Subgroups — C₂²×C₃×#487 #10679210

Order 768.

2398 conjugacy classes of subgroups.

The subgroup lattice has 218 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
2	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#487	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#103	C₂	1	4	yes	maximal
5	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×#487	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂³×C₃×Q₈	—	2	1	no	—
8	C₂³×C₃×Q₈	C₂²	1	2	yes	—
9	C₂³×C₃×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
11	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
12	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	—
13	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	—
14	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
15	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
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×#487	C₂²	1	64	yes	direct product factor
20	C₂×C₃×#103	C₂²	1	48	yes	—
21	C₂×C₃×C₄×C₈	C₂²	1	12	yes	—
22	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
24	C₂×#487	C₂×C₃	1	24	yes	direct product factor
25	C₂²×#103	C₂×C₃	1	4	yes	—
26	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
27	C₂²×C₃×Q₈	—	2	12	no	—
28	C₂²×C₃×Q₈	—	2	6	no	—
29	C₂²×C₃×Q₈	—	2	16	no	—
30	C₂²×C₃×Q₈	—	4	8	no	—
31	C₂²×C₃×Q₈	C₂³	1	12	yes	—
32	C₂³×C₃×C₄	—	2	2	no	—
33	C₂³×C₃×C₄	—	2	2	no	—
34	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
35	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
36	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₂²×C₃×C₈	—	2	8	no	—
38	C₂²×C₃×C₈	C₂³	1	12	yes	—
39	C₃×C₄⋊Q₈	—	2	24	no	—
40	C₃×C₄⋊Q₈	—	2	24	no	—
41	C₃×C₄⋊Q₈	C₂³	1	16	yes	—
42	C₃×C₄⋊Q₈	C₂³	1	16	yes	—
43	C₂×C₃×C₄⋊C₄	—	2	12	no	—
44	C₂×C₃×C₄⋊C₄	—	2	12	no	—
45	C₂×C₃×C₄⋊C₄	—	2	12	no	—
46	C₂×C₃×C₄⋊C₄	—	2	6	no	—
47	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
48	C₂×C₃×C₄²	D₄	1	6	yes	—
49	C₂×C₃×C₄²	C₂³	1	6	yes	—
50	C₃×#103	C₂³	1	64	yes	—
51	C₃×C₄×C₈	C₂³	1	16	yes	—
52	C₂³×Q₈	—	2	1	no	—
53	C₂³×Q₈	C₂²×C₃	1	2	yes	—
54	C₂³×C₈	C₂²×C₃	1	2	yes	—
55	C₂×C₄⋊Q₈	—	2	6	no	—
56	C₂×C₄⋊Q₈	—	2	6	no	—
57	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	—
58	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	—
59	C₂²×C₄⋊C₄	—	2	1	no	—
60	C₂²×C₄⋊C₄	—	2	1	no	—
61	C₂²×C₄⋊C₄	—	2	1	no	—
62	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
63	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
64	#487	C₂²×C₃	1	64	yes	direct product factor
65	C₂×#103	C₂²×C₃	1	48	yes	—
66	C₂×C₄×C₈	C₂²×C₃	1	12	yes	—
67	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×Q₈	—	2	12	no	—
69	C₂×C₃×Q₈	—	2	16	no	—
70	C₂×C₃×Q₈	—	2	48	no	—
71	C₂×C₃×Q₈	—	4	24	no	—
72	C₂×C₃×Q₈	—	4	48	no	—
73	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
