Subgroups — C₂²×Dic₉⋊Dic₃ #13151961

Order 1728.

1490 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₉⋊Dic₃	C₂	1	12	yes	maximal, direct product factor
5	C₂²×Dic₃⋊Dic₃	—	3	1	no	maximal
6	C₂²×Dic₉⋊C₄	—	3	1	no	maximal
7	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	—	2	4	no	—
9	C₂²×C₃⋊Dic₉	C₂²	1	6	yes	—
10	C₂²×C₉×Dic₃	—	2	4	no	—
11	C₂²×C₉×Dic₃	C₂²	1	6	yes	—
12	C₂²×C₃×Dic₉	C₄	1	8	yes	complemented
13	C₂²×C₃×Dic₉	C₂²	1	6	yes	—
14	Dic₉⋊Dic₃	C₂²	1	16	yes	direct product factor
15	C₂³×C₃⋊Dic₃	—	3	1	no	—
16	C₂³×C₃×Dic₃	—	3	1	no	—
17	C₂³×C₃×Dic₃	D₃	1	1	yes	characteristic
18	C₂×Dic₃⋊Dic₃	—	3	12	no	—
19	C₂³×C₄×C₉	—	3	1	no	—
20	C₂³×Dic₉	—	3	1	no	—
21	C₂³×Dic₉	—	6	1	no	—
22	C₂³×Dic₉	D₃	1	1	yes	characteristic
23	C₂×Dic₉⋊C₄	—	3	12	no	—
24	C₂³×C₃×C₉	C₂×C₄	1	4	yes	—
25	C₂³×C₃×C₉	D₄	1	4	yes	—
26	C₂³×C₃×C₉	Q₈	1	4	yes	—
27	C₂³×C₃×C₉	C₂³	1	3	yes	—
28	C₂×C₃⋊Dic₉	—	2	12	no	—
29	C₂×C₃⋊Dic₉	C₂³	1	4	yes	—
30	C₂×C₉×Dic₃	—	2	12	no	—
31	C₂×C₉×Dic₃	C₂³	1	4	yes	—
32	C₂×C₃×Dic₉	C₂×C₄	1	24	yes	complemented
33	C₂×C₃×Dic₉	C₂³	1	4	yes	—
34	C₂²×C₄⋊Dic₃	—	9	1	no	—
35	C₂²×Dic₃⋊C₄	—	9	1	no	—
36	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃⋊Dic₃	—	3	6	no	—
38	C₂²×C₃⋊Dic₃	—	6	4	no	—
39	C₂²×C₃×Dic₃	—	2	4	no	—
40	C₂²×C₃×Dic₃	—	3	8	no	—
41	C₂²×C₃×Dic₃	—	3	6	no	—
42	C₂²×C₃×Dic₃	C₂×D₃	1	6	yes	—
43	C₂⁴×C₉	—	2	1	no	—
44	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
45	Dic₃⋊Dic₃	—	3	16	no	—
46	C₂²×C₄×C₉	—	3	6	no	—
47	C₂²×C₄×C₉	—	6	4	no	—
48	C₂²×Dic₉	—	3	6	no	—
49	C₂²×Dic₉	—	6	6	no	—
50	C₂²×Dic₉	—	6	8	no	—
51	C₂²×Dic₉	—	6	4	no	—
52	C₂²×Dic₉	Dic₃	1	8	yes	complemented
53	C₂²×Dic₉	C₂×D₃	1	6	yes	—
54	Dic₉⋊C₄	—	3	16	no	—
55	C₂²×C₃×C₉	C₄⋊C₄	1	16	yes	complemented
56	C₂²×C₃×C₉	C₂²×C₄	1	6	yes	—
57	C₂²×C₃×C₉	C₂×D₄	1	6	yes	—
58	C₂²×C₃×C₉	C₂×Q₈	1	6	yes	—
59	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
60	C₃⋊Dic₉	—	2	4	no	—
61	C₉×Dic₃	—	2	4	no	—
62	C₃×Dic₉	C₂²×C₄	1	8	yes	complemented
63	C₂³×C₃×C₄	—	3	1	no	—
64	C₂³×C₃×C₄	—	9	1	no	—
65	C₂³×Dic₃	—	3	1	no	—
66	C₂³×Dic₃	—	9	1	no	—
67	C₂³×Dic₃	—	9	1	no	—
68	C₂³×Dic₃	—	18	1	no	—
69	C₂³×Dic₃	D₉	1	1	yes	characteristic
70	C₂×C₄⋊Dic₃	—	9	12	no	—
