Subgroups — #491481

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33731	C₂	1	2	yes	maximal
2	#33726	C₂	1	1	yes	maximal, characteristic
3	C₂×#2252	C₂	1	1	yes	maximal, characteristic
4	#19664	C₂	1	2	yes	maximal
5	#12524	C₂	1	4	yes	maximal, complemented
6	#12522	C₂	1	2	yes	maximal, complemented
7	#12521	C₂	1	2	yes	maximal, complemented
8	#11399	C₂	1	1	yes	maximal, characteristic
9	#3230	—	2	2	no	—
10	#3209	—	2	2	no	—
11	#3160	C₂²	1	2	yes	—
12	C₂×#500	C₂²	1	2	yes	—
13	#2255	—	2	4	no	—
14	#2254	C₂²	1	2	yes	—
15	#2254	C₂²	1	4	yes	—
16	#2253	—	2	2	no	—
17	#2252	—	2	2	no	—
18	#2252	C₂²	1	2	yes	—
19	#2252	C₂²	1	2	yes	—
20	#2252	C₂²	1	1	yes	characteristic
21	#2252	C₂²	1	1	yes	characteristic
22	#2252	C₂²	1	4	yes	—
23	#2159	—	2	4	no	—
24	#2158	—	2	2	no	—
25	C₂×#366	—	2	2	no	—
26	C₄×M₃₂	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
28	#1723	—	2	4	no	—
29	#1713	C₂²	1	4	yes	—
30	#1710	C₂²	1	4	yes	—
31	#1450	C₂²	1	1	yes	characteristic
32	#1311	C₂²	1	2	yes	—
33	C₈×C₁₆	C₂²	1	2	yes	—
34	#533	—	2	2	no	—
35	C₂×C₄⋊Q₈	—	2	2	no	—
36	C₂×M₃₂	D₄	1	2	yes	—
37	C₂²×C₁₆	D₄	1	2	yes	—
38	#500	—	2	2	no	—
39	#500	—	2	2	no	—
40	#500	—	2	2	no	—
41	#500	—	2	4	no	—
42	#500	C₂³	1	4	yes	—
43	#500	C₂³	1	2	yes	—
44	#500	C₂³	1	2	yes	—
45	#499	—	2	2	no	—
46	#499	—	2	2	no	—
47	#498	—	2	2	no	—
48	#494	—	2	4	no	—
49	#487	—	2	4	no	—
50	#487	—	2	4	no	—
51	#484	—	4	4	no	—
52	#477	—	4	4	no	—
53	#462	—	4	4	no	—
54	#451	—	4	4	no	—
55	#431	D₄	1	1	yes	characteristic
56	#431	D₄	1	1	yes	characteristic
57	#427	—	2	2	no	—
58	#427	—	2	2	no	—
59	C₂×#107	—	2	2	no	—
60	C₂×#107	—	2	2	no	—
61	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
62	#367	—	2	4	no	—
63	#367	—	2	8	no	—
64	#366	—	2	4	no	—
65	#366	—	2	4	no	—
66	#366	—	2	2	no	—
67	#366	—	2	2	no	—
68	#366	—	2	8	no	—
69	#346	D₄	1	2	yes	—
70	C₄×C₁₆	D₄	1	2	yes	—
71	C₄×C₁₆	C₂³	1	1	yes	characteristic
72	C₄×C₁₆	C₂³	1	2	yes	—
73	C₄×C₁₆	C₂³	1	1	yes	characteristic
74	#322	D₄	1	2	yes	—
75	C₈²	C₂³	1	2	yes	—
76	C₂²×Q₈	—	4	2	no	—
77	C₂×Q₁₆	—	4	8	no	—
78	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
80	C₄⋊Q₈	—	2	4	no	—
81	C₄⋊Q₈	—	2	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	4	2	no	—
84	C₄⋊Q₈	—	4	2	no	—
85	C₄⋊Q₈	—	4	2	no	—
86	#125	—	4	2	no	—
87	C₂²⋊Q₈	—	4	2	no	—
88	C₂²⋊Q₈	—	4	2	no	—
89	C₄×Q₈	—	4	2	no	—
90	C₄×Q₈	—	4	2	no	—
91	#117	—	4	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄⋊C₄	—	4	2	no	—
94	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
95	M₃₂	Q₁₆	1	8	yes	complemented
96	C₂×C₁₆	Q₁₆	1	8	yes	complemented
97	C₂×C₁₆	C₂×D₄	1	2	yes	—
98	C₂×C₁₆	C₂×D₄	1	2	yes	—
99	C₂×C₁₆	C₂×D₄	1	4	yes	—
100	#107	—	2	4	no	—
