Subgroups — #479949

Order 512.

580 conjugacy classes of subgroups.

The subgroup lattice has 132 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	2	yes	maximal
2	C₈×#131	C₂	1	1	yes	maximal, characteristic
3	#11411	C₂	1	3	yes	maximal, complemented
4	#11408	C₂	1	6	yes	maximal, complemented
5	C₂×#1312	C₂	1	3	yes	maximal, complemented
6	C₈×#41	C₄	1	1	yes	characteristic
7	C₈×#41	C₂²	1	1	yes	characteristic
8	C₄×#131	C₄	1	1	yes	characteristic
9	Q₈×C₁₆	C₄	1	4	yes	—
10	D₄×C₁₆	C₄	1	12	yes	—
11	#2162	C₂²	1	6	yes	—
12	C₂×C₄×C₁₆	C₂²	1	6	yes	—
13	#1450	C₄	1	3	yes	—
14	C₈×M₁₆	C₄	1	3	yes	—
15	C₈×M₁₆	C₂²	1	3	yes	—
16	C₂×C₈²	C₂²	1	3	yes	—
17	#1312	C₂²	1	1	yes	characteristic
18	#1312	C₂²	1	6	yes	—
19	#1312	C₂²	1	3	yes	—
20	#1311	C₂²	1	6	yes	—
21	C₂×#131	C₈	1	2	yes	—
22	C₄×#41	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂×C₄	1	12	yes	—
24	C₈×Q₈	C₈	1	2	yes	—
25	C₈×Q₈	C₂×C₄	1	2	yes	—
26	C₈×D₄	C₈	1	6	yes	—
27	C₈×D₄	C₂×C₄	1	6	yes	—
28	#431	C₂×C₄	1	3	yes	—
29	#431	C₂×C₄	1	3	yes	—
30	#405	C₈	1	6	yes	—
31	C₄×M₁₆	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
33	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
34	C₂×C₄×C₈	C₂³	1	3	yes	—
35	C₄⋊C₁₆	C₂×C₄	1	12	yes	—
36	C₂²⋊C₁₆	C₂×C₄	1	12	yes	—
37	C₄×C₁₆	C₂×C₄	1	12	yes	—
38	C₄×C₁₆	C₂³	1	6	yes	—
39	C₄×C₁₆	C₂³	1	2	yes	—
40	#322	C₂×C₄	1	6	yes	—
41	C₈²	C₂×C₄	1	3	yes	—
42	C₈²	C₂×C₄	1	3	yes	—
43	C₈²	C₂³	1	3	yes	—
44	C₈²	C₂³	1	1	yes	characteristic
45	C₂×#41	C₂×C₈	1	1	yes	characteristic
46	#131	—	2	8	no	—
47	C₂×M₁₆	C₂×C₈	1	6	yes	—
48	C₂²×C₈	C₂×C₈	1	6	yes	—
49	C₂²×C₈	C₂×C₈	1	6	yes	—
50	C₂²×C₈	C₂²×C₄	1	6	yes	—
51	C₄×Q₈	C₄²	1	1	yes	characteristic
52	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
53	C₄×D₄	C₄²	1	3	yes	—
54	C₄×D₄	C₂×C₈	1	3	yes	—
55	#117	C₂×C₈	1	3	yes	—
56	C₂×C₄²	C₂²×C₄	1	3	yes	—
57	C₂×C₁₆	—	4	12	no	—
58	C₂×C₁₆	C₂²×C₄	1	4	yes	—
59	C₂×C₁₆	C₂²×C₄	1	12	yes	—
60	C₄⋊C₈	C₄²	1	6	yes	—
61	C₄⋊C₈	C₂×C₈	1	6	yes	—
62	C₂²⋊C₈	C₄²	1	6	yes	—
63	C₂²⋊C₈	C₂×C₈	1	6	yes	—
64	#97	C₂×C₈	1	6	yes	—
65	C₄×C₈	C₂×C₈	1	6	yes	—
66	C₄×C₈	C₂×C₈	1	6	yes	—
67	C₄×C₈	C₂²×C₄	1	6	yes	—
68	C₄×C₈	C₂²×C₄	1	3	yes	—
69	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂²×C₄	1	3	yes	—
72	C₄×C₈	C₂²×C₄	1	3	yes	—
73	C₄×C₈	C₂⁴	1	1	yes	characteristic
74	#41	M₃₂	1	8	yes	—
75	C₂×Q₈	C₄×C₈	1	1	yes	characteristic
76	C₂×D₄	C₄×C₈	1	3	yes	—
77	C₂²×C₄	C₂×C₄²	1	3	yes	—
78	C₂²×C₄	C₂²×C₈	1	3	yes	—
79	C₂²×C₄	C₂²×C₈	1	3	yes	—
80	M₁₆	—	2	12	no	—
81	C₂×C₈	—	2	12	no	—
82	C₂×C₈	—	2	12	no	—
83	C₂×C₈	—	2	12	no	—
84	C₂×C₈	C₂×C₄²	1	6	yes	—
85	C₂×C₈	C₂²×C₈	1	2	yes	—
86	C₂×C₈	C₂²×C₈	1	6	yes	—
87	C₂×C₈	C₂²×C₈	1	6	yes	—
88	C₂×C₈	C₂²×C₈	1	2	yes	—
89	C₂×C₈	C₂³×C₄	1	2	yes	—
90	C₄⋊C₄	C₄×C₈	1	3	yes	—
91	C₄⋊C₄	C₄×C₈	1	3	yes	—
92	C₂²⋊C₄	C₄×C₈	1	6	yes	—
93	C₄²	C₂×C₄²	1	3	yes	—
94	C₄²	C₂²×C₈	1	3	yes	—
95	C₄²	C₂²×C₈	1	3	yes	—
96	C₄²	C₂³×C₄	1	1	yes	characteristic
97	C₁₆	—	2	4	no	—
98	C₁₆	—	2	12	no	—
99	C₂³	C₂×C₄×C₈	1	3	yes	—
100	Q₈	#346	1	4	yes	—
101	D₄	#346	1	12	yes	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	C₂×C₄×C₈	1	3	yes	—
105	C₂×C₄	C₂×C₄×C₈	1	3	yes	—
106	C₂×C₄	C₂×C₄×C₈	1	3	yes	—
107	C₂×C₄	C₂×M₃₂	1	12	yes	—
108	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
109	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
110	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
111	C₈	—	2	6	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	6	no	—
114	C₈	—	2	6	no	—
115	C₈	#504	1	4	yes	—
116	C₈	#568	1	4	yes	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂×#346	1	6	yes	—
120	C₂²	C₂²×C₄×C₈	1	1	yes	characteristic
121	C₄	—	2	3	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂×#346	1	6	yes	—
124	C₄	#2875	1	2	yes	—
125	C₄	C₂²×M₃₂	1	2	yes	—
126	C₄	C₂×#504	1	2	yes	—
127	C₂	—	2	6	no	—
128	C₂	#17316	1	1	yes	characteristic
129	C₂	C₂²×#346	1	1	yes	characteristic
130	C₂	C₄×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂²×C₄×C₈
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ 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. #11411	C₂	semidirect
4. #11408	C₂	semidirect
5. C₂×#1312	C₂	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
C₈×#131	2
#11411	2
#11408	2
C₂×#1312	2