Subgroups — #484111

Order 512.

476 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33733	C₂	1	1	yes	maximal, characteristic
2	#18066	C₂	1	2	yes	maximal, complemented
3	#12081	C₂	1	2	yes	maximal, complemented
4	#12069	C₂	1	4	yes	maximal, complemented
5	C₈×D₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×#1371	C₂	1	1	yes	maximal, characteristic, complemented
7	#11645	C₂	1	2	yes	maximal, complemented
8	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#3169	C₂²	1	2	yes	—
10	#2935	—	2	2	no	—
11	#2252	C₄	1	1	yes	characteristic
12	#2246	C₄	1	1	yes	characteristic
13	#2240	C₄	1	2	yes	—
14	#2215	C₄	1	2	yes	—
15	#2215	C₄	1	2	yes	—
16	C₄×D₁₆	C₂²	1	2	yes	—
17	C₄×D₁₆	C₂²	1	2	yes	—
18	#2162	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	#1589	C₂²	1	4	yes	—
21	#1581	—	2	4	no	—
22	C₈×D₈	C₂²	1	4	yes	—
23	C₂×#335	C₂²	1	2	yes	—
24	#1480	—	2	2	no	—
25	C₈×M₁₆	C₂²	1	1	yes	characteristic
26	#1371	C₂²	1	1	yes	characteristic
27	#1371	C₂²	1	2	yes	—
28	#1371	C₂²	1	1	yes	characteristic
29	#1331	C₂²	1	4	yes	—
30	#1331	C₂²	1	4	yes	—
31	#1312	C₂²	1	2	yes	—
32	C₈×C₁₆	C₂²	1	2	yes	—
33	C₂×#131	—	4	2	no	—
34	#532	—	2	2	no	—
35	C₂×D₁₆	C₂×C₄	1	2	yes	—
36	C₂×D₁₆	C₂×C₄	1	2	yes	—
37	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
39	#500	C₂×C₄	1	2	yes	—
40	#493	C₂×C₄	1	2	yes	—
41	#486	—	2	2	no	—
42	#466	—	2	2	no	—
43	#466	C₂×C₄	1	4	yes	—
44	C₄×D₈	—	2	2	no	—
45	C₄×D₈	C₂³	1	4	yes	—
46	#436	—	2	4	no	—
47	#435	—	4	4	no	—
48	C₈×D₄	—	2	4	no	—
49	#431	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₈	—	2	2	no	—
51	#408	—	4	2	no	—
52	C₄×M₁₆	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#366	C₂×C₄	1	2	yes	—
55	#366	C₂×C₄	1	2	yes	—
56	C₄⋊C₁₆	—	2	1	no	—
57	#357	C₂×C₄	1	4	yes	—
58	#357	C₂×C₄	1	4	yes	—
59	C₂²⋊C₁₆	—	2	1	no	—
60	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
61	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₆	C₂³	1	2	yes	—
63	C₄×C₁₆	C₂³	1	1	yes	characteristic
64	C₄×C₁₆	C₂³	1	1	yes	characteristic
65	#335	—	2	2	no	—
66	#335	C₂³	1	2	yes	—
67	#335	C₂³	1	2	yes	—
68	D₄⋊C₈	—	2	4	no	—
69	D₄⋊C₈	—	2	4	no	—
70	#322	D₄	1	2	yes	—
71	C₈²	C₂³	1	2	yes	—
72	C₂×#41	—	4	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₂×D₈	C₂²×C₄	1	4	yes	—
75	#131	—	8	8	no	—
76	C₂×M₁₆	—	4	4	no	—
77	C₂×M₁₆	D₈	1	1	yes	characteristic
78	C₂×M₁₆	D₈	1	1	yes	characteristic
79	C₂²×C₈	—	4	2	no	—
80	C₂²×C₈	—	4	4	no	—
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₈	#41	1	1	yes	characteristic
83	C₄⋊Q₈	—	2	2	no	—
84	#127	—	2	2	no	—
85	#124	—	4	2	no	—
86	#121	—	2	4	no	—
87	#121	—	4	2	no	—
88	C₄×D₄	—	2	4	no	—
89	C₄×D₄	—	4	2	no	—
90	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
91	D₁₆	—	2	4	no	—
92	D₁₆	—	4	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	4	1	no	—
96	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
100	#107	—	2	2	no	—
101	#107	C₂²×C₄	1	4	yes	—
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	2	no	—
106	#102	—	2	4	no	—
107	#102	—	2	4	no	—
108	C₂²⋊C₈	—	2	1	no	—
109	C₂²⋊C₈	—	4	4	no	—
