Finite Group Databasebeta

← #13138265

Subgroups — #13138265

Order 1728.

338 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 83 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#11720971C211yesmaximal, characteristic
2#11720930C211yesmaximal, characteristic
3C8×C3⋊Dic9C211yesmaximal, characteristic
4C9×#816C211yesmaximal, characteristic
5C3×#63376C211yesmaximal, characteristic
6#11720297C211yesmaximal, characteristic
7#11720268C211yesmaximal, characteristic
8#10590901—31nomaximal
9#10588472—31nomaximal
10C2×C9×Dic6C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C2×#6189—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C4×C3⋊Dic9C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18#88442—21no—
19#88442—21no—
20#88442—21no—
21#63919—31no—
22#63873—31no—
23C8×C3⋊Dic3—31no—
24C3×#816—31no—
25C3×#816D311yescharacteristic
26#63565—31no—
27#63565—31no—
28#63506—31no—
29#63450—31no—
30C9×#103—31no—
31#63395—31no—
32#63376D311yescharacteristic
33C8×Dic9—31no—
34C8×Dic9—61no—
35#63368—31no—
36C2×C3×C4×C9C2311yescharacteristic
37C2×C3⋊Dic9D411yescharacteristic
38C2×C3⋊Dic9D411yescharacteristic
39C2×C9×Dic3—21no—
40C2×C9×Dic3—21no—
41C2×C3×Dic9—21no—
42C2×C3×Dic9—21no—
43C9×Dic6—21no—
44C9×Dic6—21no—
45C3×Dic18—21no—
46C3×Dic18—21no—
47#6189—42no—
48C3×C8×C9—21no—
49C3⋊(C9⋊C8)—21no—
50#4792—91no—
51#4792—91no—
52C2×C3×Dic6—31no—
53C2×C3×Dic6C2×D311yescharacteristic
54C2×#627—61no—
55C2×C32×C8C2×D311yescharacteristic
56C4×C3⋊Dic3—31no—
57C2×C3⋊(C3⋊C8)—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60#3724—61no—
61#3724—61no—
62#3724—61no—
63C2×Q8×C9—31no—
64C2×Dic18—61no—
65C2×Dic18C2×D311yescharacteristic
66C2×C8×C9—21no—
67C2×C8×C9C2×D311yescharacteristic
68C9×C4⋊C4—31no—
69C4⋊Dic9C2×D311yescharacteristic
70Dic9⋊C4—61no—
71Dic9⋊C4—61no—
72Dic9⋊C4—61no—
73C4×Dic9—31no—
74C4×Dic9—61no—
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C22×C3×C9C2×D411yescharacteristic
78C3×C4×C9#4111yescharacteristic
79C3×C4×C9#4111yescharacteristic
80C3⋊Dic9SD1612yes—
81C3⋊Dic9Q1612yes—
82C9×Dic3—41no—
83C9×Dic3—41no—
84C3×Dic9—41no—
85C3×Dic9—41no—
86#944—91no—
87#944—91no—
88#888—91no—
89#888—91no—
90C3×#103—31no—
91C3×#103—91no—
92#835—91no—
93#835—91no—
94#816—31no—
95#816D911yescharacteristic
96C8×Dic3—91no—
97C8×Dic3—91no—
98C8×Dic3—181no—
99#810—91no—
100#810—91no—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—31no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C3×Dic6—21no—
109C3×Dic6—21no—
110C3×Dic6—61no—
111C3×Dic6—61no—
112#627—122no—
113C32×C8—21no—
114C3⋊(C3⋊C8)—61no—
115Q8×C9—61no—
116Q8×C9—61no—
117C2×C4×C9—21no—
118C2×C4×C9—61no—
119C2×C4×C9—61no—
120C2×C4×C9C22×D311yescharacteristic
121C2×Dic9—21no—
122C2×Dic9—21no—
123C2×Dic9—31no—
124C2×Dic9—31no—
125C2×Dic9—61no—
126C2×Dic9—61no—
127Dic18—21no—
128Dic18—21no—
129Dic18—122no—
130C8×C9—22no—
131C8×C9—21no—
132C9⋊C8—62no—
133C9⋊C8—61no—
134#487—271noSylow
135C2×C3×C9#12411yescharacteristic
136C2×C3×C9C2×SD1611yescharacteristic
137C2×C3×C9C2×Q1611yescharacteristic
138C2×C3×Q8—31no—
139C2×C3×Q8—91no—
140C2×Dic6—31no—
141C2×Dic6—181no—
142C2×Dic6—181no—
143C2×Dic6C2×D911yescharacteristic
144C2×C3×C8—21no—
145C2×C3×C8C2×D911yescharacteristic
146C2×C3×C8D3211yescharacteristic
147C3×C4⋊C4—31no—
148C3×C4⋊C4—91no—
149C4⋊Dic3—31no—
150C4⋊Dic3C2×D911yescharacteristic
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155Dic3⋊C4—181no—
156Dic3⋊C4—181no—
