Subgroups — C₂×C₃²×#469 #11770101

Order 1152.

1788 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#121	C₂	1	2	yes	maximal, complemented
4	C₃²×#469	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#102	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#469	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃²×D₄	—	2	2	no	—
9	C₂²×C₃²×D₄	C₂²	1	2	yes	—
10	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃²×D₈	—	2	4	no	—
12	C₂×C₃²×D₈	C₂²	1	4	yes	complemented
13	C₂×C₃²×M₁₆	—	2	4	no	—
14	C₂×C₃²×M₁₆	C₂²	1	4	yes	complemented
15	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
17	C₃²×#121	—	2	4	no	—
18	C₃²×#121	C₂²	1	8	yes	complemented
19	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃²×C₂²⋊C₄	—	2	2	no	—
21	C₃²×#106	C₂²	1	4	yes	complemented
22	C₃²×#102	C₂²	1	8	yes	complemented
23	C₂²×C₃×D₈	C₂×C₃	1	4	yes	complemented
24	C₂²×C₃×M₁₆	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×#121	C₂×C₃	1	8	yes	complemented
26	C₃×#469	C₂×C₃	1	32	yes	direct product factor
27	C₂×C₃×#106	C₂×C₃	1	4	yes	complemented
28	C₂×C₃×#102	C₂×C₃	1	8	yes	complemented
29	C₂⁴×C₃²	—	2	2	no	—
30	C₂⁴×C₃²	D₄	1	1	yes	characteristic
31	C₂×C₃²×D₄	—	2	8	no	—
32	C₂×C₃²×D₄	—	2	4	no	—
33	C₂×C₃²×D₄	—	2	2	no	—
34	C₂×C₃²×D₄	—	2	4	no	—
35	C₂×C₃²×D₄	—	4	8	no	—
36	C₂×C₃²×D₄	C₂³	1	4	yes	—
37	C₂²×C₃²×C₄	—	2	2	no	—
38	C₂²×C₃²×C₄	—	2	2	no	—
39	C₂²×C₃²×C₄	—	2	2	no	—
40	C₂²×C₃²×C₄	D₄	1	2	yes	—
41	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂³	1	2	yes	—
43	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
44	C₃²×D₈	—	2	8	no	—
45	C₃²×M₁₆	—	2	8	no	—
46	C₂×C₃²×C₈	—	2	2	no	—
47	C₂×C₃²×C₈	D₄	1	4	yes	complemented
48	C₂×C₃²×C₈	C₂³	1	2	yes	—
49	C₂×C₃²×C₈	C₂³	1	2	yes	—
50	C₃²×C₄⋊C₄	—	2	2	no	—
51	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
52	C₃²×C₂²⋊C₄	—	2	8	no	—
53	C₂×#469	C₃²	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₃×D₄	—	2	8	no	—
55	C₂²×C₃×D₄	C₂²×C₃	1	8	yes	—
56	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₃×D₈	—	2	16	no	—
58	C₂×C₃×D₈	C₂²×C₃	1	16	yes	complemented
59	C₂×C₃×M₁₆	—	2	16	no	—
60	C₂×C₃×M₁₆	C₂²×C₃	1	16	yes	complemented
61	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
62	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
63	C₃×#121	—	2	16	no	—
64	C₃×#121	C₂²×C₃	1	32	yes	complemented
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
66	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
67	C₃×#106	C₂²×C₃	1	16	yes	complemented
68	C₃×#102	C₂²×C₃	1	32	yes	complemented
69	C₂³×C₃²	—	2	2	no	—
70	C₂³×C₃²	—	2	4	no	—
71	C₂³×C₃²	—	2	4	no	—
72	C₂³×C₃²	—	4	4	no	—
73	C₂³×C₃²	—	4	4	no	—
74	C₂³×C₃²	—	4	4	no	—
75	C₂³×C₃²	C₂×D₄	1	2	yes	—
76	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
77	C₃²×D₄	—	2	4	no	—
78	C₃²×D₄	—	2	4	no	—
79	C₃²×D₄	—	4	16	no	—
80	C₃²×D₄	—	4	4	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	4	4	no	—
87	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
88	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃²×C₄	#41	1	2	yes	—
91	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
92	C₃²×C₈	—	2	2	no	—
93	C₃²×C₈	C₂×D₄	1	4	yes	complemented
94	C₂²×D₈	C₂×C₃²	1	1	yes	characteristic, complemented
95	C₂²×M₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
96	C₂×#121	C₂×C₃²	1	2	yes	complemented
97	#469	C₂×C₃²	1	8	yes	direct product factor
98	C₂×#106	C₂×C₃²	1	1	yes	characteristic, complemented
99	C₂×#102	C₂×C₃²	1	2	yes	complemented
100	C₂⁴×C₃	—	2	8	no	—
101	C₂⁴×C₃	C₃×D₄	1	4	yes	—
102	C₂×C₃×D₄	—	2	32	no	—
