Subgroups — #129041

Order 512.

368 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1311	C₂	1	2	yes	maximal
4	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#7136	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₄	1	2	yes	—
8	C₂×C₄×C₁₆	C₄	1	2	yes	—
9	C₂×C₄×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₈	C₄	1	2	yes	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈²	C₂²	1	2	yes	—
15	#1374	C₄	1	2	yes	—
16	#1311	—	2	4	no	—
17	#1273	C₄	1	2	yes	—
18	C₂²×C₁₆	C₂×C₄	1	2	yes	—
19	C₂²×C₁₆	C₂×C₄	1	4	yes	—
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₈	C₈	1	4	yes	—
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	4	yes	—
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	C₄⋊C₁₆	C₈	1	8	yes	complemented
31	C₄×C₁₆	—	2	4	no	—
32	C₄×C₁₆	—	2	2	no	—
33	C₄×C₁₆	—	2	1	no	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	—	2	2	no	—
36	C₄×C₁₆	C₈	1	8	yes	complemented
37	C₄×C₁₆	D₄	1	2	yes	—
38	C₄×C₁₆	Q₈	1	2	yes	—
39	#336	C₈	1	4	yes	—
40	C₈²	—	2	2	no	—
41	C₈²	—	2	2	no	—
42	C₂²×C₈	C₄²	1	1	yes	characteristic
43	C₂²×C₈	C₄²	1	1	yes	characteristic
44	C₂²×C₈	C₂×C₈	1	8	yes	—
45	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄²	C₄²	1	1	yes	characteristic
48	C₂×C₄²	C₄²	1	1	yes	characteristic
49	C₂×C₄²	C₂×C₈	1	2	yes	—
50	C₂×C₄²	C₂×C₈	1	2	yes	—
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	4	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
57	C₂×C₁₆	C₂×C₈	1	4	yes	—
58	C₂×C₁₆	C₂×C₈	1	8	yes	—
59	C₂×C₁₆	C₂×C₈	1	4	yes	—
60	C₂×C₁₆	M₁₆	1	4	yes	—
61	C₄⋊C₈	—	2	8	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	4	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×D₄	1	1	yes	characteristic
70	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₄×C₈	#41	1	1	yes	characteristic
72	C₄×C₈	#41	1	1	yes	characteristic
73	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
74	C₂²×C₄	C₄×C₈	1	2	yes	—
75	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
77	C₂²×C₄	C₂²×C₈	1	2	yes	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	—	2	8	no	—
84	C₂×C₈	—	2	8	no	—
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	1	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	4	no	—
98	C₄²	C₄×C₈	1	2	yes	—
99	C₄²	C₄×C₈	1	2	yes	—
100	C₄²	M₃₂	1	8	yes	—
101	C₄²	C₄×D₄	1	1	yes	characteristic
102	C₄²	C₄×Q₈	1	1	yes	characteristic
103	C₄²	#131	1	2	yes	—
104	C₁₆	—	2	4	no	—
105	C₁₆	—	2	8	no	—
106	C₁₆	C₄⋊C₈	1	8	yes	complemented
107	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	C₈²	1	2	yes	—
112	C₂×C₄	C₈²	1	2	yes	—
113	C₂×C₄	#346	1	4	yes	—
114	C₂×C₄	#346	1	4	yes	—
115	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
119	C₂×C₄	#405	1	1	yes	characteristic
120	C₂×C₄	#405	1	1	yes	characteristic
121	C₂×C₄	C₈×D₄	1	2	yes	—
122	C₂×C₄	C₈×Q₈	1	2	yes	—
123	C₂×C₄	C₂×M₃₂	1	4	yes	—
124	C₂×C₄	#504	1	4	yes	—
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₈	—	4	8	no	—
128	C₈	C₂×C₄⋊C₈	1	2	yes	—
129	C₈	#431	1	2	yes	—
130	C₈	C₈×D₄	1	2	yes	—
131	C₈	C₈×Q₈	1	2	yes	—
132	C₂²	C₂×C₈²	1	1	yes	characteristic
133	C₂²	C₈×M₁₆	1	1	yes	characteristic
134	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
135	C₂²	C₂×#346	1	2	yes	—
136	C₂²	#2108	1	2	yes	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	#1311	1	4	yes	complemented
140	C₄	C₈×M₁₆	1	1	yes	characteristic
141	C₄	#1450	1	1	yes	characteristic
142	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
143	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
144	C₄	#2169	1	4	yes	—
145	C₄	#2183	1	4	yes	—
146	C₂	C₈×C₄⋊C₈	1	1	yes	characteristic
147	C₂	C₂×#1311	1	1	yes	characteristic
148	C₂	#11399	1	1	yes	characteristic
149	C₂	#11408	1	2	yes	—
150	C₂	#12662	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂×C₈²
socle, prime radical	C₂³	C₂×C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ 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. C₄⋊C₁₆	C₈	semidirect
36. C₄×C₁₆	C₈	semidirect
106. C₁₆	C₄⋊C₈	semidirect
139. C₄	#1311	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₁₆	2
C₄²×C₁₆	2
C₂×#1311	2
C₈×C₄⋊C₈	2
#7136	2