Subgroups — C₂²×C₃×#492 #10679235

Order 768.

3950 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×SD₁₆	C₂	1	4	yes	maximal, complemented
2	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#127	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#492	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#492	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂³×C₃×Q₈	C₂²	1	2	yes	—
8	C₂³×C₃×D₄	—	2	2	no	—
9	C₂³×C₃×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×SD₁₆	—	2	32	no	—
11	C₂²×C₃×SD₁₆	C₂²	1	48	yes	complemented
12	C₂³×C₃×C₈	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
14	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
15	C₂×C₃×#127	—	2	6	no	—
16	C₂×C₃×#127	C₂²	1	12	yes	—
17	C₂²×C₃×C₄⋊C₄	—	2	2	no	—
18	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₃×#492	C₂²	1	64	yes	direct product factor
20	C₂×C₃×C₄×C₈	C₂²	1	12	yes	complemented
21	C₂³×SD₁₆	C₂×C₃	1	4	yes	complemented
22	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂²×#127	C₂×C₃	1	1	yes	characteristic
24	C₂×#492	C₂×C₃	1	24	yes	direct product factor
25	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂⁵×C₃	—	2	2	no	—
27	C₂²×C₃×Q₈	—	2	6	no	—
28	C₂²×C₃×Q₈	—	2	16	no	—
29	C₂²×C₃×Q₈	C₂³	1	12	yes	—
30	C₂²×C₃×D₄	—	2	16	no	—
31	C₂²×C₃×D₄	—	2	24	no	—
32	C₂²×C₃×D₄	—	2	6	no	—
33	C₂²×C₃×D₄	—	4	16	no	—
34	C₂²×C₃×D₄	C₂³	1	12	yes	—
35	C₂³×C₃×C₄	—	2	2	no	—
36	C₂³×C₃×C₄	D₄	1	2	yes	—
37	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃×SD₁₆	—	2	192	no	—
39	C₂×C₃×SD₁₆	C₂³	1	64	yes	complemented
40	C₂²×C₃×C₈	D₄	1	16	yes	complemented
41	C₂²×C₃×C₈	C₂³	1	12	yes	—
42	C₃×C₄⋊Q₈	—	2	24	no	—
43	C₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
44	C₃×#127	—	2	24	no	—
45	C₃×#127	C₂³	1	16	yes	—
46	C₂×C₃×C₄⋊C₄	—	2	24	no	—
47	C₂×C₃×C₄²	D₄	1	6	yes	—
48	C₂×C₃×C₄²	C₂³	1	6	yes	—
49	C₃×C₄×C₈	C₂³	1	16	yes	complemented
50	C₂³×Q₈	C₂²×C₃	1	2	yes	—
51	C₂³×D₄	—	2	2	no	—
52	C₂³×D₄	C₂²×C₃	1	2	yes	—
53	C₂²×SD₁₆	—	2	32	no	—
54	C₂²×SD₁₆	C₂²×C₃	1	48	yes	complemented
55	C₂³×C₈	C₂²×C₃	1	2	yes	—
56	C₂×C₄⋊Q₈	—	2	6	no	—
57	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	complemented
58	C₂×#127	—	2	6	no	—
59	C₂×#127	C₂²×C₃	1	12	yes	—
60	C₂²×C₄⋊C₄	—	2	2	no	—
61	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
62	#492	C₂²×C₃	1	64	yes	direct product factor
63	C₂×C₄×C₈	C₂²×C₃	1	12	yes	complemented
64	C₂⁴×C₃	—	2	12	no	—
65	C₂⁴×C₃	—	4	8	no	—
66	C₂⁴×C₃	—	4	16	no	—
67	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×Q₈	—	2	12	no	—
69	C₂×C₃×Q₈	—	2	48	no	—
70	C₂×C₃×Q₈	—	4	24	no	—
71	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
72	C₂×C₃×D₄	—	2	48	no	—
73	C₂×C₃×D₄	—	2	12	no	—
74	C₂×C₃×D₄	—	2	32	no	—
75	C₂×C₃×D₄	—	4	24	no	—
76	C₂×C₃×D₄	—	4	96	no	—
77	C₂×C₃×D₄	C₂⁴	1	8	yes	—
78	C₂²×C₃×C₄	—	2	12	no	—
79	C₂²×C₃×C₄	—	4	8	no	—
80	C₂²×C₃×C₄	SD₁₆	1	16	yes	complemented
81	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
82	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
