Finite Group Databasebeta

← D9×#876 #13147730

Subgroups — D9×#876 #13147730

Order 1728.

798 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 169 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#11721713C211yesmaximal, characteristic, complemented
2C9×#876C211yesmaximal, characteristic, complemented
3C3×C42×D9C211yesmaximal, characteristic, complemented
4D9×(C2×D3)⋊C4C213yesmaximal
5D9×Dic3⋊C4C213yesmaximal, complemented
6#11720963C213yesmaximal, complemented
7#11720934C213yesmaximal, complemented
8D3×#876—31nomaximal
9D9×#126—31nomaximal
10C22×D3×D9C2211yescharacteristic
11C2×C3×C4×D9C2213yes—
12C2×Dic3×D9C2213yes—
13C3×C42×C9C2211yescharacteristic, complemented
14#88536C2213yes—
15#88534C2213yescomplemented
16C9×(C2×D3)⋊C4C2213yes—
17C9×Dic3⋊C4C2213yescomplemented
18C3×C4×Dic9C2213yescomplemented
19(C2×D3)⋊Dic9C2213yes—
20#88445C2213yes—
21Dic3⋊Dic9C2213yescomplemented
22#88442C2213yescomplemented
23#64085—31no—
24C3×#876D311yescharacteristic
25C3×C42×D3—31no—
26D3×(C2×D3)⋊C4—33no—
27D3×Dic3⋊C4—33no—
28#63868—33no—
29#63859—33no—
30C9×#126—31no—
31#63460—33no—
32C4⋊C4×D9—33no—
33C22⋊C4×D9—33no—
34#63441—33no—
35#63438—31no—
36#63438—61no—
37C42×D9D311yescharacteristic, complemented
38C22×C3⋊D9C2311yescharacteristic
39C22×D3×C9C2311yescharacteristic
40C22×C3×D9C2311yescharacteristic
41C2×D3×D9—26no—
42C2×C3×C4×C9C2313yes—
43C2×C3⋊Dic9C2313yes—
44C2×C9×Dic3C2313yes—
45C2×C3×Dic9C2313yes—
46C3×C4×D9—26no—
47Dic3×D9—26no—
48D3×#126—91no—
49C2×#876—91no—
50C22×D32—31no—
51C2×C3×C4×D3—33no—
52C2×D3×Dic3—33no—
53C23×D9—31no—
54C32×C42C2×D311yescharacteristic
55#3752—33no—
56#3750—33no—
57C3×(C2×D3)⋊C4C2×D313yes—
58C3×Dic3⋊C4C2×D313yes—
59C3×C4×Dic3—33no—
60#3724—33no—
61Dic3⋊Dic3—33no—
62#3722—33no—
63(C2×D3)⋊Dic3—33no—
64C2×C4×D9—33no—
65C2×C4×D9C2×D313yes—
66C9×C4⋊C4—33no—
67C9×C22⋊C4—33no—
68C42×C9—21no—
69C42×C9C2×D311yescharacteristic, complemented
70#3676—33no—
71(C2×D9)⋊C4—33no—
72(C2×D9)⋊C4—33no—
73(C2×D9)⋊C4—63no—
74C4⋊Dic9—33no—
75Dic9⋊C4—33no—
76Dic9⋊C4—33no—
77Dic9⋊C4—63no—
78C4×Dic9C2×D313yescomplemented
79C22×C3×C9C2411yescharacteristic
80C2×C3⋊D9—23no—
81C2×D3×C9—23no—
82C2×C3×D9#4116yes—
83D3×D9—44no—
84C3×C4×C9—23no—
85C3⋊Dic9—23no—
86C9×Dic3—23no—
87C3×Dic9—23no—
88C3×#126—31no—
89C2×C3×C42—91no—
90C2×(C2×D3)⋊C4—93no—
91C2×Dic3⋊C4—93no—
92#898—93no—
93D3×C4⋊C4—93no—
94D3×C22⋊C4—93no—
95#879—93no—
