Subgroups — #496145

Order 512.

476 conjugacy classes of subgroups.

The subgroup lattice has 197 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33475	C₂	1	2	yes	maximal
2	#21735	C₂	1	1	yes	maximal, characteristic
3	#13266	C₂	1	2	yes	maximal
4	#13162	C₂	1	2	yes	maximal
5	#13161	C₂	1	2	yes	maximal
6	#12901	C₂	1	1	yes	maximal, characteristic
7	#12665	C₂	1	2	yes	maximal
8	#12664	C₂	1	2	yes	maximal
9	#12539	C₂	1	1	yes	maximal, characteristic
10	#3162	C₄	1	1	yes	characteristic
11	#2960	C₂²	1	2	yes	—
12	#2183	—	2	8	no	—
13	C₂×#364	C₂²	1	2	yes	—
14	C₂×#364	C₂²	1	2	yes	—
15	#2151	C₂²	1	4	yes	—
16	#2150	C₂²	1	2	yes	—
17	#2108	C₂²	1	4	yes	—
18	C₂×#346	C₂²	1	1	yes	characteristic
19	C₂×#346	C₂²	1	1	yes	characteristic
20	C₂×#346	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	#1995	C₄	1	2	yes	—
25	#1946	C₄	1	2	yes	—
26	#1946	C₄	1	2	yes	—
27	#1849	C₄	1	1	yes	characteristic
28	#1772	C₂²	1	2	yes	—
29	#1772	C₂²	1	2	yes	—
30	#1726	C₂²	1	1	yes	characteristic
31	#1379	C₂²	1	4	yes	—
32	#1379	C₂²	1	4	yes	—
33	#533	C₂×C₄	1	2	yes	—
34	C₂×M₃₂	C₂³	1	4	yes	—
35	C₂²×C₁₆	C₂³	1	1	yes	characteristic
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	C₂²×C₁₆	C₂³	1	1	yes	characteristic
38	C₂²×C₁₆	C₂³	1	1	yes	characteristic
39	#501	C₈	1	8	yes	—
40	#446	—	2	4	no	—
41	C₈×Q₈	—	2	4	no	—
42	C₂×#108	C₂×C₄	1	2	yes	—
43	C₂×#108	C₂×C₄	1	2	yes	—
44	#427	C₂×C₄	1	2	yes	—
45	#427	C₂×C₄	1	2	yes	—
46	#424	C₂×C₄	1	4	yes	—
47	#423	C₂×C₄	1	2	yes	—
48	#405	C₂³	1	4	yes	—
49	C₂×#97	C₂×C₄	1	1	yes	characteristic
50	C₂×#97	C₂×C₄	1	1	yes	characteristic
51	C₂×#97	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	#376	C₂×C₄	1	1	yes	characteristic
56	#364	—	2	4	no	—
57	#364	—	2	4	no	—
58	C₄⋊C₁₆	—	2	8	no	—
59	C₄⋊C₁₆	—	2	4	no	—
60	#346	—	2	4	no	—
61	#346	—	2	2	no	—
62	#346	D₄	1	4	yes	—
63	#346	Q₈	1	4	yes	complemented
64	C₄×C₁₆	—	2	4	no	—
65	C₄×C₁₆	—	2	2	no	—
66	C₄×C₁₆	—	2	2	no	—
67	C₄×C₁₆	Q₈	1	4	yes	complemented
68	#339	C₂×C₄	1	4	yes	—
69	#339	C₂×C₄	1	4	yes	—
70	C₂×M₁₆	C₂²×C₄	1	4	yes	—
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂⁴	1	1	yes	characteristic
75	C₄⋊Q₈	C₂×C₈	1	4	yes	—
76	#125	C₂×C₈	1	4	yes	—
77	C₂²⋊Q₈	—	2	4	no	—
78	C₄×Q₈	—	2	4	no	—
79	#117	C₂²×C₄	1	4	yes	—
80	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
83	M₃₂	—	2	4	no	—
84	M₃₂	—	2	4	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	2	4	no	—
92	C₂×C₁₆	C₂×D₄	1	2	yes	—
93	C₂×C₁₆	C₂×D₄	1	2	yes	—
94	C₂×C₁₆	C₂×Q₈	1	2	yes	—
95	C₂×C₁₆	C₂×Q₈	1	2	yes	—
96	C₂×C₁₆	C₂×Q₈	1	2	yes	—