83	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
84	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
85	C₃×SD₁₆	—	2	128	no	—
86	C₂×C₃×C₈	C₂×D₄	1	48	yes	complemented
87	C₂×C₃×C₈	C₂⁴	1	8	yes	—
88	C₃×C₄⋊C₄	—	2	32	no	—
89	C₃×C₄²	C₂×D₄	1	12	yes	—
90	C₃×C₄²	C₂⁴	1	4	yes	—
91	C₂⁵	—	2	2	no	—
92	C₂²×Q₈	—	2	6	no	—
93	C₂²×Q₈	—	2	16	no	—
94	C₂²×Q₈	C₂³×C₃	1	12	yes	—
95	C₂²×D₄	—	2	16	no	—
96	C₂²×D₄	—	2	6	no	—
97	C₂²×D₄	—	2	24	no	—
98	C₂²×D₄	—	4	16	no	—
99	C₂²×D₄	C₂³×C₃	1	12	yes	—
100	C₂³×C₄	—	2	2	no	—
101	C₂³×C₄	C₃×D₄	1	2	yes	—
102	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂×SD₁₆	—	2	192	no	—
104	C₂×SD₁₆	C₂³×C₃	1	64	yes	complemented
105	C₂²×C₈	C₃×D₄	1	16	yes	complemented
106	C₂²×C₈	C₂³×C₃	1	12	yes	—
107	C₄⋊Q₈	—	2	24	no	—
108	C₄⋊Q₈	C₂³×C₃	1	16	yes	complemented
109	#127	—	2	24	no	—
110	#127	C₂³×C₃	1	16	yes	—
111	C₂×C₄⋊C₄	—	2	24	no	—
112	C₂×C₄²	C₃×D₄	1	6	yes	—
113	C₂×C₄²	C₂³×C₃	1	6	yes	—
114	C₄×C₈	C₂³×C₃	1	16	yes	complemented
115	C₂³×C₃	—	2	8	no	—
116	C₂³×C₃	—	4	24	no	—
117	C₂³×C₃	—	4	48	no	—
118	C₂³×C₃	—	8	32	no	—
119	C₂³×C₃	#127	1	4	yes	—
120	C₂³×C₃	C₂×SD₁₆	1	8	yes	—
121	C₂³×C₃	C₂²×D₄	1	3	yes	—
122	C₃×Q₈	—	2	16	no	—
123	C₃×Q₈	—	4	24	no	—
124	C₃×D₄	—	2	16	no	—
125	C₃×D₄	—	4	24	no	—
126	C₃×D₄	—	4	64	no	—
127	C₂×C₃×C₄	—	2	8	no	—
128	C₂×C₃×C₄	—	4	24	no	—
129	C₂×C₃×C₄	#127	1	12	yes	—
130	C₂×C₃×C₄	C₂×SD₁₆	1	48	yes	complemented
131	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
132	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
133	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
134	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
135	C₃×C₈	C₂²×D₄	1	16	yes	complemented
136	C₂⁴	—	2	12	no	—
137	C₂⁴	—	4	8	no	—
138	C₂⁴	—	4	16	no	—
139	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂×Q₈	—	2	12	no	—
141	C₂×Q₈	—	2	48	no	—
142	C₂×Q₈	—	4	24	no	—
143	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
144	C₂×D₄	—	2	48	no	—
145	C₂×D₄	—	2	12	no	—
146	C₂×D₄	—	2	32	no	—
147	C₂×D₄	—	4	24	no	—
148	C₂×D₄	—	4	96	no	—
149	C₂×D₄	C₂⁴×C₃	1	8	yes	—
150	C₂²×C₄	—	2	12	no	—
151	C₂²×C₄	—	4	8	no	—
152	C₂²×C₄	C₃×SD₁₆	1	16	yes	complemented
153	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
154	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
155	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
156	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
157	SD₁₆	—	2	128	no	—
158	C₂×C₈	C₂×C₃×D₄	1	48	yes	complemented
159	C₂×C₈	C₂⁴×C₃	1	8	yes	—
160	C₄⋊C₄	—	2	32	no	—
161	C₄²	C₂×C₃×D₄	1	12	yes	—
162	C₄²	C₂⁴×C₃	1	4	yes	—
163	C₂²×C₃	—	4	8	no	—
164	C₂²×C₃	—	4	16	no	—
165	C₂²×C₃	—	8	48	no	—
166	C₂²×C₃	#492	1	16	yes	direct product factor
167	C₂²×C₃	C₂×#127	1	6	yes	—
168	C₂²×C₃	C₂²×SD₁₆	1	12	yes	—
169	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
170	C₃×C₄	—	4	8	no	—
171	C₃×C₄	C₂×#127	1	6	yes	—
172	C₃×C₄	C₂²×SD₁₆	1	16	yes	complemented
173	C₃×C₄	C₂³×D₄	1	2	yes	—
174	C₂³	—	2	8	no	—
