Finite Group Databasebeta

← #13137558

Subgroups — #13137558

Order 1728.

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

normal subgroups only

This group has 340 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
1#11721743C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic
3C9×#807C211yesmaximal, characteristic
4C3×#63367C211yesmaximal, characteristic
5#11720277C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic
7#11720269C211yesmaximal, characteristic
8#10590434—31nomaximal
9#10588557—31nomaximal
10C2×#6226C2211yescharacteristic
11C3×C9×C4⋊C4C2211yescharacteristic
12#88534—21no—
13C4×C3⋊Dic9—21no—
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#64096—31no—
22#63872—31no—
23C3×#808—31no—
24C3×#807D311yescharacteristic
25#63572—31no—
26#63567—31no—
27#63567—31no—
28#63452—31no—
29#63449—31no—
30#63449—61no—
31C9×#107—31no—
32#63376—31no—
33Dic9⋊C8—31no—
34#63368—31no—
35#63367D311yescharacteristic
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C3⋊Dic9D411yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41#6226—21no—
42#6226Q812yes—
43C2×C9×Dic3—21no—
44C2×C3×Dic9—21no—
45C9×C3⋊C8—21no—
46C3×C9⋊C8—21no—
47#4877—91no—
48#4857—91no—
49C2×#634—31no—
50C32×C4⋊C4C2×D311yescharacteristic
51#3750—61no—
52C4×C3⋊Dic3—31no—
53C4×C3⋊Dic3—61no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8C2×D311yescharacteristic
58Dic3⋊Dic3—61no—
59Dic3⋊Dic3—61no—
60C2×Dic18—31no—
61C2×Dic18—61no—
62C2×C8×C9—31no—
63C9×C4⋊C4—21no—
64C9×C4⋊C4—31no—
65C9×C4⋊C4C2×D311yescharacteristic
66C4⋊Dic9—61no—
67C4⋊Dic9C2×D311yescharacteristic
68Dic9⋊C4—61no—
69Dic9⋊C4—61no—
70Dic9⋊C4—62no—
71C4×Dic9—31no—
72C4×Dic9—61no—
73C4×Dic9—61no—
74C4×Dic9—62no—
75C2×C9⋊C8C2×D311yescharacteristic
76C22×C3×C9C2×D411yescharacteristic
77C3×C4×C9—41no—
78C3×C4×C9C2×Q811yescharacteristic
79C3×C4×C9#4111yescharacteristic
80C3⋊Dic9—22no—
81C3⋊Dic9—21no—
82C3⋊Dic9—21no—
83C9×Dic3—41no—
84C3×Dic9—41no—
85#890—91no—
86#890—91no—
87#887—91no—
88#887—91no—
89#887—181no—
90C3×#107—31no—
91C3×#106—91no—
92#816—91no—
93#816—91no—
94Dic3⋊C8—91no—
95Dic3⋊C8—91no—
96#810—91no—
97#810—91no—
98#808—31no—
99#807D911yescharacteristic
100C2×C32×C4—21no—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—61no—
105#634—32no—
106#634—61no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—61no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—61no—
111C2×C4×C9—22no—
112C2×C4×C9—21no—
113C2×C4×C9—21no—
114C2×C4×C9—61no—
115C2×C4×C9C22×D311yescharacteristic
116C2×Dic9—21no—
117C2×Dic9—31no—
118C2×Dic9—31no—
119C2×Dic9—61no—
120C2×Dic9—61no—
121C2×Dic9—62no—
122C2×Dic9—61no—
123Dic18—32no—
124Dic18—62no—
125Dic18—62no—
126Dic18—61no—
127C8×C9—61no—
128C9⋊C8—21no—
129#479—271noSylow
130C2×C3×C9C22⋊Q811yescharacteristic
131C2×C3×C9#13511yescharacteristic
132C2×C3×C9#13711yescharacteristic
133C2×Dic6—91no—
134C2×Dic6—91no—
135C2×Dic6—181no—
136C2×C3×C8—31no—
137C2×C3×C8—91no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4C2×D911yescharacteristic
142C3×C4⋊C4D3211yescharacteristic
143C4⋊Dic3—31no—
144C4⋊Dic3—181no—
145C4⋊Dic3—181no—
146C4⋊Dic3C2×D911yescharacteristic
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149Dic3⋊C4—181no—
150Dic3⋊C4—181no—
151Dic3⋊C4—182no—
152C4×Dic3—91no—
153C4×Dic3—91no—
154C4×Dic3—181no—
155C4×Dic3—181no—
156C4×Dic3—181no—
157C4×Dic3—182no—
158C2×C3⋊C8—31no—
159C2×C3⋊C8C2×D911yescharacteristic
160C22×C32D3×D411yescharacteristic