110	C₂²⋊C₈	—	4	4	no	—
111	#97	D₈	1	2	yes	—
112	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₄×C₈	C₂×D₄	1	2	yes	—
114	C₄×C₈	C₂×D₄	1	1	yes	characteristic
115	C₄×C₈	C₂×D₄	1	2	yes	—
116	C₄×C₈	#41	1	1	yes	characteristic
117	C₄×C₈	C₂⁴	1	1	yes	characteristic
118	#41	—	8	4	no	—
119	#41	—	8	4	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂×D₄	—	2	4	no	—
122	C₂×D₄	—	4	4	no	—
123	C₂×D₄	—	4	2	no	—
124	C₂×D₄	—	4	2	no	—
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
128	C₂²×C₄	#135	1	1	yes	characteristic
129	D₈	—	2	4	no	—
130	D₈	—	2	4	no	—
131	D₈	—	8	2	no	—
132	M₁₆	—	2	2	no	—
133	M₁₆	—	8	8	no	—
134	M₁₆	D₁₆	1	4	yes	complemented
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	8	4	no	—
142	C₂×C₈	—	8	8	no	—
143	C₂×C₈	C₄×D₄	1	1	yes	characteristic
144	C₂×C₈	C₄×D₄	1	1	yes	characteristic
145	C₂×C₈	C₂×D₈	1	2	yes	—
146	C₂×C₈	C₂×D₈	1	2	yes	—
147	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₈	C₂×#41	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₂²⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	C₄×D₄	1	1	yes	characteristic
154	C₄²	C₂×D₈	1	1	yes	characteristic
155	C₄²	#135	1	1	yes	characteristic
156	C₄²	C₂²×D₄	1	1	yes	characteristic
157	C₁₆	—	2	1	no	—
158	C₁₆	—	2	1	no	—
159	C₁₆	—	2	1	no	—
160	C₁₆	—	4	2	no	—
161	C₁₆	#131	1	2	yes	complemented
162	C₂³	—	4	4	no	—
163	C₂³	C₄×D₈	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	D₄	—	4	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	8	8	no	—
168	D₄	—	8	4	no	—
169	D₄	—	16	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	8	4	no	—
175	C₂×C₄	—	8	4	no	—
176	C₂×C₄	C₄×D₈	1	1	yes	characteristic
177	C₂×C₄	C₄×D₈	1	1	yes	characteristic
178	C₂×C₄	C₄×D₈	1	1	yes	characteristic
179	C₂×C₄	C₂×D₁₆	1	2	yes	—
180	C₂×C₄	#508	1	2	yes	—
181	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
183	C₂×C₄	C₂×#135	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	4	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	8	2	no	—
191	C₈	C₂×D₁₆	1	4	yes	—
192	C₈	C₂×#131	1	1	yes	characteristic
193	C₈	#568	1	1	yes	characteristic
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	4	no	—
199	C₂²	C₄×D₁₆	1	2	yes	—
200	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	4	no	—
205	C₄	C₄×D₁₆	1	2	yes	—
206	C₄	#2935	1	1	yes	characteristic
207	C₄	#2957	1	1	yes	characteristic
208	C₄	C₂²×D₁₆	1	1	yes	characteristic
209	C₄	C₂×#508	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	16	4	no	—
212	C₂	#18066	1	1	yes	characteristic
213	C₂	C₂×C₄×D₁₆	1	1	yes	characteristic
214	C₂	#33465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33733	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #18066	C₂	semidirect
3. #12081	C₂	semidirect
4. #12069	C₂	semidirect
5. C₈×D₁₆	C₂	semidirect
6. C₂×#1371	C₂	semidirect
7. #11645	C₂	semidirect
8. C₁₆×M₁₆	C₂	semidirect
134. M₁₆	D₁₆	semidirect
161. C₁₆	#131	semidirect

Maximal subgroups

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

subgroup	index
#33733	2
#18066	2
#12081	2
#12069	2
C₈×D₁₆	2
C₂×#1371	2
#11645	2
C₁₆×M₁₆	2