103	C₂×C₃×D₄	—	2	16	no	—
104	C₂×C₃×D₄	—	2	8	no	—
105	C₂×C₃×D₄	—	2	16	no	—
106	C₂×C₃×D₄	—	4	32	no	—
107	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
108	C₂²×C₃×C₄	—	2	8	no	—
109	C₂²×C₃×C₄	—	2	8	no	—
110	C₂²×C₃×C₄	—	2	8	no	—
111	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
112	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
113	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
114	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
115	C₃×D₈	—	2	32	no	—
116	C₃×M₁₆	—	2	32	no	—
117	C₂×C₃×C₈	—	2	8	no	—
118	C₂×C₃×C₈	C₃×D₄	1	16	yes	complemented
119	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
120	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
121	C₃×C₄⋊C₄	—	2	8	no	—
122	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
123	C₃×C₂²⋊C₄	—	2	32	no	—
124	C₂²×C₃²	—	2	2	no	—
125	C₂²×C₃²	—	2	4	no	—
126	C₂²×C₃²	—	4	4	no	—
127	C₂²×C₃²	—	4	4	no	—
128	C₂²×C₃²	—	4	4	no	—
129	C₂²×C₃²	—	4	4	no	—
130	C₂²×C₃²	—	8	8	no	—
131	C₂²×C₃²	#121	1	2	yes	—
132	C₂²×C₃²	AGL(1,ℤ/8)	1	4	yes	—
133	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
134	C₃²×C₄	—	2	2	no	—
135	C₃²×C₄	—	4	4	no	—
136	C₃²×C₄	#121	1	2	yes	—
137	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₃²×C₄	C₂×#41	1	1	yes	characteristic
139	C₂²×D₄	—	2	2	no	—
140	C₂²×D₄	C₂²×C₃²	1	2	yes	—
141	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
142	C₂×D₈	—	2	4	no	—
143	C₂×D₈	C₂²×C₃²	1	4	yes	complemented
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	C₂²×C₃²	1	4	yes	complemented
146	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
147	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
148	#121	—	2	4	no	—
149	#121	C₂²×C₃²	1	8	yes	complemented
150	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
151	C₂×C₂²⋊C₄	—	2	2	no	—
152	#106	C₂²×C₃²	1	4	yes	complemented
153	#102	C₂²×C₃²	1	8	yes	complemented
154	C₂³×C₃	—	2	8	no	—
155	C₂³×C₃	—	2	16	no	—
156	C₂³×C₃	—	2	16	no	—
157	C₂³×C₃	—	4	16	no	—
158	C₂³×C₃	—	4	16	no	—
159	C₂³×C₃	—	4	16	no	—
160	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
161	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
162	C₃×D₄	—	2	16	no	—
163	C₃×D₄	—	2	16	no	—
164	C₃×D₄	—	4	64	no	—
165	C₃×D₄	—	4	16	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	2	16	no	—
168	C₂×C₃×C₄	—	2	8	no	—
169	C₂×C₃×C₄	—	2	8	no	—
170	C₂×C₃×C₄	—	2	16	no	—
171	C₂×C₃×C₄	—	4	16	no	—
172	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
173	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
174	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
175	C₂×C₃×C₄	C₃×#41	1	8	yes	—
176	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
177	C₃×C₈	—	2	8	no	—
178	C₃×C₈	C₂×C₃×D₄	1	16	yes	complemented
179	C₂×C₃²	—	4	2	no	—
180	C₂×C₃²	—	8	4	no	—
181	C₂×C₃²	#469	1	4	yes	direct product factor
182	C₂×C₃²	C₂×#121	1	1	yes	characteristic
183	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	2	yes	—
184	C₂⁴	—	2	2	no	—
185	C₂⁴	C₃²×D₄	1	1	yes	characteristic
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	4	8	no	—
191	C₂×D₄	C₂³×C₃²	1	4	yes	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₃²×D₄	1	2	yes	—
197	C₂²×C₄	C₂³×C₃²	1	2	yes	—
198	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
199	D₈	—	2	8	no	—
200	M₁₆	—	2	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₃²×D₄	1	4	yes	complemented
203	C₂×C₈	C₂³×C₃²	1	2	yes	—
204	C₂×C₈	C₂³×C₃²	1	2	yes	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
207	C₂²⋊C₄	—	2	8	no	—
208	C₂²×C₃	—	2	16	no	—
