Subgroups — C₂²×C₃×#491 #10679215

Order 768.

2142 conjugacy classes of subgroups.

The subgroup lattice has 182 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×#125	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#491	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#103	C₂	1	4	yes	maximal
5	C₂²×C₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×#491	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂³×C₃×Q₈	C₂²	1	2	yes	—
8	C₂³×C₃×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
10	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	—
11	C₂×C₃×#125	—	2	6	no	—
12	C₂×C₃×#125	C₂²	1	12	yes	—
13	C₂²×C₃×C₄⋊C₄	—	2	2	no	—
14	C₂²×C₃×C₄⋊C₄	—	2	2	no	—
15	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₃×#491	C₂²	1	64	yes	direct product factor
18	C₂×C₃×#103	C₂²	1	48	yes	—
19	C₂×C₃×#97	C₂²	1	12	yes	—
20	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
21	C₂²×#125	C₂×C₃	1	1	yes	characteristic
22	C₂×#491	C₂×C₃	1	24	yes	direct product factor
23	C₂²×#103	C₂×C₃	1	4	yes	—
24	C₂²×#97	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₃×Q₈	—	2	6	no	—
26	C₂²×C₃×Q₈	—	2	16	no	—
27	C₂²×C₃×Q₈	C₂³	1	12	yes	—
28	C₂³×C₃×C₄	—	2	2	no	—
29	C₂³×C₃×C₄	—	2	2	no	—
30	C₂³×C₃×C₄	D₄	1	2	yes	—
31	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₈	—	2	8	no	—
33	C₂²×C₃×C₈	C₂³	1	12	yes	—
34	C₃×C₄⋊Q₈	—	2	24	no	—
35	C₃×C₄⋊Q₈	C₂³	1	16	yes	—
36	C₃×#125	—	2	24	no	—
37	C₃×#125	C₂³	1	16	yes	—
38	C₂×C₃×C₄⋊C₄	—	2	24	no	—
39	C₂×C₃×C₄⋊C₄	—	2	24	no	—
40	C₂×C₃×C₄⋊C₄	—	2	6	no	—
41	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
42	C₂×C₃×C₄²	D₄	1	6	yes	—
43	C₂×C₃×C₄²	C₂³	1	6	yes	—
44	C₃×#103	C₂³	1	64	yes	—
45	C₃×#97	C₂³	1	16	yes	—
46	C₂³×Q₈	C₂²×C₃	1	2	yes	—
47	C₂³×C₈	C₂²×C₃	1	2	yes	—
48	C₂×C₄⋊Q₈	—	2	6	no	—
49	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	—
50	C₂×#125	—	2	6	no	—
51	C₂×#125	C₂²×C₃	1	12	yes	—
52	C₂²×C₄⋊C₄	—	2	2	no	—
53	C₂²×C₄⋊C₄	—	2	2	no	—
54	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
56	#491	C₂²×C₃	1	64	yes	direct product factor
57	C₂×#103	C₂²×C₃	1	48	yes	—
58	C₂×#97	C₂²×C₃	1	12	yes	—
59	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×Q₈	—	2	12	no	—
61	C₂×C₃×Q₈	—	2	48	no	—
62	C₂×C₃×Q₈	—	4	24	no	—
63	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
64	C₂²×C₃×C₄	—	2	8	no	—
65	C₂²×C₃×C₄	—	2	12	no	—
66	C₂²×C₃×C₄	—	2	12	no	—
67	C₂²×C₃×C₄	—	4	8	no	—
68	C₂²×C₃×C₄	—	4	8	no	—
69	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
70	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
71	C₂²×C₃×C₄	#41	1	8	yes	—
72	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
73	C₂×C₃×C₈	—	2	24	no	—
74	C₂×C₃×C₈	C₂⁴	1	8	yes	—
75	C₃×C₄⋊C₄	—	2	32	no	—
76	C₃×C₄⋊C₄	—	2	32	no	—
77	C₃×C₄⋊C₄	—	2	12	no	—
78	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
79	C₃×C₄²	C₂×D₄	1	12	yes	—
80	C₃×C₄²	C₂⁴	1	4	yes	—
81	C₂²×Q₈	—	2	6	no	—
82	C₂²×Q₈	—	2	16	no	—
83	C₂²×Q₈	C₂³×C₃	1	12	yes	—
84	C₂³×C₄	—	2	2	no	—
85	C₂³×C₄	—	2	2	no	—
86	C₂³×C₄	C₃×D₄	1	2	yes	—
87	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₂²×C₈	—	2	8	no	—
89	C₂²×C₈	C₂³×C₃	1	12	yes	—
90	C₄⋊Q₈	—	2	24	no	—
91	C₄⋊Q₈	C₂³×C₃	1	16	yes	—
92	#125	—	2	24	no	—
93	#125	C₂³×C₃	1	16	yes	—
94	C₂×C₄⋊C₄	—	2	24	no	—
95	C₂×C₄⋊C₄	—	2	24	no	—
96	C₂×C₄⋊C₄	—	2	6	no	—
97	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
