Subgroups — C₂²×C₃×#486 #10679209

Order 768.

3086 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#486	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#102	C₂	1	4	yes	maximal, complemented
5	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#486	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂³×C₃×Q₈	—	2	1	no	—
8	C₂³×C₃×D₄	—	2	1	no	—
9	C₂³×C₃×D₄	—	2	1	no	—
10	C₂³×C₃×D₄	C₂²	1	2	yes	—
11	C₂³×C₃×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
13	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
14	C₂×C₃×#127	—	2	6	no	—
15	C₂×C₃×#127	C₂²	1	12	yes	—
16	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
17	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₃×#486	C₂²	1	64	yes	direct product factor
20	C₂×C₃×#102	C₂²	1	48	yes	complemented
21	C₂×C₃×C₄×C₈	C₂²	1	12	yes	complemented
22	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂²×#127	C₂×C₃	1	1	yes	characteristic
24	C₂×#486	C₂×C₃	1	24	yes	direct product factor
25	C₂²×#102	C₂×C₃	1	4	yes	complemented
26	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂⁵×C₃	—	2	2	no	—
28	C₂²×C₃×Q₈	—	2	12	no	—
29	C₂²×C₃×Q₈	—	4	8	no	—
30	C₂²×C₃×D₄	—	2	12	no	—
31	C₂²×C₃×D₄	—	2	12	no	—
32	C₂²×C₃×D₄	—	2	6	no	—
33	C₂²×C₃×D₄	—	2	16	no	—
34	C₂²×C₃×D₄	—	4	8	no	—
35	C₂²×C₃×D₄	—	4	8	no	—
36	C₂²×C₃×D₄	C₂³	1	12	yes	—
37	C₂³×C₃×C₄	—	2	2	no	—
38	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
39	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
40	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂²×C₃×C₈	—	2	8	no	—
42	C₂²×C₃×C₈	C₂³	1	12	yes	—
43	C₃×C₄⋊Q₈	—	2	24	no	—
44	C₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
45	C₃×#127	—	2	24	no	—
46	C₃×#127	C₂³	1	16	yes	—
47	C₂×C₃×C₄⋊C₄	—	2	12	no	—
48	C₂×C₃×C₄⋊C₄	—	2	6	no	—
49	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
50	C₂×C₃×C₄²	D₄	1	6	yes	—
51	C₂×C₃×C₄²	C₂³	1	6	yes	—
52	C₃×#102	C₂³	1	64	yes	complemented
53	C₃×C₄×C₈	C₂³	1	16	yes	complemented
54	C₂³×Q₈	—	2	1	no	—
55	C₂³×D₄	—	2	1	no	—
56	C₂³×D₄	—	2	1	no	—
57	C₂³×D₄	C₂²×C₃	1	2	yes	—
58	C₂³×C₈	C₂²×C₃	1	2	yes	—
59	C₂×C₄⋊Q₈	—	2	6	no	—
60	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	complemented
61	C₂×#127	—	2	6	no	—
62	C₂×#127	C₂²×C₃	1	12	yes	—
63	C₂²×C₄⋊C₄	—	2	1	no	—
64	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
66	#486	C₂²×C₃	1	64	yes	direct product factor
67	C₂×#102	C₂²×C₃	1	48	yes	complemented
68	C₂×C₄×C₈	C₂²×C₃	1	12	yes	complemented
69	C₂⁴×C₃	—	2	12	no	—
70	C₂⁴×C₃	—	4	8	no	—
71	C₂⁴×C₃	—	4	8	no	—
72	C₂⁴×C₃	—	4	8	no	—
73	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×Q₈	—	2	16	no	—
75	C₂×C₃×Q₈	—	4	48	no	—
76	C₂×C₃×D₄	—	2	12	no	—
77	C₂×C₃×D₄	—	2	16	no	—
78	C₂×C₃×D₄	—	2	16	no	—
79	C₂×C₃×D₄	—	2	48	no	—
80	C₂×C₃×D₄	—	4	24	no	—
81	C₂×C₃×D₄	—	4	48	no	—
82	C₂×C₃×D₄	—	4	48	no	—
83	C₂×C₃×D₄	C₂⁴	1	8	yes	—
84	C₂²×C₃×C₄	—	2	12	no	—
85	C₂²×C₃×C₄	—	4	8	no	—
86	C₂²×C₃×C₄	D₈	1	8	yes	—
87	C₂²×C₃×C₄	SD₁₆	1	8	yes	—
88	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
89	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
90	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
