Subgroups — #495437

Order 512.

490 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20516	C₂	1	1	yes	maximal, characteristic
2	#13263	C₂	1	2	yes	maximal
3	#13159	C₂	1	2	yes	maximal
4	#13105	C₂	1	4	yes	maximal
5	#12654	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	#12592	C₂	1	2	yes	maximal
8	C₄×#346	C₂	1	1	yes	maximal, characteristic
9	C₂×#446	C₄	1	1	yes	characteristic
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	#2295	C₄	1	1	yes	characteristic
12	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
13	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₂×#346	C₂²	1	2	yes	—
15	C₂×#346	C₂²	1	1	yes	characteristic
16	C₂×#346	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	2	yes	—
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	#1771	C₄	1	2	yes	—
20	#1770	C₄	1	1	yes	characteristic
21	C₈×C₄⋊C₄	C₂²	1	2	yes	—
22	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	#1768	C₄	1	2	yes	—
24	#1767	C₄	1	1	yes	characteristic
25	#1726	C₂²	1	2	yes	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1374	C₂²	1	4	yes	—
28	#1374	C₂²	1	4	yes	—
29	#1374	C₂²	1	2	yes	—
30	#1374	C₂²	1	4	yes	—
31	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₆	C₂³	1	4	yes	—
33	C₂²×C₁₆	C₂³	1	2	yes	—
34	C₂²×C₁₆	C₂³	1	2	yes	—
35	#446	—	2	4	no	—
36	C₈×Q₈	—	2	4	no	—
37	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×#97	C₂×C₄	1	2	yes	—
42	C₂×#97	C₂×C₄	1	2	yes	—
43	C₂×#97	C₂×C₄	1	2	yes	—
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	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	2	yes	—
49	C₂×C₄×C₈	C₂³	1	2	yes	—
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	#391	C₈	1	2	yes	—
53	#389	C₈	1	2	yes	—
54	#387	C₈	1	4	yes	—
55	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	#376	C₂×C₄	1	2	yes	—
58	C₄³	C₂×C₄	1	1	yes	characteristic
59	C₄⋊C₁₆	—	2	8	no	—
60	C₄⋊C₁₆	—	2	4	no	—
61	#346	—	2	4	no	—
62	#346	—	2	2	no	—
63	#346	Q₈	1	4	yes	complemented
64	C₄×C₁₆	—	2	4	no	—
65	C₄×C₁₆	—	2	4	no	—
66	#336	C₂×C₄	1	2	yes	—
67	#336	C₂×C₄	1	2	yes	—
68	#336	C₂×C₄	1	2	yes	—
69	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₈	C₂²×C₄	1	2	yes	—
71	C₂²×C₈	C₂²×C₄	1	2	yes	—
72	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂⁴	1	1	yes	characteristic
76	C₄×Q₈	—	2	2	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
81	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
82	C₂×C₄²	C₂×C₈	1	2	yes	—
83	C₂×C₄²	C₂×C₈	1	2	yes	—
84	C₂×C₄²	C₂²×C₄	1	2	yes	—
85	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂²×C₄	1	2	yes	—
87	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	2	4	no	—
95	C₂×C₁₆	—	2	4	no	—
96	C₂×C₁₆	C₂×Q₈	1	4	yes	—
97	C₂×C₁₆	#41	1	4	yes	—
98	C₄⋊C₈	—	2	4	no	—
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	#97	—	2	4	no	—
103	#97	—	2	4	no	—
104	#97	—	2	4	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	4	no	—
108	C₄×C₈	—	2	4	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	M₁₆	1	4	yes	—
111	C₄×C₈	C₂×Q₈	1	2	yes	—
112	C₄×C₈	#41	1	4	yes	—
113	C₄×C₈	#41	1	2	yes	—
114	#95	C₂×C₈	1	4	yes	—
115	#95	C₂×C₈	1	4	yes	—
116	#95	C₂×C₈	1	2	yes	—
117	#95	C₂×C₈	1	4	yes	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂²×C₄	C₂²×C₈	1	4	yes	—
121	C₂²×C₄	C₂²×C₈	1	2	yes	—
122	C₂²×C₄	C₂²×C₈	1	2	yes	—
123	C₂²×C₄	C₂²×C₈	1	2	yes	—
124	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₂²×C₄	C₂²×C₈	1	2	yes	—
126	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	#117	1	2	yes	—
139	C₂×C₈	C₄×Q₈	1	2	yes	—
140	C₂×C₈	C₂×M₁₆	1	2	yes	—
141	C₂×C₈	C₂×M₁₆	1	2	yes	—
142	C₂×C₈	#131	1	2	yes	—
143	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₈	C₂×#41	1	1	yes	characteristic
145	C₂×C₈	C₂×#41	1	2	yes	—
146	C₂×C₈	C₂×#41	1	2	yes	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	4	2	no	—
159	C₄²	#117	1	2	yes	—
160	C₄²	C₄×Q₈	1	2	yes	—
161	C₄²	C₂×M₁₆	1	2	yes	—
162	C₄²	#131	1	2	yes	—
163	C₁₆	—	2	4	no	—
164	C₁₆	—	2	4	no	—
165	C₁₆	—	4	4	no	—
166	C₂³	C₂³×C₈	1	1	yes	characteristic
167	Q₈	—	4	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	#431	1	2	yes	—
182	C₂×C₄	#431	1	2	yes	—
183	C₂×C₄	C₈×Q₈	1	4	yes	—
184	C₂×C₄	#504	1	4	yes	—
185	C₂×C₄	C₂×#117	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₂×C₄	C₄×#41	1	1	yes	characteristic
188	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
189	C₂×C₄	C₂×#131	1	1	yes	characteristic
190	C₂×C₄	C₂×#131	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	2	2	no	—
193	C₈	—	4	2	no	—
194	C₈	—	4	2	no	—
195	C₈	#446	1	4	yes	—
196	C₈	#529	1	2	yes	—
197	C₈	#533	1	2	yes	—
198	C₂²	C₂×#431	1	1	yes	characteristic
199	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
200	C₂²	C₈×#41	1	1	yes	characteristic
201	C₂²	C₂×#504	1	2	yes	—
202	C₂²	#3407	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	4	no	—
209	C₄	#2164	1	4	yes	—
210	C₄	#2295	1	1	yes	characteristic
211	C₄	#2922	1	1	yes	characteristic
212	C₄	C₂×#446	1	1	yes	characteristic
213	C₄	#2965	1	1	yes	characteristic
214	C₂	#21339	1	1	yes	characteristic
215	C₂	C₂×#2164	1	1	yes	characteristic
216	C₂	#33421	1	1	yes	characteristic
217	C₂	#33476	1	2	yes	—
218	C₂	#33483	1	2	yes	—

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

Maximal subgroups

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

subgroup	index
#20516	2
#13263	2
#13159	2
#13105	2
#12654	2
C₄×C₄⋊C₁₆	2
#12592	2
C₄×#346	2