Finite Group Databasebeta

← #13048644

Subgroups — #13048644

Order 1296.

520 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C4×C3⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C3×C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×C4×D3×C9C211yesmaximal, characteristic, complemented
4#10619880C211yesmaximal, characteristic, complemented
5#10619879C211yesmaximal, characteristic, complemented
6#10619874C211yesmaximal, characteristic, complemented
7#10619868C211yesmaximal, characteristic, complemented
8#89020—31nomaximal
9#88736—31nomaximal
10#88640—33nomaximal
11C2×C3⋊(C3⋊D9)C2211yescharacteristic
12C2×C3×C3⋊D9C2211yescharacteristic
13C2×C3×D3×C9C2211yescharacteristic
14C32×C4×C9C2211yescharacteristic
15C3⋊(C3⋊Dic9)C2211yescharacteristic
16C3×C3⋊Dic9C2211yescharacteristic
17C3×C9×Dic3C2211yescharacteristic
18C4×C3⋊(C3⋊D3)—31no—
19C3×C4×C3⋊D3—31no—
20C32×C4×D3D311yescharacteristic
21#6293—31no—
22#6292—31no—
23#6291—31no—
24#6290—31no—
25C2×C3×C4×C9—31no—
26#6233—31no—
27#6233—31no—
28C4⋊(C3⋊D9)—31no—
29C4×C3⋊D9—31no—
30C4×C3⋊D9—33no—
31C4×C3⋊D9—63no—
32C4×C3⋊D9—61no—
33C4×C3⋊D9D311yescharacteristic, complemented
34#6226—31no—
35C4×D3×C9D313yescomplemented
36C3×C4×D9—33no—
37#6195—33no—
38#6194—33no—
39#6192—33no—
40#6189—33no—
41C2×C32×C9C2311yescharacteristic
42C3⋊(C3⋊D9)—21no—
43C3×C3⋊D9—21no—
44C3×D3×C9—21no—
45#3828—91no—
46#3798—93no—
47#3798—91no—
48#3697—93no—
49C2×C3⋊(C3⋊D3)—31no—
50C2×C3×C3⋊D3—31no—
51C2×C32×D3C2×D311yescharacteristic
52C33×C4C2×D311yescharacteristic
53C3⋊(C3⋊Dic3)—31no—
54C3×C3⋊Dic3—31no—
55C32×Dic3C2×D311yescharacteristic
56C22×C3×C9—31no—
57C2×C3⋊D9—31no—
58C2×C3⋊D9—33no—
59C2×C3⋊D9—63no—
60C2×C3⋊D9—61no—
61C2×C3⋊D9C2×D311yescharacteristic
62C2×D3×C9C2×D313yes—
63C2×C3×D9—33no—
64C3×C4×C9—21no—
65C3×C4×C9—23no—
66C3×C4×C9—31no—
67C3×C4×C9C2×D313yes—
68C3×C4×C9C2×D311yescharacteristic
69C3⋊Dic9—31no—
70C3⋊Dic9—33no—
71C3⋊Dic9—63no—
72C3⋊Dic9—61no—
73C3⋊Dic9C2×D311yescharacteristic
74C9×Dic3C2×D313yes—
75C3×Dic9—33no—
76C32×C9#4111yescharacteristic, complemented, Sylow
77C2×C32×C4—31no—
78#638—91no—
79#638—91no—
80C4⋊(C3⋊D3)—91no—
81C4×C3⋊D3—31no—
82C4×C3⋊D3—91no—
83C4×C3⋊D3—93no—
84C4×C3⋊D3—91no—
85C4×C3⋊D3—181no—
86C4×C3⋊D3—183no—
87#634—91no—
88C3×C4×D3—93no—
89C3×C4×D3—91no—
90C3×C4×D3D913yescomplemented
91C3×C4×D3C3⋊D311yescharacteristic
92#627—93no—
93#627—91no—
94#626—93no—
95#626—91no—
96#626—93no—
97#626—91no—
98#625—93no—
99#625—91no—
100C2×C4×C9—33no—
101#611—93no—
102#611—93no—
103D36—93no—
104C4×D9—33no—
105C4×D9—93no—
106C4×D9—183no—
107Dic18—93no—
108C2×C33C22×D311yescharacteristic
109C3⋊(C3⋊D3)—61no—
110C3×C3⋊D3—61no—
111C32×D3—21no—
112C2×C3×C9—21no—
113C2×C3×C9—23no—
114C2×C3×C9—61no—
115C2×C3×C9C22×D313yes—
116C2×C3×C9C22×D311yescharacteristic
117C3⋊D9—21no—
118C3⋊D9—62no—
119C3⋊D9—66no—
120C3⋊D9—63no—
121C3⋊D9—61no—
122D3×C9—23no—
123C3×D9—63no—
124#235—271no—
125#235—273no—
126#235—271no—
127C22×C32—31no—
128C2×C3⋊D3—31no—
129C2×C3⋊D3—91no—
130C2×C3⋊D3—93no—
131C2×C3⋊D3—91no—
132C2×C3⋊D3—181no—
133C2×C3⋊D3—183no—
134C2×C3×D3—93no—
135C2×C3×D3—91no—
136C2×C3×D3C2×D913yes—
137C2×C3×D3C2×C3⋊D311yescharacteristic
138C32×C4—23no—
139C32×C4—21no—
140C32×C4—31no—
141C32×C4C2×D913yes—
142C32×C4D3211yescharacteristic
143C32×C4C2×C3⋊D311yescharacteristic
144C3⋊Dic3—31no—
145C3⋊Dic3—91no—
146C3⋊Dic3—93no—
147C3⋊Dic3—91no—
148C3⋊Dic3—181no—
149C3⋊Dic3—183no—
