Finite Group Databasebeta

← #13144102

Subgroups — #13144102

Order 1728.

530 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 133 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#11721751C211yesmaximal, characteristic
2#11721748C211yesmaximal, characteristic, complemented
3C3×C9×#106C211yesmaximal, characteristic, complemented
4#11720543C211yesmaximal, characteristic, complemented
5#11720542C211yesmaximal, characteristic, complemented
6#11720528C211yesmaximal, characteristic, complemented
7#11720525C211yesmaximal, characteristic, complemented
8#10592565—31nomaximal
9#10588546—33nomaximal
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16#88536—21no—
17#88536—21no—
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C4×C3⋊Dic9—21no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64104—31no—
22#64101—31no—
23C32×#106D311yescharacteristic
24#63648—31no—
25#63647—31no—
26#63636—31no—
27#63634—31no—
28#63457—33no—
29#63454—33no—
30C9×#106D313yescomplemented
31#63380—33no—
32(C2×D9)⋊C8—33no—
33#63369—33no—
34#63366—33no—
35C22×C3⋊D9—21no—
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—42no—
45C4×C3⋊D9—21no—
46C4×C3⋊D9—21no—
47C3×C8×C9—21no—
48C3⋊(C9⋊C8)—21no—
49#4866—93no—
50#4866—91no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4⋊(C3⋊D3)—61no—
53C2×C4×C3⋊D3—31no—
54C2×C32×C8C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57#3752—61no—
58#3752—61no—
59C4⋊(C3⋊Dic3)—31no—
60C4×C3⋊Dic3—61no—
61C2×C3⋊(C3⋊C8)—31no—
62C2×D36—33no—
63C2×D36—63no—
64C2×C4×D9—33no—
65C2×C8×C9C2×D313yes—
66C9×C4⋊C4C2×D313yes—
67C9×C4⋊C4C2×D313yes—
68(C2×D9)⋊C4—63no—
69(C2×D9)⋊C4—63no—
70C4⋊Dic9—33no—
71C4×Dic9—63no—
72C2×C9⋊C8—33no—
73C22×C3×C9C2×D411yescharacteristic
74C2×C3⋊D9—21no—
75C2×C3⋊D9—21no—
76C2×C3⋊D9—21no—
77C2×C3⋊D9—41no—
78C2×C3⋊D9—41no—
79C2×C3⋊D9—41no—
80C3×C4×C9—41no—
81C3×C4×C9—41no—
82C3×C4×C9#4111yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—21no—
85C3⋊Dic9—41no—
86#895—93no—
87#895—91no—
88#892—93no—
89#892—91no—
90C3×#106D913yescomplemented
91C3×#106C3⋊D311yescharacteristic
92#821—93no—
93#821—91no—
94(C2×D3)⋊C8—93no—
95(C2×D3)⋊C8—91no—
96#809—93no—
97#809—91no—
98#807—93no—
99#807—91no—
100C22×C3⋊D3—31no—
101C22×C3⋊D3—61no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—31no—
106C2×C3⋊Dic3—61no—
107C4⋊(C3⋊D3)—61no—
108C4⋊(C3⋊D3)—61no—
109C4⋊(C3⋊D3)—122no—
110C4×C3⋊D3—61no—
111C4×C3⋊D3—61no—
112C22×D9—33no—
113C22×D9—63no—
114C32×C8—21no—
115C3⋊(C3⋊C8)—61no—
116C2×C4×C9—23no—
117C2×C4×C9—23no—
118C2×C4×C9C22×D313yes—
119C2×Dic9—33no—
120C2×Dic9—63no—
121D36—63no—
122D36—63no—
123D36—126no—
124C4×D9—63no—
125C4×D9—63no—
126C8×C9—23no—
127C9⋊C8—63no—
128#482—271noSylow
129C2×C3×C9#12311yescharacteristic
130C2×C3×C9#13511yescharacteristic
131C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
132C3⋊D9—41no—
133C3⋊D9—81no—
134C2×D12—93no—
135C2×D12—91no—
136C2×D12—181no—
137C2×D12—183no—
138C2×C4×D3—93no—
139C2×C4×D3—91no—
140C2×C3×C8C2×D913yes—
141C2×C3×C8C2×C3⋊D311yescharacteristic
142C3×C4⋊C4C2×D913yes—
143C3×C4⋊C4C2×D913yes—
144C3×C4⋊C4C2×C3⋊D311yescharacteristic
145C3×C4⋊C4C2×C3⋊D311yescharacteristic
146(C2×D3)⋊C4—181no—
147(C2×D3)⋊C4—183no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—183no—
150C4⋊Dic3—93no—
151C4⋊Dic3—91no—
152C4×Dic3—181no—
153C4×Dic3—183no—
154C2×C3⋊C8—93no—
155C2×C3⋊C8—91no—
156C22×C32D3×D411yescharacteristic
157C2×C3⋊D3—61no—
158C2×C3⋊D3—61no—
159C2×C3⋊D3—61no—
160C2×C3⋊D3—121no—
161C2×C3⋊D3—121no—
162C2×C3⋊D3—121no—
163C32×C4—41no—
164C32×C4—41no—
165C32×C4#23511yescharacteristic
