Subgroups — C₂×#63469⋊C₃ #13153394

Order 1728.

832 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3697⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×3₋¹⁺²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#63469⋊C₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×#3663⋊C₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×AGL(1,ℤ/9)	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#907	—	3	1	no	maximal
8	C₂×#63469	C₃	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₂×#611⋊C₃	C₄	1	2	yes	—
11	#3697⋊C₃	C₄	1	4	yes	—
12	#3697⋊C₃	C₂²	1	4	yes	—
13	C₂×C₄⋊AGL(1,ℤ/9)	C₄	1	1	yes	characteristic
14	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	2	yes	—
15	C₂×3₋¹⁺²⋊Q₈	C₄	1	1	yes	characteristic
16	M₁₆×3₋¹⁺²	C₂²	1	4	yes	complemented
17	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
18	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×3₋¹⁺²⋊C₈	C₂²	1	4	yes	complemented
21	#3663⋊C₃	C₂²	1	8	yes	complemented
22	C₈×AGL(1,ℤ/9)	C₂²	1	8	yes	complemented
23	C₂×C₃×#235	—	3	1	no	—
24	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
25	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
26	C₃×#907	—	3	8	no	—
27	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	2	no	—
28	C₂×C₃×D₃×C₈	—	3	2	no	—
29	C₂×#3697	C₂×C₃	1	1	yes	characteristic
30	C₂×C₉×M₁₆	—	2	1	no	—
31	C₂×C₉×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂²×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#63469	C₂×C₃	1	8	yes	complemented
34	C₂×#3663	C₂×C₃	1	2	yes	complemented
35	C₂×C₈×D₉	C₂×C₃	1	2	yes	complemented
36	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
37	C₂²×AGL(1,ℤ/9)	C₂×C₄	1	2	yes	—
38	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
40	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
41	C₂×C₄×3₋¹⁺²	C₂³	1	2	yes	—
42	#611⋊C₃	C₂×C₄	1	8	yes	—
43	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	2	yes	—
44	C₄⋊AGL(1,ℤ/9)	C₂×C₄	1	4	yes	—
45	C₄×AGL(1,ℤ/9)	C₂×C₄	1	4	yes	—
46	C₄×AGL(1,ℤ/9)	C₂³	1	4	yes	—
47	3₋¹⁺²⋊Q₈	C₂×C₄	1	4	yes	—
48	C₈×3₋¹⁺²	C₂³	1	4	yes	—
49	3₋¹⁺²⋊C₈	C₂³	1	2	yes	—
50	3₋¹⁺²⋊C₈	C₂³	1	2	yes	—
51	C₂×C₃×#131	—	9	1	no	—
52	C₂×#907	—	3	1	no	—
53	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×#67	—	3	2	no	—
55	C₃×#235	—	3	4	no	—
56	C₃×#235	—	3	4	no	—
57	C₂×C₃×D₁₂	—	3	1	no	—
58	C₂×C₃×C₄×D₃	—	3	2	no	—
59	C₂×C₃×Dic₆	—	3	1	no	—
60	C₃²×M₁₆	C₂×D₃	1	4	yes	—
61	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	C₂×C₃×C₃⋊C₈	—	3	4	no	—
65	C₃×ΓL(1,𝔽₂₅)	—	3	8	no	—
66	C₃×D₃×C₈	—	3	8	no	—
67	C₂²×C₄×C₉	—	2	1	no	—
68	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
69	C₂×#611	C₃×C₄	1	2	yes	—
70	#3697	C₃×C₄	1	4	yes	—
71	#3697	C₂²×C₃	1	4	yes	—
72	C₂×D₃₆	C₃×C₄	1	1	yes	characteristic
73	C₂×C₄×D₉	C₂²×C₃	1	2	yes	—
74	C₂×Dic₁₈	C₃×C₄	1	1	yes	characteristic
75	C₉×M₁₆	—	2	4	no	—
76	C₉×M₁₆	C₂²×C₃	1	4	yes	complemented
77	C₂×C₈×C₉	—	2	2	no	—
78	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
79	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₉⋊C₈	C₂²×C₃	1	4	yes	complemented
