Finite Group Databasebeta

← #13137520

Subgroups — #13137520

Order 1728.

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

normal subgroups only

This group has 330 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
1C9×#890C211yesmaximal, characteristic, complemented
2#11720948C211yesmaximal, characteristic
3#11720527C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic
5#11720269C211yesmaximal, characteristic, complemented
6#11720266C211yesmaximal, characteristic
7Dic3×C9⋊C8C211yesmaximal, characteristic, complemented
8#10590533—31nomaximal
9#10588749—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3—21no—
15C9×C4⋊Dic3C2211yescharacteristic
16C9×Dic3⋊C4—21no—
17C4×C9×Dic3C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20(C2×D9)⋊Dic3—21no—
21C3×#890D311yescharacteristic
22#63860—31no—
23#63637—31no—
24C3×#809—31no—
25#63567—31no—
26#63562—31no—
27Dic3×C3⋊C8—31no—
28C9×#125—31no—
29#63437—31no—
30#63369—31no—
31#63369D311yescharacteristic
32#63368—31no—
33#63368—31no—
34#63368—61no—
35C4×C9⋊C8—31no—
36C22×C3×D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40#6226—21no—
41#6226—21no—
42C2×C9×Dic3—21no—
43C2×C9×Dic3D411yescharacteristic
44C2×C9×Dic3D411yescharacteristic
45C3×D36—21no—
46C3×D36—21no—
47C3⋊(C9⋊C8)—21no—
48C3×C9⋊C8—21no—
49#5068—91no—
50#4768—91no—
51C2×#634—31no—
52C2×C3×D12—31no—
53C32×C4⋊C4C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)—31no—
55C3×C4⋊Dic3—21no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C3×Dic3⋊C4—21no—
58C3×C4×Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60(C2×D3)⋊Dic3—61no—
61(C2×D3)⋊Dic3—61no—
62C2×D36C2×D311yescharacteristic
63C2×Dic18—31no—
64C2×Dic18—61no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4—31no—
67C9×C4⋊C4—61no—
68C9×C4⋊C4—61no—
69C9×C4⋊C4C2×D311yescharacteristic
70C42×C9—31no—
71(C2×D9)⋊C4—61no—
72(C2×D9)⋊C4—61no—
73C2×C9⋊C8—31no—
74C2×C9⋊C8—61no—
75C2×C9⋊C8C2×D311yescharacteristic
76C22×C3×C9C2×D411yescharacteristic
77C2×C3×D9—41no—
78C2×C3×D9—41no—
79C2×C3×D9—41no—
80C3×C4×C9—41no—
81C3×C4×C9#4111yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—41no—
84C9×Dic3—21no—
85C9×Dic3—21no—
86C9×Dic3—41no—
87C3×#125—31no—
88#936—91no—
89#890D911yescharacteristic, complemented
90#875—91no—
91C3×#102—91no—
92#832—91no—
93#816—91no—
94C8×Dic3—91no—
95#810—91no—
96#810—91no—
97#810—91no—
98#810—181no—
99#809—31no—
100#809—91no—
101C4×C3⋊C8—91no—
102C22×C3×D3—61no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—61no—
106#634—61no—
107#634—61no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3#6711yescharacteristic
110C2×C3×Dic3#6711yescharacteristic
111C3×D12—61no—
112C3×D12—61no—
113C22×D9—21no—
114C3⋊(C3⋊C8)—61no—
115C3×C3⋊C8—61no—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—31no—
119C2×C4×C9—31no—
120C2×C4×C9—41no—
121C2×C4×C9—61no—
122C2×C4×C9C22×D311yescharacteristic
123C2×Dic9—61no—
124C2×Dic9—121no—
125D36—21no—
126D36—21no—
127Dic18—62no—
128Dic18—61no—
129Dic18—61no—
130Dic18—121no—
131C9⋊C8—21no—
132C9⋊C8—61no—
133C9⋊C8—121no—
134#488—271noSylow
135C2×C3×C9#12411yescharacteristic
136C2×C3×C9#13511yescharacteristic
137C2×C3×C9#13511yescharacteristic
138C3×D9—81no—
139C2×C3×D4—91no—
140C2×D12—31no—
141C2×Dic6—91no—
142C2×Dic6—91no—
143C2×Dic6—181no—
144C2×C3×C8—91no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—61no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4C2×D911yescharacteristic
150C3×C4⋊C4D3211yescharacteristic
151C3×C42—31no—
152#217—181no—
153#217—181no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—181no—
156C4⋊Dic3—21no—
157C4⋊Dic3C2×D911yescharacteristic
158Dic3⋊C4—21no—
159C4×Dic3C2×D911yescharacteristic
160C2×C3⋊C8—31no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—181no—
164C22×C32C2×#6711yescharacteristic
