Subgroups — #146985

Order 512.

710 conjugacy classes of subgroups.

The subgroup lattice has 177 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1846	C₂	1	2	yes	maximal
3	C₂×#1380	C₂	1	4	yes	maximal
4	C₂³×C₁₆	C₂²	1	2	yes	—
5	C₂²×#107	C₂²	1	4	yes	—
6	C₂²×#106	—	2	2	no	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×#382	—	2	2	no	—
9	C₂×C₄×C₁₆	—	2	4	no	—
10	C₂×C₄×C₁₆	C₄	1	4	yes	complemented
11	#1846	—	2	4	no	—
12	#1846	C₄	1	8	yes	complemented
13	#1380	—	2	8	no	—
14	C₂³×C₈	D₄	1	1	yes	characteristic
15	C₂³×C₈	C₂³	1	1	yes	characteristic
16	C₂²×C₄⋊C₄	—	2	4	no	—
17	C₂²×C₄²	D₄	1	1	yes	characteristic
18	C₂²×C₁₆	—	2	4	no	—
19	C₂²×C₁₆	—	2	4	no	—
20	C₂²×C₁₆	C₂×C₄	1	4	yes	—
21	C₂²×C₁₆	D₄	1	4	yes	—
22	C₂²×C₁₆	Q₈	1	4	yes	—
23	C₂×#107	—	2	8	no	—
24	C₂×#107	—	2	8	no	—
25	C₂×#107	—	2	4	no	—
26	C₂×#107	C₂×C₄	1	8	yes	—
27	C₂×#106	—	2	8	no	—
28	C₂×#106	—	4	4	no	—
29	C₂×#106	—	4	4	no	—
30	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
32	C₂×C₄×C₈	D₄	1	4	yes	—
33	C₂×C₄×C₈	Q₈	1	2	yes	—
34	#382	—	2	8	no	—
35	#382	—	4	4	no	—
36	C₂×#95	—	4	4	no	—
37	C₄×C₁₆	—	2	8	no	—
38	C₂³×C₄	—	4	4	no	—
39	C₂³×C₄	D₈	1	2	yes	—
40	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₈	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₈	C₄⋊C₄	1	2	yes	—
43	C₂²×C₈	D₈	1	2	yes	—
44	C₂²×C₈	SD₁₆	1	4	yes	—
45	C₂²×C₈	Q₁₆	1	2	yes	—
46	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂²×C₄	1	2	yes	—
48	C₂²×C₈	C₂×D₄	1	2	yes	—
49	C₂²×C₈	C₂×D₄	1	2	yes	—
50	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₈	C₂×Q₈	1	2	yes	—
52	C₂²×C₈	#41	1	4	yes	—
53	C₂×C₄⋊C₄	—	2	8	no	—
54	C₂×C₄⋊C₄	—	4	4	no	—
55	C₂×C₄⋊C₄	—	4	4	no	—
56	C₂×C₄⋊C₄	—	4	4	no	—
57	C₂×C₄⋊C₄	—	4	4	no	—
58	C₂×C₄⋊C₄	—	4	4	no	—
59	C₂×C₄²	C₂²⋊C₄	1	4	yes	—
60	C₂×C₄²	C₄⋊C₄	1	2	yes	—
61	C₂×C₄²	SD₁₆	1	4	yes	—
62	C₂×C₄²	Q₁₆	1	2	yes	—
63	C₂×C₁₆	—	2	4	no	—
64	C₂×C₁₆	—	2	4	no	—
65	C₂×C₁₆	—	2	8	no	—
66	C₂×C₁₆	—	2	8	no	—
67	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
68	#107	—	2	16	no	—
69	#107	—	4	8	no	—
70	#106	—	4	16	no	—
71	C₄×C₈	C₄²	1	4	yes	—
72	C₄×C₈	C₂²⋊C₄	1	4	yes	—
73	C₄×C₈	C₄⋊C₄	1	8	yes	—
74	#95	—	4	16	no	—
75	C₂⁴	C₂×D₈	1	1	yes	characteristic
76	C₂²×C₄	—	2	8	no	—
77	C₂²×C₄	—	4	8	no	—
78	C₂²×C₄	—	8	4	no	—
79	C₂²×C₄	—	8	4	no	—
80	C₂²×C₄	—	8	16	no	—
81	C₂²×C₄	#102	1	8	yes	—
82	C₂²×C₄	#107	1	4	yes	—
83	C₂²×C₄	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄	#124	1	2	yes	—
86	C₂²×C₄	#127	1	1	yes	characteristic
87	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
89	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
