Subgroups — #483468

Order 512.

460 conjugacy classes of subgroups.

The subgroup lattice has 141 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×#41	C₂	1	1	yes	maximal, characteristic
2	#17828	C₂	1	2	yes	maximal
3	#12032	C₂	1	6	yes	maximal, complemented
4	#12022	C₂	1	3	yes	maximal, complemented
5	C₂×#1371	C₂	1	3	yes	maximal, complemented
6	C₈×#41	C₂²	1	1	yes	characteristic
7	#2906	—	2	2	no	—
8	Q₈×C₁₆	C₄	1	1	yes	characteristic
9	Q₈×C₁₆	C₄	1	1	yes	characteristic
10	D₄×C₁₆	C₄	1	3	yes	—
11	D₄×C₁₆	C₄	1	3	yes	—
12	#2162	C₂²	1	3	yes	—
13	C₂×C₄×C₁₆	C₂²	1	3	yes	—
14	#1571	—	2	6	no	—
15	#1567	C₂²	1	6	yes	—
16	C₂×#335	C₂²	1	6	yes	—
17	#1371	C₂²	1	6	yes	—
18	#1371	C₂²	1	6	yes	—
19	#1371	C₂²	1	1	yes	characteristic
20	#1371	C₂²	1	3	yes	—
21	C₂×#131	—	4	2	no	—
22	C₄×#41	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂×C₄	1	3	yes	—
24	C₂²×C₁₆	C₂×C₄	1	3	yes	—
25	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₈×Q₈	Q₈	1	1	yes	characteristic
27	C₈×D₄	C₂×C₄	1	3	yes	—
28	C₈×D₄	Q₈	1	3	yes	—
29	#431	D₄	1	3	yes	—
30	#424	—	4	6	no	—
31	#423	—	2	6	no	—
32	C₂×C₄⋊C₈	—	2	6	no	—
33	C₂×C₄×C₈	C₂³	1	3	yes	—
34	C₄⋊C₁₆	C₂×C₄	1	3	yes	—
35	C₄⋊C₁₆	C₂×C₄	1	3	yes	—
36	C₂²⋊C₁₆	C₂×C₄	1	6	yes	—
37	C₄×C₁₆	C₂×C₄	1	3	yes	—
38	C₄×C₁₆	C₂×C₄	1	3	yes	—
39	C₄×C₁₆	C₂³	1	1	yes	characteristic
40	C₄×C₁₆	C₂³	1	3	yes	—
41	#335	—	2	6	no	—
42	#335	—	2	6	no	—
43	#335	C₂³	1	2	yes	—
44	#335	C₂³	1	6	yes	—
45	C₂×#41	D₈	1	1	yes	characteristic
46	#131	—	8	8	no	—
47	C₂×M₁₆	—	4	6	no	—
48	C₂²×C₈	—	4	6	no	—
49	C₂²×C₈	C₂²×C₄	1	3	yes	—
50	C₂²×C₈	C₂×Q₈	1	3	yes	—
51	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Q₈	Q₁₆	1	1	yes	characteristic
53	C₄×D₄	C₄⋊C₄	1	3	yes	—
54	C₄×D₄	Q₁₆	1	3	yes	—
55	#117	D₈	1	3	yes	—
56	C₂×C₄²	C₂×D₄	1	3	yes	—
57	C₂×C₁₆	—	2	6	no	—
58	C₂×C₁₆	—	4	3	no	—
59	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₁₆	C₂²×C₄	1	3	yes	—
61	C₂×C₁₆	C₂²×C₄	1	3	yes	—
62	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	2	6	no	—
65	C₄⋊C₈	—	4	6	no	—
66	C₄⋊C₈	—	4	6	no	—
67	C₄⋊C₈	C₄⋊C₄	1	3	yes	—
68	C₄⋊C₈	C₄⋊C₄	1	3	yes	—
69	C₂²⋊C₈	C₄⋊C₄	1	6	yes	—
70	C₄×C₈	C₂²×C₄	1	3	yes	—
71	C₄×C₈	C₂×D₄	1	3	yes	—
72	C₄×C₈	C₂×Q₈	1	3	yes	—
73	C₄×C₈	C₂⁴	1	1	yes	characteristic
74	#41	D₁₆	1	4	yes	—
75	#41	Q₃₂	1	4	yes	—
76	C₂×Q₈	#107	1	1	yes	characteristic
77	C₂×D₄	#107	1	3	yes	—
78	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
79	C₂²×C₄	C₂×D₈	1	3	yes	—
80	C₂²×C₄	C₂×Q₁₆	1	3	yes	—
81	M₁₆	—	8	12	no	—
82	C₂×C₈	—	2	3	no	—
83	C₂×C₈	—	2	3	no	—
84	C₂×C₈	—	4	3	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	6	no	—
87	C₂×C₈	—	8	12	no	—
88	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
89	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
92	C₄⋊C₄	#107	1	3	yes	—
93	C₄⋊C₄	#107	1	3	yes	—
94	C₂²⋊C₄	#107	1	6	yes	—
95	C₄²	C₂×C₄⋊C₄	1	3	yes	—
96	C₄²	C₂×D₈	1	3	yes	—
97	C₄²	C₂×Q₁₆	1	3	yes	—
98	C₄²	C₂²×D₄	1	1	yes	characteristic
99	C₁₆	—	2	3	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	2	3	no	—
102	C₁₆	#131	1	2	yes	complemented
103	C₂³	C₂×#107	1	3	yes	—
104	Q₈	#366	1	4	yes	—
105	D₄	#366	1	12	yes	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	4	3	no	—
109	C₂×C₄	C₂×#107	1	3	yes	—
110	C₂×C₄	C₂×#107	1	3	yes	—
111	C₂×C₄	C₂×#107	1	3	yes	—
112	C₂×C₄	C₂×D₁₆	1	6	yes	—
113	C₂×C₄	C₂×Q₃₂	1	6	yes	—
114	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
116	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
117	C₈	—	2	3	no	—
118	C₈	—	2	1	no	—
119	C₈	—	2	3	no	—
120	C₈	—	8	2	no	—
121	C₈	—	8	6	no	—
122	C₈	C₂×#131	1	1	yes	characteristic
123	C₈	#568	1	1	yes	characteristic
124	C₂²	—	2	3	no	—
125	C₂²	—	2	3	no	—
126	C₂²	C₂×#366	1	6	yes	—
127	C₂²	C₂²×#107	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	3	no	—
130	C₄	—	2	3	no	—
131	C₄	C₂×#366	1	6	yes	—
132	C₄	#2906	1	1	yes	characteristic
133	C₄	#2916	1	1	yes	characteristic
134	C₄	C₂²×D₁₆	1	1	yes	characteristic
135	C₄	C₂²×Q₃₂	1	1	yes	characteristic
136	C₂	—	2	6	no	—
137	C₂	#17828	1	1	yes	characteristic
138	C₂	C₂²×#366	1	1	yes	characteristic
139	C₂	#33414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12032	C₂	semidirect
4. #12022	C₂	semidirect
5. C₂×#1371	C₂	semidirect
102. C₁₆	#131	semidirect

Maximal subgroups

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

subgroup	index
C₁₆×#41	2
#17828	2
#12032	2
#12022	2
C₂×#1371	2