Subgroups — #129043

Order 512.

324 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#364	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#11399	C₂	1	2	yes	maximal
4	#8266	C₂	1	1	yes	maximal, characteristic
5	#7141	C₂	1	2	yes	maximal
6	C₂×#364	C₄	1	2	yes	—
7	C₄×M₃₂	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	C₄	1	2	yes	—
9	C₂×C₄×C₁₆	C₄	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	#1564	C₄	1	2	yes	—
14	#1450	C₂²	1	2	yes	—
15	#1376	C₄	1	2	yes	—
16	#1311	—	2	2	no	—
17	C₈×C₁₆	—	2	2	no	—
18	#1276	C₄	1	2	yes	—
19	C₂×M₃₂	C₂×C₄	1	4	yes	—
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₁₆	C₂×C₄	1	2	yes	—
22	#431	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	C₄³	C₂×C₄	1	1	yes	characteristic
30	#364	C₈	1	8	yes	complemented
31	#346	—	2	2	no	—
32	C₄×C₁₆	—	2	4	no	—
33	C₄×C₁₆	—	2	2	no	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	—	2	1	no	—
36	C₄×C₁₆	C₈	1	8	yes	complemented
37	C₄×C₁₆	D₄	1	1	yes	characteristic
38	C₄×C₁₆	D₄	1	1	yes	characteristic
39	C₄×C₁₆	Q₈	1	2	yes	—
40	#335	C₈	1	4	yes	—
41	#334	C₈	1	4	yes	—
42	#322	—	2	2	no	—
43	C₈²	—	2	2	no	—
44	C₂²×C₈	C₄²	1	1	yes	characteristic
45	C₂²×C₈	C₄²	1	1	yes	characteristic
46	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄²	—	2	2	no	—
49	C₂×C₄²	C₄²	1	1	yes	characteristic
50	C₂×C₄²	C₄²	1	1	yes	characteristic
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	M₃₂	—	2	4	no	—
53	M₃₂	C₂×C₈	1	8	yes	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
58	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
59	C₂×C₁₆	C₂×C₈	1	4	yes	—
60	C₂×C₁₆	C₂×C₈	1	4	yes	—
61	C₂×C₁₆	M₁₆	1	4	yes	—
62	C₄⋊C₈	C₂×C₈	1	8	yes	—
63	C₂²⋊C₈	—	2	4	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	—	2	4	no	—
67	C₄×C₈	C₂×C₈	1	4	yes	—
68	C₄×C₈	C₂×C₈	1	4	yes	—
69	C₄×C₈	C₂×C₈	1	2	yes	—
70	C₄×C₈	C₂×C₈	1	2	yes	—
71	C₄×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₄×C₈	#41	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
78	C₂²×C₄	#131	1	2	yes	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	C₄×C₈	1	2	yes	—
83	C₂×C₈	C₄×C₈	1	1	yes	characteristic
84	C₂×C₈	C₄×C₈	1	1	yes	characteristic
85	C₂×C₈	C₄×C₈	1	2	yes	—
86	C₂×C₈	C₄×C₈	1	2	yes	—
87	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₈	#117	1	1	yes	characteristic
89	C₂×C₈	C₄×D₄	1	1	yes	characteristic
90	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₈	C₂²×C₈	1	2	yes	—
92	C₂×C₈	C₂²×C₈	1	2	yes	—
93	C₂×C₈	C₂×M₁₆	1	2	yes	—
94	C₂×C₈	#131	1	2	yes	—
95	C₄²	—	2	8	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₄²	C₄×C₈	1	2	yes	—
99	C₄²	C₄×C₈	1	2	yes	—
100	C₄²	C₄×D₄	1	1	yes	characteristic
101	C₄²	C₄×Q₈	1	1	yes	characteristic
102	C₄²	C₂²×C₈	1	2	yes	—
103	C₁₆	—	2	4	no	—
104	C₁₆	—	2	8	no	—
105	C₁₆	C₄⋊C₈	1	4	yes	—
106	C₁₆	C₄⋊C₈	1	4	yes	—
107	C₂³	#405	1	1	yes	characteristic
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
118	C₂×C₄	#405	1	1	yes	characteristic
119	C₂×C₄	C₈×D₄	1	2	yes	—
120	C₂×C₄	C₈×Q₈	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	—	4	8	no	—
123	C₈	C₈²	1	2	yes	—
124	C₈	C₈²	1	2	yes	—
125	C₈	C₂×C₄⋊C₈	1	2	yes	—
126	C₈	#431	1	2	yes	—
127	C₈	C₈×D₄	1	2	yes	—
128	C₈	C₈×Q₈	1	2	yes	—
129	C₂²	—	2	2	no	—
130	C₂²	C₈×M₁₆	1	1	yes	characteristic
131	C₂²	#1450	1	1	yes	characteristic
132	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
133	C₄	—	2	4	no	—
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	C₂×C₈²	1	1	yes	characteristic
137	C₄	C₈×M₁₆	1	1	yes	characteristic
138	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
139	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
140	C₄	#2170	1	4	yes	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	C₈×C₄⋊C₈	1	1	yes	characteristic
144	C₂	#12664	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	D₄
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₄⋊C₄
prime radical	C₂³	#405
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
30. #364	C₈	semidirect
36. C₄×C₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#364	2
C₄²×C₁₆	2
#11399	2
#8266	2
#7141	2