Subgroups — #127358

Order 512.

460 conjugacy classes of subgroups.

The subgroup lattice has 195 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	#20818	C₂	1	2	yes	maximal
3	#7230	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
5	#2283	—	2	2	no	—
6	#2162	—	2	2	no	—
7	#2162	—	2	2	no	—
8	C₂×C₄⋊C₁₆	C₄	1	2	yes	complemented
9	C₂×C₂²⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
11	#1846	—	2	2	no	—
12	#1835	C₄	1	4	yes	complemented
13	#1808	C₂²	1	4	yes	—
14	#1807	—	2	2	no	—
15	C₄×#107	—	2	2	no	—
16	C₄×#106	C₄	1	4	yes	complemented
17	C₂³×C₈	D₄	1	1	yes	characteristic
18	C₂³×C₈	C₂³	1	1	yes	characteristic
19	C₂²×C₄²	D₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	4	yes	—
21	C₂×#107	—	2	4	no	—
22	C₂×#106	—	2	4	no	—
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₈	D₄	1	2	yes	—
28	C₂×C₄×C₈	D₄	1	2	yes	—
29	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
30	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
31	#383	—	4	2	no	—
32	#382	—	2	4	no	—
33	#381	—	4	2	no	—
34	#380	—	2	4	no	—
35	C₄×C₄⋊C₄	—	2	4	no	—
36	C₄×C₂²⋊C₄	—	4	2	no	—
37	#376	—	4	2	no	—
38	C₄⋊C₁₆	—	2	4	no	—
39	C₂²⋊C₁₆	—	2	4	no	—
40	C₄×C₁₆	—	2	2	no	—
41	C₄×C₁₆	—	2	2	no	—
42	#340	—	2	4	no	—
43	#340	C₂×C₄	1	8	yes	—
44	C₂³×C₄	D₈	1	1	yes	characteristic
45	C₂³×C₄	D₈	1	1	yes	characteristic
46	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₈	—	4	2	no	—
48	C₂²×C₈	—	4	2	no	—
49	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²×C₄	1	2	yes	—
55	C₂²×C₈	C₂×D₄	1	2	yes	—
56	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄⋊C₄	—	2	8	no	—
58	C₂×C₂²⋊C₄	—	4	4	no	—
59	C₂×C₄²	—	4	4	no	—
60	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄²	SD₁₆	1	2	yes	—
65	C₂×C₄²	SD₁₆	1	2	yes	—
66	C₂×C₄²	Q₁₆	1	1	yes	characteristic
67	C₂×C₄²	Q₁₆	1	1	yes	characteristic
68	C₂×C₁₆	—	2	4	no	—
69	C₂×C₁₆	—	4	4	no	—
70	#107	—	4	8	no	—
71	#106	—	4	8	no	—
72	C₄×C₈	C₄²	1	1	yes	characteristic
73	C₄×C₈	C₄²	1	2	yes	—
74	C₄×C₈	C₄²	1	1	yes	characteristic
75	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₈	C₂²⋊C₄	1	2	yes	—
78	C₄×C₈	C₄⋊C₄	1	2	yes	—
79	C₄×C₈	C₄⋊C₄	1	2	yes	—
80	C₄×C₈	C₄⋊C₄	1	2	yes	—
81	C₄×C₈	C₄⋊C₄	1	2	yes	—
82	#95	—	4	4	no	—
83	#95	—	4	4	no	—
84	#95	—	4	4	no	—
85	#95	—	4	4	no	—
86	C₂⁴	C₂×D₈	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	—	4	8	no	—
92	C₂²×C₄	#102	1	2	yes	—
93	C₂²×C₄	#102	1	2	yes	—
94	C₂²×C₄	#102	1	2	yes	—
95	C₂²×C₄	#102	1	2	yes	—
96	C₂²×C₄	#107	1	1	yes	characteristic
97	C₂²×C₄	#107	1	1	yes	characteristic
98	C₂²×C₄	#107	1	1	yes	characteristic
99	C₂²×C₄	#107	1	1	yes	characteristic
100	C₂²×C₄	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
103	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	#95	1	1	yes	characteristic
113	C₂×C₈	#95	1	2	yes	—
114	C₂×C₈	#95	1	1	yes	characteristic
115	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
118	C₄⋊C₄	—	4	8	no	—
119	C₄⋊C₄	—	4	8	no	—
120	C₂²⋊C₄	—	8	8	no	—
121	C₄²	—	8	8	no	—
122	C₄²	#95	1	1	yes	characteristic
123	C₄²	#95	1	2	yes	—
124	C₄²	#95	1	1	yes	characteristic
125	C₄²	#103	1	2	yes	—
126	C₄²	#103	1	2	yes	—
127	C₄²	#103	1	2	yes	—
128	C₄²	#103	1	2	yes	—
129	C₄²	#106	1	2	yes	—
130	C₄²	#106	1	1	yes	characteristic
131	C₄²	#106	1	1	yes	characteristic
132	C₁₆	—	4	4	no	—
133	C₂³	—	4	1	no	—
134	C₂³	—	4	1	no	—
135	C₂³	C₂×#102	1	2	yes	—
136	C₂³	C₂×#107	1	1	yes	characteristic
137	C₂³	#508	1	4	yes	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	8	8	no	—
153	C₂×C₄	—	8	8	no	—
154	C₂×C₄	—	8	8	no	—
155	C₂×C₄	#340	1	1	yes	characteristic
156	C₂×C₄	#340	1	2	yes	—
157	C₂×C₄	#340	1	1	yes	characteristic
158	C₂×C₄	#340	1	2	yes	—
159	C₂×C₄	#340	1	1	yes	characteristic
160	C₂×C₄	#340	1	1	yes	characteristic
161	C₂×C₄	C₂×#95	1	1	yes	characteristic
162	C₂×C₄	C₂×#103	1	2	yes	—
163	C₂×C₄	C₂×#106	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	2	no	—
168	C₈	H(3,ℤ/4)	1	2	yes	—
169	C₈	#410	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	4	4	no	—
175	C₂²	C₂×#340	1	1	yes	characteristic
176	C₂²	#2138	1	4	yes	—
177	C₂²	#2158	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	2	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	1	no	—
185	C₄	—	16	8	no	—
186	C₄	#1384	1	2	yes	—
187	C₄	C₂×H(3,ℤ/4)	1	1	yes	characteristic
188	C₄	#1740	1	1	yes	characteristic
189	C₂	—	2	4	no	—
190	C₂	—	2	2	no	—
191	C₂	#7981	1	1	yes	characteristic
192	C₂	#12556	1	1	yes	characteristic
193	C₂	C₂×#1384	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. C₂×C₄⋊C₁₆	C₄	semidirect
10. C₂×C₄×C₁₆	C₄	semidirect
12. #1835	C₄	semidirect
16. C₄×#106	C₄	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
#20818	2
#7230	2