Subgroups — #483390

Order 512.

475 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33505	C₂	1	2	yes	maximal
2	C₈×#131	C₂	1	1	yes	maximal, characteristic
3	#12018	C₂	1	6	yes	maximal, complemented
4	#12012	C₂	1	3	yes	maximal, complemented
5	#12004	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	—	2	2	no	—
10	#2169	—	2	6	no	—
11	#2162	C₂²	1	6	yes	—
12	#2150	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	#1370	C₂²	1	3	yes	—
18	#1370	C₂²	1	6	yes	—
19	#1370	C₂²	1	1	yes	characteristic
20	#1367	C₂²	1	6	yes	—
21	C₂×#131	—	2	1	no	—
22	C₄×#41	C₂×C₄	1	1	yes	characteristic
23	C₂×M₃₂	—	2	6	no	—
24	C₈×Q₈	C₈	1	1	yes	characteristic
25	C₈×Q₈	C₈	1	1	yes	characteristic
26	C₈×Q₈	D₄	1	1	yes	characteristic
27	C₈×Q₈	Q₈	1	1	yes	characteristic
28	C₈×D₄	C₈	1	3	yes	—
29	C₈×D₄	C₈	1	3	yes	—
30	C₈×D₄	D₄	1	3	yes	—
31	C₈×D₄	Q₈	1	3	yes	—
32	#431	C₂×C₄	1	3	yes	—
33	#431	C₂×C₄	1	3	yes	—
34	#405	—	2	3	no	—
35	C₄×M₁₆	C₂×C₄	1	3	yes	—
36	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
38	C₂×C₄×C₈	C₂³	1	3	yes	—
39	C₄⋊C₁₆	—	2	6	no	—
40	C₄⋊C₁₆	C₂³	1	2	yes	—
41	C₄⋊C₁₆	C₂³	1	6	yes	—
42	C₂²⋊C₁₆	—	2	6	no	—
43	C₄×C₁₆	—	2	6	no	—
44	#322	C₂×C₄	1	6	yes	—
45	C₈²	C₂×C₄	1	3	yes	—
46	C₈²	C₂×C₄	1	3	yes	—
47	C₈²	C₂³	1	3	yes	—
48	C₈²	C₂³	1	1	yes	characteristic
49	C₂×#41	M₁₆	1	1	yes	characteristic
50	#131	—	2	8	no	—
51	C₂×M₁₆	—	2	3	no	—
52	C₂²×C₈	—	2	3	no	—
53	C₂²×C₈	C₂×C₈	1	3	yes	—
54	C₂²×C₈	C₂×C₈	1	3	yes	—
55	C₂²×C₈	C₂×D₄	1	3	yes	—
56	C₂²×C₈	C₂×Q₈	1	3	yes	—
57	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
59	C₄×D₄	C₄⋊C₄	1	3	yes	—
60	C₄×D₄	C₂×C₈	1	3	yes	—
61	#117	M₁₆	1	3	yes	—
62	C₂×C₄²	C₂²×C₄	1	3	yes	—
63	M₃₂	—	8	6	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	6	no	—
66	C₄⋊C₈	C₄⋊C₄	1	3	yes	—
67	C₄⋊C₈	C₄⋊C₄	1	3	yes	—
68	C₄⋊C₈	C₂×C₈	1	3	yes	—
69	C₄⋊C₈	C₂×C₈	1	3	yes	—
70	C₂²⋊C₈	C₄⋊C₄	1	6	yes	—
71	C₂²⋊C₈	C₂×C₈	1	6	yes	—
72	#97	—	2	3	no	—
73	C₄×C₈	—	2	3	no	—
74	C₄×C₈	C₂×C₈	1	3	yes	—
75	C₄×C₈	C₂×C₈	1	3	yes	—
76	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₄×C₈	C₂²×C₄	1	3	yes	—
78	C₄×C₈	C₂²×C₄	1	3	yes	—
79	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₄×C₈	C₂²×C₄	1	3	yes	—
81	C₄×C₈	C₂×D₄	1	3	yes	—
82	C₄×C₈	C₂×Q₈	1	3	yes	—
83	C₄×C₈	C₂⁴	1	1	yes	characteristic
84	#41	—	2	4	no	—
85	C₂×Q₈	C₄⋊C₈	1	1	yes	characteristic
86	C₂×D₄	C₄⋊C₈	1	3	yes	—
87	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
88	C₂²×C₄	C₂²×C₈	1	3	yes	—
89	C₂²×C₄	C₂×M₁₆	1	3	yes	—
90	M₁₆	—	2	12	no	—
91	C₂×C₈	—	2	12	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	3	no	—
94	C₂×C₈	—	2	3	no	—
95	C₂×C₈	—	2	3	no	—
96	C₂×C₈	—	2	6	no	—
97	C₂×C₈	—	4	3	no	—
98	C₂×C₈	—	4	3	no	—
99	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
100	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
101	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₈	C₂²×C₈	1	3	yes	—
103	C₂×C₈	C₂²×C₈	1	3	yes	—
104	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
107	C₄⋊C₄	C₄⋊C₈	1	3	yes	—
108	C₄⋊C₄	C₄⋊C₈	1	3	yes	—
109	C₂²⋊C₄	C₄⋊C₈	1	6	yes	—
110	C₄²	C₂×C₄⋊C₄	1	3	yes	—
111	C₄²	C₂²×C₈	1	3	yes	—
112	C₄²	C₂×M₁₆	1	3	yes	—
113	C₄²	C₂³×C₄	1	1	yes	characteristic
114	C₁₆	—	4	6	no	—
115	C₁₆	—	4	2	no	—
116	C₂³	C₂×C₄⋊C₈	1	3	yes	—
117	Q₈	#364	1	4	yes	—
118	D₄	—	2	6	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	2	3	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	2	3	no	—
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
125	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
126	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
127	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
129	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	—	2	6	no	—
132	C₈	—	2	3	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	3	no	—
135	C₈	—	2	3	no	—
136	C₈	—	2	1	no	—
137	C₈	—	2	3	no	—
138	C₈	#504	1	2	yes	—
139	C₈	#576	1	1	yes	characteristic
140	C₈	#578	1	1	yes	characteristic
141	C₂²	—	2	3	no	—
142	C₂²	—	2	3	no	—
143	C₂²	—	2	3	no	—
144	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
145	C₄	—	2	3	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	3	no	—
148	C₄	C₂×#364	1	6	yes	—
149	C₄	#2913	1	1	yes	characteristic
150	C₄	#2916	1	1	yes	characteristic
151	C₄	C₂×#504	1	1	yes	characteristic
152	C₄	#3407	1	1	yes	characteristic
153	C₂	—	4	3	no	—
154	C₂	#17789	1	1	yes	characteristic
155	C₂	#33398	1	1	yes	characteristic
156	C₂	C₂²×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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. #12018	C₂	semidirect
4. #12012	C₂	semidirect
5. #12004	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33505	2
C₈×#131	2
#12018	2
#12012	2
#12004	2