157C4×Dic3—91no—
158C4×Dic3—91no—
159C4×Dic3—181no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—181no—
163C22×C32D3×D411yescharacteristic
164C32×C4#23511yescharacteristic
165C32×C4#23711yescharacteristic
166C3⋊Dic3—32no—
167C3⋊Dic3—32no—
168C3×Dic3—41no—
169C3×Dic3—41no—
170C3×Dic3—121no—
171C3×Dic3—121no—
172C22×C9—21no—
173C22×C9D3×D411yescharacteristic
174C4×C9—21no—
175C4×C9—21no—
176C4×C9—121no—
177C4×C9—121no—
178C4×C9#23511yescharacteristic
179C4×C9#23711yescharacteristic
180Dic9—32no—
181Dic9—32no—
182Dic9—41no—
183Dic9—41no—
184Dic9—62no—
185Dic9—62no—
186C4⋊Q8—271no—
187C4⋊Q8—271no—
188#103—31no—
189#103—91no—
190#103—271no—
191#103—271no—
192C4×C8—271no—
193C3×C9#48711yescharacteristic, complemented, Sylow
194C3×Q8—61no—
195C3×Q8—61no—
196C3×Q8—181no—
197C3×Q8—181no—
198C2×C3×C4—21no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4C22×D911yescharacteristic
204C2×C3×C4C2×D3211yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—91no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—91no—
213C2×Dic3—181no—
214C2×Dic3—181no—
215Dic6—21no—
216Dic6—21no—
217Dic6—61no—
218Dic6—61no—
219Dic6—362no—
220Dic6—362no—
221C3×C8—22no—
222C3×C8—21no—
223C3×C8—21no—
224C3⋊C8—182no—
225C3⋊C8—181no—
226C3⋊C8—181no—
227C2×C32#88511yescharacteristic
228C2×C32D3×SD1611yescharacteristic
229C2×C32D3×Q1611yescharacteristic
230C2×C9—21no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9#88511yescharacteristic
234C2×C9D3×SD1611yescharacteristic
235C2×C9D3×Q1611yescharacteristic
236C2×Q8—31no—
237C2×Q8—91no—
238C2×Q8—541no—
239C2×C8—271no—
240C2×C8D3×D911yescharacteristic
241C4⋊C4—31no—
242C4⋊C4—91no—
243C4⋊C4—541no—
244C4⋊C4—541no—
245C4⋊C4—541no—
246C42—271no—
247C22×C3—21no—
248C22×C3D4×D911yescharacteristic
249C22×C3#380211yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—121no—
253C3×C4—121no—
254C3×C4—361no—
255C3×C4—361no—
256C3×C4#369711yescharacteristic
257C3×C4#369911yescharacteristic
258C3×C4#379811yescharacteristic
259C3×C4#380511yescharacteristic
260Dic3—41no—
261Dic3—41no—
262Dic3—92no—
263Dic3—92no—
264Dic3—92no—
265Dic3—92no—
266Dic3—121no—
267Dic3—121no—
268Dic3—182no—
269Dic3—182no—
270C32#479211yescharacteristic
271C9—21no—
272C9#479211yescharacteristic, complemented
273Q8—61no—
274Q8—61no—
275Q8—181no—
276Q8—181no—
277Q8—1082no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—271no—
283C2×C4—271no—
284C2×C4C2×D3×D911yescharacteristic
285C8—21no—
286C8—541no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3#6344711yescharacteristic
291C2×C3SD16×D911yescharacteristic
292C2×C3Q16×D911yescharacteristic
293C2×C3#6379611yescharacteristic
294C2×C3#6380211yescharacteristic
295C2×C3#6391511yescharacteristic
296C22#8864611yescharacteristic
297C4—121no—
298C4—121no—
299C4—272no—
300C4—272no—
301C4—361no—
302C4—361no—
303C4#8864011yescharacteristic
304C4#8866311yescharacteristic
305C3—21no—
306C3#1058847211yescharacteristic, complemented
307C3#1059090111yescharacteristic
308C2#1172081711yescharacteristic
309C2#1172082911yescharacteristic
310C2#1172104411yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#487
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
193. C3×C9#487semidirect
272. C9#4792semidirect
306. C3#10588472semidirect

Maximal subgroups

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

subgroupindex
#117209712
#117209302
C8×C3⋊Dic92
C9×#8162
C3×#633762
#117202972
#117202682
#105909013
#105884723