91	C₂²×C₃×C₄	#41	1	8	yes	—
92	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
93	C₂×C₃×C₈	—	2	24	no	—
94	C₂×C₃×C₈	C₂⁴	1	8	yes	—
95	C₃×C₄⋊C₄	—	2	16	no	—
96	C₃×C₄⋊C₄	—	2	12	no	—
97	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
98	C₃×C₄²	C₂×D₄	1	12	yes	—
99	C₃×C₄²	C₂⁴	1	4	yes	—
100	C₂⁵	—	2	2	no	—
101	C₂²×Q₈	—	2	12	no	—
102	C₂²×Q₈	—	4	8	no	—
103	C₂²×D₄	—	2	6	no	—
104	C₂²×D₄	—	2	12	no	—
105	C₂²×D₄	—	2	12	no	—
106	C₂²×D₄	—	2	16	no	—
107	C₂²×D₄	—	4	8	no	—
108	C₂²×D₄	—	4	8	no	—
109	C₂²×D₄	C₂³×C₃	1	12	yes	—
110	C₂³×C₄	—	2	2	no	—
111	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
112	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
113	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂²×C₈	—	2	8	no	—
115	C₂²×C₈	C₂³×C₃	1	12	yes	—
116	C₄⋊Q₈	—	2	24	no	—
117	C₄⋊Q₈	C₂³×C₃	1	16	yes	complemented
118	#127	—	2	24	no	—
119	#127	C₂³×C₃	1	16	yes	—
120	C₂×C₄⋊C₄	—	2	12	no	—
121	C₂×C₄⋊C₄	—	2	6	no	—
122	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
123	C₂×C₄²	C₃×D₄	1	6	yes	—
124	C₂×C₄²	C₂³×C₃	1	6	yes	—
125	#102	C₂³×C₃	1	64	yes	complemented
126	C₄×C₈	C₂³×C₃	1	16	yes	complemented
127	C₂³×C₃	—	2	8	no	—
128	C₂³×C₃	—	4	24	no	—
129	C₂³×C₃	—	4	24	no	—
130	C₂³×C₃	—	4	24	no	—
131	C₂³×C₃	—	8	32	no	—
132	C₂³×C₃	#124	1	4	yes	—
133	C₂³×C₃	C₂×D₈	1	4	yes	—
134	C₂³×C₃	C₂×SD₁₆	1	4	yes	—
135	C₂³×C₃	C₂²×D₄	1	3	yes	—
136	C₃×Q₈	—	4	32	no	—
137	C₃×D₄	—	2	16	no	—
138	C₃×D₄	—	4	24	no	—
139	C₃×D₄	—	4	32	no	—
140	C₃×D₄	—	4	32	no	—
141	C₂×C₃×C₄	—	2	8	no	—
142	C₂×C₃×C₄	—	4	24	no	—
143	C₂×C₃×C₄	#124	1	12	yes	—
144	C₂×C₃×C₄	C₂×D₈	1	24	yes	—
145	C₂×C₃×C₄	C₂×SD₁₆	1	24	yes	—
146	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
147	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
148	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
149	C₂×C₃×C₄	C₂×#41	1	12	yes	—
150	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
151	C₃×C₈	—	2	8	no	—
152	C₂⁴	—	2	12	no	—
153	C₂⁴	—	4	8	no	—
154	C₂⁴	—	4	8	no	—
155	C₂⁴	—	4	8	no	—
156	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×Q₈	—	2	16	no	—
158	C₂×Q₈	—	4	48	no	—
159	C₂×D₄	—	2	12	no	—
160	C₂×D₄	—	2	16	no	—
161	C₂×D₄	—	2	16	no	—
162	C₂×D₄	—	2	48	no	—
163	C₂×D₄	—	4	24	no	—
164	C₂×D₄	—	4	48	no	—
165	C₂×D₄	—	4	48	no	—
166	C₂×D₄	C₂⁴×C₃	1	8	yes	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	4	8	no	—
169	C₂²×C₄	C₃×D₈	1	8	yes	—
170	C₂²×C₄	C₃×SD₁₆	1	8	yes	—
171	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
172	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
173	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
174	C₂²×C₄	C₃×#41	1	8	yes	—
175	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
176	C₂×C₈	—	2	24	no	—
177	C₂×C₈	C₂⁴×C₃	1	8	yes	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	2	12	no	—
180	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
181	C₄²	C₂×C₃×D₄	1	12	yes	—
182	C₄²	C₂⁴×C₃	1	4	yes	—
183	C₂²×C₃	—	4	8	no	—
184	C₂²×C₃	—	4	8	no	—
185	C₂²×C₃	—	4	8	no	—
186	C₂²×C₃	—	8	48	no	—
187	C₂²×C₃	#486	1	16	yes	direct product factor
188	C₂²×C₃	C₂×#124	1	6	yes	—
189	C₂²×C₃	C₂²×D₈	1	6	yes	—
