Finite Group Databasebeta

← D3×#126⋊C9 #13147224

Subgroups — D3×#126⋊C9 #13147224

Order 1728.

254 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 41 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
1C3×#126⋊C9C211yesmaximal, characteristic, complemented
2D3×C42⋊C9C211yesmaximal, characteristic, complemented
3C3⋊(#126⋊C9)C211yesmaximal, characteristic, complemented
4C3×D3×#126C311yesmaximal, characteristic
5C2×#126⋊C9—31nomaximal
6C2×D3×C22⋊C9—41nomaximal
7C3×C42⋊C9C2211yescharacteristic, complemented
8C32×#126C2×C311yescharacteristic
9C3×#898—31no—
10C3×D3×C4⋊C4—31no—
11C3×D3×C22⋊C4—31no—
12C3×#879—31no—
13C3×#876C2×C311yescharacteristic
14C3×C42×D3C2×C311yescharacteristic
15#126⋊C9—21no—
16#126⋊C9—31no—
17#126⋊C9D311yescharacteristic, direct product factor
18C2×C42⋊C9—31no—
19C2×C3×C22⋊C9—41no—
20D3×C22⋊C9—41no—
21D3×C22⋊C9—41no—
22C2×C3×#126—31no—
23D3×#126C911yescharacteristic, complemented
24C23×C3×D3A411yescharacteristic
25C2×C3×C4×D3—31no—
26C2×C3×C4×D3—31no—
27C22×C22⋊C9—121no—
28C32×C4⋊C4—31no—
29C32×C22⋊C4—31no—
30C32×C42C22×C311yescharacteristic
31C3×#217—31no—
32C3×(C2×D3)⋊C4—31no—
33C3×(C2×D3)⋊C4—31no—
34C3×C4⋊Dic3—31no—
35C3×Dic3⋊C4—31no—
36C3×Dic3⋊C4—31no—
37C3×C4×Dic3—31no—
38C42⋊C9—21no—
39C42⋊C9C2×D311yescharacteristic, complemented
40C2×D3×C9—161no—
41C3×C22⋊C9—41no—
42C3×#126—21no—
43C3×#126—31no—
44C3×#126—91no—
45C3×#126—91no—
46C3×#126C2×C911yescharacteristic, complemented
47C3×#126C3×D311yescharacteristic
48C2×C3×C4⋊C4—91no—
49C2×C3×C22⋊C4—91no—
50C2×C3×C42—31no—
51#898—31no—
52D3×C4⋊C4—31no—
53D3×C22⋊C4—31no—
54#879—31no—
55#876C2×C911yescharacteristic, complemented
56C42×D3C2×C911yescharacteristic, complemented
57C23×C32C2×A411yescharacteristic
58C22×C3×D3—61no—
59C22×C3×D3—61no—
60C22×C3×D3C2×A411yescharacteristic
61C22×C3×D3C2×A411yescharacteristic
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—31no—
66C3×C4×D3—61no—
67C3×C4×D3—61no—
68C3×C4×D3—61no—
69C3×C4×D3—61no—
70C2×C22⋊C9—41no—
71C2×C22⋊C9—81no—
72C2×C22⋊C9—121no—
73C2×C22⋊C9—121no—
74C2×#126—31noSylow
75C2×C3×C9—161no—
76D3×C9—161no—
77D3×C9—161no—
78C24×C3—31no—
79C23×D3C22⋊C911yescharacteristic
80C22×C3×C4—91no—
81C22×C3×C4—91no—
82C2×C4×D3—31no—
83C2×C4×D3—31no—
84C3×C4⋊C4—31no—
85C3×C4⋊C4—31no—
86C3×C4⋊C4—61no—
87C3×C4⋊C4—91no—
88C3×C4⋊C4—91no—
89C3×C4⋊C4—91no—
90C3×C22⋊C4—31no—
91C3×C22⋊C4—31no—
92C3×C22⋊C4—61no—
93C3×C22⋊C4—91no—
94C3×C22⋊C4—91no—
95C3×C22⋊C4—91no—
96C3×C42—21no—
97C3×C42—91no—
98C3×C42C22×C911yescharacteristic, complemented
99C3×C42C2×C3×D311yescharacteristic
100#217—31no—
101(C2×D3)⋊C4—31no—
102(C2×D3)⋊C4—31no—
103C4⋊Dic3—31no—
104Dic3⋊C4—31no—
105Dic3⋊C4—31no—
106C4×Dic3—31no—
107C22×C32—61no—
108C22×C32C22×A411yescharacteristic
109C2×C3×D3—31no—
110C2×C3×D3—31no—
111C2×C3×D3—41no—
112C2×C3×D3—61no—
113C2×C3×D3—121no—
114C32×C4—61no—
115C32×C4—61no—
116C3×Dic3—61no—
117C3×Dic3—61no—
118C22×C9—481no—
119C22⋊C9—41no—
120C22⋊C9—81no—
121#126—31no—
122#126—91no—
123#126—91no—
124#126D3×C911yescharacteristic, complemented
