Subgroups — #494157

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2158	C₂	1	4	yes	maximal
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20797	C₂	1	2	yes	maximal
4	#12907	C₂	1	4	yes	maximal, complemented
5	#12906	C₂	1	2	yes	maximal, complemented
6	#12905	C₂	1	2	yes	maximal, complemented
7	C₂³×C₁₆	C₂²	1	2	yes	—
8	C₂×#427	—	2	2	no	—
9	C₂×#427	C₂²	1	4	yes	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2285	—	2	2	no	—
12	#2158	—	2	16	no	—
13	C₂×#367	C₂²	1	8	yes	—
14	C₂×#366	C₂²	1	8	yes	—
15	C₂×C₄×C₁₆	C₄	1	8	yes	complemented
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	#1846	—	2	2	no	—
20	#1846	C₂²	1	4	yes	—
21	#1845	C₂²	1	4	yes	—
22	#1844	—	2	2	no	—
23	C₂³×C₈	D₄	1	1	yes	characteristic
24	C₂³×C₈	C₂³	1	1	yes	characteristic
25	C₂×#117	—	2	4	no	—
26	C₂²×C₄²	D₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	8	yes	—
28	C₂²×C₁₆	D₄	1	8	yes	—
29	C₂²×C₁₆	Q₈	1	8	yes	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	2	yes	—
32	#427	—	2	16	no	—
33	#427	—	4	8	no	—
34	C₂×#107	—	2	4	no	—
35	C₂×#107	C₂³	1	8	yes	—
36	C₂×#106	—	2	4	no	—
37	C₂×#106	—	2	4	no	—
38	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	Q₈	1	4	yes	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#384	—	2	4	no	—
45	#382	—	2	4	no	—
46	#380	—	4	4	no	—
47	C₄×C₄⋊C₄	—	4	4	no	—
48	#367	—	2	16	no	—
49	#366	—	2	16	no	—
50	C₄×C₁₆	C₂×C₄	1	4	yes	—
51	C₄×C₁₆	C₂×C₄	1	4	yes	—
52	C₄×C₁₆	C₂×C₄	1	8	yes	—
53	C₂³×C₄	D₈	1	2	yes	—
54	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₈	C₄⋊C₄	1	4	yes	—
56	C₂²×C₈	D₈	1	4	yes	—
57	C₂²×C₈	Q₁₆	1	4	yes	—
58	C₂²×C₈	C₂²×C₄	1	2	yes	—
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	4	yes	—
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	C₂×Q₈	1	4	yes	—
64	C₂²×C₈	C₂×Q₈	1	2	yes	—
65	C₂²×C₈	C₂⁴	1	1	yes	characteristic
66	#117	—	4	8	no	—
67	#117	—	4	8	no	—
68	C₂×C₄⋊C₄	—	2	8	no	—
69	C₂×C₄⋊C₄	—	4	8	no	—
70	C₂×C₂²⋊C₄	—	4	4	no	—
71	C₂×C₄²	—	4	4	no	—
72	C₂×C₄²	C₄⋊C₄	1	4	yes	—
73	C₂×C₄²	D₈	1	2	yes	—
74	C₂×C₄²	Q₁₆	1	4	yes	—
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₂×C₁₆	C₄⋊C₄	1	32	yes	complemented
78	C₂×C₁₆	C₂²×C₄	1	4	yes	—
79	C₂×C₁₆	C₂²×C₄	1	4	yes	—
80	C₂×C₁₆	C₂×D₄	1	4	yes	—
81	C₂×C₁₆	C₂×D₄	1	4	yes	—
82	C₂×C₁₆	C₂×Q₈	1	4	yes	—
83	C₂×C₁₆	C₂×Q₈	1	4	yes	—
84	#107	—	2	16	no	—
85	#107	—	4	8	no	—
86	#106	—	4	8	no	—
87	#106	—	4	8	no	—
88	C₄×C₈	C₄⋊C₄	1	4	yes	—
89	C₄×C₈	C₄⋊C₄	1	4	yes	—
90	C₄×C₈	C₂²×C₄	1	2	yes	—
91	C₄×C₈	C₂²×C₄	1	2	yes	—
92	C₄×C₈	C₂×Q₈	1	2	yes	—
93	C₄×C₈	C₂×Q₈	1	2	yes	—
94	#95	—	4	8	no	—
95	C₂⁴	C₂×D₈	1	1	yes	characteristic
96	C₂²×C₄	—	2	8	no	—
97	C₂²×C₄	—	4	8	no	—
98	C₂²×C₄	#107	1	8	yes	—
99	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
100	C₂²×C₄	#127	1	2	yes	—
101	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
102	C₂²×C₄	C₂×D₈	1	2	yes	—
103	C₂²×C₄	C₂×D₈	1	2	yes	—
