Subgroups — C₂×#63481⋊C₃ #13153469

Order 1728.

828 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88724	C₂	1	2	yes	maximal, complemented
2	C₂×Q₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#3675⋊C₃	C₂	1	2	yes	maximal, complemented
4	#63481⋊C₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×D₇₂⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#919	—	3	1	no	maximal
8	C₂×#63481	C₃	1	1	yes	maximal, characteristic, complemented
9	C₂×Q₈×3₋¹⁺²	C₂²	1	2	yes	—
10	#88724	—	2	4	no	—
11	#88724	C₂²	1	8	yes	complemented
12	C₂×C₄⋊AGL(1,ℤ/9)	—	2	2	no	—
13	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	2	yes	—
14	C₂×C₄×AGL(1,ℤ/9)	—	2	2	no	—
15	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
16	Q₁₆×3₋¹⁺²	C₂²	1	4	yes	complemented
17	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
18	#3675⋊C₃	C₂²	1	8	yes	complemented
19	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
20	D₇₂⋊C₃	C₂²	1	4	yes	complemented
21	C₈×AGL(1,ℤ/9)	C₂²	1	4	yes	complemented
22	C₂×C₃×#239	—	3	2	no	—
23	C₂×C₃²×Q₁₆	D₃	1	1	yes	characteristic
24	C₂×C₃×#215	—	3	2	no	—
25	C₃×#919	—	3	8	no	—
26	C₂×C₃×D₂₄	—	3	1	no	—
27	C₂×C₃×D₃×C₈	—	3	1	no	—
28	C₂×#3701	C₂×C₃	1	2	yes	complemented
29	C₂×C₉×Q₁₆	—	2	1	no	—
30	C₂×C₉×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#3675	C₂×C₃	1	2	yes	complemented
32	#63481	C₂×C₃	1	8	yes	complemented
33	C₂×D₇₂	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₈×D₉	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×AGL(1,ℤ/9)	—	2	2	no	—
36	C₂²×AGL(1,ℤ/9)	D₄	1	1	yes	characteristic
37	Q₈×3₋¹⁺²	—	2	2	no	—
38	Q₈×3₋¹⁺²	C₂³	1	4	yes	—
39	C₂×C₄×3₋¹⁺²	—	2	2	no	—
40	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
41	C₂×3₋¹⁺²⋊C₄	D₄	1	1	yes	characteristic
42	C₄⋊AGL(1,ℤ/9)	—	2	8	no	—
43	C₄⋊AGL(1,ℤ/9)	—	2	2	no	—
44	C₄⋊AGL(1,ℤ/9)	C₂³	1	4	yes	—
45	C₄×AGL(1,ℤ/9)	—	2	8	no	—
46	C₄×AGL(1,ℤ/9)	D₄	1	2	yes	—
47	C₄×AGL(1,ℤ/9)	C₂³	1	2	yes	—
48	C₈×3₋¹⁺²	C₂³	1	2	yes	—
49	3₋¹⁺²⋊C₈	C₂³	1	2	yes	—
50	C₂×C₃×#135	—	9	1	no	—
51	C₂×#919	—	3	1	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
53	C₃×#239	—	3	8	no	—
54	C₃×#239	—	6	4	no	—
55	C₂×C₃×D₁₂	—	3	2	no	—
56	C₂×C₃×D₁₂	—	6	2	no	—
57	C₂×C₃×C₄×D₃	—	3	1	no	—
58	C₂×C₃×C₄×D₃	—	6	2	no	—
59	C₃²×Q₁₆	C₂×D₃	1	4	yes	—
60	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×#215	—	3	8	no	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₃×D₂₄	—	3	4	no	—
64	C₃×D₃×C₈	—	3	4	no	—
65	C₂×Q₈×C₉	—	2	2	no	—
66	C₂×Q₈×C₉	C₂²×C₃	1	2	yes	—
67	#3701	—	2	4	no	—
68	#3701	C₂²×C₃	1	8	yes	complemented
69	C₂×D₃₆	—	2	2	no	—
70	C₂×D₃₆	C₂²×C₃	1	2	yes	—
71	C₂×C₄×D₉	—	2	2	no	—
72	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
73	C₉×Q₁₆	—	2	4	no	—
74	C₉×Q₁₆	C₂²×C₃	1	4	yes	complemented
75	C₂×C₈×C₉	—	2	1	no	—
76	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