96#876—91no—
97#876—91no—
98#876—93no—
99#876—93no—
100#876—181no—
101#876D911yescharacteristic, direct product factor
102C42×D3—31no—
103C22×C3⋊D3—31no—
104C22×C3×D3—31no—
105C22×C3×D3C22×D311yescharacteristic
106C2×D32—66no—
107C2×C32×C4C22×D313yes—
108C2×C3⋊Dic3—33no—
109C2×C3×Dic3—33no—
110C2×C3×Dic3C22×D313yes—
111C3×C4×D3—66no—
112D3×Dic3—66no—
113C23×C9—31no—
114C22×D9—31no—
115C22×D9—61no—
116C22×D9—66no—
117C22×D9C22×D311yescharacteristic
118C2×C4×C9—23no—
119C2×C4×C9—33no—
120C2×C4×C9C22×D313yes—
121C2×Dic9—33no—
122C2×Dic9—63no—
123C2×Dic9C22×D313yes—
124C4×D9—26no—
125C4×D9—66no—
126C2×#126—271noSylow
127C2×C3×C9C2×#4113yes—
128C3⋊D9—41no—
129D3×C9—41no—
130C3×D9#12614yescomplemented
131C23×D3—91no—
132C23×D3—91no—
133C22×C3×C4—93no—
134C22×Dic3—93no—
135C2×C4×D3—33no—
136C2×C4×D3—93no—
137C3×C4⋊C4—33no—
138C3×C22⋊C4—33no—
139C3×C42—21no—
140C3×C42—93no—
141C3×C42C2×D911yescharacteristic, complemented
142C3×C42D3211yescharacteristic
143#217—93no—
144(C2×D3)⋊C4—93no—
145(C2×D3)⋊C4—93no—
146(C2×D3)⋊C4—93no—
147(C2×D3)⋊C4—93no—
148(C2×D3)⋊C4—93no—
149(C2×D3)⋊C4—183no—
150(C2×D3)⋊C4C2×D913yes—
151C4⋊Dic3—93no—
152Dic3⋊C4—93no—
153Dic3⋊C4—93no—
154Dic3⋊C4—93no—
155Dic3⋊C4—93no—
156Dic3⋊C4—93no—
157Dic3⋊C4—183no—
158Dic3⋊C4C2×D913yescomplemented
159C4×Dic3—33no—
160C22×C32C23×D311yescharacteristic
161C2×C3⋊D3—63no—
162C2×C3×D3—23no—
163C2×C3×D3—36no—
164D32—124no—
165C32×C4—23no—
166C3⋊Dic3—63no—
167C3×Dic3—23no—
168C3×Dic3—63no—
169C22×C9—21no—
170C22×C9—63no—
171C22×C9C23×D311yescharacteristic
172C2×D9—63no—
173C2×D9—123no—
174C2×D9—124no—
175C2×D9#23516yes—
176C4×C9—23no—
177C4×C9—43no—
178C4×C9—63no—
179Dic9—23no—
180Dic9—63no—
181Dic9—123no—
182#126—31no—
183#126—273no—
184#126—273no—
185#126—271no—
186C2×C4⋊C4—273no—
187C2×C22⋊C4—273no—
188C2×C42—91no—
189C3×C9C2×#12611yescharacteristic, complemented, Sylow
190C23×C3—31no—
191C23×C3—91no—
192C22×D3—31no—
193C22×D3—91no—
194C22×D3—91no—
195C22×D3—181no—
196C22×D3—186no—
197C22×D3—186no—
198C22×D3C22×D911yescharacteristic
199C2×C3×C4—23no—
200C2×C3×C4—33no—
201C2×C3×C4—93no—
202C2×C3×C4—186no—
203C2×C3×C4C22×D913yes—
204C2×C3×C4C2×D3213yes—
205C2×Dic3—33no—
206C2×Dic3—93no—
207C2×Dic3—93no—
208C2×Dic3—183no—
209C2×Dic3—186no—
210C2×Dic3C22×D913yes—
211C4×D3—66no—
212C4×D3—186no—
213C2×C32D3×#4113yes—
214C3⋊D3—121no—
215C3×D3—34no—
216C3×D3—41no—
