Subgroups — #135651

Order 512.

292 conjugacy classes of subgroups.

The subgroup lattice has 153 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#8806	C₂	1	2	yes	maximal
3	#7154	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆	C₄	1	2	yes	—
5	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	#1842	—	2	2	no	—
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×#335	—	2	2	no	—
11	C₂×#335	C₂²	1	4	yes	—
12	C₂²×C₁₆	C₂×C₄	1	2	yes	—
13	C₂²×C₁₆	C₂×C₄	1	2	yes	—
14	C₂×C₄⋊C₈	—	2	4	no	—
15	C₂×C₄⋊C₈	—	4	2	no	—
16	C₂×C₄⋊C₈	—	4	2	no	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
21	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
23	C₄³	D₄	1	1	yes	characteristic
24	C₄×C₁₆	—	2	4	no	—
25	C₄×C₁₆	—	2	2	no	—
26	C₄×C₁₆	—	2	4	no	—
27	C₄×C₁₆	C₈	1	4	yes	complemented
28	#335	—	2	8	no	—
29	#335	—	4	4	no	—
30	C₂²×C₈	—	4	4	no	—
31	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄²	D₈	1	1	yes	characteristic
38	C₂×C₄²	D₈	1	1	yes	characteristic
39	C₂×C₄²	Q₁₆	1	1	yes	characteristic
40	C₂×C₄²	Q₁₆	1	1	yes	characteristic
41	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	—	2	2	no	—
44	C₂×C₁₆	—	2	2	no	—
45	C₂×C₁₆	—	2	2	no	—
46	C₂×C₁₆	C₂×C₈	1	4	yes	—
47	C₂×C₁₆	M₁₆	1	4	yes	—
48	C₄⋊C₈	—	4	4	no	—
49	C₄⋊C₈	—	4	4	no	—
50	C₄⋊C₈	—	8	4	no	—
51	C₄⋊C₈	—	8	4	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	C₂×C₈	1	2	yes	—
56	C₄×C₈	M₁₆	1	2	yes	—
57	C₄×C₈	SD₁₆	1	4	yes	—
58	C₄×C₈	#41	1	4	yes	—
59	C₂²×C₄	#107	1	1	yes	characteristic
60	C₂²×C₄	#107	1	1	yes	characteristic
61	C₂²×C₄	#107	1	1	yes	characteristic
62	C₂²×C₄	#107	1	1	yes	characteristic
63	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
65	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	8	4	no	—
71	C₂×C₈	—	8	4	no	—
72	C₂×C₈	—	8	4	no	—
73	C₂×C₈	C₄⋊C₈	1	2	yes	—
74	C₂×C₈	#106	1	2	yes	—
75	C₂×C₈	#106	1	2	yes	—
76	C₂×C₈	#108	1	2	yes	—
77	C₂×C₈	#117	1	2	yes	—
78	C₂×C₈	#125	1	2	yes	—
79	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₈	C₂×M₁₆	1	2	yes	—
81	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
82	C₄²	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	2	1	no	—
85	C₄²	—	2	1	no	—
86	C₄²	C₄⋊C₈	1	2	yes	—
87	C₄²	#108	1	2	yes	—
88	C₄²	D₁₆	1	2	yes	—
89	C₄²	SD₃₂	1	4	yes	—
90	C₄²	Q₃₂	1	2	yes	—
91	C₄²	#124	1	2	yes	—
92	C₄²	C₂×SD₁₆	1	2	yes	—
93	C₁₆	—	2	4	no	—
94	C₁₆	—	2	4	no	—
95	C₂³	C₂×#107	1	1	yes	characteristic
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	#335	1	2	yes	—
103	C₂×C₄	#335	1	2	yes	—
104	C₂×C₄	#366	1	2	yes	—
105	C₂×C₄	#366	1	2	yes	—
106	C₂×C₄	#367	1	2	yes	—
107	C₂×C₄	#367	1	2	yes	—
108	C₂×C₄	#368	1	2	yes	—
109	C₂×C₄	#368	1	2	yes	—
110	C₂×C₄	#382	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₄	#423	1	1	yes	characteristic
113	C₂×C₄	C₂×#106	1	1	yes	characteristic
114	C₂×C₄	C₂×#108	1	1	yes	characteristic
115	C₂×C₄	C₂×#108	1	1	yes	characteristic
116	C₂×C₄	#486	1	2	yes	—
117	C₂×C₄	#487	1	2	yes	—
118	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
119	C₂×C₄	C₂×SD₃₂	1	2	yes	—
120	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	2	no	—
123	C₈	—	16	4	no	—
124	C₈	#334	1	4	yes	—
125	C₈	#431	1	2	yes	—
126	C₈	#432	1	2	yes	—
127	C₂²	C₂×#335	1	1	yes	characteristic
128	C₂²	#1567	1	1	yes	characteristic
129	C₂²	#1834	1	1	yes	characteristic
130	C₂²	C₂×#366	1	1	yes	characteristic
131	C₂²	C₂×#367	1	1	yes	characteristic
132	C₂²	C₂×#368	1	1	yes	characteristic
133	C₂²	C₂×#368	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	#1371	1	4	yes	—
138	C₄	#1372	1	4	yes	—
139	C₄	C₂×#334	1	1	yes	characteristic
140	C₄	#1566	1	1	yes	characteristic
141	C₄	#1831	1	1	yes	characteristic
142	C₄	#1837	1	1	yes	characteristic
143	C₄	#2240	1	2	yes	—
144	C₄	#2241	1	2	yes	—
145	C₂	#8784	1	1	yes	characteristic
146	C₂	C₂×#1371	1	1	yes	characteristic
147	C₂	C₂×#1372	1	1	yes	characteristic
148	C₂	#12022	1	1	yes	characteristic
149	C₂	#12023	1	1	yes	characteristic
150	C₂	#12888	1	1	yes	characteristic
151	C₂	#12891	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×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
27. 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
C₄²×C₁₆	2
#8806	2
#7154	2