161C32×C4—41no—
162C32×C4#23511yescharacteristic
163C32×C4D3×Q811yescharacteristic
164C3⋊Dic3—62no—
165C3⋊Dic3—61no—
166C3⋊Dic3—61no—
167C3×Dic3—41no—
168C3×Dic3—121no—
169C22×C9—21no—
170C22×C9D3×D411yescharacteristic
171C4×C9—21no—
172C4×C9—21no—
173C4×C9—42no—
174C4×C9—41no—
175C4×C9—121no—
176C4×C9#23511yescharacteristic
177C4×C9D3×Q811yescharacteristic
178Dic9—41no—
179Dic9—64no—
180Dic9—62no—
181Dic9—62no—
182Dic9—61no—
183Dic9—61no—
184Dic9—121no—
185#125—271no—
186C4×Q8—271no—
187#107—31no—
188#106—91no—
189C4⋊C8—271no—
190#103—271no—
191#103—271no—
192C3×C9#47911yescharacteristic, complemented, Sylow
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—61no—
198C2×C3×C4—181no—
199C2×C3×C4C22×D911yescharacteristic
200C2×C3×C4C2×D3211yescharacteristic
201C2×Dic3—21no—
202C2×Dic3—61no—
203C2×Dic3—91no—
204C2×Dic3—91no—
205C2×Dic3—91no—
206C2×Dic3—91no—
207C2×Dic3—181no—
208C2×Dic3—181no—
209C2×Dic3—182no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212Dic6—92no—
213Dic6—92no—
214Dic6—182no—
215Dic6—182no—
216Dic6—181no—
217Dic6—181no—
218C3×C8—61no—
219C3×C8—181no—
220C3⋊C8—21no—
221C3⋊C8—61no—
222C2×C32#89611yescharacteristic
223C2×C32#91211yescharacteristic
224C2×C32#91811yescharacteristic
225C2×C9—21no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9#89611yescharacteristic
229C2×C9#91511yescharacteristic
230C2×C9#91611yescharacteristic
231C2×Q8—271no—
232C2×C8—31no—
233C2×C8—91no—
234C4⋊C4—31no—
235C4⋊C4—91no—
236C4⋊C4—541no—
237C4⋊C4—541no—
238C4⋊C4—541no—
239C4⋊C4D3×D911yescharacteristic
240C42—271no—
241C42—541no—
242C22×C3—21no—
243C22×C3D4×D911yescharacteristic
244C22×C3#381111yescharacteristic
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—42no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—121no—
251C3×C4—361no—
252C3×C4#369711yescharacteristic
253C3×C4Q8×D911yescharacteristic
254C3×C4#379711yescharacteristic
255C3×C4#379811yescharacteristic
256Dic3—41no—
257Dic3—121no—
258Dic3—184no—
259Dic3—182no—
260Dic3—182no—
261Dic3—181no—
262Dic3—181no—
263Dic3—182no—
264Dic3—181no—
265Dic3—181no—
266Dic3—361no—
267C32#487711yescharacteristic
268C9—21no—
269C9#485711yescharacteristic, complemented
270Q8—272no—
271Q8—541no—
272C2×C4—21no—
273C2×C4—61no—
274C2×C4—181no—
275C2×C4—271no—
276C2×C4—271no—
277C2×C4—541no—
278C2×C4C2×D3×D911yescharacteristic
279C8—61no—
280C8—181no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3#6345811yescharacteristic
285C2×C3#6347411yescharacteristic
286C2×C3#6348011yescharacteristic
287C2×C3#6386511yescharacteristic
288C2×C3#6393611yescharacteristic
289C2×C3#6395111yescharacteristic
290C22#8866811yescharacteristic
291C4—41no—
292C4—121no—
293C4—361no—
294C4—541no—
295C4—541no—
296C4—542no—
297C4(C3⋊D9)⋊Q811yescharacteristic
298C4#8864011yescharacteristic
299C3—21no—
300C3#1058855711yescharacteristic, complemented
301C3#1059043411yescharacteristic
302C2#1172092311yescharacteristic
303C2#1172117811yescharacteristic
304C2#1172122511yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socle, prime radicalC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#479
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
192. C3×C9#479semidirect
269. C9#4857semidirect
300. C3#10588557semidirect

Maximal subgroups

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

subgroupindex
#117217432
#117209372
C9×#8072
C3×#633672
#117202772
#117202732
#117202692
#105904343
#105885573