Finite Group Databasebeta

← #13140701

Subgroups — #13140701

Order 1728.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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×Q8×Dic9C211yesmaximal, characteristic
2#11720935C211yesmaximal, characteristic
3#11720582C211yesmaximal, characteristic
4C9×#835C211yesmaximal, characteristic
5#11720279C211yesmaximal, characteristic
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720265C211yesmaximal, characteristic
8#10591547—31nomaximal
9#10588476—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C4⋊(C3⋊Dic9)C2211yescharacteristic
12C2×C3⋊(C9⋊C8)C2211yescharacteristic
13C9×C4⋊Dic3C2211yescharacteristic
14C3×C4⋊Dic9—21no—
15C3×C4⋊Dic9C2211yescharacteristic
16C3×C4×Dic9—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20#88442—21no—
21C3×Q8×Dic3—31no—
22#63869—31no—
23#63662—31no—
24C3×#835D311yescharacteristic
25#63575—31no—
26#63574—31no—
27Dic3⋊(C3⋊C8)—31no—
28Q8×Dic9D311yescharacteristic
29#63452—31no—
30C9×#103—31no—
31#63395—31no—
32#63395—61no—
33#63378—31no—
34Dic9⋊C8—31no—
35#63367—31no—
36C3×Q8×C9—21no—
37C3×Q8×C9Q812yes—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45C3⋊(C9⋊C8)—21no—
46C9×C3⋊C8—21no—
47#5022—91no—
48#4796—91no—
49C2×C32×Q8C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)—31no—
51C2×C3⋊(C3⋊C8)—31no—
52C3×C4⋊Dic3—31no—
53C3×C4⋊Dic3—61no—
54C3×C4⋊Dic3C2×D311yescharacteristic
55C3×C4×Dic3—31no—
56C3×C4×Dic3—61no—
57C2×C3×C3⋊C8C2×D311yescharacteristic
58#3724—61no—
59Dic3⋊Dic3—61no—
60C2×Q8×C9—21no—
61C2×Q8×C9C2×D311yescharacteristic
62C2×C8×C9—31no—
63C9×C4⋊C4—31no—
64C4⋊Dic9—21no—
65C4⋊Dic9—31no—
66C4⋊Dic9—61no—
67C4⋊Dic9—61no—
68C4⋊Dic9C2×D311yescharacteristic
69Dic9⋊C4—61no—
70C4×Dic9—21no—
71C4×Dic9C2×D311yescharacteristic
72C2×C9⋊C8—31no—
73C2×C9⋊C8—61no—
74C22×C3×C9C2×D411yescharacteristic
75C3×C4×C9—22no—
76C3×C4×C9C2×Q811yescharacteristic
77C3×C4×C9#4111yescharacteristic
78C3⋊Dic9—41no—
79C9×Dic3—41no—
80C3×Dic9—21no—
81C3×Dic9—21no—
82C3×Dic9—41no—
83C3×C4×Q8—91no—
84Q8×Dic3—31no—
85#890—91no—
86#870—91no—
87C3×#103—31no—
88#835—91no—
89#835—91no—
90#835—181no—
91#835D911yescharacteristic
92#818—91no—
93Dic3⋊C8—91no—
94#808—91no—
95#808—91no—
96#807—91no—
97C4⋊(C3⋊C8)—91no—
98C32×Q8—21no—
99C32×Q8Dic612yes—
100C2×C32×C4—21no—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—61no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—31no—
105C2×C3×Dic3—31no—
106C2×C3×Dic3—61no—
107C3⋊(C3⋊C8)—61no—
108C3×C3⋊C8—21no—
109Q8×C9—21no—
110Q8×C9—22no—
111Q8×C9—41no—
112Q8×C9Dic612yes—
113C2×C4×C9—21no—
114C2×C4×C9—21no—
115C2×C4×C9—41no—
116C2×C4×C9—61no—
117C2×C4×C9C22×D311yescharacteristic
118C2×Dic9—21no—
119C2×Dic9—61no—
120C2×Dic9—121no—
121C2×Dic9#6711yescharacteristic
122C2×Dic9#6711yescharacteristic
123C8×C9—61no—
124C9⋊C8—61no—
125C9⋊C8—121no—
126#479—271noSylow
127C2×C3×C9C22⋊Q811yescharacteristic
128C2×C3×C9#13511yescharacteristic
129C2×C3×C9#13711yescharacteristic
130C2×C3×Q8—21no—
131C2×C3×Q8C2×D911yescharacteristic
132C2×C3×Q8D3211yescharacteristic
133C2×C3×C8—31no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—181no—
137C3×C42—91no—
138C3×C42—181no—
139C4⋊Dic3—31no—
140C4⋊Dic3—61no—
141C4⋊Dic3—91no—
142C4⋊Dic3—91no—
143C4⋊Dic3—181no—
144C4⋊Dic3—181no—
145C4⋊Dic3C2×D911yescharacteristic
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149C4×Dic3—31no—
150C4×Dic3—61no—
