Finite Group Databasebeta

← #13138279

Subgroups — #13138279

Order 1728.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#11720937C211yesmaximal, characteristic
2#11720930C211yesmaximal, characteristic
3#11720538C211yesmaximal, characteristic
4C9×#816C211yesmaximal, characteristic
5C3×#63376C211yesmaximal, characteristic
6#11720297C211yesmaximal, characteristic
7#11720268C211yesmaximal, characteristic
8#10590904—31nomaximal
9#10588477—31nomaximal
10C2×C9×Dic6C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C4×C3⋊Dic9C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17Dic3⋊Dic9—21no—
18#88442—21no—
19#88442—21no—
20Dic9⋊Dic3—21no—
21#63873—31no—
22#63872—31no—
23#63642—31no—
24C3×#816—31no—
25C3×#816D311yescharacteristic
26#63565—31no—
27#63565—31no—
28#63506—31no—
29#63452—31no—
30C9×#103—31no—
31#63395—31no—
32#63376D311yescharacteristic
33#63375—31no—
34#63375—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—
47C3×C8×C9—21no—
48C3⋊(C9⋊C8)—21no—
49#4797—91no—
50#4797—91no—
51C2×C3×Dic6—31no—
52C2×C3×Dic6C2×D311yescharacteristic
53C2×C32×C8C2×D311yescharacteristic
54C4×C3⋊Dic3—31no—
55C2×C3⋊(C3⋊C8)—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58#3724—61no—
59#3724—61no—
60Dic3⋊Dic3—61no—
61Dic3⋊Dic3—61no—
62C2×Q8×C9—31no—
63C2×Dic18C2×D311yescharacteristic
64C2×C8×C9—21no—
65C2×C8×C9C2×D311yescharacteristic
66C9×C4⋊C4—31no—
67C4⋊Dic9—61no—
68C4⋊Dic9C2×D311yescharacteristic
69Dic9⋊C4—61no—
70Dic9⋊C4—61no—
71Dic9⋊C4—61no—
72C4×Dic9—31no—
73C4×Dic9—61no—
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C22×C3×C9C2×D411yescharacteristic
77C3×C4×C9#4111yescharacteristic
78C3×C4×C9#4111yescharacteristic
79C3⋊Dic9—21no—
80C3⋊Dic9—21no—
81C9×Dic3—41no—
82C9×Dic3—41no—
83C3×Dic9—41no—
84C3×Dic9—41no—
85#944—91no—
86#944—91no—
87#890—91no—
88#890—91no—
89C3×#103—31no—
90C3×#103—91no—
91#835—91no—
92#835—91no—
93#816—31no—
94#816D911yescharacteristic
95#815—91no—
96#815—91no—
97#815—181no—
98#810—91no—
99#810—91no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—31no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C3×Dic6—21no—
108C3×Dic6—21no—
109C3×Dic6—61no—
110C3×Dic6—61no—
111C32×C8—21no—
112C3⋊(C3⋊C8)—61no—
113Q8×C9—61no—
114Q8×C9—61no—
115C2×C4×C9—21no—
116C2×C4×C9—61no—
117C2×C4×C9—61no—
118C2×C4×C9C22×D311yescharacteristic
119C2×Dic9—21no—
120C2×Dic9—21no—
121C2×Dic9—31no—
122C2×Dic9—31no—
123C2×Dic9—61no—
124C2×Dic9—61no—
125Dic18—21no—
126Dic18—21no—
127C8×C9—22no—
128C8×C9—21no—
129C9⋊C8—62no—
130C9⋊C8—61no—
131#491—271noSylow
132C2×C3×C9#12411yescharacteristic
133C2×C3×C9#13711yescharacteristic
134C2×C3×C9#13711yescharacteristic
135C2×C3×Q8—31no—
136C2×C3×Q8—91no—
137C2×Dic6—31no—
138C2×Dic6C2×D911yescharacteristic
139C2×C3×C8—21no—
140C2×C3×C8C2×D911yescharacteristic
141C2×C3×C8D3211yescharacteristic
142C3×C4⋊C4—31no—
143C3×C4⋊C4—91no—
144C4⋊Dic3—31no—
145C4⋊Dic3—181no—
146C4⋊Dic3—181no—
147C4⋊Dic3C2×D911yescharacteristic
148Dic3⋊C4—181no—
149Dic3⋊C4—181no—
150Dic3⋊C4—181no—
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154C4×Dic3—91no—
155C4×Dic3—91no—
156C4×Dic3—181no—
157C2×C3⋊C8—91no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—181no—
160C22×C32D3×D411yescharacteristic
161C32×C4#23511yescharacteristic
