Subgroups — #496142

Order 512.

484 conjugacy classes of subgroups.

The subgroup lattice has 211 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	#21732	C₂	1	1	yes	maximal, characteristic
3	#13267	C₂	1	2	yes	maximal
4	#13162	C₂	1	2	yes	maximal
5	#13161	C₂	1	2	yes	maximal
6	#12900	C₂	1	1	yes	maximal, characteristic
7	#12665	C₂	1	2	yes	maximal
8	#12664	C₂	1	2	yes	maximal
9	C₄×#346	C₂	1	1	yes	maximal, characteristic
10	#3160	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×#346	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	#1994	C₄	1	2	yes	—
25	#1945	C₄	1	2	yes	—
26	#1945	C₄	1	2	yes	—
27	#1848	C₄	1	1	yes	characteristic
28	#1772	C₂²	1	2	yes	—
29	#1772	C₂²	1	2	yes	—
30	C₄²×C₈	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	#500	C₈	1	2	yes	—
40	#499	C₈	1	2	yes	—
41	#499	C₈	1	2	yes	—
42	#498	C₈	1	2	yes	—
43	#446	—	2	4	no	—
44	C₈×Q₈	—	2	4	no	—
45	C₂×#108	C₂×C₄	1	2	yes	—
46	C₂×#108	C₂×C₄	1	2	yes	—
47	#427	C₂×C₄	1	2	yes	—
48	#427	C₂×C₄	1	2	yes	—
49	#424	C₂×C₄	1	4	yes	—
50	#423	C₂×C₄	1	2	yes	—
51	#405	C₂³	1	4	yes	—
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
58	C₄³	C₂×C₄	1	1	yes	characteristic
59	#364	—	2	4	no	—
60	#364	—	2	4	no	—
61	C₄⋊C₁₆	—	2	8	no	—
62	C₄⋊C₁₆	—	2	4	no	—
63	#346	—	2	4	no	—
64	#346	—	2	2	no	—
65	#346	—	2	2	no	—
66	#346	D₄	1	4	yes	—
67	#346	Q₈	1	4	yes	complemented
68	C₄×C₁₆	—	2	4	no	—
69	C₄×C₁₆	—	2	2	no	—
70	C₄×C₁₆	Q₈	1	4	yes	complemented
71	#339	C₂×C₄	1	4	yes	—
72	#339	C₂×C₄	1	4	yes	—
73	C₂×M₁₆	C₂²×C₄	1	4	yes	—
74	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₈	C₂⁴	1	1	yes	characteristic
78	C₄⋊Q₈	C₂×C₈	1	4	yes	—
79	#125	C₂×C₈	1	4	yes	—
80	C₂²⋊Q₈	—	2	4	no	—
81	C₄×Q₈	—	2	4	no	—
82	#117	C₂²×C₄	1	4	yes	—
83	C₂×C₄²	—	2	2	no	—
84	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
87	M₃₂	—	2	4	no	—
88	M₃₂	—	2	4	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	2	4	no	—
96	C₂×C₁₆	C₂×D₄	1	2	yes	—
97	C₂×C₁₆	C₂×D₄	1	2	yes	—
98	C₂×C₁₆	C₂×Q₈	1	2	yes	—
99	C₂×C₁₆	C₂×Q₈	1	2	yes	—
100	C₂×C₁₆	C₂×Q₈	1	2	yes	—
101	C₂×C₁₆	C₂×Q₈	1	2	yes	—
102	#108	—	2	4	no	—
103	#108	—	2	4	no	—
104	#107	C₂×C₈	1	4	yes	—
105	#107	C₂×C₈	1	4	yes	—
106	#106	C₂×C₈	1	4	yes	—
107	#106	C₂×C₈	1	4	yes	—
108	C₄⋊C₈	—	2	8	no	—
109	C₄⋊C₈	—	2	4	no	—
110	#97	—	2	4	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	2	4	no	—
114	C₄×C₈	C₂×C₈	1	1	yes	characteristic
115	C₄×C₈	C₂×C₈	1	1	yes	characteristic
116	C₄×C₈	C₂×C₈	1	1	yes	characteristic
117	C₄×C₈	C₂×C₈	1	1	yes	characteristic
118	C₄×C₈	C₂×D₄	1	2	yes	—
119	C₄×C₈	C₂×Q₈	1	2	yes	—
120	C₄×C₈	C₂×Q₈	1	2	yes	—
121	C₄×C₈	#41	1	2	yes	—
122	C₄×C₈	#41	1	2	yes	—
123	C₄×C₈	#41	1	2	yes	—
124	C₂×Q₈	—	2	4	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	#131	1	2	yes	—
128	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
129	M₁₆	—	2	4	no	—
130	M₁₆	—	2	4	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	C₄×D₄	1	2	yes	—
135	C₂×C₈	C₄×Q₈	1	2	yes	—
136	C₂×C₈	C₄×Q₈	1	2	yes	—
137	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₈	#131	1	2	yes	—
142	C₂×C₈	#131	1	2	yes	—
143	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₈	C₂×#41	1	1	yes	characteristic
147	C₂×C₈	C₂×#41	1	1	yes	characteristic
148	C₂×C₈	C₂×#41	1	1	yes	characteristic
149	C₄⋊C₄	—	2	8	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	C₂²×C₈	1	8	yes	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	4	4	no	—
157	C₄²	C₄×D₄	1	2	yes	—
158	C₄²	C₄×Q₈	1	2	yes	—
159	C₄²	C₄×Q₈	1	2	yes	—
160	C₄²	C₂²×C₈	1	1	yes	characteristic
161	C₄²	C₂²×C₈	1	1	yes	characteristic
162	C₁₆	—	2	2	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	—	4	4	no	—
166	C₁₆	C₄⋊Q₈	1	4	yes	—
167	C₂³	C₂×#131	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₈×D₄	1	2	yes	—
176	C₂×C₄	C₈×D₄	1	2	yes	—
177	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
181	C₂×C₄	C₂×#131	1	1	yes	characteristic
182	C₂×C₄	C₂×#131	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	4	4	no	—
185	C₈	C₈×Q₈	1	2	yes	—
186	C₈	C₈×Q₈	1	2	yes	—
187	C₈	C₈×Q₈	1	2	yes	—
188	C₈	C₈×Q₈	1	2	yes	—
189	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
190	C₈	#532	1	1	yes	characteristic
191	C₈	#533	1	1	yes	characteristic
192	C₈	#533	1	1	yes	characteristic
193	C₂²	—	2	2	no	—
194	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
195	C₂²	#2960	1	1	yes	characteristic
196	C₂²	#2960	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	8	no	—
202	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
203	C₄	#2931	1	1	yes	characteristic
204	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
205	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	C₈×C₄⋊Q₈	1	1	yes	characteristic
209	C₂	#33443	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
67. #346	Q₈	semidirect
70. 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
#21732	2
#13267	2
#13162	2
#13161	2
#12900	2
#12665	2
#12664	2
C₄×#346	2