209	C₂²×C₃	—	2	8	no	—
210	C₂²×C₃	—	4	16	no	—
211	C₂²×C₃	—	4	16	no	—
212	C₂²×C₃	—	4	16	no	—
213	C₂²×C₃	—	4	16	no	—
214	C₂²×C₃	—	8	32	no	—
215	C₂²×C₃	C₃×#121	1	8	yes	—
216	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	16	yes	—
217	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
218	C₃×C₄	—	2	8	no	—
219	C₃×C₄	—	4	16	no	—
220	C₃×C₄	C₃×#121	1	8	yes	—
221	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
222	C₃×C₄	C₂×C₃×#41	1	4	yes	—
223	C₃²	C₂×#469	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	—	2	2	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	4	no	—
230	C₂³	C₂×C₃²×D₄	1	2	yes	—
231	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	16	no	—
235	D₄	—	4	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
243	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
244	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
245	C₂×C₄	C₃²×#41	1	2	yes	—
246	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	C₂×C₃²×D₄	1	4	yes	complemented
249	C₂×C₃	—	4	8	no	—
250	C₂×C₃	—	8	16	no	—
251	C₂×C₃	C₃×#469	1	16	yes	direct product factor
252	C₂×C₃	C₂×C₃×#121	1	4	yes	—
253	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	8	yes	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	4	4	no	—
260	C₂²	—	8	8	no	—
261	C₂²	C₃²×#121	1	2	yes	—
262	C₂²	C₃²×AGL(1,ℤ/8)	1	4	yes	—
263	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	4	4	no	—
266	C₄	C₃²×#121	1	2	yes	—
267	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
268	C₄	C₂×C₃²×#41	1	1	yes	characteristic
269	C₃	C₂×C₃×#469	1	4	yes	direct product factor
270	C₂	—	4	2	no	—
271	C₂	—	8	4	no	—
272	C₂	C₃²×#469	1	4	yes	direct product factor
273	C₂	C₂×C₃²×#121	1	1	yes	characteristic
274	C₂	C₂×C₃²×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂³×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×D₈	C₂	semidirect
2. C₂²×C₃²×M₁₆	C₂	semidirect
3. C₂×C₃²×#121	C₂	semidirect
4. C₃²×#469	C₂	direct (N × H)
5. C₂×C₃²×#106	C₂	semidirect
6. C₂×C₃²×#102	C₂	semidirect
7. C₂×C₃×#469	C₃	direct (N × H)
12. C₂×C₃²×D₈	C₂²	semidirect
14. C₂×C₃²×M₁₆	C₂²	semidirect
18. C₃²×#121	C₂²	semidirect
21. C₃²×#106	C₂²	semidirect
22. C₃²×#102	C₂²	semidirect
23. C₂²×C₃×D₈	C₂×C₃	semidirect
24. C₂²×C₃×M₁₆	C₂×C₃	semidirect
25. C₂×C₃×#121	C₂×C₃	semidirect
26. C₃×#469	C₂×C₃	direct (N × H)
27. C₂×C₃×#106	C₂×C₃	semidirect
28. C₂×C₃×#102	C₂×C₃	semidirect
47. C₂×C₃²×C₈	D₄	semidirect
53. C₂×#469	C₃²	direct (N × H)
58. C₂×C₃×D₈	C₂²×C₃	semidirect
60. C₂×C₃×M₁₆	C₂²×C₃	semidirect
64. C₃×#121	C₂²×C₃	semidirect
67. C₃×#106	C₂²×C₃	semidirect
68. C₃×#102	C₂²×C₃	semidirect
93. C₃²×C₈	C₂×D₄	semidirect
94. C₂²×D₈	C₂×C₃²	semidirect
95. C₂²×M₁₆	C₂×C₃²	semidirect
96. C₂×#121	C₂×C₃²	semidirect
97. #469	C₂×C₃²	direct (N × H)
98. C₂×#106	C₂×C₃²	semidirect
99. C₂×#102	C₂×C₃²	semidirect
118. C₂×C₃×C₈	C₃×D₄	semidirect
143. C₂×D₈	C₂²×C₃²	semidirect
145. C₂×M₁₆	C₂²×C₃²	semidirect
149. #121	C₂²×C₃²	semidirect
152. #106	C₂²×C₃²	semidirect
153. #102	C₂²×C₃²	semidirect
178. C₃×C₈	C₂×C₃×D₄	semidirect
181. C₂×C₃²	#469	direct (N × H)
202. C₂×C₈	C₃²×D₄	semidirect
223. C₃²	C₂×#469	direct (N × H)
248. C₈	C₂×C₃²×D₄	semidirect
251. C₂×C₃	C₃×#469	direct (N × H)
269. C₃	C₂×C₃×#469	direct (N × H)
272. C₂	C₃²×#469	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×D₈	2
C₂²×C₃²×M₁₆	2
C₂×C₃²×#121	2
C₃²×#469	2
C₂×C₃²×#106	2
C₂×C₃²×#102	2
C₂×C₃×#469	3

Subgroups — C2×C32×#469 #11770101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#469 #11770101