150C3×Dic3—93no—
151C3×Dic3—91no—
152C3×Dic3C2×D913yes—
153C3×Dic3C2×C3⋊D311yescharacteristic
154C22×C9—33no—
155C2×D9—33no—
156C2×D9—93no—
157C2×D9—183no—
158C4×C9—23no—
159C4×C9—33no—
160C4×C9D3213yes—
161Dic9—33no—
162Dic9—93no—
163Dic9—183no—
164C33#23511yescharacteristic
165C3×C9—21no—
166C3×C9—23no—
167C3×C9#23513yescomplemented
168C3×C9#23511yescharacteristic, complemented
169C2×C3×C4—33no—
170C2×C3×C4—31no—
171C2×C3×C4—271no—
172#67—271no—
173#67—273no—
174#67—271no—
175#67—273no—
176#67—271no—
177#67—271no—
178D12—271no—
179D12—273no—
180D12—271no—
181C4×D3—93no—
182C4×D3—91no—
183C4×D3—273no—
184C4×D3—271no—
185C4×D3—271no—
186C4×D3—543no—
187C4×D3—541no—
188C4×D3C3⋊D911yescharacteristic, complemented
189Dic6—271no—
190Dic6—273no—
191Dic6—271no—
192C2×C32—23no—
193C2×C32—21no—
194C2×C32—61no—
195C2×C32C22×D913yes—
196C2×C32C2×D3211yescharacteristic
197C2×C32C22×C3⋊D311yescharacteristic
198C3⋊D3—61no—
199C3⋊D3—182no—
200C3⋊D3—186no—
201C3⋊D3—183no—
202C3⋊D3—181no—
203C3⋊D3—181no—
204C3×D3—23no—
205C3×D3—21no—
206C3×D3—183no—
207C3×D3—181no—
208C2×C9—23no—
209C2×C9—63no—
210C2×C9C2×D3213yes—
211D9—63no—
212D9—186no—
213D9—183no—
214#41—811noSylow
215C22×C3—33no—
216C22×C3—31no—
217C22×C3—271no—
218C2×D3—93no—
219C2×D3—91no—
220C2×D3—273no—
221C2×D3—271no—
222C2×D3—271no—
223C2×D3—543no—
224C2×D3—541no—
225C2×D3C2×C3⋊D911yescharacteristic
226C3×C4—23no—
227C3×C4—21no—
228C3×C4—33no—
229C3×C4—31no—
230C3×C4—271no—
231C3×C4D3×D913yes—
232C3×C4C2×C3⋊D911yescharacteristic
233C3×C4D3×C3⋊D311yescharacteristic
234Dic3—93no—
235Dic3—91no—
236Dic3—273no—
237Dic3—271no—
238Dic3—271no—
239Dic3—543no—
240Dic3—541no—
241Dic3C2×C3⋊D911yescharacteristic
242C32—23no—
243C32—21no—
244C32#369713yescomplemented
245C32#379811yescharacteristic
246C32#382811yescharacteristic
247C9—23no—
248C9#379813yescomplemented
249Q8—811no—
250D4—811no—
251D4—811no—
252D4—811no—
253C2×C4—31no—
254C2×C4—271no—
255C2×C4—811no—
256C2×C3—23no—
257C2×C3—21no—
258C2×C3—63no—
259C2×C3—61no—
260C2×C3—541no—
261C2×C3C2×D3×D913yes—
262C2×C3C22×C3⋊D911yescharacteristic
263C2×C3C2×D3×C3⋊D311yescharacteristic
264D3—21no—
265D3—183no—
266D3—181no—
267D3—546no—
268D3—542no—
269D3—543no—
270D3—541no—
271D3—541no—
272C22—31no—
273C22—271no—
274C22—811no—
275C4—31no—
276C4—271no—
277C4—811no—
278C4D3×C3⋊D911yescharacteristic
279C3—23no—
280C3—21no—
281C3#8864013yescomplemented
282C3#8873611yescharacteristic, complemented
283C3#8902011yescharacteristic
284C2—61no—
285C2—541no—
286C2—1621no—
287C2C2×D3×C3⋊D911yescharacteristic
subgroupG/H
center, hypercenterC4D3×C3⋊D9
commutator, cosocleC2×C32×C9C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC32×C4×C9C22
socleC2×C33C22×D3
hypocenterC32×C9#41
LCSG ▷ C2×C32×C9 ▷ C32×C9
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×C3⋊(C3⋊D9)C2semidirect
2. C3×C4×C3⋊D9C2semidirect
3. C3×C4×D3×C9C2semidirect
4. #10619880C2semidirect
5. #10619879C2semidirect
6. #10619874C2semidirect
7. #10619868C2semidirect
33. C4×C3⋊D9D3semidirect
35. C4×D3×C9D3semidirect
76. C32×C9#41semidirect
90. C3×C4×D3D9semidirect
167. C3×C9#235semidirect
168. C3×C9#235semidirect
188. C4×D3C3⋊D9semidirect
244. C32#3697semidirect
248. C9#3798semidirect
281. C3#88640semidirect
282. C3#88736semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊(C3⋊D9)2
C3×C4×C3⋊D92
C3×C4×D3×C92
#106198802
#106198792
#106198742
#106198682
#890203
#887363
#886403