Finite Group Databasebeta

← #13137529

Subgroups — #13137529

Order 1728.

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

normal subgroups only

This group has 326 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#11721746C211yesmaximal, characteristic
2#11720910C211yesmaximal, characteristic
3C3×#63368C211yesmaximal, characteristic
4C9×#807C211yesmaximal, characteristic
5#11720276C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic
7#11720264C211yesmaximal, characteristic
8#10590494—31nomaximal
9#10588557—31nomaximal
10C2×C3×Dic18C2211yescharacteristic
11C3×C9×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊Dic9)—21no—
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14#88534—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#64099—31no—
22#63843—31no—
23C3×#810—31no—
24C3×#807D311yescharacteristic
25#63573—31no—
26#63572—31no—
27#63566—31no—
28#63452—31no—
29#63452—61no—
30#63449—31no—
31C9×#107—31no—
32#63376—31no—
33Dic9⋊C8—31no—
34#63368D311yescharacteristic
35#63367—31no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C3⋊Dic9D411yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C3×Dic18—21no—
44C3×Dic18Q812yes—
45C9×C3⋊C8—21no—
46C3×C9⋊C8—21no—
47#4877—91no—
48#4799—91no—
49C2×C3×Dic6—31no—
50C32×C4⋊C4C2×D311yescharacteristic
51C4⋊(C3⋊Dic3)—31no—
52C4⋊(C3⋊Dic3)—61no—
53#3750—61no—
54C4×C3⋊Dic3—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8C2×D311yescharacteristic
58Dic3⋊Dic3—61no—
59Dic32—61no—
60C2×Dic18C2×D311yescharacteristic
61C2×C8×C9—31no—
62C9×C4⋊C4—21no—
63C9×C4⋊C4—31no—
64C9×C4⋊C4C2×D311yescharacteristic
65C4⋊Dic9—31no—
66C4⋊Dic9—61no—
67C4⋊Dic9—61no—
68C4⋊Dic9—62no—
69Dic9⋊C4—61no—
70Dic9⋊C4—61no—
71Dic9⋊C4—62no—
72C4×Dic9—31no—
73C4×Dic9—61no—
74C4×Dic9—61no—
75C2×C9⋊C8C2×D311yescharacteristic
76C22×C3×C9C2×D411yescharacteristic
77C3×C4×C9—41no—
78C3×C4×C9C2×Q811yescharacteristic
79C3×C4×C9#4111yescharacteristic
80C3⋊Dic9—21no—
81C3⋊Dic9—21no—
82C3⋊Dic9—41no—
83C9×Dic3—41no—
84C3×Dic9—22no—
85Q8×Dic3—91no—
86#890—91no—
87#890—91no—
88#890—181no—
89#887—91no—
90C3×#107—31no—
91C3×#103—91no—
92#835—91no—
93#817—91no—
94#816—91no—
95Dic3⋊C8—91no—
96Dic3⋊C8—91no—
97#810—31no—
98#808—91no—
99#807D911yescharacteristic
100C2×C32×C4—21no—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—61no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—61no—
107C3×Dic6—32no—
108C3×Dic6—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—21no—
124Dic18Dic612yes—
125C8×C9—61no—
126C9⋊C8—21no—
127#479—271noSylow
128C2×C3×C9C22⋊Q811yescharacteristic
129C2×C3×C9#13511yescharacteristic
130C2×C3×C9#13711yescharacteristic
131C2×C3×Q8—91no—
132C2×Dic6—31no—
133C2×C3×C8—31no—
134C2×C3×C8—91no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4C2×D911yescharacteristic
138C3×C4⋊C4D3211yescharacteristic
139C4⋊Dic3—91no—
140C4⋊Dic3—91no—
141C4⋊Dic3—181no—
142C4⋊Dic3—181no—
143C4⋊Dic3—181no—
144C4⋊Dic3—181no—
145C4⋊Dic3—182no—
146C4⋊Dic3C2×D911yescharacteristic
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149Dic3⋊C4—181no—
150Dic3⋊C4—182no—
151C4×Dic3—91no—
152C4×Dic3—91no—