74	C₂²×C₃×C₄	—	2	12	no	—
75	C₂²×C₃×C₄	—	2	12	no	—
76	C₂²×C₃×C₄	—	4	8	no	—
77	C₂²×C₃×C₄	—	4	8	no	—
78	C₂²×C₃×C₄	SD₁₆	1	8	yes	—
79	C₂²×C₃×C₄	Q₁₆	1	8	yes	—
80	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
81	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
82	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
83	C₂²×C₃×C₄	#41	1	8	yes	—
84	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
85	C₂×C₃×C₈	—	2	24	no	—
86	C₂×C₃×C₈	C₂⁴	1	8	yes	—
87	C₃×C₄⋊C₄	—	2	16	no	—
88	C₃×C₄⋊C₄	—	2	16	no	—
89	C₃×C₄⋊C₄	—	2	16	no	—
90	C₃×C₄⋊C₄	—	2	12	no	—
91	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
92	C₃×C₄²	C₂×D₄	1	12	yes	—
93	C₃×C₄²	C₂⁴	1	4	yes	—
94	C₂²×Q₈	—	2	6	no	—
95	C₂²×Q₈	—	2	12	no	—
96	C₂²×Q₈	—	2	16	no	—
97	C₂²×Q₈	—	4	8	no	—
98	C₂²×Q₈	C₂³×C₃	1	12	yes	—
99	C₂³×C₄	—	2	2	no	—
100	C₂³×C₄	—	2	2	no	—
101	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
102	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
103	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
104	C₂²×C₈	—	2	8	no	—
105	C₂²×C₈	C₂³×C₃	1	12	yes	—
106	C₄⋊Q₈	—	2	24	no	—
107	C₄⋊Q₈	—	2	24	no	—
108	C₄⋊Q₈	C₂³×C₃	1	16	yes	—
109	C₄⋊Q₈	C₂³×C₃	1	16	yes	—
110	C₂×C₄⋊C₄	—	2	12	no	—
111	C₂×C₄⋊C₄	—	2	12	no	—
112	C₂×C₄⋊C₄	—	2	12	no	—
113	C₂×C₄⋊C₄	—	2	6	no	—
114	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
115	C₂×C₄²	C₃×D₄	1	6	yes	—
116	C₂×C₄²	C₂³×C₃	1	6	yes	—
117	#103	C₂³×C₃	1	64	yes	—
118	C₄×C₈	C₂³×C₃	1	16	yes	—
119	C₂³×C₃	#124	1	4	yes	—
120	C₂³×C₃	C₂×SD₁₆	1	4	yes	—
121	C₂³×C₃	C₂×Q₁₆	1	4	yes	—
122	C₂³×C₃	C₂²×D₄	1	3	yes	—
123	C₃×Q₈	—	2	16	no	—
124	C₃×Q₈	—	4	24	no	—
125	C₃×Q₈	—	4	32	no	—
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	2	8	no	—
128	C₂×C₃×C₄	—	4	24	no	—
129	C₂×C₃×C₄	—	4	24	no	—
130	C₂×C₃×C₄	#124	1	12	yes	—
131	C₂×C₃×C₄	C₂×SD₁₆	1	24	yes	—
132	C₂×C₃×C₄	C₂×Q₁₆	1	24	yes	—
133	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
134	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
135	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
136	C₂×C₃×C₄	C₂×#41	1	12	yes	—
137	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
138	C₃×C₈	—	2	8	no	—
139	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂×Q₈	—	2	12	no	—
141	C₂×Q₈	—	2	16	no	—
142	C₂×Q₈	—	2	48	no	—
143	C₂×Q₈	—	4	24	no	—
144	C₂×Q₈	—	4	48	no	—
145	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
146	C₂²×C₄	—	2	12	no	—
147	C₂²×C₄	—	2	12	no	—
148	C₂²×C₄	—	4	8	no	—
149	C₂²×C₄	—	4	8	no	—
150	C₂²×C₄	C₃×SD₁₆	1	8	yes	—
151	C₂²×C₄	C₃×Q₁₆	1	8	yes	—
152	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
153	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
154	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