104	C₂²×C₄	C₂×D₈	1	2	yes	—
105	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
107	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
108	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₈	#107	1	16	yes	—
110	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
111	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₈	#127	1	2	yes	—
115	C₂×C₈	C₄⋊Q₈	1	8	yes	—
116	C₂×C₈	C₄⋊Q₈	1	2	yes	—
117	C₂×C₈	C₂×D₈	1	2	yes	—
118	C₂×C₈	C₂×D₈	1	2	yes	—
119	C₂×C₈	C₂×Q₁₆	1	2	yes	—
120	C₂×C₈	C₂×Q₁₆	1	2	yes	—
121	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₈	C₂²×D₄	1	2	yes	—
123	C₂×C₈	C₂²×Q₈	1	2	yes	—
124	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
125	C₄⋊C₄	—	4	8	no	—
126	C₄⋊C₄	—	4	8	no	—
127	C₄⋊C₄	—	8	16	no	—
128	C₂²⋊C₄	—	8	8	no	—
129	C₄²	—	8	8	no	—
130	C₄²	#107	1	8	yes	—
131	C₄²	C₂×C₄⋊C₄	1	2	yes	—
132	C₄²	C₂×C₄⋊C₄	1	2	yes	—
133	C₄²	C₂×Q₁₆	1	2	yes	—
134	C₄²	C₂×Q₁₆	1	2	yes	—
135	C₁₆	C₂×C₄⋊C₄	1	8	yes	—
136	C₁₆	C₂×C₄⋊C₄	1	8	yes	—
137	C₂³	—	2	4	no	—
138	C₂³	C₂×#107	1	2	yes	—
139	C₂³	#493	1	2	yes	—
140	C₂³	#500	1	2	yes	—
141	C₂³	C₂²×D₈	1	1	yes	characteristic
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	16	no	—
144	C₂×C₄	—	8	8	no	—
145	C₂×C₄	—	8	16	no	—
146	C₂×C₄	#384	1	4	yes	—
147	C₂×C₄	C₂×#107	1	4	yes	—
148	C₂×C₄	C₂×#107	1	2	yes	—
149	C₂×C₄	C₂×#107	1	4	yes	—
150	C₂×C₄	C₂×#107	1	4	yes	—
151	C₂×C₄	#493	1	2	yes	—
152	C₂×C₄	#494	1	4	yes	—
153	C₂×C₄	#500	1	2	yes	—
154	C₂×C₄	#500	1	4	yes	—
155	C₂×C₄	#508	1	8	yes	—
156	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄	C₂×#127	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
160	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
161	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
162	C₈	#384	1	4	yes	—
163	C₈	C₂×#107	1	4	yes	—
164	C₈	C₂×#107	1	4	yes	—
165	C₈	C₂²×C₄⋊C₄	1	2	yes	—
166	C₈	C₂×C₄⋊Q₈	1	2	yes	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	4	no	—
169	C₂²	—	2	4	no	—
170	C₂²	#1845	1	4	yes	—
171	C₂²	C₂²×#107	1	1	yes	characteristic
172	C₂²	C₂×#493	1	1	yes	characteristic
173	C₂²	C₂×#500	1	1	yes	characteristic
174	C₂²	C₂×#508	1	4	yes	—
175	C₄	—	2	4	no	—
176	C₄	—	16	8	no	—
177	C₄	#1845	1	4	yes	—
178	C₄	#2158	1	8	yes	complemented
179	C₄	C₂×#384	1	1	yes	characteristic
180	C₄	C₂²×#107	1	1	yes	characteristic
181	C₄	C₂×#494	1	1	yes	characteristic
182	C₄	C₂×#500	1	1	yes	characteristic
183	C₂	—	2	4	no	—
184	C₂	C₂×#1845	1	1	yes	characteristic
185	C₂	C₂×#2158	1	2	yes	—
186	C₂	#33713	1	2	yes	—
187	C₂	#33726	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×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
4. #12907	C₂	semidirect
5. #12906	C₂	semidirect
6. #12905	C₂	semidirect
15. C₂×C₄×C₁₆	C₄	semidirect
77. C₂×C₁₆	C₄⋊C₄	semidirect
178. C₄	#2158	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2158	2
C₂²×C₄×C₁₆	2
#20797	2
#12907	2
#12906	2
#12905	2