Finite Group Databasebeta

← #13144091

Subgroups — #13144091

Order 1728.

510 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 135 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#11721749C211yesmaximal, characteristic
2#11721746C211yesmaximal, characteristic, complemented
3C3×C9×#106C211yesmaximal, characteristic, complemented
4#11720543C211yesmaximal, characteristic
5#11720537C211yesmaximal, characteristic, complemented
6#11720528C211yesmaximal, characteristic
7#11720525C211yesmaximal, characteristic, complemented
8#10592570—31nomaximal
9#10588551—33nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C2×C3×C8×C9C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88536—21no—
16C4⋊(C3⋊Dic9)—21no—
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18#88534—21no—
19C4×C3⋊Dic9C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64102—31no—
22#64099—31no—
23C32×#106D311yescharacteristic
24#63648—31no—
25#63641—31no—
26#63636—31no—
27#63634—31no—
28#63455—33no—
29#63452—33no—
30C9×#106D313yescomplemented
31#63380—33no—
32Dic9⋊C8—33no—
33#63369—33no—
34#63366—33no—
35C22×C3⋊D9—21no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)Q812yes—
44C4×C3⋊D9—42no—
45C3×C8×C9—21no—
46C3⋊(C9⋊C8)—21no—
47#4871—93no—
48#4871—91no—
49C2×C4⋊(C3⋊D3)—31no—
50C2×C4×C3⋊D3—61no—
51C2×C32×C8C2×D311yescharacteristic
52C32×C4⋊C4C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54#3752—61no—
55C4⋊(C3⋊Dic3)—31no—
56C4⋊(C3⋊Dic3)—61no—
57#3750—61no—
58C4×C3⋊Dic3—31no—
59C2×C3⋊(C3⋊C8)—31no—
60C2×D36—33no—
61C2×C4×D9—63no—
62C2×C8×C9C2×D313yes—
63C9×C4⋊C4C2×D313yes—
64C9×C4⋊C4C2×D313yes—
65(C2×D9)⋊C4—63no—
66C4⋊Dic9—33no—
67C4⋊Dic9—63no—
68Dic9⋊C4—63no—
69C4×Dic9—33no—
70C2×C9⋊C8—33no—
71C22×C3×C9C2×D411yescharacteristic
72C2×C3⋊D9—22no—
73C2×C3⋊D9—41no—
74C2×C3⋊D9—41no—
75C3×C4×C9—41no—
76C3×C4×C9—41no—
77C3×C4×C9C2×Q811yescharacteristic
78C3×C4×C9#4111yescharacteristic
79C3⋊Dic9—21no—
80C3⋊Dic9—21no—
81C3⋊Dic9—41no—
82#893—93no—
83#893—91no—
84#890—93no—
85#890—91no—
86C3×#106D913yescomplemented
87C3×#106C3⋊D311yescharacteristic
88#821—93no—
89#821—91no—
90Dic3⋊C8—93no—
91Dic3⋊C8—91no—
92#809—93no—
93#809—91no—
94#807—93no—
95#807—91no—
96C22×C3⋊D3—61no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—31no—
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—61no—
103C4⋊(C3⋊D3)—32no—
104C4⋊(C3⋊D3)—61no—
105C4×C3⋊D3—122no—
106C22×D9—63no—
107C32×C8—21no—
108C3⋊(C3⋊C8)—61no—
109C2×C4×C9—23no—
110C2×C4×C9—23no—
111C2×C4×C9C22×D313yes—
112C2×Dic9—33no—
113C2×Dic9—33no—
114C2×Dic9—63no—
115D36—36no—
116D36—63no—
117C4×D9—126no—
118C8×C9—23no—
119C9⋊C8—63no—
120#478—271noSylow
121C2×C3×C9C22⋊Q811yescharacteristic
122C2×C3×C9#13511yescharacteristic
123C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
124C3⋊D9—42no—
125C2×D12—93no—
126C2×D12—91no—
127C2×C4×D3—181no—
128C2×C4×D3—183no—
129C2×C3×C8C2×D913yes—
130C2×C3×C8C2×C3⋊D311yescharacteristic
131C3×C4⋊C4C2×D913yes—
132C3×C4⋊C4C2×D913yes—
133C3×C4⋊C4C2×C3⋊D311yescharacteristic
134C3×C4⋊C4C2×C3⋊D311yescharacteristic
135(C2×D3)⋊C4—181no—
136(C2×D3)⋊C4—183no—
137C4⋊Dic3—93no—
138C4⋊Dic3—91no—
139C4⋊Dic3—181no—
140C4⋊Dic3—183no—
141Dic3⋊C4—181no—
142Dic3⋊C4—183no—
143C4×Dic3—93no—
144C4×Dic3—91no—
145C2×C3⋊C8—93no—
146C2×C3⋊C8—91no—
147C22×C32D3×D411yescharacteristic