153C4×Dic3—181no—
154C4×Dic3—181no—
155C4×Dic3—181no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8C2×D911yescharacteristic
158C22×C32D3×D411yescharacteristic
159C32×C4—41no—
160C32×C4#23511yescharacteristic
161C32×C4D3×Q811yescharacteristic
162C3⋊Dic3—61no—
163C3⋊Dic3—61no—
164C3⋊Dic3—121no—
165C3×Dic3—41no—
166C3×Dic3—62no—
167C22×C9—21no—
168C22×C9D3×D411yescharacteristic
169C4×C9—21no—
170C4×C9—21no—
171C4×C9—42no—
172C4×C9—41no—
173C4×C9—121no—
174C4×C9C2×Dic611yescharacteristic
175C4×C9#23711yescharacteristic
176Dic9—22no—
177Dic9—61no—
178Dic9—61no—
179Dic9—121no—
180Dic9—121no—
181Dic9—122no—
182Dic9—121no—
183#125—271no—
184C4×Q8—271no—
185#107—31no—
186#106—271no—
187C4⋊C8—271no—
188#103—91no—
189#103—271no—
190C3×C9#47911yescharacteristic, complemented, Sylow
191C3×Q8—92no—
192C3×Q8—181no—
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—32no—
213Dic6—61no—
214C3×C8—61no—
215C3×C8—181no—
216C3⋊C8—21no—
217C3⋊C8—61no—
218C2×C32#89611yescharacteristic
219C2×C32#91211yescharacteristic
220C2×C32#91811yescharacteristic
221C2×C9—21no—
222C2×C9—21no—
223C2×C9—21no—
224C2×C9#87811yescharacteristic
225C2×C9#91611yescharacteristic
226C2×C9#91811yescharacteristic
227C2×Q8—91no—
228C2×C8—31no—
229C2×C8—91no—
230C4⋊C4—31no—
231C4⋊C4—271no—
232C4⋊C4—541no—
233C4⋊C4—541no—
234C4⋊C4—541no—
235C4⋊C4D3×D911yescharacteristic
236C42—271no—
237C42—541no—
238C22×C3—21no—
239C22×C3D4×D911yescharacteristic
240C22×C3#381111yescharacteristic
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—42no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—121no—
247C3×C4—182no—
248C3×C4#369711yescharacteristic
249C3×C4Q8×D911yescharacteristic
250C3×C4D3×Dic611yescharacteristic
251C3×C4#379511yescharacteristic
252Dic3—41no—
253Dic3—62no—
254Dic3—181no—
255Dic3—181no—
256Dic3—181no—
257Dic3—181no—
258Dic3—361no—
259Dic3—361no—
260Dic3—362no—
261Dic3—361no—
262Dic3—361no—
263C32#487711yescharacteristic
264C9—21no—
265C9#479911yescharacteristic, complemented
266Q8—92no—
267Q8—181no—
268C2×C4—21no—
269C2×C4—61no—
270C2×C4—181no—
271C2×C4—271no—
272C2×C4—271no—
273C2×C4—541no—
274C2×C4C2×D3×D911yescharacteristic
275C8—61no—
276C8—181no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3#6345811yescharacteristic
281C2×C3#6347411yescharacteristic
282C2×C3#6348011yescharacteristic
283C2×C3#6389611yescharacteristic
284C2×C3#6393611yescharacteristic
285C2×C3#6394511yescharacteristic
286C22#8866811yescharacteristic
287C4—41no—
288C4—121no—
289C4—182no—
290C4—541no—
291C4—541no—
292C4—1081no—
293C4D9×Dic611yescharacteristic
294C4#8864111yescharacteristic
295C3—21no—
296C3#1058855711yescharacteristic, complemented
297C3#1059049411yescharacteristic
298C2#1172099611yescharacteristic
299C2#1172117811yescharacteristic
300C2#1172122111yescharacteristic
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
190. C3×C9#479semidirect
265. C9#4799semidirect
296. C3#10588557semidirect

Maximal subgroups

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

subgroupindex
#117217462
#117209102
C3×#633682
C9×#8072
#117202762
#117202732
#117202642
#105904943
#105885573