71	C₂×Dic₃⋊C₄	—	9	12	no	—
72	C₂³×C₃²	Dic₆	1	4	yes	—
73	C₂³×C₃²	C₄×D₃	1	4	yes	—
74	C₂³×C₃²	#67	1	4	yes	—
75	C₂³×C₃²	C₂²×D₃	1	3	yes	—
76	C₂×C₃⋊Dic₃	—	3	4	no	—
77	C₂×C₃⋊Dic₃	—	6	12	no	—
78	C₂×C₃×Dic₃	—	2	12	no	—
79	C₂×C₃×Dic₃	—	3	24	no	—
80	C₂×C₃×Dic₃	—	3	4	no	—
81	C₂×C₃×Dic₃	C₂²×D₃	1	4	yes	—
82	C₂³×C₉	—	2	4	no	—
83	C₂³×C₉	—	2	4	no	—
84	C₂³×C₉	—	2	4	no	—
85	C₂³×C₉	—	2	3	no	—
86	C₂³×C₉	Dic₆	1	4	yes	—
87	C₂³×C₉	D₁₂	1	4	yes	—
88	C₂³×C₉	C₂×Dic₃	1	4	yes	—
89	C₂³×C₉	C₂²×D₃	1	3	yes	—
90	C₂×C₄×C₉	—	3	4	no	—
91	C₂×C₄×C₉	—	6	12	no	—
92	C₂×Dic₉	—	3	4	no	—
93	C₂×Dic₉	—	6	4	no	—
94	C₂×Dic₉	—	6	24	no	—
95	C₂×Dic₉	—	6	12	no	—
96	C₂×Dic₉	C₂×Dic₃	1	24	yes	complemented
97	C₂×Dic₉	C₂²×D₃	1	4	yes	—
98	C₂²×C₄⋊C₄	—	27	1	no	Sylow
99	C₂×C₃×C₉	C₂×C₄⋊C₄	1	12	yes	complemented
100	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
103	C₂⁴×C₃	—	2	1	no	—
104	C₂⁴×C₃	C₂×D₉	1	1	yes	characteristic
105	C₂⁴×C₃	D₃²	1	1	yes	characteristic
106	C₂²×C₃×C₄	—	3	6	no	—
107	C₂²×C₃×C₄	—	6	4	no	—
108	C₂²×C₃×C₄	—	9	8	no	—
109	C₂²×C₃×C₄	—	9	6	no	—
110	C₂²×Dic₃	—	2	4	no	—
111	C₂²×Dic₃	—	3	8	no	—
112	C₂²×Dic₃	—	3	6	no	—
113	C₂²×Dic₃	—	9	6	no	—
114	C₂²×Dic₃	—	9	6	no	—
115	C₂²×Dic₃	—	18	6	no	—
116	C₂²×Dic₃	—	18	8	no	—
117	C₂²×Dic₃	—	18	4	no	—
118	C₂²×Dic₃	—	18	4	no	—
119	C₂²×Dic₃	C₂×D₉	1	6	yes	—
120	C₄⋊Dic₃	—	9	16	no	—
121	Dic₃⋊C₄	—	9	16	no	—
122	C₂²×C₃²	Dic₃⋊C₄	1	16	yes	—
123	C₂²×C₃²	C₂×Dic₆	1	6	yes	—
124	C₂²×C₃²	C₂×C₄×D₃	1	6	yes	—
125	C₂²×C₃²	C₂×#67	1	6	yes	—
126	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
127	C₃⋊Dic₃	—	6	4	no	—
128	C₃×Dic₃	—	2	4	no	—
129	C₃×Dic₃	—	3	8	no	—
130	C₂²×C₉	—	2	1	no	—
131	C₂²×C₉	—	2	16	no	—
132	C₂²×C₉	—	2	6	no	—
133	C₂²×C₉	—	2	6	no	—
134	C₂²×C₉	—	2	6	no	—
135	C₂²×C₉	C₄⋊Dic₃	1	16	yes	complemented
136	C₂²×C₉	C₂×Dic₆	1	6	yes	—
137	C₂²×C₉	C₂×D₁₂	1	6	yes	—
138	C₂²×C₉	C₂²×Dic₃	1	6	yes	—
139	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
140	C₄×C₉	—	6	4	no	—
141	Dic₉	—	6	8	no	—
142	Dic₉	—	6	4	no	—
143	Dic₉	C₂²×Dic₃	1	8	yes	complemented
144	C₂³×C₄	—	3	1	no	—
145	C₂³×C₄	—	9	1	no	—
146	C₂³×C₄	—	27	1	no	—
147	C₂×C₄⋊C₄	—	27	12	no	—
148	C₃×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	—	2	4	no	—