217C2×C9—23no—
218C2×C9—121no—
219C2×C9C2×#23513yes—
220D9—121no—
221D9—241no—
222D9#87614yesdirect product factor
223C24—271no—
224C22×C4—93no—
225C22×C4—273no—
226C4⋊C4—33no—
227C4⋊C4—273no—
228C4⋊C4—273no—
229C4⋊C4—273no—
230C22⋊C4—33no—
231C22⋊C4—273no—
232C22⋊C4—273no—
233C22⋊C4—273no—
234C42—93no—
235C42D3×D911yescharacteristic, complemented
236C22×C3—21no—
237C22×C3—63no—
238C22×C3—96no—
239C22×C3C23×D911yescharacteristic
240C22×C3C22×D3211yescharacteristic
241C2×D3—23no—
242C2×D3—36no—
243C2×D3—183no—
244C2×D3—183no—
245C2×D3—363no—
246C2×D3—364no—
247C2×D3—364no—
248C3×C4—23no—
249C3×C4—23no—
250C3×C4—43no—
251C3×C4—63no—
252C3×C4—183no—
253Dic3—23no—
254Dic3—63no—
255Dic3—183no—
256Dic3—183no—
257Dic3—363no—
258C32D3×#12611yescharacteristic
259C9—21no—
260C9C2×#87611yescharacteristic, complemented
261C23—31no—
262C23—91no—
263C23—271no—
264C23—546no—
265C2×C4—33no—
266C2×C4—93no—
267C2×C4—186no—
268C2×C4—273no—
269C2×C4—546no—
270C2×C4C2×D3×D913yes—
271C2×C3—23no—
272C2×C3—94no—
273C2×C3—121no—
274C2×C3#41×D913yes—
275C2×C3D3×#23513yes—
276D3—34no—
277D3—41no—
278D3—361no—
279D3—361no—
280D3—721no—
281C22—63no—
282C22—96no—
283C22—543no—
284C22—1084no—
285C22C22×D3×D911yescharacteristic
286C4—23no—
287C4—63no—
288C4—183no—
289C4—543no—
290C3—21no—
291C3D9×#12611yescharacteristic, complemented
292C3D3×#87611yescharacteristic
293C2—94no—
294C2—121no—
295C2—1081no—
296C2D9×#23513yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×D3×D9C22
hypercenterC42D3×D9
hypocenterC3×C9C2×#126
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721713C2semidirect
2. C9×#876C2semidirect
3. C3×C42×D9C2semidirect
5. D9×Dic3⋊C4C2semidirect
6. #11720963C2semidirect
7. #11720934C2semidirect
13. C3×C42×C9C22semidirect
15. #88534C22semidirect
17. C9×Dic3⋊C4C22semidirect
18. C3×C4×Dic9C22semidirect
21. Dic3⋊Dic9C22semidirect
22. #88442C22semidirect
37. C42×D9D3semidirect
69. C42×C9C2×D3semidirect
78. C4×Dic9C2×D3semidirect
101. #876D9direct (N × H)
130. C3×D9#126semidirect
141. C3×C42C2×D9semidirect
158. Dic3⋊C4C2×D9semidirect
189. C3×C9C2×#126semidirect
222. D9#876direct (N × H)
235. C42D3×D9semidirect
260. C9C2×#876semidirect
291. C3D9×#126semidirect

Maximal subgroups

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

subgroupindex
#117217132
C9×#8762
C3×C42×D92
D9×(C2×D3)⋊C42
D9×Dic3⋊C42
#117209632
#117209342
D3×#8763
D9×#1263