101	#107	—	2	2	no	—
102	#107	—	2	4	no	—
103	#107	—	2	2	no	—
104	#107	—	2	2	no	—
105	#107	—	2	2	no	—
106	#107	—	4	2	no	—
107	#107	—	4	2	no	—
108	#107	—	4	8	no	—
109	#106	—	4	2	no	—
110	#106	—	4	2	no	—
111	C₄⋊C₈	D₈	1	2	yes	—
112	C₄⋊C₈	D₈	1	2	yes	—
113	#103	—	4	8	no	—
114	#103	—	4	8	no	—
115	C₂²⋊C₈	D₈	1	2	yes	—
116	C₂²⋊C₈	D₈	1	2	yes	—
117	C₄×C₈	C₂×D₄	1	1	yes	characteristic
118	C₄×C₈	C₂×D₄	1	1	yes	characteristic
119	C₄×C₈	C₂×D₄	1	1	yes	characteristic
120	C₄×C₈	C₂×D₄	1	2	yes	—
121	C₄×C₈	C₂×D₄	1	2	yes	—
122	C₄×C₈	C₂⁴	1	1	yes	characteristic
123	C₂×Q₈	—	4	2	no	—
124	C₂×Q₈	—	4	4	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	8	2	no	—
127	C₂×Q₈	—	16	4	no	—
128	C₂²×C₄	—	8	2	no	—
129	C₂²×C₄	#127	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
132	Q₁₆	—	16	8	no	—
133	C₂×C₈	#127	1	1	yes	characteristic
134	C₂×C₈	#127	1	1	yes	characteristic
135	C₂×C₈	C₂×D₈	1	4	yes	—
136	C₂×C₈	C₂×D₈	1	4	yes	—
137	C₂×C₈	C₂×Q₁₆	1	4	yes	—
138	C₂×C₈	C₂×Q₁₆	1	4	yes	—
139	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	8	2	no	—
148	C₄⋊C₄	—	8	2	no	—
149	C₄⋊C₄	—	8	2	no	—
150	C₄⋊C₄	—	8	2	no	—
151	C₂²⋊C₄	—	8	2	no	—
152	C₄²	—	8	2	no	—
153	C₄²	#127	1	1	yes	characteristic
154	C₄²	C₂×D₈	1	1	yes	characteristic
155	C₄²	C₂×D₈	1	1	yes	characteristic
156	C₄²	C₂²×D₄	1	1	yes	characteristic
157	C₁₆	C₂×Q₁₆	1	4	yes	—
158	C₁₆	C₂×Q₁₆	1	4	yes	—
159	C₁₆	C₂×Q₁₆	1	8	yes	—
160	C₂³	#493	1	1	yes	characteristic
161	Q₈	—	8	4	no	—
162	Q₈	—	8	8	no	—
163	Q₈	—	8	4	no	—
164	Q₈	—	16	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	16	4	no	—
171	C₂×C₄	#493	1	1	yes	characteristic
172	C₂×C₄	#493	1	1	yes	characteristic
173	C₂×C₄	#493	1	1	yes	characteristic
174	C₂×C₄	#494	1	2	yes	—
175	C₂×C₄	#494	1	2	yes	—
176	C₂×C₄	C₂×#127	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	—	2	4	no	—
181	C₈	#494	1	2	yes	—
182	C₈	#494	1	2	yes	—
183	C₈	C₂²×Q₁₆	1	2	yes	—
184	C₈	C₂²×Q₁₆	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#1713	1	2	yes	—
188	C₂²	C₂×#493	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	16	4	no	—
192	C₄	—	16	2	no	—
193	C₄	—	16	2	no	—
194	C₄	#1713	1	2	yes	—
195	C₄	C₂×#494	1	1	yes	characteristic
196	C₄	C₂×#494	1	1	yes	characteristic
197	C₄	#3415	1	2	yes	—
198	C₄	#3417	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	C₂×#1713	1	1	yes	characteristic
201	C₂	#33720	1	1	yes	characteristic
202	C₂	#33721	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #12524	C₂	semidirect
6. #12522	C₂	semidirect
7. #12521	C₂	semidirect
95. M₃₂	Q₁₆	semidirect
96. C₂×C₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#33731	2
#33726	2
C₂×#2252	2
#19664	2
#12524	2
#12522	2
#12521	2
#11399	2