Finite Group Databasebeta

← #13137525

Subgroups — #13137525

Order 1728.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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×#63450C211yesmaximal, characteristic
2#11720913C211yesmaximal, characteristic
3#11720527C211yesmaximal, characteristic
4C9×#808C211yesmaximal, characteristic
5#11720277C211yesmaximal, characteristic
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720265C211yesmaximal, characteristic
8#10590487—31nomaximal
9#10588534—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×Dic18—21no—
12C3×C9×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C3×C4⋊Dic9C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88442—21no—
20Dic3×Dic9—21no—
21C3×#888—31no—
22#63844—31no—
23#63637—31no—
24C3×#808D311yescharacteristic
25#63574—31no—
26Dic3⋊(C3⋊C8)—31no—
27#63567—31no—
28#63450D311yescharacteristic
29#63449—31no—
30C9×#106—31no—
31#63376—31no—
32Dic9⋊C8—31no—
33#63368—31no—
34#63368—61no—
35#63367—31no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39#6226—21no—
40#6226Q812yescomplemented
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45C3×Dic18—42no—
46C3⋊(C9⋊C8)—21no—
47C9×C3⋊C8—21no—
48#5094—91no—
49#4854—91no—
50C2×#634—31no—
51C2×C3×Dic6—61no—
52C32×C4⋊C4C2×D311yescharacteristic
53C2×C3⋊(C3⋊C8)—31no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C3×Dic3⋊C4—61no—
57C3×C4×Dic3—31no—
58C2×C3×C3⋊C8C2×D311yescharacteristic
59#3724—61no—
60Dic32—61no—
61C2×Dic18—21no—
62C2×Dic18—31no—
63C2×Dic18—61no—
64C2×C8×C9—31no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4—31no—
67C9×C4⋊C4C2×D311yescharacteristic
68C4⋊Dic9C2×D311yescharacteristic
69Dic9⋊C4—21no—
70Dic9⋊C4—61no—
71C4×Dic9—61no—
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—22no—
80C9×Dic3—41no—
81C3×Dic9—21no—
82C3×Dic9—41no—
83C3×Dic9SD1612yes—
84C3×C4⋊Q8—91no—
85#888—31no—
86#887—91no—
87C4×Dic6—91no—
88C3×#106—31no—
89#816—91no—
90Dic3⋊C8—91no—
91#810—91no—
92#810—91no—
93#810—91no—
94#810—181no—
95#808—91no—
96#808—91no—
97#808D911yescharacteristic
98C4⋊(C3⋊C8)—91no—
99C2×C32×C4—21no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—61no—
102#634—32no—
103#634—61no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—31no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—61no—
108C3×Dic6—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—32no—
122Dic18—42no—
123Dic18—62no—
124Dic18—61no—
125Dic18—121no—
126C8×C9—61no—
127C9⋊C8—61no—
128C9⋊C8—121no—
129#475—271noSylow
130C2×C3×C9C22⋊Q811yescharacteristic
131C2×C3×C9C2×SD1611yescharacteristic
132C2×C3×C9#13711yescharacteristic
133C2×C3×Q8—181no—
134C2×Dic6—61no—
135C2×Dic6—91no—
136C2×Dic6—91no—
137C2×Dic6—181no—
138C2×C3×C8—31no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—181no—
143C3×C4⋊C4C2×D911yescharacteristic
144C3×C4⋊C4D3211yescharacteristic
145C3×C42—91no—
146C4⋊Dic3—31no—
147C4⋊Dic3C2×D911yescharacteristic
148Dic3⋊C4—61no—
149Dic3⋊C4—181no—
150Dic3⋊C4—181no—
151C4×Dic3—31no—
152C4×Dic3—181no—
153C4×Dic3—181no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—181no—
157C2×C3⋊C8C2×D911yescharacteristic
158C22×C32D3×D411yescharacteristic
159C32×C4—41no—
160C32×C4#23511yescharacteristic
