Subgroups — #497937

Order 512.

510 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22727	C₂	1	2	yes	maximal
3	#13178	C₂	1	4	yes	maximal
4	#13176	C₂	1	2	yes	maximal
5	#12907	C₂	1	2	yes	maximal
6	#12906	C₂	1	1	yes	maximal, characteristic
7	C₄×#367	C₂	1	1	yes	maximal, characteristic
8	C₄×#367	C₂	1	2	yes	maximal
9	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
10	#2355	—	2	2	no	—
11	Q₈×C₁₆	C₄	1	8	yes	complemented
12	C₂×#367	C₂²	1	2	yes	—
13	C₂×#367	C₂²	1	1	yes	characteristic
14	C₂×#367	C₂²	1	2	yes	—
15	C₂×#367	C₂²	1	1	yes	characteristic
16	C₂×#366	C₂²	1	4	yes	—
17	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	2	yes	—
21	#1949	—	2	4	no	—
22	#1948	—	2	2	no	—
23	#1846	C₂²	1	4	yes	—
24	#1845	C₂²	1	2	yes	—
25	C₄×#107	C₂²	1	2	yes	—
26	C₄×#107	C₂²	1	4	yes	—
27	#1380	C₂²	1	4	yes	—
28	#1380	C₂²	1	2	yes	—
29	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	C₈×Q₈	C₂×C₄	1	4	yes	—
34	C₈×Q₈	Q₈	1	4	yes	—
35	C₂×#107	—	2	2	no	—
36	C₂×#107	C₂³	1	4	yes	—
37	C₂×#107	C₂³	1	2	yes	—
38	C₂×#107	C₂³	1	2	yes	—
39	C₂×#106	—	2	4	no	—
40	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₈	D₄	1	2	yes	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	2	yes	—
44	#391	—	4	2	no	—
45	#387	—	4	4	no	—
46	#384	—	2	2	no	—
47	#382	—	2	4	no	—
48	C₄×C₄⋊C₄	—	2	2	no	—
49	C₄×C₄⋊C₄	—	2	4	no	—
50	C₄×C₄⋊C₄	—	4	2	no	—
51	#367	—	2	4	no	—
52	#367	—	2	2	no	—
53	#367	—	4	2	no	—
54	#367	Q₈	1	4	yes	complemented
55	#366	—	4	4	no	—
56	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
57	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
58	C₄×C₁₆	C₂×C₄	1	4	yes	—
59	C₄×C₁₆	C₂×C₄	1	8	yes	—
60	#340	—	2	4	no	—
61	#340	—	2	2	no	—
62	C₂²×Q₈	D₈	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₄	1	2	yes	—
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₈	C₂⁴	1	1	yes	characteristic
66	C₄×Q₈	C₄⋊C₄	1	4	yes	—
67	C₄×Q₈	Q₁₆	1	4	yes	—
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	2	4	no	—
71	C₂×C₄⋊C₄	—	4	2	no	—
72	C₂×C₄⋊C₄	—	4	2	no	—
73	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
74	C₂×C₄⋊C₄	D₈	1	2	yes	—
75	C₂×C₄²	—	4	2	no	—
76	C₂×C₄²	—	4	4	no	—
77	C₂×C₄²	C₂×D₄	1	2	yes	—
78	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	C₂²×C₄	1	4	yes	—
84	C₂×C₁₆	C₂²×C₄	1	2	yes	—
85	C₂×C₁₆	C₂²×C₄	1	2	yes	—
86	C₂×C₁₆	C₂×Q₈	1	2	yes	—
87	C₂×C₁₆	#41	1	2	yes	—
88	#107	—	2	2	no	—
89	#107	—	2	4	no	—
90	#107	—	2	2	no	—
91	#107	—	2	2	no	—
92	#107	—	2	4	no	—
93	#107	—	8	2	no	—
94	#107	C₂×Q₈	1	4	yes	—
95	#106	—	8	4	no	—
96	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
97	C₄⋊C₈	C₄⋊C₄	1	8	yes	—
98	C₄×C₈	C₂²×C₄	1	2	yes	—
99	C₄×C₈	C₂²×C₄	1	4	yes	—
100	C₄×C₈	C₂×Q₈	1	2	yes	—
101	C₄×C₈	C₂×Q₈	1	4	yes	—
102	#95	—	4	2	no	—
103	#95	—	4	4	no	—
104	#95	—	4	4	no	—
105	#95	—	4	4	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	#107	1	4	yes	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	—	4	2	no	—
110	C₂²×C₄	—	4	4	no	—
111	C₂²×C₄	C₂×D₈	1	2	yes	—
112	C₂²×C₄	C₂×D₈	1	2	yes	—
113	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₈	C₂×#41	1	1	yes	characteristic
126	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	8	4	no	—
135	C₄⋊C₄	—	16	2	no	—
136	C₄⋊C₄	#107	1	8	yes	—
137	C₄⋊C₄	#107	1	4	yes	—
138	C₄²	—	8	4	no	—
139	C₄²	—	16	4	no	—
140	C₄²	C₂×C₄⋊C₄	1	4	yes	—
141	C₄²	C₂×C₄⋊C₄	1	2	yes	—
142	C₄²	C₂×Q₁₆	1	4	yes	—
143	C₄²	C₂×Q₁₆	1	2	yes	—
144	C₁₆	—	2	4	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	C₄×Q₈	1	4	yes	—
147	C₂³	C₂²×D₈	1	1	yes	characteristic
148	Q₈	—	2	8	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	4	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	4	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	C₂×#107	1	4	yes	—
162	C₂×C₄	C₂×#107	1	2	yes	—
163	C₂×C₄	C₂×#107	1	4	yes	—
164	C₂×C₄	C₂×#107	1	2	yes	—
165	C₂×C₄	#508	1	4	yes	—
166	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
168	C₂×C₄	#550	1	1	yes	characteristic
169	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
170	C₂×C₄	#576	1	1	yes	characteristic
171	C₂×C₄	#578	1	1	yes	characteristic
172	C₈	—	2	4	no	—
173	C₈	—	2	2	no	—
174	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₈	#520	1	1	yes	characteristic
176	C₈	#557	1	1	yes	characteristic
177	C₈	Q₈²	1	1	yes	characteristic
178	C₂²	C₂²×#107	1	1	yes	characteristic
179	C₂²	Q₈×D₈	1	1	yes	characteristic
180	C₂²	#3391	1	1	yes	characteristic
181	C₂²	C₂×#508	1	2	yes	—
182	C₂²	#3416	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	2	2	no	—
186	C₄	—	8	4	no	—
187	C₄	—	16	2	no	—
188	C₄	—	16	4	no	—
189	C₄	#2158	1	4	yes	—
190	C₄	Q₈×C₄⋊C₄	1	1	yes	characteristic
191	C₄	#2913	1	1	yes	characteristic
192	C₄	Q₈×Q₁₆	1	1	yes	characteristic
193	C₄	#3395	1	1	yes	characteristic
194	C₂	Q₈×#107	1	1	yes	characteristic
195	C₂	C₂×#2158	1	1	yes	characteristic
196	C₂	#33411	1	1	yes	characteristic
197	C₂	#33948	1	2	yes	—
198	C₂	#33961	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. Q₈×C₁₆	C₄	semidirect
54. #367	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₆	2
#22727	2
#13178	2
#13176	2
#12907	2
#12906	2
C₄×#367	2