Finite Group Databasebeta

← #13144088

Subgroups — #13144088

Order 1728.

540 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 137 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#11721744C211yesmaximal, characteristic, complemented
3C3×C9×#106C211yesmaximal, characteristic, complemented
4#11720543C211yesmaximal, characteristic
5#11720537C211yesmaximal, characteristic, complemented
6#11720528C211yesmaximal, characteristic
7#11720526C211yesmaximal, characteristic, complemented
8#10592569—31nomaximal
9#10588550—33nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C2×#6226—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16#88536—21no—
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18#88534—21no—
19C4×C3⋊Dic9C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64102—31no—
22#64097—31no—
23C32×#106D311yescharacteristic
24#63648—31no—
25#63641—31no—
26#63636—31no—
27#63635—31no—
28#63455—33no—
29#63450—33no—
30C9×#106D313yescomplemented
31#63380—33no—
32Dic9⋊C8—33no—
33#63369—33no—
34#63367—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)Q812yescomplemented
44C4×C3⋊D9—42no—
45#6226—42no—
46C3×C8×C9—21no—
47C3⋊(C9⋊C8)—21no—
48#4870—93no—
49#4870—91no—
50C2×C4⋊(C3⋊D3)—31no—
51C2×C4×C3⋊D3—61no—
52C2×#634—61no—
53C2×C32×C8C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56#3752—61no—
57C4⋊(C3⋊Dic3)—31no—
58#3750—61no—
59C4×C3⋊Dic3—31no—
60C2×C3⋊(C3⋊C8)—31no—
61C2×D36—33no—
62C2×C4×D9—63no—
63C2×Dic18—63no—
64C2×C8×C9C2×D313yes—
65C9×C4⋊C4C2×D313yes—
66C9×C4⋊C4C2×D313yes—
67(C2×D9)⋊C4—63no—
68C4⋊Dic9—33no—
69Dic9⋊C4—63no—
70C4×Dic9—33no—
71C2×C9⋊C8—33no—
72C22×C3×C9C2×D411yescharacteristic
73C2×C3⋊D9—22no—
74C2×C3⋊D9—41no—
75C2×C3⋊D9—41no—
76C3×C4×C9—41no—
77C3×C4×C9—41no—
78C3×C4×C9C2×Q811yescharacteristic
79C3×C4×C9#4111yescharacteristic
80C3⋊Dic9—21no—
81C3⋊Dic9—41no—
82C3⋊Dic9SD1612yes—
83#893—93no—
84#893—91no—
85#888—93no—
86#888—91no—
87C3×#106D913yescomplemented
88C3×#106C3⋊D311yescharacteristic
89#821—93no—
90#821—91no—
91Dic3⋊C8—93no—
92Dic3⋊C8—91no—
93#809—93no—
94#809—91no—
95#808—93no—
96#808—91no—
97C22×C3⋊D3—61no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—61no—
104C4⋊(C3⋊D3)—32no—
105C4⋊(C3⋊D3)—61no—
106C4×C3⋊D3—122no—
107#634—122no—
108C22×D9—63no—
109C32×C8—21no—
110C3⋊(C3⋊C8)—61no—
111C2×C4×C9—23no—
112C2×C4×C9—23no—
113C2×C4×C9C22×D313yes—
114C2×Dic9—33no—
115C2×Dic9—33no—
116C2×Dic9—63no—
117D36—36no—
118D36—63no—
119C4×D9—126no—
120Dic18—126no—
121C8×C9—23no—
122C9⋊C8—63no—
123#476—271noSylow
124C2×C3×C9C22⋊Q811yescharacteristic
125C2×C3×C9C2×SD1611yescharacteristic
126C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
127C3⋊D9—42no—
128C2×D12—93no—
129C2×D12—91no—
130C2×C4×D3—181no—
131C2×C4×D3—183no—
132C2×Dic6—181no—
133C2×Dic6—183no—
134C2×C3×C8C2×D913yes—
135C2×C3×C8C2×C3⋊D311yescharacteristic
136C3×C4⋊C4C2×D913yes—
137C3×C4⋊C4C2×D913yes—
138C3×C4⋊C4C2×C3⋊D311yescharacteristic
139C3×C4⋊C4C2×C3⋊D311yescharacteristic
140(C2×D3)⋊C4—181no—
141(C2×D3)⋊C4—183no—
142C4⋊Dic3—93no—
143C4⋊Dic3—91no—
144Dic3⋊C4—181no—
145Dic3⋊C4—183no—
146C4×Dic3—93no—
147C4×Dic3—91no—
148C2×C3⋊C8—93no—
149C2×C3⋊C8—91no—
150C22×C32D3×D411yescharacteristic
151C2×C3⋊D3—62no—
152C2×C3⋊D3—121no—
153C2×C3⋊D3—121no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4#23511yescharacteristic
