Subgroups — #213369

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 227 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#499	C₂	1	1	yes	maximal, characteristic
2	C₈×#125	C₂	1	2	yes	maximal
3	#9871	C₂	1	2	yes	maximal
4	#9596	C₂	1	2	yes	maximal
5	#9595	C₂	1	2	yes	maximal
6	#8807	C₂	1	1	yes	maximal, characteristic
7	#8285	C₂	1	2	yes	maximal
8	#8284	C₂	1	2	yes	maximal
9	C₄×#322	C₂	1	1	yes	maximal, characteristic
10	C₂×#499	C₄	1	1	yes	characteristic
11	C₄×#125	C₂²	1	2	yes	—
12	#1991	C₄	1	2	yes	—
13	#1942	C₄	1	2	yes	—
14	#1941	C₄	1	2	yes	—
15	#1939	C₂²	1	4	yes	—
16	#1916	—	2	4	no	—
17	#1846	C₄	1	1	yes	characteristic
18	#1831	C₂²	1	2	yes	—
19	C₄×#107	C₂²	1	2	yes	—
20	C₄×#106	C₂²	1	2	yes	—
21	C₈×C₄⋊C₄	—	2	4	no	—
22	C₈×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄²×C₈	C₂²	1	1	yes	characteristic
24	C₄²×C₈	C₂²	1	1	yes	characteristic
25	C₄²×C₈	C₂²	1	1	yes	characteristic
26	C₂×#335	C₂²	1	2	yes	—
27	C₂×#334	C₂²	1	2	yes	—
28	C₂×#322	C₂²	1	1	yes	characteristic
29	C₂×#322	C₂²	1	1	yes	characteristic
30	C₂×#322	C₂²	1	1	yes	characteristic
31	C₂×#322	C₂²	1	1	yes	characteristic
32	#1279	C₂²	1	4	yes	—
33	#1279	C₂²	1	4	yes	—
34	C₂×#125	C₂×C₄	1	2	yes	—
35	#499	C₈	1	8	yes	complemented
36	C₂×#107	C₂×C₄	1	2	yes	—
37	C₂×#107	C₂×C₄	1	2	yes	—
38	C₂×#106	C₂×C₄	1	2	yes	—
39	C₂×#106	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₈	—	2	4	no	—
41	C₂×C₄⋊C₈	C₂³	1	4	yes	—
42	C₂×C₄×C₈	—	2	4	no	—
43	C₂×C₄×C₈	—	2	2	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	4	yes	—
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	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	#389	C₂×C₄	1	4	yes	—
56	#387	—	2	4	no	—
57	#382	C₂×C₄	1	2	yes	—
58	C₄×C₄⋊C₄	—	2	4	no	—
59	C₄×C₄⋊C₄	C₂³	1	4	yes	—
60	C₄³	C₂³	1	1	yes	characteristic
61	#340	C₂×C₄	1	4	yes	—
62	#340	C₂×C₄	1	4	yes	—
63	#336	—	2	4	no	—
64	#336	—	2	4	no	—
65	#336	—	2	4	no	—
66	#335	—	2	4	no	—
67	#334	—	2	4	no	—
68	#322	—	2	2	no	—
69	#322	—	2	2	no	—
70	#322	Q₈	1	4	yes	—
71	#322	Q₈	1	4	yes	—
72	C₂²×C₈	—	2	1	no	—
73	C₂²×C₈	—	2	1	no	—
74	C₂²×C₈	—	2	4	no	—
75	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₄	1	2	yes	—
80	C₂²×C₈	C₂×D₄	1	2	yes	—
81	#125	—	2	4	no	—
82	#125	C₂×C₈	1	8	yes	—
83	C₂×C₄⋊C₄	—	2	4	no	—
84	C₂×C₄⋊C₄	—	2	4	no	—
85	C₂×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₂×C₄⋊C₄	C₂²×C₄	1	4	yes	—
87	C₂×C₄²	—	2	4	no	—
88	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₄	1	2	yes	—
91	C₂×C₄²	#41	1	2	yes	—
92	C₂×C₄²	C₂⁴	1	1	yes	characteristic
93	#107	—	2	4	no	—
94	#107	C₂×C₈	1	8	yes	—
95	#106	—	2	4	no	—
96	#106	C₂×C₈	1	8	yes	—
97	C₄⋊C₈	—	2	4	no	—
98	C₄⋊C₈	—	2	4	no	—
99	C₄⋊C₈	—	4	8	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	4	4	no	—
108	C₄×C₈	—	4	4	no	—