125C2×C4⋊C4—91no—
126C2×C22⋊C4—91no—
127C2×C42—31no—
128C3×C9—161noSylow
129C23×C3—21no—
130C23×C3—31no—
131C23×C3—31no—
132C23×C3—181no—
133C23×C3—181no—
134C23×C3C2×C22⋊C911yescharacteristic
135C23×C3D3×A411yescharacteristic
136C22×D3—61no—
137C22×D3—61no—
138C22×D3C2×C22⋊C911yescharacteristic
139C22×D3C2×C22⋊C911yescharacteristic
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—31no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4—181no—
152C2×Dic3—31no—
153C2×Dic3—31no—
154C4×D3—61no—
155C4×D3—61no—
156C4×D3—61no—
157C4×D3—61no—
158C2×C32—31no—
159C2×C32—41no—
160C3×D3—31no—
161C3×D3—41no—
162C3×D3#126⋊C311yescharacteristic
163C2×C9—161no—
164C2×C9—321no—
165C2×C9—481no—
166C2×C9—481no—
167C24—31no—
168C22×C4—91no—
169C22×C4—91no—
170C4⋊C4—31no—
171C4⋊C4—91no—
172C4⋊C4—91no—
173C4⋊C4—91no—
174C22⋊C4—31no—
175C22⋊C4—91no—
176C22⋊C4—91no—
177C22⋊C4—91no—
178C42—91no—
179C42C2×D3×C911yescharacteristic, complemented
180C22×C3—21no—
181C22×C3—61no—
182C22×C3—61no—
183C22×C3—91no—
184C22×C3—91no—
185C22×C3—121no—
186C22×C3—121no—
187C22×C3—181no—
188C22×C3—361no—
189C22×C3C22×C22⋊C911yescharacteristic
190C22×C3C2×D3×A411yescharacteristic
191C2×D3—31no—
192C2×D3—31no—
193C2×D3—41no—
194C2×D3—61no—
195C2×D3—121no—
196C3×C4—61no—
197C3×C4—61no—
198C3×C4—61no—
199C3×C4—61no—
200C3×C4—121no—
201C3×C4—121no—
202C3×C4—181no—
203C3×C4—181no—
204Dic3—61no—
205Dic3—61no—
206C32C2×#126⋊C311yescharacteristic
207C9—161no—
208C9—321no—
209C23—31no—
210C23—31no—
211C23—181no—
212C23—181no—
213C23D3×C22⋊C911yescharacteristic
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—91no—
217C2×C4—91no—
218C2×C4—181no—
219C2×C4—181no—
220C2×C4—181no—
221C2×C4—181no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—31no—
225C2×C3—41no—
226C2×C3—41no—
227C2×C3—61no—
228C2×C3—81no—
229C2×C3—91no—
230C2×C3—121no—
231D3—31no—
232D3—41no—
233D3#126⋊C911yescharacteristic, direct product factor
234C22—61no—
235C22—91no—
236C22—91no—
237C22—121no—
238C22—181no—
239C22—361no—
240C22C2×D3×C22⋊C911yescharacteristic
241C4—61no—
242C4—61no—
243C4—181no—
244C4—181no—
245C3—21no—
246C3C2×#126⋊C911yescharacteristic, complemented
247C3D3×#126⋊C311yescharacteristic
248C2—31no—
249C2—31no—
250C2—41no—
251C2—91no—
252C2—121no—
subgroupG/H
center, hypercenterC3D3×#126⋊C3
commutator, hypocenterC3×C42C22×C9
FrattiniC22×C3C2×D3×A4
FittingC32×#126C2×C3
socleC22×C32C22×A4
prime radicalC23×C3×D3A4
cosocleC32×C42C22×C3
LCSG ▷ C3×C42
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×#126⋊C9C2semidirect
2. D3×C42⋊C9C2semidirect
3. C3⋊(#126⋊C9)C2semidirect
7. C3×C42⋊C9C22semidirect
17. #126⋊C9D3direct (N × H)
23. D3×#126C9semidirect
39. C42⋊C9C2×D3semidirect
46. C3×#126C2×C9semidirect
55. #876C2×C9semidirect
56. C42×D3C2×C9semidirect
98. C3×C42C22×C9semidirect
124. #126D3×C9semidirect
179. C42C2×D3×C9semidirect
233. D3#126⋊C9direct (N × H)
246. C3C2×#126⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×#126⋊C92
D3×C42⋊C92
C3⋊(#126⋊C9)2
C3×D3×#1263
C2×#126⋊C93
C2×D3×C22⋊C94