155	C₂²×C₄	C₃×#41	1	8	yes	—
156	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
157	C₂×C₈	—	2	24	no	—
158	C₂×C₈	C₂⁴×C₃	1	8	yes	—
159	C₄⋊C₄	—	2	16	no	—
160	C₄⋊C₄	—	2	16	no	—
161	C₄⋊C₄	—	2	16	no	—
162	C₄⋊C₄	—	2	12	no	—
163	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
164	C₄²	C₂×C₃×D₄	1	12	yes	—
165	C₄²	C₂⁴×C₃	1	4	yes	—
166	C₂²×C₃	#487	1	16	yes	direct product factor
167	C₂²×C₃	C₂×#124	1	6	yes	—
168	C₂²×C₃	C₂²×SD₁₆	1	6	yes	—
169	C₂²×C₃	C₂²×Q₁₆	1	6	yes	—
170	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
171	C₃×C₄	—	4	8	no	—
172	C₃×C₄	—	4	8	no	—
173	C₃×C₄	C₂×#124	1	6	yes	—
174	C₃×C₄	C₂²×SD₁₆	1	8	yes	—
175	C₃×C₄	C₂²×Q₁₆	1	8	yes	—
176	C₃×C₄	C₂²×#41	1	2	yes	—
177	C₂³	C₃×#124	1	4	yes	—
178	C₂³	C₂×C₃×SD₁₆	1	4	yes	—
179	C₂³	C₂×C₃×Q₁₆	1	4	yes	—
180	C₂³	C₂²×C₃×D₄	1	3	yes	—
181	Q₈	—	2	16	no	—
182	Q₈	—	4	32	no	—
183	Q₈	—	4	24	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	4	24	no	—
187	C₂×C₄	—	4	24	no	—
188	C₂×C₄	C₃×#124	1	12	yes	—
189	C₂×C₄	C₂×C₃×SD₁₆	1	24	yes	—
190	C₂×C₄	C₂×C₃×Q₁₆	1	24	yes	—
191	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
192	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
193	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
194	C₂×C₄	C₂×C₃×#41	1	12	yes	—
195	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
196	C₈	—	2	8	no	—
197	C₂×C₃	C₂×#487	1	12	yes	direct product factor
198	C₂×C₃	C₂²×#124	1	1	yes	characteristic
199	C₂×C₃	C₂³×SD₁₆	1	1	yes	characteristic
200	C₂×C₃	C₂³×Q₁₆	1	1	yes	characteristic
201	C₂²	C₃×#487	1	16	yes	direct product factor
202	C₂²	C₂×C₃×#124	1	6	yes	—
203	C₂²	C₂²×C₃×SD₁₆	1	6	yes	—
204	C₂²	C₂²×C₃×Q₁₆	1	6	yes	—
205	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
206	C₄	—	4	8	no	—
207	C₄	—	4	8	no	—
208	C₄	C₂×C₃×#124	1	6	yes	—
209	C₄	C₂²×C₃×SD₁₆	1	8	yes	—
210	C₄	C₂²×C₃×Q₁₆	1	8	yes	—
211	C₄	C₂²×C₃×#41	1	2	yes	—
212	C₃	C₂²×#487	1	1	yes	characteristic, direct product factor, Sylow
213	C₂	C₂×C₃×#487	1	12	yes	direct product factor
214	C₂	C₂²×C₃×#124	1	1	yes	characteristic
215	C₂	C₂³×C₃×SD₁₆	1	1	yes	characteristic
216	C₂	C₂³×C₃×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
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
3. C₂×C₃×#487	C₂	direct (N × H)
6. C₂²×#487	C₃	direct (N × H)
19. C₃×#487	C₂²	direct (N × H)
24. C₂×#487	C₂×C₃	direct (N × H)
64. #487	C₂²×C₃	direct (N × H)
166. C₂²×C₃	#487	direct (N × H)
197. C₂×C₃	C₂×#487	direct (N × H)
201. C₂²	C₃×#487	direct (N × H)
212. C₃	C₂²×#487	direct (N × H)
213. C₂	C₂×C₃×#487	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
C₂×C₃×#487	2
C₂²×C₃×#103	2
C₂²×C₃×C₄×C₈	2
C₂²×#487	3

Subgroups — C22×C3×#487 #10679210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#487 #10679210