Subgroups — #483456

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2162	C₂	1	1	yes	maximal, characteristic
2	#17793	C₂	1	2	yes	maximal
3	#12036	C₂	1	8	yes	maximal, complemented
4	#12024	C₂	1	4	yes	maximal, complemented
5	#2920	—	2	2	no	—
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	#2162	C₄	1	4	yes	—
8	#2162	C₂²	1	4	yes	—
9	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
10	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	#1568	—	2	2	no	—
13	#1568	C₂²	1	4	yes	—
14	#1567	C₂²	1	4	yes	—
15	#1566	—	2	2	no	—
16	#1564	—	2	2	no	—
17	#1564	C₂²	1	4	yes	—
18	#1372	C₂²	1	8	yes	complemented
19	#1371	C₂²	1	8	yes	complemented
20	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₈	Q₈	1	1	yes	characteristic
22	C₂²×C₄²	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂×C₄	1	4	yes	—
24	#432	—	2	4	no	—
25	#432	—	4	2	no	—
26	#431	—	2	4	no	—
27	#431	—	4	2	no	—
28	#423	—	2	4	no	—
29	#407	—	4	2	no	—
30	C₄×M₁₆	—	4	2	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₈	D₄	1	2	yes	—
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
36	C₂×C₄×C₈	Q₈	1	2	yes	—
37	C₂×C₄×C₈	C₂³	1	2	yes	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
40	C₂²⋊C₁₆	C₂×C₄	1	8	yes	—
41	C₄×C₁₆	C₂×C₄	1	4	yes	—
42	C₄×C₁₆	C₂³	1	4	yes	—
43	#335	—	2	4	no	—
44	#335	C₂³	1	8	yes	—
45	#334	—	2	4	no	—
46	#334	—	2	4	no	—
47	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
48	C₂³×C₄	D₈	1	1	yes	characteristic
49	C₂³×C₄	Q₁₆	1	1	yes	characteristic
50	C₂×M₁₆	—	4	4	no	—
51	C₂²×C₈	—	4	2	no	—
52	C₂²×C₈	—	4	2	no	—
53	C₂²×C₈	C₄⋊C₄	1	2	yes	—
54	C₂²×C₈	C₄⋊C₄	1	2	yes	—
55	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₄	1	2	yes	—
58	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₈	C₂×Q₈	1	2	yes	—
61	C₂×C₄²	C₄⋊C₄	1	2	yes	—
62	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄²	D₈	1	2	yes	—
64	C₂×C₄²	D₈	1	1	yes	characteristic
65	C₂×C₄²	Q₁₆	1	1	yes	characteristic
66	C₂×C₄²	Q₁₆	1	2	yes	—
67	C₂×C₄²	C₂×D₄	1	2	yes	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₂×C₁₆	—	4	4	no	—
70	C₂×C₁₆	C₂²×C₄	1	8	yes	—
71	C₄⋊C₈	—	2	8	no	—
72	C₄⋊C₈	—	4	4	no	—
73	C₂²⋊C₈	—	4	4	no	—
74	C₂²⋊C₈	—	4	4	no	—
75	#97	—	4	4	no	—
76	C₄×C₈	—	4	4	no	—
77	C₄×C₈	C₄⋊C₄	1	2	yes	—
78	C₄×C₈	C₄⋊C₄	1	2	yes	—
79	C₄×C₈	C₄⋊C₄	1	2	yes	—
80	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₄×C₈	C₂²×C₄	1	2	yes	—
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	C₂×D₄	1	2	yes	—
84	C₄×C₈	C₂×Q₈	1	2	yes	—
85	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₄×C₈	C₂⁴	1	1	yes	characteristic
87	C₂⁴	#107	1	1	yes	characteristic
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	1	no	—
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	#107	1	2	yes	—
93	C₂²×C₄	#107	1	2	yes	—
94	C₂²×C₄	#107	1	1	yes	characteristic
95	C₂²×C₄	#107	1	2	yes	—
96	C₂²×C₄	#107	1	1	yes	characteristic
97	C₂²×C₄	#107	1	1	yes	characteristic
98	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
99	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
101	C₂²×C₄	C₂×D₈	1	2	yes	—
102	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
103	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
104	M₁₆	—	8	8	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	8	no	—
114	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
120	C₄²	#107	1	2	yes	—
121	C₄²	#107	1	2	yes	—
122	C₄²	#107	1	2	yes	—
123	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₄²	C₂×C₄⋊C₄	1	2	yes	—
125	C₄²	C₂×D₈	1	2	yes	—
126	C₄²	C₂×D₈	1	1	yes	characteristic
127	C₄²	C₂×Q₁₆	1	2	yes	—
128	C₄²	C₂×Q₁₆	1	1	yes	characteristic
129	C₄²	C₂²×D₄	1	1	yes	characteristic
130	C₁₆	—	2	8	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	4	1	no	—
133	C₂³	—	4	1	no	—
134	C₂³	C₂×#107	1	2	yes	—
135	C₂³	C₂×#107	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	#368	1	4	yes	—
150	C₂×C₄	C₂×#107	1	1	yes	characteristic
151	C₂×C₄	C₂×#107	1	1	yes	characteristic
152	C₂×C₄	C₂×#107	1	2	yes	—
153	C₂×C₄	C₂×#107	1	2	yes	—
154	C₂×C₄	C₂×#107	1	1	yes	characteristic
155	C₂×C₄	C₂×#107	1	2	yes	—
156	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
158	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	8	no	—
164	C₈	#430	1	2	yes	—
165	C₈	#568	1	2	yes	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	4	4	no	—
172	C₂²	C₂×#368	1	2	yes	—
173	C₂²	C₂²×#107	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	2	no	—
180	C₄	C₂×#368	1	4	yes	—
181	C₄	#2905	1	1	yes	characteristic
182	C₄	C₂×#430	1	1	yes	characteristic
183	C₄	#3414	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	4	2	no	—
186	C₂	#17814	1	1	yes	characteristic
187	C₂	#33409	1	1	yes	characteristic
188	C₂	C₂²×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ 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
3. #12036	C₂	semidirect
4. #12024	C₂	semidirect
18. #1372	C₂²	semidirect
19. #1371	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2162	2
#17793	2
#12036	2
#12024	2