109	C₄×C₈	—	4	8	no	—
110	C₄×C₈	C₂×C₈	1	4	yes	—
111	C₄×C₈	C₂×Q₈	1	2	yes	—
112	C₄×C₈	C₂×Q₈	1	2	yes	—
113	C₄×C₈	C₂×Q₈	1	2	yes	—
114	C₄×C₈	C₂×Q₈	1	2	yes	—
115	C₄×C₈	C₂×Q₈	1	2	yes	—
116	C₄×C₈	C₂×Q₈	1	2	yes	—
117	C₄×C₈	#41	1	2	yes	—
118	C₄×C₈	#41	1	2	yes	—
119	#95	—	2	4	no	—
120	#95	—	2	4	no	—
121	#95	—	2	4	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	C₄×D₄	1	2	yes	—
125	C₂²×C₄	C₄×D₄	1	2	yes	—
126	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₄	C₂×#41	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	—	4	4	no	—
143	C₂×C₈	C₄×Q₈	1	2	yes	—
144	C₂×C₈	C₄×Q₈	1	2	yes	—
145	C₂×C₈	C₄×Q₈	1	2	yes	—
146	C₂×C₈	C₄×Q₈	1	2	yes	—
147	C₂×C₈	C₄⋊Q₈	1	4	yes	—
148	C₂×C₈	C₂²×C₈	1	2	yes	—
149	C₂×C₈	C₂²×C₈	1	2	yes	—
150	C₂×C₈	#131	1	2	yes	—
151	C₂×C₈	#131	1	2	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	16	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	8	no	—
157	C₄⋊C₄	C₂²×C₈	1	8	yes	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	4	no	—
162	C₄²	—	4	4	no	—
163	C₄²	—	4	4	no	—
164	C₄²	C₂²×C₈	1	2	yes	—
165	C₄²	#131	1	2	yes	—
166	C₄²	C₂²×Q₈	1	1	yes	characteristic
167	C₄²	C₂²×Q₈	1	1	yes	characteristic
168	C₄²	C₂×#41	1	1	yes	characteristic
169	C₄²	C₂×#41	1	1	yes	characteristic
170	C₂³	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₈×D₄	1	4	yes	—
183	C₂×C₄	#444	1	4	yes	—
184	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₄	#532	1	1	yes	characteristic
188	C₂×C₄	#533	1	1	yes	characteristic
189	C₂×C₄	#533	1	1	yes	characteristic
190	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
191	C₂×C₄	C₂×#131	1	1	yes	characteristic
192	C₂×C₄	C₂×#131	1	1	yes	characteristic
193	C₂×C₄	C₂×#131	1	1	yes	characteristic
194	C₂×C₄	#575	1	4	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	4	2	no	—
199	C₈	—	4	2	no	—
200	C₈	—	4	2	no	—
201	C₈	—	8	4	no	—
202	C₈	C₈×Q₈	1	4	yes	—
203	C₈	C₈×Q₈	1	4	yes	—
204	C₂²	C₄×C₄⋊Q₈	1	1	yes	characteristic
205	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
206	C₂²	#2931	1	1	yes	characteristic
207	C₂²	#2943	1	2	yes	—
208	C₂²	C₂×#444	1	2	yes	—
209	C₄	—	2	4	no	—
210	C₄	—	2	4	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	8	no	—
214	C₄	—	8	4	no	—
215	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
216	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
217	C₄	#2960	1	1	yes	characteristic
218	C₄	#2960	1	1	yes	characteristic
219	C₄	#3166	1	2	yes	—
220	C₄	#3183	1	2	yes	—
221	C₂	#17849	1	2	yes	—
222	C₂	#17963	1	2	yes	—
223	C₂	C₈×C₄⋊Q₈	1	1	yes	characteristic
224	C₂	#21728	1	1	yes	characteristic
225	C₂	#21734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
35. #499	C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#499	2
C₈×#125	2
#9871	2
#9596	2
#9595	2
#8807	2
#8285	2
#8284	2
C₄×#322	2