165C2×C3×D3—121no—
166C2×C3×D3—121no—
167C2×C3×D3—121no—
168C32×C4—41no—
169C32×C4#23911yescharacteristic
170C32×C4#23911yescharacteristic
171C3⋊Dic3—121no—
172C3×Dic3—21no—
173C3×Dic3—21no—
174C3×Dic3—41no—
175C22×C9—21no—
176C22×C9D3×D411yescharacteristic
177C2×D9—41no—
178C2×D9—41no—
179C2×D9—41no—
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—41no—
183C4×C9—61no—
184C4×C9—61no—
185C4×C9—81no—
186C4×C9—121no—
187C4×C9#23511yescharacteristic
188C4×C9#23711yescharacteristic
189Dic9—122no—
190Dic9—121no—
191#125—31no—
192#124—271no—
193#103—271no—
194#103—271no—
195#102—91no—
196#102—271no—
197C4×C8—271no—
198C3×C9#48811yescharacteristic, complemented, Sylow
199C23×C3—181no—
200C22×D3—61no—
201C3×D4—181no—
202C3×D4—181no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—41no—
209C2×C3×C4—61no—
210C2×C3×C4C22×D911yescharacteristic
211C2×C3×C4C2×D3211yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—181no—
214C2×Dic3—181no—
215C2×Dic3—361no—
216C2×Dic3#61111yescharacteristic
217C2×Dic3#61111yescharacteristic
218D12—61no—
219D12—61no—
220Dic6—182no—
221Dic6—181no—
222Dic6—181no—
223Dic6—181no—
224Dic6—181no—
225Dic6—361no—
226C3×C8—181no—
227C3⋊C8—61no—
228C3⋊C8—181no—
229C3⋊C8—181no—
230C3⋊C8—361no—
231C2×C32#94711yescharacteristic
232C2×C32#95011yescharacteristic
233C2×C32#95011yescharacteristic
234C3×D3—241no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9#88511yescharacteristic
239C2×C9#91211yescharacteristic
240C2×C9#91611yescharacteristic
241D9—81no—
242C2×Q8—271no—
243C2×D4—91no—
244C2×C8—91no—
245C2×C8—271no—
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4D3×D911yescharacteristic
250C22⋊C4—541no—
251C22⋊C4—541no—
252C42—31no—
253C22×C3—21no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3—361no—
257C22×C3C2×#61111yescharacteristic
258C22×C3D3×#6711yescharacteristic
259C2×D3—121no—
260C2×D3—121no—
261C2×D3—121no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—81no—
269C3×C4—121no—
270C3×C4#370111yescharacteristic
271C3×C4#370111yescharacteristic
272C3×C4#379611yescharacteristic
273C3×C4#379911yescharacteristic
274Dic3—21no—
275Dic3—21no—
276Dic3—41no—
277Dic3—362no—
278Dic3—361no—
279Dic3—361no—
280C32#506811yescharacteristic
281C9—21no—
282C9#476811yescharacteristic, complemented
283C23—181no—
284Q8—541no—
285Q8—541no—
286D4—181no—
287D4—181no—
288C2×C4—21no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—61no—
292C2×C4—541no—
293C2×C4C2×D3×D911yescharacteristic
294C8—181no—
295C8—541no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—721no—
300C2×C3#6350911yescharacteristic
301C2×C3#6351311yescharacteristic
302C2×C3#6351311yescharacteristic
303C2×C3#6390711yescharacteristic
304C2×C3#6393311yescharacteristic
305C2×C3#6394711yescharacteristic
306D3—241no—
307C22—361no—
308C22—361no—
309C22—361no—
310C22D3×#61111yescharacteristic
311C4—41no—
312C4—61no—
313C4—61no—
314C4—121no—
315C4—1081no—
316C4#8863411yescharacteristic
317C4#8863511yescharacteristic
318C3—21no—
319C3#1058874911yescharacteristic, complemented
320C3#1059053311yescharacteristic
321C2—721no—
322C2#1172100311yescharacteristic
323C2#1172116911yescharacteristic
324C2#1172120711yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
prime radicalC2×C3×D36C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#488
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
1. C9×#890C2semidirect
3. #11720527C2semidirect
5. #11720269C2semidirect
7. Dic3×C9⋊C8C2semidirect
89. #890D9semidirect
198. C3×C9#488semidirect
282. C9#4768semidirect
319. C3#10588749semidirect

Maximal subgroups

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

subgroupindex
C9×#8902
#117209482
#117205272
C3×#633692
#117202692
#117202662
Dic3×C9⋊C82
#105905333
#105887493