157C32×C4D3×Q811yescharacteristic
158C3⋊Dic3—32no—
159C3⋊Dic3—61no—
160C3⋊Dic3—121no—
161C22×C9D3×D413yes—
162C2×D9—66no—
163C2×D9—123no—
164C2×D9—123no—
165C4×C9—43no—
166C4×C9—43no—
167C4×C9#23513yes—
168C4×C9D3×Q813yes—
169Dic9—36no—
170Dic9—63no—
171Dic9—123no—
172C4⋊Q8—271no—
173C4×D4—271no—
174#106—271no—
175#106C3⋊D911yescharacteristic, complemented
176C4⋊C8—271no—
177#102—271no—
178#102—271no—
179C3×C9#47611yescharacteristic, complemented, Sylow
180C22×D3—183no—
181C22×D3—181no—
182C2×C3×C4—23no—
183C2×C3×C4—23no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4C22×D913yes—
187C2×C3×C4C22×C3⋊D311yescharacteristic
188C2×Dic3—93no—
189C2×Dic3—91no—
190C2×Dic3—93no—
191C2×Dic3—91no—
192C2×Dic3—183no—
193C2×Dic3—181no—
194D12—96no—
195D12—92no—
196D12—183no—
197D12—181no—
198C4×D3—362no—
199C4×D3—366no—
200Dic6—362no—
201Dic6—366no—
202C3×C8—23no—
203C3×C8—21no—
204C3⋊C8—183no—
205C3⋊C8—181no—
206C2×C32#89611yescharacteristic
207C2×C32D3×SD1611yescharacteristic
208C2×C32#91411yescharacteristic
209C3⋊D3—122no—
210C2×C9#89613yes—
211C2×C9D3×SD1613yes—
212C2×C9#91413yes—
213D9—126no—
214C2×Q8—541no—
215C2×D4—271no—
216C22×C4—541no—
217C2×C8—271no—
218C2×C8C2×C3⋊D911yescharacteristic
219C4⋊C4—271no—
220C4⋊C4—541no—
221C4⋊C4C2×C3⋊D911yescharacteristic
222C4⋊C4C2×C3⋊D911yescharacteristic
223C22⋊C4—541no—
224C42—271no—
225C22×C3D4×D913yes—
226C22×C3D4×C3⋊D311yescharacteristic
227C2×D3—182no—
228C2×D3—186no—
229C2×D3—361no—
230C2×D3—361no—
231C2×D3—363no—
232C2×D3—363no—
233C3×C4—41no—
234C3×C4—41no—
235C3×C4—43no—
236C3×C4—43no—
237C3×C4#369713yes—
238C3×C4Q8×D913yes—
239C3×C4#382811yescharacteristic
240C3×C4Q8×C3⋊D311yescharacteristic
241Dic3—92no—
242Dic3—96no—
243Dic3—181no—
244Dic3—183no—
245Dic3—361no—
246Dic3—363no—
247C32#487011yescharacteristic
248C9#487013yescomplemented
249C23—541no—
250Q8—1082no—
251D4—272no—
252D4—541no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—271no—
256C2×C4—271no—
257C2×C4—541no—
258C2×C4—1082no—
259C2×C4C22×C3⋊D911yescharacteristic
260C8—21no—
261C8—541no—
262C2×C3#6345813yes—
263C2×C3SD16×D913yes—
264C2×C3#6347613yes—
265C2×C3#6410511yescharacteristic
266C2×C3SD16×C3⋊D311yescharacteristic
267C2×C3#6412311yescharacteristic
268D3—362no—
269D3—366no—
270C22—542no—
271C22—1081no—
272C22—1081no—
273C22D4×C3⋊D911yescharacteristic
274C4—41no—
275C4—41no—
276C4—272no—
277C4—541no—
278C4—1081no—
279C4#8873611yescharacteristic
280C4Q8×C3⋊D911yescharacteristic
281C3#1058855013yescomplemented
282C3#1059256911yescharacteristic
283C2—1082no—
284C2#1172175211yescharacteristic
285C2SD16×C3⋊D911yescharacteristic
286C2#1172179611yescharacteristic
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#476
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. #11721744C2semidirect
3. C3×C9×#106C2semidirect
5. #11720537C2semidirect
7. #11720526C2semidirect
30. C9×#106D3semidirect
43. C4⋊(C3⋊D9)Q8semidirect
87. C3×#106D9semidirect
175. #106C3⋊D9semidirect
179. C3×C9#476semidirect
248. C9#4870semidirect
281. C3#10588550semidirect

Maximal subgroups

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

subgroupindex
#117217492
#117217442
C3×C9×#1062
#117205432
#117205372
#117205282
#117205262
#105925693
#105885503