77	#3675	C₂²×C₃	1	8	yes	complemented
78	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
79	D₇₂	C₂²×C₃	1	4	yes	complemented
80	C₈×D₉	C₂²×C₃	1	4	yes	complemented
81	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
82	C₂×AGL(1,ℤ/9)	—	2	2	no	—
83	C₂×AGL(1,ℤ/9)	—	2	4	no	—
84	C₂×AGL(1,ℤ/9)	—	4	4	no	—
85	C₂×AGL(1,ℤ/9)	C₂×D₄	1	2	yes	—
86	C₄×3₋¹⁺²	—	2	4	no	—
87	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
88	C₄×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
89	3₋¹⁺²⋊C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×#41	—	9	2	no	—
91	C₂×#239	—	3	2	no	—
92	C₃×#135	—	9	8	no	—
93	C₂×C₃×Q₁₆	—	3	1	no	—
94	C₂×C₃×Q₁₆	C₃×D₃	1	1	yes	characteristic
95	C₂×C₃×SD₁₆	—	9	2	no	—
96	C₂×C₃×D₈	—	9	1	no	—
97	C₂²×C₃×C₈	—	9	1	no	—
98	C₂×#215	—	3	2	no	—
99	#919	—	3	8	no	—
100	C₂×D₂₄	—	3	1	no	—
101	C₂×D₃×C₈	—	3	1	no	—
102	C₂²×C₃×D₃	—	3	1	no	—
103	C₂²×C₃×D₃	—	6	2	no	—
104	C₃²×Q₈	—	2	2	no	—
105	C₃²×Q₈	C₂²×D₃	1	4	yes	—
106	C₂×C₃²×C₄	—	2	2	no	—
107	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₂×C₃×Dic₃	—	3	1	no	—
109	C₃×D₁₂	—	3	4	no	—
110	C₃×D₁₂	—	6	8	no	—
111	C₃×D₁₂	—	6	2	no	—
112	C₃×C₄×D₃	—	3	2	no	—
113	C₃×C₄×D₃	—	3	2	no	—
114	C₃×C₄×D₃	—	6	8	no	—
115	C₂²×D₉	—	2	2	no	—
116	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
117	C₃²×C₈	C₂²×D₃	1	2	yes	—
118	C₃×C₃⋊C₈	—	3	2	no	—
119	Q₈×C₉	—	2	2	no	—
120	Q₈×C₉	—	2	4	no	—
121	Q₈×C₉	—	4	2	no	—
122	Q₈×C₉	C₂³×C₃	1	4	yes	—
123	C₂×C₄×C₉	—	2	2	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	4	2	no	—
126	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
127	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
128	D₃₆	—	2	8	no	—
129	D₃₆	—	2	2	no	—
130	D₃₆	C₂³×C₃	1	4	yes	—
131	C₄×D₉	—	2	8	no	—
132	C₄×D₉	C₃×D₄	1	2	yes	—
133	C₄×D₉	C₂³×C₃	1	2	yes	—
134	C₈×C₉	—	2	2	no	—
135	C₈×C₉	C₂³×C₃	1	2	yes	—
136	C₉⋊C₈	C₂³×C₃	1	2	yes	—
137	C₂×#135	—	9	1	no	Sylow
138	C₂×3₋¹⁺²	#135	1	2	yes	complemented
139	C₂×3₋¹⁺²	C₂²×D₄	1	1	yes	characteristic
140	AGL(1,ℤ/9)	—	2	2	no	—
141	AGL(1,ℤ/9)	—	4	4	no	—
142	C₃×#41	—	9	8	no	—
143	C₃×#41	—	18	4	no	—
144	C₂×C₃×Q₈	—	3	2	no	—
145	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
146	C₂×C₃×D₄	—	9	2	no	—
147	C₂×C₃×D₄	—	18	2	no	—
148	C₂²×C₃×C₄	—	9	1	no	—
149	C₂²×C₃×C₄	—	18	2	no	—
150	#239	—	3	8	no	—
151	#239	—	6	4	no	—
152	C₂×D₁₂	—	3	2	no	—
153	C₂×D₁₂	—	6	2	no	—
154	C₂×C₄×D₃	—	3	1	no	—
155	C₂×C₄×D₃	—	6	2	no	—
156	C₃×Q₁₆	—	3	4	no	—
157	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	—
158	C₃×SD₁₆	—	9	8	no	—
159	C₃×D₈	—	9	4	no	—
160	C₂×C₃×C₈	—	3	1	no	—
161	C₂×C₃×C₈	—	9	4	no	—
162	C₂×C₃×C₈	—	9	1	no	—
163	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
164	#215	—	3	8	no	—
165	C₂×C₃⋊C₈	—	3	1	no	—
166	D₂₄	—	3	4	no	—
167	D₃×C₈	—	3	4	no	—