166C32×C4#23911yescharacteristic
167C3⋊Dic3—61no—
168C3⋊Dic3—121no—
169C22×C9D3×D413yes—
170C2×D9—63no—
171C2×D9—63no—
172C2×D9—63no—
173C2×D9—123no—
174C2×D9—123no—
175C2×D9—123no—
176C4×C9—43no—
177C4×C9—43no—
178C4×C9#23513yes—
179C4×C9#23913yes—
180Dic9—63no—
181Dic9—123no—
182#121—271no—
183#117—271no—
184#107—271no—
185#106C3⋊D911yescharacteristic, complemented
186#102—271no—
187#102—271no—
188C22⋊C8—271no—
189C3×C9#48211yescharacteristic, complemented, Sylow
190C22×D3—93no—
191C22×D3—91no—
192C22×D3—183no—
193C22×D3—181no—
194C2×C3×C4—23no—
195C2×C3×C4—23no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4C22×D913yes—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—93no—
201C2×Dic3—91no—
202C2×Dic3—183no—
203C2×Dic3—181no—
204D12—183no—
205D12—183no—
206D12—181no—
207D12—181no—
208D12—362no—
209D12—366no—
210C4×D3—183no—
211C4×D3—183no—
212C4×D3—181no—
213C4×D3—181no—
214C3×C8—23no—
215C3×C8—21no—
216C3⋊C8—183no—
217C3⋊C8—181no—
218C2×C32#89411yescharacteristic
219C2×C32#91411yescharacteristic
220C2×C32#91611yescharacteristic
221C3⋊D3—121no—
222C3⋊D3—241no—
223C2×C9#89413yes—
224C2×C9#91413yes—
225C2×C9#91613yes—
226D9—123no—
227D9—243no—
228C2×D4—271no—
229C2×D4—541no—
230C22×C4—271no—
231C2×C8—271no—
232C2×C8C2×C3⋊D911yescharacteristic
233C4⋊C4—271no—
234C4⋊C4C2×C3⋊D911yescharacteristic
235C4⋊C4C2×C3⋊D911yescharacteristic
236C22⋊C4—541no—
237C22⋊C4—541no—
238C42—541no—
239C22×C3D4×D913yes—
240C22×C3D4×C3⋊D311yescharacteristic
241C2×D3—181no—
242C2×D3—181no—
243C2×D3—181no—
244C2×D3—183no—
245C2×D3—183no—
246C2×D3—183no—
247C2×D3—361no—
248C2×D3—361no—
249C2×D3—361no—
250C2×D3—363no—
251C2×D3—363no—
252C2×D3—363no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—43no—
256C3×C4—43no—
257C3×C4#369713yes—
258C3×C4#370113yes—
259C3×C4#382811yescharacteristic
260C3×C4#383211yescharacteristic
261Dic3—181no—
262Dic3—183no—
263Dic3—361no—
264Dic3—363no—
265C32#486611yescharacteristic
266C9#486613yescomplemented
267C23—271no—
268C23—541no—
269D4—541no—
270D4—541no—
271D4—1082no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—271no—
275C2×C4—541no—
276C2×C4—541no—
277C2×C4—541no—
278C2×C4C22×C3⋊D911yescharacteristic
279C8—21no—
280C8—541no—
281C2×C3#6345613yes—
282C2×C3#6347613yes—
283C2×C3#6347813yes—
284C2×C3#6410311yescharacteristic
285C2×C3#6412311yescharacteristic
286C2×C3#6412511yescharacteristic
287D3—361no—
288D3—363no—
289D3—721no—
290D3—723no—
291C22—541no—
292C22—541no—
293C22—541no—
294C22—1081no—
295C22—1081no—
296C22—1081no—
297C22D4×C3⋊D911yescharacteristic
298C4—41no—
299C4—41no—
300C4—541no—
301C4—1081no—
302C4#8873611yescharacteristic
303C4#8874611yescharacteristic
304C3#1058854613yescomplemented
305C3#1059256511yescharacteristic
306C2—1081no—
307C2—2161no—
308C2#1172175011yescharacteristic
309C2#1172179611yescharacteristic
310C2#1172179811yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socleC22×C32D3×D4
prime radical#11721751C2
hypercenter#106C3⋊D9
hypocenterC3×C9#482
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11721748C2semidirect
3. C3×C9×#106C2semidirect
4. #11720543C2semidirect
5. #11720542C2semidirect
6. #11720528C2semidirect
7. #11720525C2semidirect
30. C9×#106D3semidirect
90. C3×#106D9semidirect
185. #106C3⋊D9semidirect
189. C3×C9#482semidirect
266. C9#4866semidirect
304. C3#10588546semidirect

Maximal subgroups

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

subgroupindex
#117217512
#117217482
C3×C9×#1062
#117205432
#117205422
#117205282
#117205252
#105925653
#105885463