150	C₂³×C₃	—	2	4	no	—
151	C₂³×C₃	—	2	4	no	—
152	C₂³×C₃	—	2	3	no	—
153	C₂³×C₃	Dic₁₈	1	4	yes	—
154	C₂³×C₃	C₄×D₉	1	4	yes	—
155	C₂³×C₃	#611	1	4	yes	—
156	C₂³×C₃	C₂²×D₉	1	3	yes	—
157	C₂³×C₃	D₃×Dic₃	1	4	yes	—
158	C₂³×C₃	#626	1	4	yes	—
159	C₂³×C₃	#627	1	4	yes	—
160	C₂³×C₃	C₂×D₃²	1	3	yes	—
161	C₂×C₃×C₄	—	3	4	no	—
162	C₂×C₃×C₄	—	6	12	no	—
163	C₂×C₃×C₄	—	9	24	no	—
164	C₂×C₃×C₄	—	9	4	no	—
165	C₂×Dic₃	—	2	12	no	—
166	C₂×Dic₃	—	3	4	no	—
167	C₂×Dic₃	—	3	24	no	—
168	C₂×Dic₃	—	9	4	no	—
169	C₂×Dic₃	—	9	4	no	—
170	C₂×Dic₃	—	18	4	no	—
171	C₂×Dic₃	—	18	24	no	—
172	C₂×Dic₃	—	18	12	no	—
173	C₂×Dic₃	—	18	12	no	—
174	C₂×Dic₃	C₂²×D₉	1	4	yes	—
175	C₂×C₃²	C₂×Dic₃⋊C₄	1	12	yes	—
176	C₂×C₃²	C₂²×Dic₆	1	1	yes	characteristic
177	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
179	C₂×C₉	—	2	1	no	—
180	C₂×C₉	—	2	1	no	—
181	C₂×C₉	—	2	1	no	—
182	C₂×C₉	—	2	12	no	—
183	C₂×C₉	C₂×C₄⋊Dic₃	1	12	yes	complemented
184	C₂×C₉	C₂²×Dic₆	1	1	yes	characteristic
185	C₂×C₉	C₂²×D₁₂	1	1	yes	characteristic
186	C₂×C₉	C₂³×Dic₃	1	1	yes	characteristic
187	C₂⁴	D₃×D₉	1	1	yes	characteristic
188	C₂²×C₄	—	3	6	no	—
189	C₂²×C₄	—	6	4	no	—
190	C₂²×C₄	—	9	8	no	—
191	C₂²×C₄	—	9	6	no	—
192	C₂²×C₄	—	27	6	no	—
193	C₂²×C₄	—	54	4	no	—
194	C₄⋊C₄	—	27	16	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	6	no	—
197	C₂²×C₃	—	2	6	no	—
198	C₂²×C₃	—	2	6	no	—
199	C₂²×C₃	—	2	16	no	—
200	C₂²×C₃	Dic₉⋊C₄	1	16	yes	complemented
201	C₂²×C₃	C₂×Dic₁₈	1	6	yes	—
202	C₂²×C₃	C₂×C₄×D₉	1	6	yes	—
203	C₂²×C₃	C₂×#611	1	6	yes	—
204	C₂²×C₃	Dic₃⋊Dic₃	1	16	yes	—
205	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
206	C₂²×C₃	C₂×D₃×Dic₃	1	6	yes	—
207	C₂²×C₃	C₂×#626	1	6	yes	—
208	C₂²×C₃	C₂×#627	1	6	yes	—
209	C₂²×C₃	C₂²×D₃²	1	1	yes	characteristic
210	C₃×C₄	—	6	4	no	—
211	C₃×C₄	—	9	8	no	—
212	Dic₃	—	2	4	no	—
213	Dic₃	—	3	8	no	—
214	Dic₃	—	18	8	no	—
215	Dic₃	—	18	4	no	—
216	Dic₃	—	18	4	no	—
217	C₃²	C₂²×Dic₃⋊C₄	1	1	yes	characteristic
218	C₉	—	2	1	no	—
219	C₉	C₂²×C₄⋊Dic₃	1	1	yes	characteristic, complemented
220	C₂³	#6189	1	4	yes	—
221	C₂³	Dic₃×D₉	1	4	yes	—
222	C₂³	#6195	1	4	yes	—
223	C₂³	C₂×D₃×D₉	1	3	yes	—
224	C₂×C₄	—	3	4	no	—
225	C₂×C₄	—	6	12	no	—