168	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
169	C₂×C₃×D₃	—	3	2	no	—
170	C₂×C₃×D₃	—	6	2	no	—
171	C₂×C₃×D₃	—	6	4	no	—
172	C₂×C₃×D₃	—	12	4	no	—
173	C₃²×C₄	—	2	4	no	—
174	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
175	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
176	C₃×Dic₃	—	3	2	no	—
177	C₂²×C₉	—	2	1	no	—
178	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×D₉	—	2	2	no	—
180	C₂×D₉	—	2	4	no	—
181	C₂×D₉	—	4	4	no	—
182	C₂×D₉	C₂×C₃×D₄	1	2	yes	—
183	C₄×C₉	—	2	4	no	—
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	4	4	no	—
187	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
188	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
189	Dic₉	C₂×C₃×D₄	1	2	yes	—
190	C₂×#41	—	9	2	no	—
191	#135	—	9	8	no	—
192	C₂×Q₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
193	C₂×SD₁₆	—	9	2	no	—
194	C₂×D₈	—	9	1	no	—
195	C₂²×C₈	—	9	1	no	—
196	3₋¹⁺²	C₂×#135	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₃	—	9	1	no	—
198	C₂³×C₃	—	18	2	no	—
199	C₂²×D₃	—	3	1	no	—
200	C₂²×D₃	—	6	2	no	—
201	C₃×Q₈	—	2	2	no	—
202	C₃×Q₈	—	3	4	no	—
203	C₃×Q₈	—	6	2	no	—
204	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
205	C₃×D₄	—	9	4	no	—
206	C₃×D₄	—	18	2	no	—
207	C₃×D₄	—	18	8	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	3	1	no	—
210	C₂×C₃×C₄	—	6	2	no	—
211	C₂×C₃×C₄	—	9	2	no	—
212	C₂×C₃×C₄	—	9	2	no	—
213	C₂×C₃×C₄	—	9	1	no	—
214	C₂×C₃×C₄	—	18	8	no	—
215	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
216	C₂×Dic₃	—	3	1	no	—
217	D₁₂	—	3	4	no	—
218	D₁₂	—	6	8	no	—
219	D₁₂	—	6	2	no	—
220	C₄×D₃	—	3	2	no	—
221	C₄×D₃	—	3	2	no	—
222	C₄×D₃	—	6	8	no	—
223	C₃×C₈	—	3	2	no	—
224	C₃×C₈	—	9	2	no	—
225	C₃×C₈	C₂²×C₃×D₃	1	2	yes	—
226	C₃⋊C₈	—	3	2	no	—
227	C₂×C₃²	#919	1	2	yes	—
228	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
229	C₃×D₃	—	6	2	no	—
230	C₃×D₃	—	12	4	no	—
231	C₂×C₉	—	2	2	no	—
232	C₂×C₉	—	2	1	no	—
233	C₂×C₉	C₃×#135	1	2	yes	complemented
234	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
235	D₉	—	2	2	no	—
236	D₉	—	4	4	no	—
237	#41	—	9	8	no	—
238	#41	—	18	4	no	—
239	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
240	C₂×D₄	—	9	2	no	—
241	C₂×D₄	—	18	2	no	—
242	C₂²×C₄	—	9	1	no	—
243	C₂²×C₄	—	18	2	no	—
244	Q₁₆	C₂×AGL(1,ℤ/9)	1	4	yes	complemented
245	SD₁₆	—	9	8	no	—
246	D₈	—	9	4	no	—
247	C₂×C₈	—	9	1	no	—
248	C₂×C₈	—	9	4	no	—
249	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
250	C₂²×C₃	—	3	1	no	—
251	C₂²×C₃	—	9	2	no	—
252	C₂²×C₃	—	18	2	no	—
253	C₂²×C₃	—	18	4	no	—
254	C₂²×C₃	—	36	4	no	—
255	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
256	C₂×D₃	—	3	2	no	—
257	C₂×D₃	—	6	2	no	—
258	C₂×D₃	—	6	4	no	—
259	C₂×D₃	—	12	4	no	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	3	1	no	—
262	C₃×C₄	—	3	1	no	—
263	C₃×C₄	—	6	4	no	—
264	C₃×C₄	—	9	2	no	—