98	C₂×C₄²	C₃×D₄	1	6	yes	—
99	C₂×C₄²	C₂³×C₃	1	6	yes	—
100	#103	C₂³×C₃	1	64	yes	—
101	#97	C₂³×C₃	1	16	yes	—
102	C₂³×C₃	#124	1	4	yes	—
103	C₂³×C₃	#137	1	8	yes	—
104	C₂³×C₃	C₂²×D₄	1	3	yes	—
105	C₃×Q₈	—	2	16	no	—
106	C₃×Q₈	—	4	24	no	—
107	C₂×C₃×C₄	—	2	24	no	—
108	C₂×C₃×C₄	—	2	8	no	—
109	C₂×C₃×C₄	—	2	8	no	—
110	C₂×C₃×C₄	—	4	24	no	—
111	C₂×C₃×C₄	—	4	24	no	—
112	C₂×C₃×C₄	#124	1	12	yes	—
113	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
114	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
115	C₂×C₃×C₄	C₂×#41	1	12	yes	—
116	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
117	C₃×C₈	—	2	8	no	—
118	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
119	C₂×Q₈	—	2	12	no	—
120	C₂×Q₈	—	2	48	no	—
121	C₂×Q₈	—	4	24	no	—
122	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	2	12	no	—
125	C₂²×C₄	—	2	12	no	—
126	C₂²×C₄	—	4	8	no	—
127	C₂²×C₄	—	4	8	no	—
128	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
129	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
130	C₂²×C₄	C₃×#41	1	8	yes	—
131	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
132	C₂×C₈	—	2	24	no	—
133	C₂×C₈	C₂⁴×C₃	1	8	yes	—
134	C₄⋊C₄	—	2	32	no	—
135	C₄⋊C₄	—	2	32	no	—
136	C₄⋊C₄	—	2	12	no	—
137	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
138	C₄²	C₂×C₃×D₄	1	12	yes	—
139	C₄²	C₂⁴×C₃	1	4	yes	—
140	C₂²×C₃	#491	1	16	yes	direct product factor
141	C₂²×C₃	C₂×#124	1	6	yes	—
142	C₂²×C₃	C₂×#137	1	12	yes	—
143	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
144	C₃×C₄	—	2	8	no	—
145	C₃×C₄	—	4	8	no	—
146	C₃×C₄	—	4	8	no	—
147	C₃×C₄	C₂×#124	1	6	yes	—
148	C₃×C₄	C₂²×#41	1	2	yes	—
149	C₂³	C₃×#124	1	4	yes	—
150	C₂³	C₃×#137	1	8	yes	—
151	C₂³	C₂²×C₃×D₄	1	3	yes	—
152	Q₈	—	2	16	no	—
153	Q₈	—	4	24	no	—
154	C₂×C₄	—	2	24	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	4	24	no	—
158	C₂×C₄	—	4	24	no	—
159	C₂×C₄	C₃×#124	1	12	yes	—
160	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
161	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
162	C₂×C₄	C₂×C₃×#41	1	12	yes	—
163	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
164	C₈	—	2	8	no	—
165	C₂×C₃	C₂×#491	1	12	yes	direct product factor
166	C₂×C₃	C₂²×#124	1	1	yes	characteristic
167	C₂×C₃	C₂²×#137	1	2	yes	—
168	C₂²	C₃×#491	1	16	yes	direct product factor
169	C₂²	C₂×C₃×#124	1	6	yes	—
170	C₂²	C₂×C₃×#137	1	12	yes	—
171	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
172	C₄	—	2	8	no	—
173	C₄	—	4	8	no	—
174	C₄	—	4	8	no	—
175	C₄	C₂×C₃×#124	1	6	yes	—
176	C₄	C₂²×C₃×#41	1	2	yes	—
177	C₃	C₂²×#491	1	1	yes	characteristic, direct product factor, Sylow
178	C₂	C₂×C₃×#491	1	12	yes	direct product factor
179	C₂	C₂²×C₃×#124	1	1	yes	characteristic
180	C₂	C₂²×C₃×#137	1	2	yes	—

	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×#491	C₂	direct (N × H)
6. C₂²×#491	C₃	direct (N × H)
17. C₃×#491	C₂²	direct (N × H)
22. C₂×#491	C₂×C₃	direct (N × H)
56. #491	C₂²×C₃	direct (N × H)
140. C₂²×C₃	#491	direct (N × H)
165. C₂×C₃	C₂×#491	direct (N × H)
168. C₂²	C₃×#491	direct (N × H)
177. C₃	C₂²×#491	direct (N × H)
178. C₂	C₂×C₃×#491	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×#125	2
C₂×C₃×#491	2
C₂²×C₃×#103	2
C₂²×C₃×#97	2
C₂²×#491	3

Subgroups — C22×C3×#491 #10679215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#491 #10679215