Subgroups — #512516

Order 512.

554 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	#33312	C₂	1	1	yes	maximal, characteristic
2	#33311	C₂	1	1	yes	maximal, characteristic
3	#13284	C₂	1	4	yes	maximal
4	#13283	C₂	1	4	yes	maximal
5	#12888	C₂	1	4	yes	maximal
6	C₄×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#500	—	2	1	no	—
8	C₂×#500	C₂²	1	2	yes	—
9	C₂×#499	—	2	1	no	—
10	C₂×#499	C₂²	1	2	yes	—
11	#2868	—	2	1	no	—
12	#2859	—	2	1	no	—
13	C₂×#346	C₂²	1	2	yes	—
14	C₂×#346	C₂²	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	#1846	—	2	1	no	—
17	#1846	C₂²	1	2	yes	—
18	#1845	—	2	1	no	—
19	#1845	—	2	1	no	—
20	#1845	C₂²	1	2	yes	—
21	#1845	C₂²	1	2	yes	—
22	#1845	C₂²	1	2	yes	—
23	#1844	—	2	1	no	—
24	C₄²×C₈	C₂²	1	1	yes	characteristic
25	#1380	C₂²	1	16	yes	—
26	C₂×C₄⋊Q₈	—	2	2	no	—
27	C₂×C₄⋊Q₈	—	4	2	no	—
28	C₂×#125	—	2	2	no	—
29	C₂²×C₁₆	C₂³	1	4	yes	—
30	#500	—	2	8	no	—
31	#500	—	4	4	no	—
32	#499	—	2	8	no	—
33	#499	—	4	4	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×#107	—	2	2	no	—
36	C₂×#107	—	2	2	no	—
37	C₂×#107	—	2	2	no	—
38	C₂×#107	C₂³	1	4	yes	—
39	C₂×#107	C₂³	1	4	yes	—
40	C₂×#106	—	2	2	no	—
41	C₂×#106	—	2	2	no	—
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	#389	—	4	2	no	—
49	#389	—	4	2	no	—
50	#384	—	2	2	no	—
51	#384	—	2	2	no	—
52	#384	—	2	2	no	—
53	#384	—	4	2	no	—
54	#382	—	2	2	no	—
55	C₄³	D₄	1	1	yes	characteristic
56	#346	—	2	4	no	—
57	#346	—	2	4	no	—
58	C₄×C₁₆	Q₈	1	8	yes	complemented
59	C₂²×Q₈	—	4	2	no	—
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	C₂⁴	1	1	yes	characteristic
64	C₄⋊Q₈	—	4	4	no	—
65	C₄⋊Q₈	—	4	4	no	—
66	C₄⋊Q₈	—	8	8	no	—
67	#125	—	4	4	no	—
68	#125	—	4	4	no	—
69	C₂×C₄⋊C₄	—	2	4	no	—
70	C₂×C₄⋊C₄	—	2	4	no	—
71	C₂×C₄⋊C₄	—	4	4	no	—
72	C₂×C₄⋊C₄	—	4	2	no	—
73	C₂×C₄⋊C₄	—	4	4	no	—
74	C₂×C₄⋊C₄	—	4	4	no	—
75	C₂×C₄⋊C₄	—	4	2	no	—
76	C₂×C₄⋊C₄	—	4	2	no	—
77	C₂×C₄²	D₈	1	2	yes	—
78	C₂×C₄²	D₈	1	2	yes	—
79	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₁₆	—	2	4	no	—
83	C₂×C₁₆	—	2	4	no	—
84	C₂×C₁₆	C₂×Q₈	1	8	yes	—
85	#107	—	2	8	no	—
86	#107	—	2	8	no	—
87	#107	—	4	4	no	—
88	#107	—	4	4	no	—
89	#107	—	4	4	no	—
90	#107	—	4	4	no	—
91	#106	—	4	4	no	—
92	#106	—	4	4	no	—
93	C₄×C₈	SD₁₆	1	4	yes	—
94	C₄×C₈	SD₁₆	1	4	yes	—
95	C₄×C₈	Q₁₆	1	4	yes	—
96	C₄×C₈	C₂×Q₈	1	4	yes	—
97	C₄×C₈	#41	1	4	yes	—
98	C₄×C₈	#41	1	4	yes	—
99	#95	—	4	2	no	—
100	#95	—	4	2	no	—
101	C₂×Q₈	—	8	8	no	—
102	C₂×Q₈	—	8	4	no	—
103	C₂²×C₄	—	4	4	no	—
104	C₂²×C₄	—	4	4	no	—
105	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
