—
113C2×C4×C9—41no—
114C2×C4×C9—61no—
115C2×C4×C9C22×D311yescharacteristic
116C2×Dic9—21no—
117C2×Dic9—61no—
118C2×Dic9—121no—
119C2×Dic9#6711yescharacteristic
120C2×Dic9#6711yescharacteristic
121Dic18—21no—
122Dic18—122no—
123Dic18Dic612yescomplemented
124C8×C9—61no—
125C9⋊C8—62no—
126C9⋊C8—61no—
127#475—271noSylow
128C2×C3×C9C22⋊Q811yescharacteristic
129C2×C3×C9C2×SD1611yescharacteristic
130C2×C3×C9#13711yescharacteristic
131C2×C3×Q8—91no—
132C2×Dic6—31no—
133C2×Dic6—181no—
134C2×Dic6—181no—
135C2×C3×C8—31no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4C2×D911yescharacteristic
140C3×C4⋊C4D3211yescharacteristic
141C3×C42—91no—
142C3×C42—181no—
143C4⋊Dic3—91no—
144C4⋊Dic3—91no—
145C4⋊Dic3—181no—
146C4⋊Dic3—181no—
147C4⋊Dic3C2×D911yescharacteristic
148Dic3⋊C4—61no—
149Dic3⋊C4—181no—
150C4×Dic3—31no—
151C4×Dic3—61no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—181no—
155C2×C3⋊C8C2×D911yescharacteristic
156C22×C32D3×D411yescharacteristic
157C32×C4—41no—
158C32×C4#23511yescharacteristic
159C32×C4D3×Q811yescharacteristic
160C3⋊Dic3—121no—
161C3×Dic3—32no—
162C3×Dic3—41no—
163C3×Dic3—62no—
164C3×Dic3—61no—
165C22×C9—21no—
166C22×C9C2×#6711yescharacteristic
167C4×C9—21no—
168C4×C9—21no—
169C4×C9—41no—
170C4×C9—81no—
171C4×C9—121no—
172C4×C9C2×Dic611yescharacteristic
173C4×C9#23511yescharacteristic
174Dic9—22no—
175Dic9—21no—
176Dic9—121no—
177Dic9—241no—
178Dic9#21512yes—
179C4⋊Q8—271no—
180C4×Q8—91no—
181#106—31no—
182#106—271no—
183C4⋊C8—271no—
184#103—271no—
185#103—271no—
186C3×C9#47511yescharacteristic, complemented, Sylow
187C3×Q8—92no—
188C3×Q8—181no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—41no—
193C2×C3×C4—61no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—181no—
197C2×C3×C4C22×D911yescharacteristic
198C2×C3×C4C2×D3211yescharacteristic
199C2×Dic3—21no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—61no—
203C2×Dic3—181no—
204C2×Dic3—181no—
205C2×Dic3—361no—
206Dic6—32no—
207Dic6—61no—
208Dic6—362no—
209Dic6—362no—
210C3×C8—61no—
211C3⋊C8—21no—
212C3⋊C8—182no—
213C3⋊C8—181no—
214C3⋊C8—181no—
215C2×C32#89611yescharacteristic
216C2×C32D3×SD1611yescharacteristic
217C2×C32#91511yescharacteristic
218C2×C9—21no—
219C2×C9—21no—
220C2×C9—21no—
221C2×C9#92411yescharacteristic
222C2×C9C2×#21511yescharacteristic
223C2×C9#95111yescharacteristic
224C2×Q8—91no—
225C2×Q8—541no—
226C2×C8—31no—
227C2×C8—271no—
228C4⋊C4—31no—
229C4⋊C4—181no—
230C4⋊C4—271no—
231C4⋊C4—541no—
232C4⋊C4D3×D911yescharacteristic
233C42—91no—
234C42—181no—
235C22×C3—21no—
236C22×C3D4×D911yescharacteristic
237C22×C3D3×#6711yescharacteristic
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—81no—
243C3×C4—92no—
244C3×C4—121no—
245C3×C4—182no—
246C3×C4—181no—
247C3×C4#369711yescharacteristic
248C3×C4Q8×D911yescharacteristic
249C3×C4D3×Dic611yescharacteristic
250C3×C4#379811yescharacteristic
251Dic3—32no—
252Dic3—41no—
253Dic3—62no—
254Dic3—61no—
255Dic3—361no—
256Dic3—361no—
257Dic3—721no—
258C32#485411yescharacteristic
259C9—21no—
260C9#502011yescharacteristic, complemented
261Q8—92no—
262Q8—181no—
263Q8—1082no—
264C2×C4—21no—
265C2×C4—61no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—181no—
269C2×C4—541no—
270C2×C4C2×D3×D911yescharacteristic
271C8—61no—
272C8—541no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3#6345811yescharacteristic
277C2×C3SD16×D911yescharacteristic
278C2×C3#6347711yescharacteristic
279C2×C3#6385111yescharacteristic
280C2×C3#6392911yescharacteristic
281C2×C3D3×#21511yescharacteristic
282C22D9×#6711yescharacteristic
283C4—41no—
284C4—92no—
285C4—121no—
286C4—181no—
287C4—182no—
288C4—1081no—
289C4D9×Dic611yescharacteristic
290C4#8864011yescharacteristic
291C3—21no—
292C3#1058853411yescharacteristic, complemented
293C3#1059055611yescharacteristic
294C2#1172093911yescharacteristic
295C2#1172118111yescharacteristic
296C2D9×#21511yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socle, prime radicalC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#475
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
44. C3×Dic18Q8semidirect
123. Dic18Dic6semidirect
186. C3×C9#475semidirect
260. C9#5020semidirect
292. C3#10588534semidirect

Maximal subgroups

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

subgroupindex
C3×#634492
#117209692
#117205262
C9×#8082
#117202762
Dic9⋊(C3⋊C8)2
#117202682
#105905563
#105885343