161C32×C4D3×Q811yescharacteristic
162C3⋊Dic3—62no—
163C3×Dic3—32no—
164C3×Dic3—41no—
165C3×Dic3—61no—
166C3×Dic3—121no—
167C22×C9—21no—
168C22×C9C2×#6711yescharacteristic
169C4×C9—21no—
170C4×C9—21no—
171C4×C9—41no—
172C4×C9—81no—
173C4×C9—121no—
174C4×C9D3×Q811yescharacteristic
175C4×C9#23911yescharacteristic
176Dic9—21no—
177Dic9—41no—
178Dic9—62no—
179Dic9—122no—
180Dic9#21412yes—
181C4⋊Q8—91no—
182C4×Q8—271no—
183#106—31no—
184#106—271no—
185C4⋊C8—271no—
186#103—271no—
187#103—271no—
188C3×C9#47511yescharacteristic, complemented, Sylow
189C3×Q8—362no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4—61no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—181no—
198C2×C3×C4C22×D911yescharacteristic
199C2×C3×C4C2×D3211yescharacteristic
200C2×Dic3—21no—
201C2×Dic3—31no—
202C2×Dic3—31no—
203C2×Dic3—61no—
204C2×Dic3—181no—
205C2×Dic3—181no—
206C2×Dic3—361no—
207Dic6—92no—
208Dic6—92no—
209Dic6—122no—
210Dic6—182no—
211Dic6—181no—
212Dic6—181no—
213Dic6—361no—
214C3×C8—61no—
215C3⋊C8—21no—
216C3⋊C8—181no—
217C3⋊C8—181no—
218C3⋊C8—361no—
219C2×C32#89611yescharacteristic
220C2×C32D3×SD1611yescharacteristic
221C2×C32#91511yescharacteristic
222C2×C9—21no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9C2×#21411yescharacteristic
226C2×C9#94211yescharacteristic
227C2×C9#94611yescharacteristic
228C2×Q8—181no—
229C2×Q8—271no—
230C2×C8—31no—
231C2×C8—271no—
232C4⋊C4—31no—
233C4⋊C4—91no—
234C4⋊C4—181no—
235C4⋊C4—541no—
236C4⋊C4D3×D911yescharacteristic
237C42—91no—
238C42—541no—
239C22×C3—21no—
240C22×C3D4×D911yescharacteristic
241C22×C3D3×#6711yescharacteristic
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—81no—
247C3×C4—92no—
248C3×C4—121no—
249C3×C4—181no—
250C3×C4—361no—
251C3×C4#369711yescharacteristic
252C3×C4Q8×D911yescharacteristic
253C3×C4#379711yescharacteristic
254C3×C4#379911yescharacteristic
255Dic3—32no—
256Dic3—41no—
257Dic3—61no—
258Dic3—121no—
259Dic3—182no—
260Dic3—182no—
261Dic3—362no—
262C32#485411yescharacteristic
263C9—21no—
264C9#509411yescharacteristic, complemented
265Q8—272no—
266Q8—362no—
267Q8—541no—
268C2×C4—21no—
269C2×C4—61no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—181no—
273C2×C4—541no—
274C2×C4C2×D3×D911yescharacteristic
275C8—61no—
276C8—541no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3#6345811yescharacteristic
281C2×C3SD16×D911yescharacteristic
282C2×C3#6347711yescharacteristic
283C2×C3#6388811yescharacteristic
284C2×C3D3×#21411yescharacteristic
285C2×C3#6394611yescharacteristic
286C22D9×#6711yescharacteristic
287C4—41no—
288C4—92no—
289C4—121no—
290C4—181no—
291C4—361no—
292C4—542no—
293C4(C3⋊D9)⋊Q811yescharacteristic
294C4#8864211yescharacteristic
295C3—21no—
296C3#1058853411yescharacteristic, complemented
297C3#1059048711yescharacteristic
298C2#1172098511yescharacteristic
299C2D9×#21411yescharacteristic
300C2#1172121411yescharacteristic
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
40. #6226Q8semidirect
188. C3×C9#475semidirect
264. C9#5094semidirect
296. C3#10588534semidirect

Maximal subgroups

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

subgroupindex
C3×#634502
#117209132
#117205272
C9×#8082
#117202772
Dic9⋊(C3⋊C8)2
#117202652
#105904873
#105885343