97	C₂×C₁₆	C₂×Q₈	1	2	yes	—
98	#108	—	2	4	no	—
99	#108	—	2	4	no	—
100	#107	C₂×C₈	1	4	yes	—
101	#107	C₂×C₈	1	4	yes	—
102	#106	C₂×C₈	1	4	yes	—
103	#106	C₂×C₈	1	4	yes	—
104	C₄⋊C₈	—	2	8	no	—
105	C₄⋊C₈	—	2	4	no	—
106	#97	—	2	2	no	—
107	#97	—	2	2	no	—
108	#97	—	2	4	no	—
109	#97	C₂×C₈	1	4	yes	—
110	C₄×C₈	—	2	4	no	—
111	C₄×C₈	C₂×D₄	1	2	yes	—
112	C₄×C₈	C₂×Q₈	1	2	yes	—
113	C₄×C₈	C₂×Q₈	1	2	yes	—
114	C₄×C₈	#41	1	2	yes	—
115	C₄×C₈	#41	1	2	yes	—
116	C₄×C₈	#41	1	2	yes	—
117	#95	—	2	2	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	#131	1	2	yes	—
122	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	2	4	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	C₄×D₄	1	2	yes	—
129	C₂×C₈	C₄×Q₈	1	2	yes	—
130	C₂×C₈	C₄×Q₈	1	2	yes	—
131	C₂×C₈	C₂²×C₈	1	2	yes	—
132	C₂×C₈	C₂²×C₈	1	2	yes	—
133	C₂×C₈	#131	1	2	yes	—
134	C₂×C₈	#131	1	2	yes	—
135	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₈	C₂×#41	1	1	yes	characteristic
139	C₂×C₈	C₂×#41	1	1	yes	characteristic
140	C₂×C₈	C₂×#41	1	1	yes	characteristic
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	C₂²×C₈	1	8	yes	—
144	C₂²⋊C₄	—	2	4	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	C₄×D₄	1	2	yes	—
149	C₄²	C₄×Q₈	1	2	yes	—
150	C₄²	C₄×Q₈	1	2	yes	—
151	C₄²	C₂²×C₈	1	2	yes	—
152	C₁₆	—	2	2	no	—
153	C₁₆	—	2	2	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	—	4	4	no	—
156	C₁₆	C₄⋊Q₈	1	4	yes	—
157	C₂³	C₂×#131	1	1	yes	characteristic
158	Q₈	—	8	4	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	C₈×D₄	1	4	yes	—
166	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
170	C₂×C₄	C₂×#131	1	1	yes	characteristic
171	C₂×C₄	C₂×#131	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	4	4	no	—
174	C₈	C₈×Q₈	1	4	yes	—
175	C₈	C₈×Q₈	1	4	yes	—
176	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
177	C₈	#532	1	1	yes	characteristic
178	C₈	#533	1	1	yes	characteristic
179	C₈	#533	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
182	C₂²	#2960	1	1	yes	characteristic
183	C₂²	#2960	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	8	no	—
188	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
189	C₄	#2931	1	1	yes	characteristic
190	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
191	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	C₈×C₄⋊Q₈	1	1	yes	characteristic
195	C₂	#33442	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
63. #346	Q₈	semidirect
67. 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
#33475	2
#21735	2
#13266	2
#13162	2
#13161	2
#12901	2
#12665	2
#12664	2
#12539	2