Subgroups — #497088

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33474	C₂	1	2	yes	maximal
2	#21732	C₂	1	1	yes	maximal, characteristic
3	#13266	C₂	1	2	yes	maximal
4	#13106	C₂	1	8	yes	maximal
5	#12883	C₂	1	1	yes	maximal, characteristic
6	#12590	C₂	1	1	yes	maximal, characteristic
7	#3160	C₄	1	1	yes	characteristic
8	#2960	C₂²	1	2	yes	—
9	#2183	—	2	2	no	—
10	#2183	—	2	2	no	—
11	Q₈×C₁₆	—	2	2	no	—
12	Q₈×C₁₆	—	2	2	no	—
13	#2151	C₂²	1	4	yes	—
14	#2150	C₂²	1	2	yes	—
15	#2108	C₂²	1	4	yes	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	#1994	C₄	1	2	yes	—
19	#1945	C₄	1	4	yes	—
20	#1848	C₄	1	1	yes	characteristic
21	#1772	C₂²	1	4	yes	—
22	C₄²×C₈	C₂²	1	1	yes	characteristic
23	#1379	C₂²	1	8	yes	—
24	#1376	C₂²	1	4	yes	—
25	#1374	C₂²	1	4	yes	—
26	#533	C₂×C₄	1	2	yes	—
27	C₂×M₃₂	C₂³	1	4	yes	—
28	C₂²×C₁₆	C₂³	1	2	yes	—
29	C₂²×C₁₆	C₂³	1	2	yes	—
30	#500	C₈	1	1	yes	characteristic
31	#500	C₈	1	1	yes	characteristic
32	#499	C₈	1	4	yes	—
33	#498	C₈	1	2	yes	—
34	#446	—	2	2	no	—
35	#446	—	2	2	no	—
36	C₈×Q₈	—	2	2	no	—
37	C₈×Q₈	—	2	2	no	—
38	C₂×#108	C₂×C₄	1	4	yes	—
39	#427	C₂×C₄	1	4	yes	—
40	#424	C₂×C₄	1	4	yes	—
41	#423	C₂×C₄	1	2	yes	—
42	#405	C₂³	1	4	yes	—
43	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	2	yes	—
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₄³	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₁₆	—	2	4	no	—
49	C₄⋊C₁₆	—	2	4	no	—
50	C₄⋊C₁₆	—	2	2	no	—
51	C₄⋊C₁₆	—	2	2	no	—
52	#346	—	2	4	no	—
53	C₄×C₁₆	—	2	4	no	—
54	C₄×C₁₆	—	2	1	no	—
55	C₄×C₁₆	—	2	1	no	—
56	C₄×C₁₆	D₄	1	1	yes	characteristic
57	C₄×C₁₆	D₄	1	2	yes	—
58	C₄×C₁₆	D₄	1	1	yes	characteristic
59	#339	C₂×C₄	1	8	yes	—
60	C₂×M₁₆	C₂²×C₄	1	4	yes	—
61	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂²×C₄	1	2	yes	—
63	C₂²×C₈	C₂⁴	1	1	yes	characteristic
64	C₄⋊Q₈	C₂×C₈	1	2	yes	—
65	C₄⋊Q₈	C₂×C₈	1	2	yes	—
66	#125	C₂×C₈	1	4	yes	—
67	C₂²⋊Q₈	—	2	2	no	—
68	C₂²⋊Q₈	—	2	2	no	—
69	C₄×Q₈	—	2	2	no	—
70	C₄×Q₈	—	2	2	no	—
71	#117	C₂²×C₄	1	4	yes	—
72	C₂×C₄²	—	2	1	no	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	C₂²×C₄	1	2	yes	—
75	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
76	M₃₂	—	2	8	no	—
77	C₂×C₁₆	—	2	4	no	—
78	C₂×C₁₆	—	2	4	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	4	no	—
81	C₂×C₁₆	C₂×D₄	1	2	yes	—
82	C₂×C₁₆	C₂×D₄	1	2	yes	—
83	#108	—	4	4	no	—
84	#107	C₂×C₈	1	4	yes	—
85	#107	C₂×C₈	1	4	yes	—
86	#106	C₂×C₈	1	8	yes	—
87	C₄⋊C₈	—	2	4	no	—
88	C₄⋊C₈	—	2	4	no	—
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	2	2	no	—
91	#97	—	2	4	no	—
92	C₄×C₈	—	2	4	no	—
93	C₄×C₈	—	2	4	no	—
94	C₄×C₈	—	4	2	no	—
95	C₄×C₈	C₂×C₈	1	1	yes	characteristic
96	C₄×C₈	C₂×C₈	1	2	yes	—
97	C₄×C₈	C₂×C₈	1	1	yes	characteristic
98	C₄×C₈	C₂×D₄	1	1	yes	characteristic
99	C₄×C₈	C₂×D₄	1	1	yes	characteristic
100	C₄×C₈	#41	1	1	yes	characteristic
101	C₄×C₈	#41	1	1	yes	characteristic
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	#131	1	1	yes	characteristic
106	C₂²×C₄	#131	1	1	yes	characteristic
107	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
108	M₁₆	—	2	8	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	C₄×D₄	1	1	yes	characteristic
114	C₂×C₈	C₄×D₄	1	1	yes	characteristic
115	C₂×C₈	C₂²×C₈	1	2	yes	—
116	C₂×C₈	C₂²×C₈	1	2	yes	—
117	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₈	C₂×#41	1	1	yes	characteristic
119	C₂×C₈	2₋¹⁺⁴	1	4	yes	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂²×C₈	1	4	yes	—
125	C₄⋊C₄	C₂²×C₈	1	4	yes	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₄²	—	2	4	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	4	2	no	—
132	C₄²	C₄×D₄	1	1	yes	characteristic
133	C₄²	C₄×D₄	1	1	yes	characteristic
134	C₄²	C₂²×C₈	1	1	yes	characteristic
135	C₄²	C₂²×C₈	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	—	2	2	no	—
138	C₁₆	—	4	4	no	—
139	C₁₆	—	4	2	no	—
140	C₂³	C₂×#131	1	1	yes	characteristic
141	Q₈	—	4	4	no	—
142	Q₈	—	4	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₈×D₄	1	1	yes	characteristic
151	C₂×C₄	C₈×D₄	1	1	yes	characteristic
152	C₂×C₄	C₈×D₄	1	1	yes	characteristic
153	C₂×C₄	C₈×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₄	#519	1	2	yes	—
156	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
157	C₂×C₄	#568	1	2	yes	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	4	4	no	—
163	C₈	#536	1	2	yes	—
164	C₈	#541	1	2	yes	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
168	C₂²	#2985	1	2	yes	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	4	no	—
172	C₄	—	4	4	no	—
173	C₄	—	4	4	no	—
174	C₄	#2170	1	2	yes	—
175	C₄	#2171	1	2	yes	—
176	C₄	#2327	1	1	yes	characteristic
177	C₄	#2932	1	1	yes	characteristic
178	C₄	#2982	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#22238	1	1	yes	characteristic
181	C₂	C₂×#2170	1	1	yes	characteristic
182	C₂	C₂×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#33474	2
#21732	2
#13266	2
#13106	2
#12883	2
#12590	2