226	C₂×C₄	—	9	4	no	—
227	C₂×C₄	—	9	24	no	—
228	C₂×C₄	—	27	4	no	—
229	C₂×C₄	—	54	12	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	12	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	C₂×Dic₉⋊C₄	1	12	yes	complemented
235	C₂×C₃	C₂²×Dic₁₈	1	1	yes	characteristic
236	C₂×C₃	C₂²×C₄×D₉	1	1	yes	characteristic
237	C₂×C₃	C₂²×#611	1	1	yes	characteristic
238	C₂×C₃	C₂×Dic₃⋊Dic₃	1	12	yes	—
239	C₂×C₃	C₂²×D₃×Dic₃	1	1	yes	characteristic
240	C₂×C₃	C₂²×#626	1	1	yes	characteristic
241	C₂×C₃	C₂²×#627	1	1	yes	characteristic
242	C₂²	Dic₉⋊Dic₃	1	16	yes	direct product factor
243	C₂²	C₂×#6189	1	6	yes	—
244	C₂²	C₂×Dic₃×D₉	1	6	yes	—
245	C₂²	C₂×#6195	1	6	yes	—
246	C₂²	C₂²×D₃×D₉	1	1	yes	characteristic
247	C₄	—	6	4	no	—
248	C₄	—	9	8	no	—
249	C₄	—	54	4	no	—
250	C₃	—	2	1	no	—
251	C₃	C₂²×Dic₉⋊C₄	1	1	yes	characteristic, complemented
252	C₃	C₂²×Dic₃⋊Dic₃	1	1	yes	characteristic
253	C₂	C₂×Dic₉⋊Dic₃	1	12	yes	direct product factor
254	C₂	C₂²×#6189	1	1	yes	characteristic
255	C₂	C₂²×Dic₃×D₉	1	1	yes	characteristic
256	C₂	C₂²×#6195	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	D₃×D₉
commutator	C₂×C₃×C₉	C₂³×C₄
Frattini	C₂²×C₃	C₂²×D₃²
Fitting	C₂⁴×C₃×C₉	C₂²
socle, prime radical	C₂⁴×C₃²	C₂×D₃
cosocle	C₂²×C₃×C₉	C₂⁴
hypocenter	C₃×C₉	C₂²×C₄⋊C₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₉⋊Dic₃	C₂	direct (N × H)
12. C₂²×C₃×Dic₉	C₄	semidirect
14. Dic₉⋊Dic₃	C₂²	direct (N × H)
32. C₂×C₃×Dic₉	C₂×C₄	semidirect
52. C₂²×Dic₉	Dic₃	semidirect
55. C₂²×C₃×C₉	C₄⋊C₄	semidirect
62. C₃×Dic₉	C₂²×C₄	semidirect
96. C₂×Dic₉	C₂×Dic₃	semidirect
99. C₂×C₃×C₉	C₂×C₄⋊C₄	semidirect
135. C₂²×C₉	C₄⋊Dic₃	semidirect
143. Dic₉	C₂²×Dic₃	semidirect
148. C₃×C₉	C₂²×C₄⋊C₄	semidirect
183. C₂×C₉	C₂×C₄⋊Dic₃	semidirect
200. C₂²×C₃	Dic₉⋊C₄	semidirect
219. C₉	C₂²×C₄⋊Dic₃	semidirect
234. C₂×C₃	C₂×Dic₉⋊C₄	semidirect
242. C₂²	Dic₉⋊Dic₃	direct (N × H)
251. C₃	C₂²×Dic₉⋊C₄	semidirect
253. C₂	C₂×Dic₉⋊Dic₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃⋊Dic₉	2
C₂³×C₉×Dic₃	2
C₂³×C₃×Dic₉	2
C₂×Dic₉⋊Dic₃	2
C₂²×Dic₃⋊Dic₃	3
C₂²×Dic₉⋊C₄	3

Subgroups — C22×Dic9⋊Dic3 #13151961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₉⋊Dic₃ #13151961