265	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
266	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
267	Dic₃	—	3	2	no	—
268	C₃²	C₂×#919	1	1	yes	characteristic
269	C₉	—	2	1	no	—
270	C₉	C₂×C₃×#135	1	1	yes	characteristic, complemented
271	C₂³	—	9	1	no	—
272	C₂³	—	18	2	no	—
273	Q₈	—	2	2	no	—
274	Q₈	C₂²×AGL(1,ℤ/9)	1	4	yes	—
275	D₄	—	9	4	no	—
276	D₄	—	18	8	no	—
277	D₄	—	18	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	9	1	no	—
280	C₂×C₄	—	9	2	no	—
281	C₂×C₄	—	9	2	no	—
282	C₂×C₄	—	18	8	no	—
283	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
284	C₈	—	9	2	no	—
285	C₈	C₂²×AGL(1,ℤ/9)	1	2	yes	—
286	C₂×C₃	—	3	2	no	—
287	C₂×C₃	—	3	1	no	—
288	C₂×C₃	—	18	2	no	—
289	C₂×C₃	—	36	4	no	—
290	C₂×C₃	C₃×#919	1	2	yes	—
291	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
292	D₃	—	6	2	no	—
293	D₃	—	12	4	no	—
294	C₂²	—	9	2	no	—
295	C₂²	—	18	2	no	—
296	C₂²	—	18	4	no	—
297	C₂²	—	36	4	no	—
298	C₂²	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
299	C₄	—	2	4	no	—
300	C₄	—	9	2	no	—
301	C₄	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
302	C₄	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
303	C₃	—	3	1	no	—
304	C₃	C₂×C₃×#919	1	1	yes	characteristic
305	C₂	—	18	2	no	—
306	C₂	—	36	4	no	—
307	C₂	#63481⋊C₃	1	2	yes	direct product factor
308	C₂	C₂×D₄×AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,ℤ/9)
commutator, cosocle	C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂×Q₁₆×3₋¹⁺²	C₂
socle	C₂²×C₃	C₃×D₃×D₄
hypercenter	C₂×Q₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×#135
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#88724	C₂	semidirect
2. C₂×Q₁₆×3₋¹⁺²	C₂	semidirect
3. C₂×#3675⋊C₃	C₂	semidirect
4. #63481⋊C₃	C₂	direct (N × H)
5. C₂×D₇₂⋊C₃	C₂	semidirect
6. C₂×C₈×AGL(1,ℤ/9)	C₂	semidirect
8. C₂×#63481	C₃	semidirect
11. #88724	C₂²	semidirect
16. Q₁₆×3₋¹⁺²	C₂²	semidirect
18. #3675⋊C₃	C₂²	semidirect
20. D₇₂⋊C₃	C₂²	semidirect
21. C₈×AGL(1,ℤ/9)	C₂²	semidirect
28. C₂×#3701	C₂×C₃	semidirect
30. C₂×C₉×Q₁₆	C₂×C₃	semidirect
31. C₂×#3675	C₂×C₃	semidirect
32. #63481	C₂×C₃	semidirect
33. C₂×D₇₂	C₂×C₃	semidirect
34. C₂×C₈×D₉	C₂×C₃	semidirect
68. #3701	C₂²×C₃	semidirect
74. C₉×Q₁₆	C₂²×C₃	semidirect
77. #3675	C₂²×C₃	semidirect
79. D₇₂	C₂²×C₃	semidirect
80. C₈×D₉	C₂²×C₃	semidirect
138. C₂×3₋¹⁺²	#135	semidirect
192. C₂×Q₁₆	AGL(1,ℤ/9)	semidirect
196. 3₋¹⁺²	C₂×#135	semidirect
233. C₂×C₉	C₃×#135	semidirect
244. Q₁₆	C₂×AGL(1,ℤ/9)	semidirect
270. C₉	C₂×C₃×#135	semidirect
307. C₂	#63481⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#88724	2
C₂×Q₁₆×3₋¹⁺²	2
C₂×#3675⋊C₃	2
#63481⋊C₃	2
C₂×D₇₂⋊C₃	2
C₂×C₈×AGL(1,ℤ/9)	2
C₂×C₃×#919	3
C₂×#63481	3

Subgroups — C2×#63481⋊C3 #13153469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#63481⋊C₃ #13153469