190	C₂²×C₃	C₂²×SD₁₆	1	6	yes	—
191	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
192	C₃×C₄	—	4	8	no	—
193	C₃×C₄	C₂×#124	1	6	yes	—
194	C₃×C₄	C₂²×D₈	1	8	yes	—
195	C₃×C₄	C₂²×SD₁₆	1	8	yes	—
196	C₃×C₄	C₂²×#41	1	2	yes	—
197	C₂³	—	2	8	no	—
198	C₂³	—	4	24	no	—
199	C₂³	—	4	24	no	—
200	C₂³	—	4	24	no	—
201	C₂³	—	8	32	no	—
202	C₂³	C₃×#124	1	4	yes	—
203	C₂³	C₂×C₃×D₈	1	4	yes	—
204	C₂³	C₂×C₃×SD₁₆	1	4	yes	—
205	C₂³	C₂²×C₃×D₄	1	3	yes	—
206	Q₈	—	4	32	no	—
207	D₄	—	2	16	no	—
208	D₄	—	4	32	no	—
209	D₄	—	4	32	no	—
210	D₄	—	4	24	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	24	no	—
213	C₂×C₄	C₃×#124	1	12	yes	—
214	C₂×C₄	C₂×C₃×D₈	1	24	yes	—
215	C₂×C₄	C₂×C₃×SD₁₆	1	24	yes	—
216	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
217	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
218	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
219	C₂×C₄	C₂×C₃×#41	1	12	yes	—
220	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
221	C₈	—	2	8	no	—
222	C₂×C₃	—	8	8	no	—
223	C₂×C₃	C₂×#486	1	12	yes	direct product factor
224	C₂×C₃	C₂²×#124	1	1	yes	characteristic
225	C₂×C₃	C₂³×D₈	1	1	yes	characteristic
226	C₂×C₃	C₂³×SD₁₆	1	1	yes	characteristic
227	C₂²	—	4	8	no	—
228	C₂²	—	4	8	no	—
229	C₂²	—	4	8	no	—
230	C₂²	—	8	48	no	—
231	C₂²	C₃×#486	1	16	yes	direct product factor
232	C₂²	C₂×C₃×#124	1	6	yes	—
233	C₂²	C₂²×C₃×D₈	1	6	yes	—
234	C₂²	C₂²×C₃×SD₁₆	1	6	yes	—
235	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
236	C₄	—	4	8	no	—
237	C₄	C₂×C₃×#124	1	6	yes	—
238	C₄	C₂²×C₃×D₈	1	8	yes	—
239	C₄	C₂²×C₃×SD₁₆	1	8	yes	—
240	C₄	C₂²×C₃×#41	1	2	yes	—
241	C₃	C₂²×#486	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	—	8	8	no	—
243	C₂	C₂×C₃×#486	1	12	yes	direct product factor
244	C₂	C₂²×C₃×#124	1	1	yes	characteristic
245	C₂	C₂³×C₃×D₈	1	1	yes	characteristic
246	C₂	C₂³×C₃×SD₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂²×C₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄⋊Q₈	C₂	semidirect
3. C₂×C₃×#486	C₂	direct (N × H)
4. C₂²×C₃×#102	C₂	semidirect
5. C₂²×C₃×C₄×C₈	C₂	semidirect
6. C₂²×#486	C₃	direct (N × H)
13. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
19. C₃×#486	C₂²	direct (N × H)
20. C₂×C₃×#102	C₂²	semidirect
21. C₂×C₃×C₄×C₈	C₂²	semidirect
22. C₂²×C₄⋊Q₈	C₂×C₃	semidirect
24. C₂×#486	C₂×C₃	direct (N × H)
25. C₂²×#102	C₂×C₃	semidirect
26. C₂²×C₄×C₈	C₂×C₃	semidirect
44. C₃×C₄⋊Q₈	C₂³	semidirect
52. C₃×#102	C₂³	semidirect
53. C₃×C₄×C₈	C₂³	semidirect
60. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
66. #486	C₂²×C₃	direct (N × H)
67. C₂×#102	C₂²×C₃	semidirect
68. C₂×C₄×C₈	C₂²×C₃	semidirect
117. C₄⋊Q₈	C₂³×C₃	semidirect
125. #102	C₂³×C₃	semidirect
126. C₄×C₈	C₂³×C₃	semidirect
187. C₂²×C₃	#486	direct (N × H)
223. C₂×C₃	C₂×#486	direct (N × H)
231. C₂²	C₃×#486	direct (N × H)
241. C₃	C₂²×#486	direct (N × H)
243. C₂	C₂×C₃×#486	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×C₃×C₄⋊Q₈	2
C₂²×C₃×#127	2
C₂×C₃×#486	2
C₂²×C₃×#102	2
C₂²×C₃×C₄×C₈	2
C₂²×#486	3

Subgroups — C22×C3×#486 #10679209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#486 #10679209