Finite Group Databasebeta

← #13137512

Subgroups — #13137512

Order 1728.

356 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 85 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#11720993C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic
4C3×#63367C211yesmaximal, characteristic, complemented
5#11720267C211yesmaximal, characteristic
6#11720263C211yesmaximal, characteristic, complemented
7Dic3⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
8#10590470—31nomaximal
9#10588747—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×(C3⋊D9)⋊C4—21no—
12C3×C9×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3—21no—
15C9×C4⋊Dic3C2211yescharacteristic
16C9×Dic3⋊C4—21no—
17C4×C9×Dic3C2211yescharacteristic
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88445—21no—
21C3×#890D311yescharacteristic
22#63894—31no—
23#63636—31no—
24C3×#808—31no—
25#63576—31no—
26Dic3⋊(C3⋊C8)—31no—
27#63564—31no—
28C9×#125—31no—
29C4×D36—31no—
30#63369—31no—
31#63369—31no—
32#63369—61no—
33#63367D311yescharacteristic, complemented
34#63366—31no—
35C4⋊(C9⋊C8)—31no—
36C22×C3⋊D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C4⋊(C3⋊D9)—21no—
40C4⋊(C3⋊D9)Q812yes—
41C2×C9×Dic3—21no—
42C2×C9×Dic3D411yescharacteristic
43C2×C9×Dic3D411yescharacteristic
44C2×C3×Dic9—21no—
45(C3⋊D9)⋊C4—42no—
46C3⋊(C9⋊C8)—21no—
47C3×C9⋊C8—21no—
48#5066—91no—
49#4871—91no—
50C2×C4⋊(C3⋊D3)—31no—
51C2×#624—61no—
52C32×C4⋊C4C2×D311yescharacteristic
53C2×C3⋊(C3⋊C8)—31no—
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C3×Dic3⋊C4—21no—
58C3×C4×Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60#3722—61no—
61C2×D36—31no—
62C2×D36—61no—
63C2×C4×D9—61no—
64C9×C4⋊C4—21no—
65C9×C4⋊C4—31no—
66C9×C4⋊C4—61no—
67C9×C4⋊C4—61no—
68C9×C4⋊C4C2×D311yescharacteristic
69C42×C9—31no—
70(C2×D9)⋊C4—61no—
71C4⋊Dic9C2×D311yescharacteristic
72C2×C9⋊C8—31no—
73C2×C9⋊C8—61no—
74C2×C9⋊C8C2×D311yescharacteristic
75C22×C3×C9C2×D411yescharacteristic
76C2×C3⋊D9—22no—
77C2×C3⋊D9—41no—
78C2×C3⋊D9—41no—
79C3×C4×C9—41no—
80C3×C4×C9C2×Q811yescharacteristic
81C3×C4×C9#4111yescharacteristic
82C9×Dic3—21no—
83C9×Dic3—21no—
84C9×Dic3—41no—
85C3×Dic9—41no—
86C3×#125—31no—
87#893—91no—
88#890D911yescharacteristic, complemented
89C4×D12—91no—
90C3×#106—91no—
91#821—91no—
92Dic3⋊C8—91no—
93#809—91no—
94#809—91no—
95#809—91no—
96#809—181no—
97#808—31no—
98#807—91no—
99#807—91no—
100C4⋊(C3⋊C8)—91no—
101C22×C3⋊D3—61no—
102C2×C32×C4—21no—
103C2×C32×C4C22×D311yescharacteristic
104C4⋊(C3⋊D3)—32no—
105C4⋊(C3⋊D3)—61no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3#6711yescharacteristic
109C2×C3×Dic3#6711yescharacteristic
110#624—122no—
111C22×D9—61no—
112C22×D9—121no—
113C3⋊(C3⋊C8)—61no—
114C3×C3⋊C8—61no—
115C2×C4×C9—21no—
116C2×C4×C9—21no—
117C2×C4×C9—31no—
118C2×C4×C9—31no—
119C2×C4×C9—41no—
120C2×C4×C9—61no—
121C2×C4×C9C22×D311yescharacteristic
122C2×Dic9—21no—
123D36—32no—
124D36—62no—
125D36—61no—
126D36—121no—
127C4×D9—122no—
128C9⋊C8—21no—
129C9⋊C8—61no—
130C9⋊C8—121no—
131#478—271noSylow
132C2×C3×C9C22⋊Q811yescharacteristic
133C2×C3×C9#13511yescharacteristic
134C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
135C3⋊D9—42no—
136C2×D12—91no—
137C2×D12—91no—
138C2×D12—181no—
139C2×C4×D3—181no—
140C2×C4×D3—181no—
141C2×C3×C8—91no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—61no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4C2×D911yescharacteristic
148C3×C4⋊C4D3211yescharacteristic
149C3×C42—31no—
150(C2×D3)⋊C4—181no—
151(C2×D3)⋊C4—181no—
152C4⋊Dic3—21no—
153C4⋊Dic3—31no—
154C4⋊Dic3C2×D911yescharacteristic
155Dic3⋊C4—21no—
156C4×Dic3C2×D911yescharacteristic
157C2×C3⋊C8—31no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—181no—
