Finite Group Databasebeta

← #13140702

Subgroups — #13140702

Order 1728.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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×#63506C211yesmaximal, characteristic
2#11720935C211yesmaximal, characteristic
3#11720582C211yesmaximal, characteristic
4C9×#835C211yesmaximal, characteristic
5#11720276C211yesmaximal, characteristic
6#11720274C211yesmaximal, characteristic
7#11720268C211yesmaximal, characteristic
8#10591613—31nomaximal
9#10588477—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C3×Dic9⋊C4—21no—
16C3×Dic9⋊C4—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20#88442—21no—
21C3×#944—31no—
22#63869—31no—
23#63662—31no—
24C3×#835D311yescharacteristic
25#63566—31no—
26#63565—31no—
27#63554—31no—
28#63506D311yescharacteristic
29#63452—31no—
30C9×#103—31no—
31#63395—31no—
32#63395—61no—
33#63376—31no—
34#63375—31no—
35#63368—31no—
36C3×Q8×C9—21no—
37C3×Q8×C9—21no—
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×Dic18—21no—
46C3×Dic18—21no—
47C3⋊(C9⋊C8)—21no—
48C9×C3⋊C8—21no—
49#5082—91no—
50#4797—91no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C3×Dic6—31no—
53C4⋊(C3⋊Dic3)—31no—
54C2×C3⋊(C3⋊C8)—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—61no—
58C3×C4×Dic3—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60#3724—61no—
61Dic3⋊Dic3—61no—
62C2×Q8×C9—21no—
63C2×Q8×C9C2×D311yescharacteristic
64C2×Dic18C2×D311yescharacteristic
65C2×C8×C9—31no—
66C9×C4⋊C4—31no—
67C4⋊Dic9—31no—
68C4⋊Dic9—61no—
69C4⋊Dic9—61no—
70Dic9⋊C4—21no—
71Dic9⋊C4—21no—
72Dic9⋊C4—61no—
73C4×Dic9C2×D311yescharacteristic
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C22×C3×C9C2×D411yescharacteristic
77C3×C4×C9—41no—
78C3×C4×C9#4111yescharacteristic
79C3×C4×C9#4111yescharacteristic
80C3⋊Dic9—41no—
81C9×Dic3—41no—
82C3×Dic9—21no—
83C3×Dic9—21no—
84C3×Dic9—41no—
85C3×C4⋊Q8—91no—
86#944—31no—
87#890—91no—
88#870—91no—
89C3×#103—31no—
90#835—91no—
91#835—91no—
92#835—91no—
93#835—91no—
94#835—181no—
95#835D911yescharacteristic
96#816—91no—
97#815—91no—
98#810—91no—
99#803—91no—
100C32×Q8—21no—
101C32×Q8—21no—
102C2×C32×C4—21no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—61no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—61no—
109C3×Dic6—61no—
110C3×Dic6—61no—
111C3⋊(C3⋊C8)—61no—
112C3×C3⋊C8—21no—
113Q8×C9—22no—
114Q8×C9—21no—
115Q8×C9—21no—
116Q8×C9—41no—
117C2×C4×C9—21no—
118C2×C4×C9—21no—
119C2×C4×C9—41no—
120C2×C4×C9—61no—
121C2×C4×C9C22×D311yescharacteristic
122C2×Dic9—21no—
123C2×Dic9—61no—
124C2×Dic9—121no—
125C2×Dic9#6711yescharacteristic
126C2×Dic9#6711yescharacteristic
127Dic18—21no—
128Dic18—21no—
129C8×C9—61no—
130C9⋊C8—61no—
131C9⋊C8—121no—
132#491—271noSylow
133C2×C3×C9#12411yescharacteristic
134C2×C3×C9#13711yescharacteristic
135C2×C3×C9#13711yescharacteristic
136C2×C3×Q8—21no—
137C2×C3×Q8—91no—
138C2×C3×Q8C2×D911yescharacteristic
139C2×C3×Q8D3211yescharacteristic
140C2×Dic6—31no—
141C2×C3×C8—31no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4—181no—
144C3×C4⋊C4—181no—
145C3×C42—91no—
146C4⋊Dic3—91no—
147C4⋊Dic3—91no—
148C4⋊Dic3—181no—
149C4⋊Dic3—181no—
150C4⋊Dic3C2×D911yescharacteristic
151Dic3⋊C4—61no—
152Dic3⋊C4—61no—
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155Dic3⋊C4—181no—
156C4×Dic3—31no—
157C2×C3⋊C8—91no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—181no—
160C2×C3⋊C8C2×D911yescharacteristic