82	#3663	C₂²×C₃	1	8	yes	complemented
83	C₈×D₉	C₂²×C₃	1	8	yes	complemented
84	C₂²×3₋¹⁺²	—	2	2	no	—
85	C₂²×3₋¹⁺²	C₂²×C₄	1	2	yes	—
86	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
87	C₂×AGL(1,ℤ/9)	—	2	4	no	—
88	C₂×AGL(1,ℤ/9)	C₂²×C₄	1	4	yes	—
89	C₄×3₋¹⁺²	C₂²×C₄	1	2	yes	—
90	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
91	C₄×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
92	3₋¹⁺²⋊C₄	C₂²×C₄	1	4	yes	—
93	C₂×C₃×#41	—	9	1	no	—
94	C₂×#235	—	3	1	no	—
95	C₃×#131	—	9	8	no	—
96	C₂×C₃×M₁₆	—	3	1	no	—
97	C₂×C₃×M₁₆	—	9	2	no	—
98	C₂×C₃×M₁₆	C₃×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₈	—	9	2	no	—
100	C₂²×C₃×C₈	—	9	1	no	—
101	C₂²×C₃⋊C₈	—	3	1	no	—
102	#907	—	3	8	no	—
103	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
104	C₂×D₃×C₈	—	3	2	no	—
105	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₃×D₃	—	3	2	no	—
107	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
109	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
111	C₃×#67	—	3	8	no	—
112	C₂×C₃×Dic₃	—	3	2	no	—
113	C₃×D₁₂	—	3	4	no	—
114	C₃×C₄×D₃	—	3	4	no	—
115	C₃×C₄×D₃	—	3	4	no	—
116	C₃×Dic₆	—	3	4	no	—
117	C₂³×C₉	—	2	1	no	—
118	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂²×D₉	C₂×C₃×C₄	1	2	yes	—
120	C₃²×C₈	C₂²×D₃	1	4	yes	—
121	C₃×C₃⋊C₈	—	3	2	no	—
122	C₃×C₃⋊C₈	—	3	2	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	2	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	2	2	no	—
127	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
128	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
129	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
131	#611	C₂×C₃×C₄	1	8	yes	—
132	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
133	D₃₆	C₂×C₃×C₄	1	4	yes	—
134	C₄×D₉	C₂×C₃×C₄	1	4	yes	—
135	C₄×D₉	C₂³×C₃	1	4	yes	—
136	Dic₁₈	C₂×C₃×C₄	1	4	yes	—
137	C₈×C₉	—	2	4	no	—
138	C₈×C₉	C₂³×C₃	1	4	yes	—
139	C₉⋊C₈	C₂³×C₃	1	2	yes	—
140	C₉⋊C₈	C₂³×C₃	1	2	yes	—
141	C₂×#131	—	9	1	no	Sylow
142	C₂×3₋¹⁺²	—	2	2	no	—
143	C₂×3₋¹⁺²	#131	1	2	yes	complemented
144	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
145	AGL(1,ℤ/9)	—	2	4	no	—
146	C₃×#41	—	9	4	no	—
147	C₃×#41	—	9	4	no	—
148	C₂×C₃×Q₈	—	9	1	no	—
149	C₂×C₃×D₄	—	9	1	no	—
150	C₂×C₃×D₄	—	9	2	no	—
151	C₂²×C₃×C₄	—	3	1	no	—
152	C₂²×C₃×C₄	—	9	2	no	—
153	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
154	C₂×#67	—	3	2	no	—
155	#235	—	3	4	no	—
156	#235	—	3	4	no	—
157	C₂×D₁₂	—	3	1	no	—
158	C₂×C₄×D₃	—	3	2	no	—
159	C₂×Dic₆	—	3	1	no	—
160	C₃×M₁₆	—	3	4	no	—
161	C₃×M₁₆	—	9	8	no	—
162	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
163	C₂×C₃×C₈	—	3	2	no	—
164	C₂×C₃×C₈	—	9	8	no	—
165	C₂×C₃×C₈	—	9	1	no	—
166	C₂×C₃×C₈	—	9	1	no	—
167	C₂×C₃×C₈	—	9	4	no	—
168	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
169	C₂×C₃⋊C₈	—	3	1	no	—
170	C₂×C₃⋊C₈	—	3	1	no	—
171	C₂×C₃⋊C₈	—	3	4	no	—
172	ΓL(1,𝔽₂₅)	—	3	8	no	—
173	D₃×C₈	—	3	8	no	—
174	C₂²×C₃²	—	2	2	no	—
175	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