151C2×C3⋊C8—91no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—181no—
154C2×C3⋊C8C2×D911yescharacteristic
155C22×C32D3×D411yescharacteristic
156C32×C4—22no—
157C32×C4C2×Dic611yescharacteristic
158C32×C4#23711yescharacteristic
159C3⋊Dic3—121no—
160C3×Dic3—41no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—121no—
164C22×C9—21no—
165C22×C9C2×#6711yescharacteristic
166C4×C9—22no—
167C4×C9—21no—
168C4×C9—21no—
169C4×C9—42no—
170C4×C9—121no—
171C4×C9C2×Dic611yescharacteristic
172C4×C9#23511yescharacteristic
173Dic9—21no—
174Dic9—21no—
175Dic9—41no—
176Dic9—121no—
177Dic9—241no—
178#125—271no—
179C4×Q8—91no—
180#107—271no—
181#106—271no—
182C4⋊C8—271no—
183#103—31no—
184#103—271no—
185C3×C9#47911yescharacteristic, complemented, Sylow
186C3×Q8—21no—
187C3×Q8—21no—
188C3×Q8—22no—
189C3×Q8—41no—
190C3×Q8Dic1812yes—
191C3×Q8#62712yes—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—41no—
196C2×C3×C4—61no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—181no—
200C2×C3×C4C22×D911yescharacteristic
201C2×C3×C4C2×D3211yescharacteristic
202C2×Dic3—21no—
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3—61no—
206C2×Dic3—181no—
207C2×Dic3—181no—
208C2×Dic3—361no—
209C3×C8—61no—
210C3⋊C8—21no—
211C3⋊C8—181no—
212C3⋊C8—181no—
213C3⋊C8—361no—
214C2×C32#87811yescharacteristic
215C2×C32#91511yescharacteristic
216C2×C32#91911yescharacteristic
217C2×C9—21no—
218C2×C9—21no—
219C2×C9—21no—
220C2×C9#92411yescharacteristic
221C2×C9#94211yescharacteristic
222C2×C9#95011yescharacteristic
223C2×Q8D3×D911yescharacteristic
224C2×C8—31no—
225C2×C8—271no—
226C4⋊C4—31no—
227C4⋊C4—91no—
228C4⋊C4—181no—
229C4⋊C4—271no—
230C4⋊C4—541no—
231C4⋊C4—541no—
232C42—91no—
233C42—181no—
234C22×C3—21no—
235C22×C3D4×D911yescharacteristic
236C22×C3D3×#6711yescharacteristic
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—42no—
242C3×C4—121no—
243C3×C4—181no—
244C3×C4—181no—
245C3×C4—361no—
246C3×C4C2×Dic1811yescharacteristic
247C3×C4#369911yescharacteristic
248C3×C4#380411yescharacteristic
249C3×C4C2×#62711yescharacteristic
250Dic3—41no—
251Dic3—61no—
252Dic3—61no—
253Dic3—121no—
254Dic3—361no—
255Dic3—361no—
256Dic3—721no—
257C32#479611yescharacteristic
258C9—21no—
259C9#502211yescharacteristic, complemented
260Q8—21no—
261Q8#618912yes—
262C2×C4—21no—
263C2×C4—61no—
264C2×C4—91no—
265C2×C4—91no—
266C2×C4—181no—
267C2×C4—541no—
268C2×C4C2×D3×D911yescharacteristic
269C8—61no—
270C8—541no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3#6344011yescharacteristic
275C2×C3#6347711yescharacteristic
276C2×C3#6348111yescharacteristic
277C2×C3#6394611yescharacteristic
278C2×C3#6394911yescharacteristic
279C2×C3#6396411yescharacteristic
280C22D9×#6711yescharacteristic
281C4—22no—
282C4—121no—
283C4—181no—
284C4—181no—
285C4—361no—
286C4—1081no—
287C4C2×#618911yescharacteristic
288C4#8866211yescharacteristic
289C3—21no—
290C3#1058847611yescharacteristic, complemented
291C3#1059154711yescharacteristic
292C2#1172121411yescharacteristic
293C2#1172122311yescharacteristic
294C2#1172125311yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#479
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
185. C3×C9#479semidirect
259. C9#5022semidirect
290. C3#10588476semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×Dic92
#117209352
#117205822
C9×#8352
#117202792
Dic9⋊(C3⋊C8)2
#117202652
#105915473
#105884763