175	C₂³	—	4	48	no	—
176	C₂³	—	4	24	no	—
177	C₂³	—	8	32	no	—
178	C₂³	C₃×#127	1	4	yes	—
179	C₂³	C₂×C₃×SD₁₆	1	8	yes	—
180	C₂³	C₂²×C₃×D₄	1	3	yes	—
181	Q₈	—	2	16	no	—
182	Q₈	—	4	24	no	—
183	D₄	—	2	16	no	—
184	D₄	—	4	64	no	—
185	D₄	—	4	24	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	4	24	no	—
188	C₂×C₄	C₃×#127	1	12	yes	—
189	C₂×C₄	C₂×C₃×SD₁₆	1	48	yes	complemented
190	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
191	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
192	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
193	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
194	C₈	C₂²×C₃×D₄	1	16	yes	complemented
195	C₂×C₃	—	8	8	no	—
196	C₂×C₃	C₂×#492	1	12	yes	direct product factor
197	C₂×C₃	C₂²×#127	1	1	yes	characteristic
198	C₂×C₃	C₂³×SD₁₆	1	2	yes	—
199	C₂²	—	4	8	no	—
200	C₂²	—	4	16	no	—
201	C₂²	—	8	48	no	—
202	C₂²	C₃×#492	1	16	yes	direct product factor
203	C₂²	C₂×C₃×#127	1	6	yes	—
204	C₂²	C₂²×C₃×SD₁₆	1	12	yes	—
205	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
206	C₄	—	4	8	no	—
207	C₄	C₂×C₃×#127	1	6	yes	—
208	C₄	C₂²×C₃×SD₁₆	1	16	yes	complemented
209	C₄	C₂³×C₃×D₄	1	2	yes	—
210	C₃	C₂²×#492	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	8	8	no	—
212	C₂	C₂×C₃×#492	1	12	yes	direct product factor
213	C₂	C₂²×C₃×#127	1	1	yes	characteristic
214	C₂	C₂³×C₃×SD₁₆	1	2	yes	—

	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. C₂³×C₃×SD₁₆	C₂	semidirect
2. C₂²×C₃×C₄⋊Q₈	C₂	semidirect
4. C₂×C₃×#492	C₂	direct (N × H)
5. C₂²×C₃×C₄×C₈	C₂	semidirect
6. C₂²×#492	C₃	direct (N × H)
11. C₂²×C₃×SD₁₆	C₂²	semidirect
14. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
19. C₃×#492	C₂²	direct (N × H)
20. C₂×C₃×C₄×C₈	C₂²	semidirect
21. C₂³×SD₁₆	C₂×C₃	semidirect
22. C₂²×C₄⋊Q₈	C₂×C₃	semidirect
24. C₂×#492	C₂×C₃	direct (N × H)
25. C₂²×C₄×C₈	C₂×C₃	semidirect
39. C₂×C₃×SD₁₆	C₂³	semidirect
40. C₂²×C₃×C₈	D₄	semidirect
43. C₃×C₄⋊Q₈	C₂³	semidirect
49. C₃×C₄×C₈	C₂³	semidirect
54. C₂²×SD₁₆	C₂²×C₃	semidirect
57. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
62. #492	C₂²×C₃	direct (N × H)
63. C₂×C₄×C₈	C₂²×C₃	semidirect
80. C₂²×C₃×C₄	SD₁₆	semidirect
86. C₂×C₃×C₈	C₂×D₄	semidirect
104. C₂×SD₁₆	C₂³×C₃	semidirect
105. C₂²×C₈	C₃×D₄	semidirect
108. C₄⋊Q₈	C₂³×C₃	semidirect
114. C₄×C₈	C₂³×C₃	semidirect
130. C₂×C₃×C₄	C₂×SD₁₆	semidirect
135. C₃×C₈	C₂²×D₄	semidirect
152. C₂²×C₄	C₃×SD₁₆	semidirect
158. C₂×C₈	C₂×C₃×D₄	semidirect
166. C₂²×C₃	#492	direct (N × H)
172. C₃×C₄	C₂²×SD₁₆	semidirect
189. C₂×C₄	C₂×C₃×SD₁₆	semidirect
194. C₈	C₂²×C₃×D₄	semidirect
196. C₂×C₃	C₂×#492	direct (N × H)
202. C₂²	C₃×#492	direct (N × H)
208. C₄	C₂²×C₃×SD₁₆	semidirect
210. C₃	C₂²×#492	direct (N × H)
212. C₂	C₂×C₃×#492	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×SD₁₆	2
C₂²×C₃×C₄⋊Q₈	2
C₂²×C₃×#127	2
C₂×C₃×#492	2
C₂²×C₃×C₄×C₈	2
C₂²×#492	3

Subgroups — C22×C3×#492 #10679235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#492 #10679235