Subgroups — #497172

Order 512.

452 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21715	C₂	1	1	yes	maximal, characteristic
2	#13263	C₂	1	4	yes	maximal
3	#13160	C₂	1	2	yes	maximal
4	#13104	C₂	1	4	yes	maximal
5	#12880	C₂	1	1	yes	maximal, characteristic
6	C₁₆×C₄⋊C₄	C₂	1	2	yes	maximal
7	#12592	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₂²	1	2	yes	—
9	#2399	C₄	1	1	yes	characteristic
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
12	C₂×#346	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	4	yes	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#1985	C₄	1	2	yes	—
16	#1940	C₄	1	2	yes	—
17	#1940	C₄	1	2	yes	—
18	#1843	C₄	1	1	yes	characteristic
19	C₈×C₄⋊C₄	C₂²	1	2	yes	—
20	C₈×C₄⋊C₄	C₂²	1	2	yes	—
21	#1726	C₂²	1	1	yes	characteristic
22	#1374	C₂²	1	2	yes	—
23	#1374	C₂²	1	2	yes	—
24	#1374	C₂²	1	2	yes	—
25	#1374	C₂²	1	4	yes	—
26	#1374	C₂²	1	4	yes	—
27	#1374	C₂²	1	2	yes	—
28	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
29	C₂²×C₁₆	C₂³	1	4	yes	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	2	yes	—
32	C₈×Q₈	—	4	4	no	—
33	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
37	C₂×#97	C₂×C₄	1	1	yes	characteristic
38	C₂×#97	C₂×C₄	1	1	yes	characteristic
39	C₂×#97	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	4	yes	—
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#398	C₈	1	4	yes	—
45	#395	C₈	1	4	yes	—
46	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	#376	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	4	no	—
50	C₄⋊C₁₆	—	4	4	no	—
51	#346	—	2	2	no	—
52	C₄×C₁₆	—	4	4	no	—
53	C₄×C₁₆	Q₈	1	4	yes	complemented
54	#336	C₂×C₄	1	4	yes	—
55	#336	C₂×C₄	1	4	yes	—
56	C₂²×Q₈	C₂×C₈	1	2	yes	—
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₄	1	4	yes	—
59	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂⁴	1	1	yes	characteristic
62	C₄×Q₈	—	4	2	no	—
63	C₄×Q₈	—	4	2	no	—
64	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
65	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
66	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
68	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄²	C₂²×C₄	1	4	yes	—
70	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	4	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	C₂×Q₈	1	4	yes	—
81	C₄⋊C₈	—	2	4	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄⋊C₈	—	4	4	no	—
85	#97	—	2	2	no	—
86	#97	—	4	1	no	—
87	#97	—	4	1	no	—
88	C₄×C₈	—	2	4	no	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	C₂×Q₈	1	2	yes	—
93	C₄×C₈	#41	1	2	yes	—
94	#95	C₂×C₈	1	4	yes	—
95	#95	C₂×C₈	1	2	yes	—
96	#95	C₂×C₈	1	2	yes	—
97	#95	C₂×C₈	1	2	yes	—
98	#95	C₂×C₈	1	2	yes	—
99	C₂×Q₈	—	2	8	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	C₂²×C₈	1	2	yes	—
102	C₂²×C₄	C₂²×C₈	1	4	yes	—
103	C₂²×C₄	C₂²×C₈	1	2	yes	—
104	C₂²×C₄	C₂²×C₈	1	2	yes	—
105	C₂²×C₄	C₂²×C₈	1	4	yes	—
106	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	C₄×Q₈	1	2	yes	—
119	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₈	C₂×#41	1	1	yes	characteristic
121	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
123	C₄⋊C₄	—	2	8	no	—
124	C₄⋊C₄	—	2	8	no	—
125	C₄⋊C₄	—	4	4	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	4	no	—
133	C₄²	C₄×Q₈	1	2	yes	—
134	C₄²	#131	1	2	yes	—
135	C₁₆	—	2	4	no	—
136	C₁₆	—	4	2	no	—
137	C₁₆	—	4	4	no	—
138	C₂³	C₂³×C₈	1	1	yes	characteristic
139	Q₈	—	4	8	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	C₈×Q₈	1	4	yes	—
151	C₂×C₄	#504	1	8	yes	—
152	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₂×C₄	#518	1	1	yes	characteristic
154	C₂×C₄	#519	1	1	yes	characteristic
155	C₂×C₄	C₂×#131	1	1	yes	characteristic
156	C₂×C₄	#568	1	1	yes	characteristic
157	C₂×C₄	#568	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	4	1	no	—
160	C₈	—	4	4	no	—
161	C₈	—	4	1	no	—
162	C₈	#542	1	2	yes	—
163	C₈	#544	1	2	yes	—
164	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
165	C₂²	#2972	1	1	yes	characteristic
166	C₂²	#2982	1	1	yes	characteristic
167	C₂²	C₂×#504	1	4	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	2	8	no	—
170	C₄	—	4	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	#2331	1	1	yes	characteristic
174	C₄	#2961	1	1	yes	characteristic
175	C₄	#2981	1	1	yes	characteristic
176	C₄	#2985	1	1	yes	characteristic
177	C₂	#22307	1	1	yes	characteristic
178	C₂	#33474	1	2	yes	—
179	C₂	#33491	1	2	yes	—
180	C₂	#33500	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
53. C₄×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#21715	2
#13263	2
#13160	2
#13104	2
#12880	2
C₁₆×C₄⋊C₄	2
#12592	2