176	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂×C₃×D₃	—	3	4	no	—
178	C₂×C₃×D₃	—	6	4	no	—
179	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
180	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
181	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
182	C₃×Dic₃	—	3	4	no	—
183	C₂²×C₉	—	2	2	no	—
184	C₂²×C₉	—	2	1	no	—
185	C₂²×C₉	—	2	2	no	—
186	C₂²×C₉	—	4	2	no	—
187	C₂²×C₉	C₂²×C₃×C₄	1	2	yes	—
188	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
189	C₂×D₉	—	2	4	no	—
190	C₂×D₉	C₂²×C₃×C₄	1	4	yes	—
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	2	no	—
193	C₄×C₉	—	2	1	no	—
194	C₄×C₉	C₂²×C₃×C₄	1	2	yes	—
195	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
196	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
197	Dic₉	C₂²×C₃×C₄	1	4	yes	—
198	C₂×#41	—	9	1	no	—
199	#131	—	9	8	no	—
200	C₂×M₁₆	—	9	2	no	—
201	C₂×M₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
202	C₂²×C₈	—	9	1	no	—
203	C₂²×C₈	—	9	2	no	—
204	3₋¹⁺²	C₂×#131	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₃	—	3	1	no	—
206	C₂³×C₃	—	9	2	no	—
207	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
208	C₂²×D₃	—	3	2	no	—
209	C₃×Q₈	—	9	4	no	—
210	C₃×D₄	—	9	4	no	—
211	C₃×D₄	—	9	8	no	—
212	C₂×C₃×C₄	—	3	1	no	—
213	C₂×C₃×C₄	—	3	2	no	—
214	C₂×C₃×C₄	—	3	2	no	—
215	C₂×C₃×C₄	—	3	1	no	—
216	C₂×C₃×C₄	—	9	4	no	—
217	C₂×C₃×C₄	—	9	4	no	—
218	C₂×C₃×C₄	—	9	2	no	—
219	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
220	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
221	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
222	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
223	#67	—	3	8	no	—
224	C₂×Dic₃	—	3	2	no	—
225	D₁₂	—	3	4	no	—
226	C₄×D₃	—	3	4	no	—
227	C₄×D₃	—	3	4	no	—
228	Dic₆	—	3	4	no	—
229	C₃×C₈	—	3	4	no	—
230	C₃×C₈	—	9	2	no	—
231	C₃×C₈	—	9	2	no	—
232	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
233	C₃⋊C₈	—	3	2	no	—
234	C₃⋊C₈	—	3	2	no	—
235	C₂×C₃²	—	2	2	no	—
236	C₂×C₃²	#907	1	2	yes	—
237	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
238	C₃×D₃	—	6	4	no	—
239	C₂×C₉	—	2	2	no	—
240	C₂×C₉	—	2	2	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	4	2	no	—
243	C₂×C₉	C₃×#131	1	2	yes	complemented
244	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
245	D₉	—	2	4	no	—
246	#41	—	9	4	no	—
247	#41	—	9	4	no	—
248	C₂×Q₈	—	9	1	no	—
249	C₂×D₄	—	9	2	no	—
250	C₂×D₄	—	9	1	no	—
251	C₂²×C₄	—	9	2	no	—
252	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
253	M₁₆	—	9	8	no	—
254	M₁₆	C₂×AGL(1,ℤ/9)	1	4	yes	complemented
255	C₂×C₈	—	9	4	no	—
256	C₂×C₈	—	9	1	no	—
257	C₂×C₈	—	9	1	no	—
258	C₂×C₈	—	9	8	no	—
259	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	3	1	no	—
262	C₂²×C₃	—	3	2	no	—
263	C₂²×C₃	—	6	2	no	—
264	C₂²×C₃	—	9	4	no	—
265	C₂²×C₃	—	18	4	no	—
266	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
267	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
268	C₂×D₃	—	3	4	no	—
269	C₂×D₃	—	6	4	no	—
270	C₃×C₄	—	3	1	no	—
271	C₃×C₄	—	3	1	no	—
272	C₃×C₄	—	3	2	no	—
273	C₃×C₄	—	9	4	no	—