148C2×C3⋊D3—62no—
149C2×C3⋊D3—121no—
150C2×C3⋊D3—121no—
151C32×C4—41no—
152C32×C4—41no—
153C32×C4#23511yescharacteristic
154C32×C4D3×Q811yescharacteristic
155C3⋊Dic3—61no—
156C3⋊Dic3—61no—
157C3⋊Dic3—121no—
158C22×C9D3×D413yes—
159C2×D9—66no—
160C2×D9—123no—
161C2×D9—123no—
162C4×C9—43no—
163C4×C9—43no—
164C4×C9#23513yes—
165C4×C9D3×Q813yes—
166Dic9—63no—
167Dic9—63no—
168Dic9—123no—
169#125—271no—
170C4×D4—271no—
171#107—271no—
172#106C3⋊D911yescharacteristic, complemented
173C4⋊C8—271no—
174#102—271no—
175#102—271no—
176C3×C9#47811yescharacteristic, complemented, Sylow
177C22×D3—183no—
178C22×D3—181no—
179C2×C3×C4—23no—
180C2×C3×C4—23no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C22×D913yes—
184C2×C3×C4C22×C3⋊D311yescharacteristic
185C2×Dic3—93no—
186C2×Dic3—93no—
187C2×Dic3—91no—
188C2×Dic3—91no—
189C2×Dic3—183no—
190C2×Dic3—181no—
191D12—96no—
192D12—92no—
193D12—183no—
194D12—181no—
195C4×D3—362no—
196C4×D3—366no—
197C3×C8—23no—
198C3×C8—21no—
199C3⋊C8—183no—
200C3⋊C8—181no—
201C2×C32#89611yescharacteristic
202C2×C32#91411yescharacteristic
203C2×C32#91611yescharacteristic
204C3⋊D3—122no—
205C2×C9#89613yes—
206C2×C9#91413yes—
207C2×C9#91613yes—
208D9—126no—
209C2×D4—271no—
210C22×C4—541no—
211C2×C8—271no—
212C2×C8C2×C3⋊D911yescharacteristic
213C4⋊C4—271no—
214C4⋊C4—541no—
215C4⋊C4—541no—
216C4⋊C4C2×C3⋊D911yescharacteristic
217C4⋊C4C2×C3⋊D911yescharacteristic
218C22⋊C4—541no—
219C42—271no—
220C22×C3D4×D913yes—
221C22×C3D4×C3⋊D311yescharacteristic
222C2×D3—182no—
223C2×D3—186no—
224C2×D3—361no—
225C2×D3—361no—
226C2×D3—363no—
227C2×D3—363no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—43no—
231C3×C4—43no—
232C3×C4#369713yes—
233C3×C4Q8×D913yes—
234C3×C4#382811yescharacteristic
235C3×C4Q8×C3⋊D311yescharacteristic
236Dic3—181no—
237Dic3—181no—
238Dic3—183no—
239Dic3—183no—
240Dic3—361no—
241Dic3—363no—
242C32#487111yescharacteristic
243C9#487113yescomplemented
244C23—541no—
245D4—272no—
246D4—541no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—271no—
250C2×C4—271no—
251C2×C4—541no—
252C2×C4—1082no—
253C2×C4C22×C3⋊D911yescharacteristic
254C8—21no—
255C8—541no—
256C2×C3#6345813yes—
257C2×C3#6347613yes—
258C2×C3#6347813yes—
259C2×C3#6410511yescharacteristic
260C2×C3#6412311yescharacteristic
261C2×C3#6412511yescharacteristic
262D3—362no—
263D3—366no—
264C22—542no—
265C22—1081no—
266C22—1081no—
267C22D4×C3⋊D911yescharacteristic
268C4—41no—
269C4—41no—
270C4—541no—
271C4—541no—
272C4—1081no—
273C4#8873611yescharacteristic
274C4Q8×C3⋊D911yescharacteristic
275C3#1058855113yescomplemented
276C3#1059257011yescharacteristic
277C2—1082no—
278C2#1172175211yescharacteristic
279C2#1172179611yescharacteristic
280C2#1172179811yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenter#106C3⋊D9
hypocenterC3×C9#478
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. #11721746C2semidirect
3. C3×C9×#106C2semidirect
5. #11720537C2semidirect
7. #11720525C2semidirect
30. C9×#106D3semidirect
86. C3×#106D9semidirect
172. #106C3⋊D9semidirect
176. C3×C9#478semidirect
243. C9#4871semidirect
275. C3#10588551semidirect

Maximal subgroups

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

subgroupindex
#117217492
#117217462
C3×C9×#1062
#117205432
#117205372
#117205282
#117205252
#105925703
#105885513