107	C₂²×C₄	C₂×D₈	1	2	yes	—
108	C₂²×C₄	C₂×D₈	1	2	yes	—
109	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₈	#124	1	2	yes	—
111	C₂×C₈	#124	1	2	yes	—
112	C₂×C₈	C₄⋊Q₈	1	2	yes	—
113	C₂×C₈	C₂×SD₁₆	1	2	yes	—
114	C₂×C₈	C₂×SD₁₆	1	2	yes	—
115	C₂×C₈	C₂×SD₁₆	1	2	yes	—
116	C₂×C₈	C₂×SD₁₆	1	2	yes	—
117	C₂×C₈	C₂×Q₁₆	1	2	yes	—
118	C₂×C₈	C₂×Q₁₆	1	2	yes	—
119	C₂×C₈	C₂²×Q₈	1	2	yes	—
120	C₂×C₈	C₂×#41	1	2	yes	—
121	C₂×C₈	C₂×#41	1	2	yes	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	4	4	no	—
124	C₄⋊C₄	—	4	4	no	—
125	C₄⋊C₄	—	4	4	no	—
126	C₄⋊C₄	—	8	8	no	—
127	C₄⋊C₄	—	8	8	no	—
128	C₄⋊C₄	—	8	4	no	—
129	C₄⋊C₄	—	8	4	no	—
130	C₄⋊C₄	—	8	8	no	—
131	C₄⋊C₄	—	8	4	no	—
132	C₄²	—	2	4	no	—
133	C₄²	—	2	4	no	—
134	C₄²	#124	1	2	yes	—
135	C₄²	#124	1	2	yes	—
136	C₄²	C₄⋊Q₈	1	2	yes	—
137	C₄²	C₂×SD₁₆	1	2	yes	—
138	C₄²	C₂×SD₁₆	1	2	yes	—
139	C₄²	C₂×Q₁₆	1	2	yes	—
140	C₁₆	—	2	8	no	—
141	C₂³	C₂²×D₈	1	1	yes	characteristic
142	Q₈	—	16	8	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	8	4	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	—	8	8	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	8	4	no	—
152	C₂×C₄	#486	1	2	yes	—
153	C₂×C₄	#486	1	4	yes	—
154	C₂×C₄	#486	1	2	yes	—
155	C₂×C₄	#500	1	4	yes	—
156	C₂×C₄	#509	1	4	yes	—
157	C₂×C₄	#510	1	4	yes	—
158	C₂×C₄	C₂×#124	1	1	yes	characteristic
159	C₂×C₄	C₂×#124	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
161	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
162	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
163	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
164	C₈	#487	1	4	yes	—
165	C₈	#487	1	4	yes	—
166	C₈	#498	1	4	yes	—
167	C₈	#533	1	4	yes	—
168	C₂²	C₂×#486	1	1	yes	characteristic
169	C₂²	C₂×#486	1	1	yes	characteristic
170	C₂²	C₂×#500	1	1	yes	characteristic
171	C₂²	C₂×#509	1	2	yes	—
172	C₂²	C₂×#510	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	4	no	—
176	C₄	—	16	4	no	—
177	C₄	—	16	4	no	—
178	C₄	#2255	1	8	yes	—
179	C₄	#2862	1	1	yes	characteristic
180	C₄	C₂×#487	1	1	yes	characteristic
181	C₄	C₂×#487	1	1	yes	characteristic
182	C₄	C₂×#498	1	1	yes	characteristic
183	C₂	#33289	1	1	yes	characteristic
184	C₂	#33704	1	1	yes	characteristic
185	C₂	#33705	1	1	yes	characteristic
186	C₂	#33707	1	2	yes	—
187	C₂	C₂×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×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
58. 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
#33312	2
#33311	2
#13284	2
#13283	2
#12888	2
C₄×#346	2