Subgroups — #483452

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#12034	C₂	1	8	yes	maximal, complemented
4	#12023	C₂	1	2	yes	maximal, complemented
5	#12022	C₂	1	2	yes	maximal, complemented
6	#2920	—	2	2	no	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#2162	C₄	1	4	yes	—
9	#2162	C₂²	1	4	yes	—
10	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
11	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1568	—	2	2	no	—
15	#1568	C₂²	1	4	yes	—
16	#1567	C₂²	1	4	yes	—
17	#1566	—	2	2	no	—
18	#1564	—	2	2	no	—
19	#1564	C₂²	1	4	yes	—
20	#1372	C₂²	1	8	yes	complemented
21	#1371	C₂²	1	8	yes	complemented
22	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₈	Q₈	1	1	yes	characteristic
24	C₂²×C₄²	D₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	C₂×C₄	1	2	yes	—
27	#432	—	2	4	no	—
28	#432	—	4	2	no	—
29	#431	—	2	4	no	—
30	#431	—	4	2	no	—
31	#423	—	2	4	no	—
32	#407	—	4	2	no	—
33	C₄×M₁₆	—	4	2	no	—
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄×C₈	D₄	1	2	yes	—
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈	Q₈	1	2	yes	—
40	C₂×C₄×C₈	C₂³	1	2	yes	—
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
43	C₂²⋊C₁₆	C₂×C₄	1	8	yes	—
44	C₄×C₁₆	C₂×C₄	1	2	yes	—
45	C₄×C₁₆	C₂×C₄	1	2	yes	—
46	C₄×C₁₆	C₂³	1	2	yes	—
47	C₄×C₁₆	C₂³	1	2	yes	—
48	#335	—	2	4	no	—
49	#335	C₂³	1	8	yes	—
50	#334	—	2	4	no	—
51	#334	—	2	4	no	—
52	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₄	D₈	1	1	yes	characteristic
54	C₂³×C₄	Q₁₆	1	1	yes	characteristic
55	C₂×M₁₆	—	4	4	no	—
56	C₂²×C₈	—	4	2	no	—
57	C₂²×C₈	—	4	2	no	—
58	C₂²×C₈	C₄⋊C₄	1	2	yes	—
59	C₂²×C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂²×C₄	1	2	yes	—
63	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₈	C₂×Q₈	1	2	yes	—
66	C₂×C₄²	C₄⋊C₄	1	2	yes	—
67	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄²	D₈	1	1	yes	characteristic
69	C₂×C₄²	D₈	1	2	yes	—
70	C₂×C₄²	Q₁₆	1	1	yes	characteristic
71	C₂×C₄²	Q₁₆	1	2	yes	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₄	1	2	yes	—
74	C₂×C₁₆	—	4	2	no	—
75	C₂×C₁₆	—	4	2	no	—
76	C₂×C₁₆	C₂²×C₄	1	4	yes	—
77	C₂×C₁₆	C₂²×C₄	1	4	yes	—
78	C₄⋊C₈	—	2	8	no	—
79	C₄⋊C₈	—	4	4	no	—
80	C₂²⋊C₈	—	4	4	no	—
81	C₂²⋊C₈	—	4	4	no	—
82	#97	—	4	4	no	—
83	C₄×C₈	—	4	4	no	—
84	C₄×C₈	C₄⋊C₄	1	2	yes	—
85	C₄×C₈	C₄⋊C₄	1	2	yes	—
86	C₄×C₈	C₄⋊C₄	1	2	yes	—
87	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₄×C₈	C₂²×C₄	1	2	yes	—
89	C₄×C₈	C₂×D₄	1	2	yes	—
90	C₄×C₈	C₂×D₄	1	1	yes	characteristic
91	C₄×C₈	C₂×Q₈	1	2	yes	—
92	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₄×C₈	C₂⁴	1	1	yes	characteristic
94	C₂⁴	#107	1	1	yes	characteristic
95	C₂²×C₄	—	2	4	no	—
96	C₂²×C₄	—	4	1	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	#107	1	2	yes	—
100	C₂²×C₄	#107	1	1	yes	characteristic
101	C₂²×C₄	#107	1	2	yes	—
102	C₂²×C₄	#107	1	1	yes	characteristic
103	C₂²×C₄	#107	1	2	yes	—
104	C₂²×C₄	#107	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
106	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄	C₂×D₈	1	2	yes	—
108	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
109	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
110	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
111	M₁₆	—	8	8	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	8	no	—
121	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
127	C₄²	#107	1	2	yes	—
128	C₄²	#107	1	2	yes	—
129	C₄²	#107	1	2	yes	—
130	C₄²	C₂×C₄⋊C₄	1	2	yes	—
131	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₄²	C₂×D₈	1	2	yes	—
133	C₄²	C₂×D₈	1	1	yes	characteristic
134	C₄²	C₂×Q₁₆	1	2	yes	—
135	C₄²	C₂×Q₁₆	1	1	yes	characteristic
136	C₄²	C₂²×D₄	1	1	yes	characteristic
137	C₁₆	—	2	4	no	—
138	C₁₆	—	2	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	4	1	no	—
141	C₂³	—	4	1	no	—
142	C₂³	C₂×#107	1	2	yes	—
143	C₂³	C₂×#107	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	C₂×#107	1	2	yes	—
158	C₂×C₄	C₂×#107	1	2	yes	—
159	C₂×C₄	C₂×#107	1	2	yes	—
160	C₂×C₄	C₂×#107	1	1	yes	characteristic
161	C₂×C₄	C₂×#107	1	1	yes	characteristic
162	C₂×C₄	C₂×#107	1	1	yes	characteristic
163	C₂×C₄	#508	1	4	yes	—
164	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
166	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	2	no	—
169	C₈	—	2	2	no	—
170	C₈	—	2	1	no	—
171	C₈	—	8	8	no	—
172	C₈	#430	1	2	yes	—
173	C₈	#568	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	4	no	—
180	C₂²	#2158	1	2	yes	—
181	C₂²	C₂²×#107	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	2	no	—
187	C₄	#2158	1	2	yes	—
188	C₄	#2161	1	2	yes	—
189	C₄	#2905	1	1	yes	characteristic
190	C₄	C₂×#430	1	1	yes	characteristic
191	C₄	C₂×#508	1	2	yes	—
192	C₂	—	2	2	no	—
193	C₂	—	4	2	no	—
194	C₂	#17814	1	1	yes	characteristic
195	C₂	C₂×#2158	1	1	yes	characteristic
196	C₂	C₂×#2161	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. #12034	C₂	semidirect
4. #12023	C₂	semidirect
5. #12022	C₂	semidirect
20. #1372	C₂²	semidirect
21. #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
#12034	2
#12023	2
#12022	2