162C32×C4#23711yescharacteristic
163C3⋊Dic3—61no—
164C3⋊Dic3—61no—
165C3×Dic3—41no—
166C3×Dic3—41no—
167C3×Dic3—121no—
168C3×Dic3—121no—
169C22×C9—21no—
170C22×C9D3×D411yescharacteristic
171C4×C9—21no—
172C4×C9—21no—
173C4×C9—121no—
174C4×C9—121no—
175C4×C9#23511yescharacteristic
176C4×C9#23711yescharacteristic
177Dic9—41no—
178Dic9—41no—
179Dic9—61no—
180Dic9—61no—
181Dic9—121no—
182Dic9—121no—
183C4⋊Q8—271no—
184#125—271no—
185#103—31no—
186#103—91no—
187#103—271no—
188#103—271no—
189#97—271no—
190C3×C9#49111yescharacteristic, complemented, Sylow
191C3×Q8—61no—
192C3×Q8—61no—
193C3×Q8—181no—
194C3×Q8—181no—
195C2×C3×C4—21no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200C2×C3×C4C22×D911yescharacteristic
201C2×C3×C4C2×D3211yescharacteristic
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—91no—
207C2×Dic3—91no—
208C2×Dic3—91no—
209C2×Dic3—91no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212Dic6—21no—
213Dic6—21no—
214Dic6—61no—
215Dic6—61no—
216C3×C8—22no—
217C3×C8—21no—
218C3×C8—21no—
219C3⋊C8—182no—
220C3⋊C8—181no—
221C3⋊C8—181no—
222C2×C32#88511yescharacteristic
223C2×C32#91511yescharacteristic
224C2×C32#91811yescharacteristic
225C2×C9—21no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9#88511yescharacteristic
229C2×C9#91511yescharacteristic
230C2×C9#91811yescharacteristic
231C2×Q8—31no—
232C2×Q8—91no—
233C2×C8—271no—
234C2×C8D3×D911yescharacteristic
235C4⋊C4—31no—
236C4⋊C4—91no—
237C4⋊C4—541no—
238C4⋊C4—541no—
239C4⋊C4—541no—
240C4⋊C4—541no—
241C42—271no—
242C22×C3—21no—
243C22×C3D4×D911yescharacteristic
244C22×C3#380211yescharacteristic
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—121no—
248C3×C4—121no—
249C3×C4—361no—
250C3×C4—361no—
251C3×C4#369711yescharacteristic
252C3×C4#369911yescharacteristic
253C3×C4#379811yescharacteristic
254C3×C4#380511yescharacteristic
255Dic3—41no—
256Dic3—41no—
257Dic3—121no—
258Dic3—121no—
259Dic3—181no—
260Dic3—181no—
261Dic3—181no—
262Dic3—181no—
263Dic3—361no—
264Dic3—361no—
265C32#479711yescharacteristic
266C9—21no—
267C9#479711yescharacteristic, complemented
268Q8—61no—
269Q8—61no—
270Q8—181no—
271Q8—181no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—181no—
275C2×C4—181no—
276C2×C4—271no—
277C2×C4—271no—
278C2×C4C2×D3×D911yescharacteristic
279C8—21no—
280C8—541no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3#6344711yescharacteristic
285C2×C3#6347711yescharacteristic
286C2×C3#6348011yescharacteristic
287C2×C3#6381111yescharacteristic
288C2×C3#6381111yescharacteristic
289C2×C3#6391511yescharacteristic
290C22#8864611yescharacteristic
291C4—121no—
292C4—121no—
293C4—361no—
294C4—361no—
295C4—541no—
296C4—541no—
297C4#8864011yescharacteristic
298C4#8866311yescharacteristic
299C3—21no—
300C3#1058847711yescharacteristic, complemented
301C3#1059090411yescharacteristic
302C2#1172082511yescharacteristic
303C2#1172083211yescharacteristic
304C2#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#491
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
190. C3×C9#491semidirect
267. C9#4797semidirect
300. C3#10588477semidirect

Maximal subgroups

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

subgroupindex
#117209372
#117209302
#117205382
C9×#8162
C3×#633762
#117202972
#117202682
#105909043
#105884773