161C22×C32C2×#6711yescharacteristic
162C2×C3⋊D3—62no—
163C2×C3⋊D3—121no—
164C2×C3⋊D3—121no—
165C32×C4—41no—
166C32×C4D3×Q811yescharacteristic
167C32×C4#23911yescharacteristic
168C3×Dic3—21no—
169C3×Dic3—21no—
170C3×Dic3—41no—
171C3×Dic3—121no—
172C22×C9—21no—
173C22×C9D3×D411yescharacteristic
174C2×D9—62no—
175C2×D9—122no—
176C2×D9—121no—
177C2×D9—121no—
178C2×D9—241no—
179C2×D9—241no—
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×C9D3×Q811yescharacteristic
189Dic9—41no—
190#125—31no—
191C4×D4—271no—
192#107—271no—
193#106—91no—
194C4⋊C8—271no—
195#102—271no—
196#102—271no—
197C3×C9#47811yescharacteristic, complemented, Sylow
198C22×D3—181no—
199C22×D3—181no—
200C22×D3—361no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—41no—
207C2×C3×C4—61no—
208C2×C3×C4—181no—
209C2×C3×C4C22×D911yescharacteristic
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—21no—
212C2×Dic3—61no—
213C2×Dic3#61111yescharacteristic
214C2×Dic3#61111yescharacteristic
215D12—92no—
216D12—92no—
217D12—182no—
218D12—181no—
219D12—181no—
220D12—361no—
221C4×D3—362no—
222C4×D3—362no—
223C3×C8—181no—
224C3⋊C8—61no—
225C3⋊C8—181no—
226C3⋊C8—181no—
227C3⋊C8—361no—
228C2×C32#94611yescharacteristic
229C2×C32#94911yescharacteristic
230C2×C32#95011yescharacteristic
231C3⋊D3—122no—
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9#89611yescharacteristic
236C2×C9#91411yescharacteristic
237C2×C9#91611yescharacteristic
238D9—122no—
239D9—242no—
240C2×D4—271no—
241C22×C4—541no—
242C2×C8—91no—
243C2×C8—271no—
244C4⋊C4—31no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—91no—
248C4⋊C4D3×D911yescharacteristic
249C22⋊C4—541no—
250C42—31no—
251C22×C3—21no—
252C22×C3C2×#61111yescharacteristic
253C22×C3D3×#6711yescharacteristic
254C2×D3—182no—
255C2×D3—182no—
256C2×D3—362no—
257C2×D3—361no—
258C2×D3—361no—
259C2×D3—361no—
260C2×D3—361no—
261C2×D3—721no—
262C2×D3—721no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—81no—
270C3×C4—121no—
271C3×C4—361no—
272C3×C4Q8×D911yescharacteristic
273C3×C4#370111yescharacteristic
274C3×C4#379711yescharacteristic
275C3×C4#379911yescharacteristic
276Dic3—21no—
277Dic3—21no—
278Dic3—41no—
279Dic3—121no—
280C32#506611yescharacteristic
281C9—21no—
282C9#487111yescharacteristic, complemented
283C23—541no—
284D4—272no—
285D4—541no—
286C2×C4—21no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—61no—
290C2×C4—181no—
291C2×C4—1082no—
292C2×C4C2×D3×D911yescharacteristic
293C8—181no—
294C8—541no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3#6350811yescharacteristic
299C2×C3#6351211yescharacteristic
300C2×C3#6351311yescharacteristic
301C2×C3#6388811yescharacteristic
302C2×C3#6393511yescharacteristic
303C2×C3#6393911yescharacteristic
304D3—362no—
305D3—362no—
306D3—722no—
307C22—542no—
308C22—1081no—
309C22—1081no—
310C22D3×#61111yescharacteristic
311C4—41no—
312C4—61no—
313C4—61no—
314C4—121no—
315C4—361no—
316C4#8863511yescharacteristic
317C4(C3⋊D9)⋊Q811yescharacteristic
318C3—21no—
319C3#1058874711yescharacteristic, complemented
320C3#1059047011yescharacteristic
321C2—1082no—
322C2#1172098611yescharacteristic
323C2#1172117111yescharacteristic
324C2#1172120511yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#478
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
4. C3×#63367C2semidirect
6. #11720263C2semidirect
7. Dic3⋊(C9⋊C8)C2semidirect
33. #63367D3semidirect
88. #890D9semidirect
197. C3×C9#478semidirect
282. C9#4871semidirect
319. C3#10588747semidirect

Maximal subgroups

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

subgroupindex
C9×#8902
#117209932
#117205282
C3×#633672
#117202672
#117202632
Dic3⋊(C9⋊C8)2
#105904703
#105887473