90	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
91	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
92	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₈	#95	1	4	yes	—
94	C₂×C₈	#102	1	8	yes	—
95	C₂×C₈	#103	1	8	yes	—
96	C₂×C₈	#106	1	4	yes	—
97	C₂×C₈	#107	1	4	yes	—
98	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₈	#117	1	2	yes	—
103	C₂×C₈	C₄×D₄	1	4	yes	—
104	C₂×C₈	C₄×Q₈	1	4	yes	—
105	C₂×C₈	#121	1	4	yes	—
106	C₂×C₈	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₈	#125	1	2	yes	—
108	C₂×C₈	C₄⋊Q₈	1	2	yes	—
109	C₄⋊C₄	—	4	8	no	—
110	C₄⋊C₄	—	4	8	no	—
111	C₄⋊C₄	—	8	4	no	—
112	C₄⋊C₄	—	8	4	no	—
113	C₄²	#95	1	4	yes	—
114	C₄²	#103	1	8	yes	—
115	C₄²	#106	1	4	yes	—
116	C₁₆	—	2	8	no	—
117	C₂³	C₂×#102	1	2	yes	—
118	C₂³	C₂×#107	1	1	yes	characteristic
119	C₂³	#486	1	2	yes	—
120	C₂³	#493	1	1	yes	characteristic
121	C₂³	#500	1	1	yes	characteristic
122	C₂³	#508	1	8	yes	—
123	C₂×C₄	—	2	16	no	—
124	C₂×C₄	—	2	8	no	—
125	C₂×C₄	—	8	16	no	—
126	C₂×C₄	—	8	8	no	—
127	C₂×C₄	—	16	8	no	—
128	C₂×C₄	—	16	8	no	—
129	C₂×C₄	#340	1	8	yes	—
130	C₂×C₄	C₂×#95	1	1	yes	characteristic
131	C₂×C₄	#382	1	1	yes	characteristic
132	C₂×C₄	#384	1	1	yes	characteristic
133	C₂×C₄	#390	1	2	yes	—
134	C₂×C₄	#391	1	2	yes	—
135	C₂×C₄	C₂×#102	1	2	yes	—
136	C₂×C₄	C₂×#103	1	2	yes	—
137	C₂×C₄	C₂×#103	1	2	yes	—
138	C₂×C₄	C₂×#106	1	1	yes	characteristic
139	C₂×C₄	C₂×#106	1	1	yes	characteristic
140	C₂×C₄	C₂×#107	1	1	yes	characteristic
141	C₂×C₄	#486	1	2	yes	—
142	C₂×C₄	#487	1	2	yes	—
143	C₂×C₄	#487	1	2	yes	—
144	C₂×C₄	#492	1	2	yes	—
145	C₂×C₄	#494	1	1	yes	characteristic
146	C₂×C₄	#498	1	2	yes	—
147	C₂×C₄	#500	1	1	yes	characteristic
148	C₈	#340	1	8	yes	—
149	C₈	C₄×C₄⋊C₄	1	2	yes	—
150	C₈	#380	1	2	yes	—
151	C₈	#389	1	4	yes	—
152	C₂²	C₂×#340	1	1	yes	characteristic
153	C₂²	#1834	1	1	yes	characteristic
154	C₂²	#1845	1	1	yes	characteristic
155	C₂²	#1957	1	2	yes	—
156	C₂²	#1990	1	2	yes	—
157	C₂²	#2138	1	8	yes	—
158	C₂²	#2158	1	4	yes	—
159	C₂²	#2242	1	8	yes	—
160	C₂²	#2249	1	4	yes	—
161	C₂²	#2253	1	4	yes	—
162	C₄	—	2	8	no	—
163	C₄	—	16	8	no	—
164	C₄	C₄⋊#95	1	1	yes	characteristic
165	C₄	C₂×#340	1	1	yes	characteristic
166	C₄	#1834	1	1	yes	characteristic
167	C₄	#1844	1	1	yes	characteristic
168	C₄	#1956	1	2	yes	—
169	C₄	#1990	1	2	yes	—
170	C₂	#10841	1	1	yes	characteristic
171	C₂	#12556	1	2	yes	—
172	C₂	#12889	1	2	yes	—
173	C₂	#12907	1	2	yes	—
174	C₂	#13204	1	4	yes	—
175	C₂	#13281	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₁₆	C₄	semidirect
12. #1846	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₆	2
C₂×#1846	2
C₂×#1380	2