161C22×C32D3×D411yescharacteristic
162C32×C4—41no—
163C32×C4#23511yescharacteristic
164C32×C4#23711yescharacteristic
165C3⋊Dic3—121no—
166C3×Dic3—41no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—121no—
170C22×C9—21no—
171C22×C9C2×#6711yescharacteristic
172C4×C9—21no—
173C4×C9—21no—
174C4×C9—42no—
175C4×C9—41no—
176C4×C9—121no—
177C4×C9#23711yescharacteristic
178C4×C9#23711yescharacteristic
179Dic9—21no—
180Dic9—21no—
181Dic9—41no—
182Dic9—121no—
183Dic9—241no—
184C4⋊Q8—91no—
185#125—271no—
186#103—31no—
187#103—271no—
188#103—271no—
189#103—271no—
190#97—271no—
191C3×C9#49111yescharacteristic, complemented, Sylow
192C3×Q8—22no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8—21no—
197C3×Q8—41no—
198C3×Q8—181no—
199C3×Q8—181no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4—61no—
205C2×C3×C4—91no—
206C2×C3×C4—91no—
207C2×C3×C4—181no—
208C2×C3×C4C22×D911yescharacteristic
209C2×C3×C4C2×D3211yescharacteristic
210C2×Dic3—21no—
211C2×Dic3—31no—
212C2×Dic3—31no—
213C2×Dic3—61no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216C2×Dic3—361no—
217Dic6—61no—
218Dic6—61no—
219C3×C8—61no—
220C3⋊C8—21no—
221C3⋊C8—181no—
222C3⋊C8—181no—
223C3⋊C8—361no—
224C2×C32#88511yescharacteristic
225C2×C32#91511yescharacteristic
226C2×C32#91811yescharacteristic
227C2×C9—21no—
228C2×C9—21no—
229C2×C9—21no—
230C2×C9#93611yescharacteristic
231C2×C9#94211yescharacteristic
232C2×C9#94211yescharacteristic
233C2×Q8—91no—
234C2×Q8D3×D911yescharacteristic
235C2×C8—31no—
236C2×C8—271no—
237C4⋊C4—31no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C4⋊C4—541no—
243C42—91no—
244C22×C3—21no—
245C22×C3D4×D911yescharacteristic
246C22×C3D3×#6711yescharacteristic
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—42no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—121no—
253C3×C4—181no—
254C3×C4—181no—
255C3×C4—361no—
256C3×C4#369711yescharacteristic
257C3×C4#369911yescharacteristic
258C3×C4#380411yescharacteristic
259C3×C4#380511yescharacteristic
260Dic3—41no—
261Dic3—61no—
262Dic3—61no—
263Dic3—121no—
264Dic3—361no—
265Dic3—361no—
266Dic3—721no—
267C32#479711yescharacteristic
268C9—21no—
269C9#508211yescharacteristic, complemented
270Q8—21no—
271Q8—21no—
272Q8—181no—
273Q8—181no—
274C2×C4—21no—
275C2×C4—61no—
276C2×C4—91no—
277C2×C4—91no—
278C2×C4—181no—
279C2×C4—541no—
280C2×C4C2×D3×D911yescharacteristic
281C8—61no—
282C8—541no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3#6344711yescharacteristic
287C2×C3#6347711yescharacteristic
288C2×C3#6348011yescharacteristic
289C2×C3#6394611yescharacteristic
290C2×C3#6394811yescharacteristic
291C2×C3#6395911yescharacteristic
292C22D9×#6711yescharacteristic
293C4—41no—
294C4—121no—
295C4—181no—
296C4—181no—
297C4—361no—
298C4—1081no—
299C4#8865811yescharacteristic
300C4#8866311yescharacteristic
301C3—21no—
302C3#1058847711yescharacteristic, complemented
303C3#1059161311yescharacteristic
304C2#1172121411yescharacteristic
305C2#1172122211yescharacteristic
306C2#1172125711yescharacteristic
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#491
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
191. C3×C9#491semidirect
269. C9#5082semidirect
302. C3#10588477semidirect

Maximal subgroups

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

subgroupindex
C3×#635062
#117209352
#117205822
C9×#8352
#117202762
#117202742
#117202682
#105916133
#105884773