Subgroups — #496161

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33726	C₂	1	1	yes	maximal, characteristic
2	#21733	C₂	1	2	yes	maximal
3	#13286	C₂	1	2	yes	maximal
4	#13170	C₂	1	2	yes	maximal
5	#13169	C₂	1	2	yes	maximal
6	#12909	C₂	1	1	yes	maximal, characteristic
7	#12671	C₂	1	2	yes	maximal
8	#12670	C₂	1	2	yes	maximal
9	#12539	C₂	1	1	yes	maximal, characteristic
10	#3160	C₂²	1	2	yes	—
11	#2961	—	2	2	no	—
12	#2254	C₄	1	2	yes	complemented
13	#2253	C₄	1	2	yes	complemented
14	#2253	C₄	1	2	yes	complemented
15	#2252	C₄	1	2	yes	complemented
16	C₂×#368	C₂²	1	2	yes	—
17	C₂×#368	C₂²	1	2	yes	—
18	#2158	C₂²	1	2	yes	—
19	#2158	C₂²	1	2	yes	—
20	C₂×#346	C₂²	1	1	yes	characteristic
21	C₂×#346	C₂²	1	1	yes	characteristic
22	C₂×#346	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
26	#1995	—	2	2	no	—
27	#1946	—	2	2	no	—
28	#1945	C₂²	1	4	yes	—
29	#1848	C₂²	1	2	yes	—
30	#1773	—	2	2	no	—
31	#1772	C₂²	1	4	yes	—
32	#1726	C₂²	1	1	yes	characteristic
33	#1382	C₂²	1	4	yes	—
34	#1382	C₂²	1	4	yes	—
35	#533	—	2	2	no	—
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	C₂²×C₁₆	C₂³	1	1	yes	characteristic
40	#500	C₂×C₄	1	4	yes	—
41	#499	—	2	2	no	—
42	#499	C₂×C₄	1	4	yes	—
43	#498	—	2	2	no	—
44	#446	—	4	8	no	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	C₂³	1	4	yes	—
47	#427	—	2	2	no	—
48	#427	C₂³	1	4	yes	—
49	#424	—	2	4	no	—
50	#423	—	2	2	no	—
51	#405	—	2	4	no	—
52	C₂×#97	D₄	1	1	yes	characteristic
53	C₂×#97	D₄	1	1	yes	characteristic
54	C₂×#97	D₄	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	#376	D₄	1	1	yes	characteristic
59	#368	—	2	4	no	—
60	#368	—	2	4	no	—
61	#367	C₂×C₄	1	4	yes	—
62	#367	C₂×C₄	1	4	yes	—
63	#366	C₂×C₄	1	4	yes	—
64	#366	C₂×C₄	1	4	yes	—
65	#346	—	2	2	no	—
66	#346	—	2	1	no	—
67	#346	—	2	1	no	—
68	#346	Q₈	1	4	yes	complemented
69	C₄×C₁₆	—	2	2	no	—
70	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
71	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
72	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
73	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
74	C₄×C₁₆	Q₈	1	4	yes	complemented
75	#339	—	2	4	no	—
76	#339	—	2	4	no	—
77	C₂×M₁₆	—	2	4	no	—
78	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₈	C₂⁴	1	1	yes	characteristic
82	C₄⋊Q₈	—	2	4	no	—
83	#125	—	2	4	no	—
84	C₂²⋊Q₈	—	4	4	no	—
85	C₄×Q₈	—	4	4	no	—
86	#117	—	2	4	no	—
87	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
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	C₂×C₁₆	#41	1	2	yes	—
103	C₂×C₁₆	#41	1	2	yes	—
104	#108	—	2	4	no	—
105	#108	—	2	4	no	—
106	#108	—	4	4	no	—
107	#107	—	2	4	no	—
108	#107	C₂²×C₄	1	8	yes	—
109	#106	—	2	4	no	—
110	#106	—	2	4	no	—
111	C₄⋊C₈	—	4	8	no	—
112	C₄⋊C₈	—	4	4	no	—
113	#97	—	2	2	no	—
114	#97	—	4	4	no	—
115	#97	D₈	1	4	yes	—
116	#97	Q₁₆	1	4	yes	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	—	2	1	no	—
119	C₄×C₈	—	4	4	no	—
120	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₄×C₈	C₂×Q₈	1	2	yes	—
123	C₄×C₈	C₂×Q₈	1	2	yes	—
124	C₄×C₈	#41	1	1	yes	characteristic
125	C₄×C₈	#41	1	1	yes	characteristic
126	#95	—	2	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	#135	1	2	yes	—
131	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
132	M₁₆	—	4	4	no	—
133	M₁₆	—	4	4	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	C₄×D₄	1	1	yes	characteristic
138	C₂×C₈	C₄×D₄	1	1	yes	characteristic
139	C₂×C₈	C₄⋊Q₈	1	2	yes	—
140	C₂×C₈	C₂×D₈	1	2	yes	—
141	C₂×C₈	C₂×D₈	1	2	yes	—
142	C₂×C₈	C₂×Q₁₆	1	2	yes	—
143	C₂×C₈	C₂×Q₁₆	1	2	yes	—
144	C₂×C₈	#135	1	2	yes	—
145	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₈	C₂×#41	1	1	yes	characteristic
149	C₂×C₈	C₂×#41	1	1	yes	characteristic
150	C₂×C₈	C₂×#41	1	1	yes	characteristic
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₂²⋊C₄	—	4	4	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	4	4	no	—
158	C₄²	C₄×D₄	1	1	yes	characteristic
159	C₄²	C₄×D₄	1	1	yes	characteristic
160	C₄²	C₄⋊Q₈	1	2	yes	—
161	C₄²	C₂×D₈	1	2	yes	—
162	C₄²	C₂×Q₁₆	1	2	yes	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	C₄×Q₈	1	2	yes	—
166	C₁₆	C₄×Q₈	1	2	yes	—
167	C₁₆	C₄×Q₈	1	2	yes	—
168	C₁₆	C₄×Q₈	1	2	yes	—
169	C₂³	C₂×#135	1	1	yes	characteristic
170	Q₈	—	16	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	8	no	—
177	C₂×C₄	C₄×D₈	1	2	yes	—
178	C₂×C₄	C₄×D₈	1	2	yes	—
179	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₂×C₄	#532	1	1	yes	characteristic
182	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
183	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
184	C₂×C₄	C₂×#135	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	8	4	no	—
187	C₈	C₄×Q₁₆	1	2	yes	—
188	C₈	C₄×Q₁₆	1	2	yes	—
189	C₈	#500	1	4	yes	—
190	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
191	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
192	C₈	#533	1	1	yes	characteristic
193	C₈	#533	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
196	C₂²	#3160	1	1	yes	characteristic
197	C₂²	#3170	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	8	8	no	—
203	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
204	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
205	C₄	C₂×#500	1	1	yes	characteristic
206	C₄	#3169	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	C₄×#500	1	1	yes	characteristic
210	C₂	#33465	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #2254	C₄	semidirect
13. #2253	C₄	semidirect
14. #2253	C₄	semidirect
15. #2252	C₄	semidirect
68. #346	Q₈	semidirect
74. 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
#33726	2
#21733	2
#13286	2
#13170	2
#13169	2
#12909	2
#12671	2
#12670	2
#12539	2