274	C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
275	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
276	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
277	Dic₃	—	3	4	no	—
278	C₃²	C₂×#907	1	1	yes	characteristic
279	C₉	—	2	1	no	—
280	C₉	C₂×C₃×#131	1	1	yes	characteristic, complemented
281	C₂³	—	9	2	no	—
282	C₂³	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
283	Q₈	—	9	4	no	—
284	D₄	—	9	8	no	—
285	D₄	—	9	4	no	—
286	C₂×C₄	—	9	2	no	—
287	C₂×C₄	—	9	4	no	—
288	C₂×C₄	—	9	4	no	—
289	C₂×C₄	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
290	C₂×C₄	C₄×AGL(1,ℤ/9)	1	2	yes	—
291	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	2	yes	—
292	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
293	C₈	—	9	2	no	—
294	C₈	—	9	2	no	—
295	C₈	C₂²×AGL(1,ℤ/9)	1	4	yes	—
296	C₂×C₃	—	2	2	no	—
297	C₂×C₃	—	3	2	no	—
298	C₂×C₃	—	3	1	no	—
299	C₂×C₃	—	6	2	no	—
300	C₂×C₃	—	18	4	no	—
301	C₂×C₃	C₃×#907	1	2	yes	—
302	C₂×C₃	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
303	D₃	—	6	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	9	4	no	—
306	C₂²	—	18	4	no	—
307	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
308	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
309	C₄	—	9	4	no	—
310	C₄	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
311	C₄	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
312	C₄	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
313	C₃	—	3	1	no	—
314	C₃	C₂×C₃×#907	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	18	4	no	—
317	C₂	#63469⋊C₃	1	2	yes	direct product factor
318	C₂	C₂²×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂³×C₃×C₄
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂×M₁₆×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×C₄×D₃
prime radical	C₂×#3697⋊C₃	C₂
cosocle	C₄×C₉	C₂⁴×C₃
hypercenter	C₂×M₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×#131
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×3₋¹⁺²	C₂	semidirect
3. C₂²×3₋¹⁺²⋊C₈	C₂	semidirect
4. #63469⋊C₃	C₂	direct (N × H)
5. C₂×#3663⋊C₃	C₂	semidirect
6. C₂×C₈×AGL(1,ℤ/9)	C₂	semidirect
8. C₂×#63469	C₃	semidirect
16. M₁₆×3₋¹⁺²	C₂²	semidirect
20. C₂×3₋¹⁺²⋊C₈	C₂²	semidirect
21. #3663⋊C₃	C₂²	semidirect
22. C₈×AGL(1,ℤ/9)	C₂²	semidirect
31. C₂×C₉×M₁₆	C₂×C₃	semidirect
32. C₂²×C₉⋊C₈	C₂×C₃	semidirect
33. #63469	C₂×C₃	semidirect
34. C₂×#3663	C₂×C₃	semidirect
35. C₂×C₈×D₉	C₂×C₃	semidirect
76. C₉×M₁₆	C₂²×C₃	semidirect
81. C₂×C₉⋊C₈	C₂²×C₃	semidirect
82. #3663	C₂²×C₃	semidirect
83. C₈×D₉	C₂²×C₃	semidirect
143. C₂×3₋¹⁺²	#131	semidirect
201. C₂×M₁₆	AGL(1,ℤ/9)	semidirect
204. 3₋¹⁺²	C₂×#131	semidirect
243. C₂×C₉	C₃×#131	semidirect
254. M₁₆	C₂×AGL(1,ℤ/9)	semidirect
280. C₉	C₂×C₃×#131	semidirect
317. C₂	#63469⋊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×#3697⋊C₃	2
C₂×M₁₆×3₋¹⁺²	2
C₂²×3₋¹⁺²⋊C₈	2
#63469⋊C₃	2
C₂×#3663⋊C₃	2
C₂×C₈×AGL(1,ℤ/9)	2
C₂×C₃×#907	3
C₂×#63469	3

Subgroups — C2×#63469⋊C3 #13153394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#63469⋊C₃ #13153394