Subgroups — #484113

Order 512.

408 conjugacy classes of subgroups.

The subgroup lattice has 203 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33734	C₂	1	1	yes	maximal, characteristic
2	#18068	C₂	1	2	yes	maximal
3	#12083	C₂	1	2	yes	maximal, complemented
4	#12072	C₂	1	4	yes	maximal, complemented
5	C₈×Q₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×#1371	C₂	1	1	yes	maximal, characteristic
7	#11646	C₂	1	2	yes	maximal
8	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic
9	#3170	C₂²	1	2	yes	—
10	#2962	—	2	2	no	—
11	#2252	C₄	1	1	yes	characteristic
12	#2247	C₄	1	1	yes	characteristic
13	#2241	C₄	1	2	yes	—
14	#2216	C₄	1	2	yes	—
15	#2216	C₄	1	2	yes	—
16	C₄×Q₃₂	C₂²	1	2	yes	—
17	C₄×Q₃₂	C₂²	1	2	yes	—
18	#2162	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	#1591	C₂²	1	4	yes	—
21	#1584	—	2	4	no	—
22	C₈×Q₁₆	C₂²	1	4	yes	—
23	C₂×#335	C₂²	1	2	yes	—
24	#1481	—	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	#1332	C₂²	1	4	yes	—
30	#1332	C₂²	1	4	yes	—
31	#1312	C₂²	1	2	yes	—
32	C₈×C₁₆	C₂²	1	2	yes	—
33	#533	—	2	2	no	—
34	C₂×Q₃₂	C₂×C₄	1	2	yes	—
35	C₂×Q₃₂	C₂×C₄	1	2	yes	—
36	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
38	#500	C₂×C₄	1	2	yes	—
39	#494	C₂×C₄	1	2	yes	—
40	#487	—	2	2	no	—
41	#467	—	2	2	no	—
42	#467	C₂×C₄	1	4	yes	—
43	#446	—	2	4	no	—
44	#446	—	4	4	no	—
45	C₈×Q₈	—	2	4	no	—
46	C₄×Q₁₆	—	2	2	no	—
47	C₄×Q₁₆	C₂³	1	4	yes	—
48	#431	D₄	1	1	yes	characteristic
49	#424	—	4	2	no	—
50	C₂×C₄⋊C₈	—	2	2	no	—
51	#405	—	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	#358	C₂×C₄	1	4	yes	—
58	#358	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	Q₈⋊C₈	—	2	4	no	—
69	Q₈⋊C₈	—	2	4	no	—
70	#322	D₄	1	2	yes	—
71	C₈²	C₂³	1	2	yes	—
72	C₂×Q₁₆	—	2	2	no	—
73	C₂×Q₁₆	C₂²×C₄	1	4	yes	—
74	C₂×M₁₆	D₈	1	1	yes	characteristic
75	C₂×M₁₆	D₈	1	1	yes	characteristic
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₈	#41	1	1	yes	characteristic
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	#125	—	4	2	no	—
82	C₂²⋊Q₈	—	2	4	no	—
83	C₂²⋊Q₈	—	4	2	no	—
84	C₄×Q₈	—	2	4	no	—
85	C₄×Q₈	—	4	2	no	—
86	#117	—	4	2	no	—
87	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
88	Q₃₂	—	2	4	no	—
89	Q₃₂	—	4	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	4	1	no	—
93	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
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	#107	—	2	2	no	—
98	#107	C₂²×C₄	1	4	yes	—
99	C₄⋊C₈	—	2	1	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	4	4	no	—
103	C₄⋊C₈	—	4	4	no	—
104	C₄⋊C₈	—	4	2	no	—
105	#103	—	2	4	no	—
106	#103	—	2	4	no	—
107	C₂²⋊C₈	—	2	1	no	—
108	#97	—	4	4	no	—
109	#97	D₈	1	2	yes	—
110	C₄×C₈	—	4	4	no	—
111	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₄×C₈	C₂×D₄	1	2	yes	—
113	C₄×C₈	C₂×D₄	1	1	yes	characteristic
114	C₄×C₈	C₂×D₄	1	2	yes	—
115	C₄×C₈	#41	1	1	yes	characteristic
116	C₄×C₈	C₂⁴	1	1	yes	characteristic
117	C₂×Q₈	—	2	4	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
120	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
121	C₂²×C₄	#135	1	1	yes	characteristic
122	Q₁₆	—	2	4	no	—
123	Q₁₆	—	2	4	no	—
124	Q₁₆	—	8	2	no	—
125	M₁₆	—	2	2	no	—
126	M₁₆	Q₃₂	1	4	yes	complemented
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	8	4	no	—
134	C₂×C₈	C₄×D₄	1	1	yes	characteristic
135	C₂×C₈	C₄×D₄	1	1	yes	characteristic
136	C₂×C₈	C₂×D₈	1	2	yes	—
137	C₂×C₈	C₂×D₈	1	2	yes	—
138	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₈	C₂×#41	1	1	yes	characteristic
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₂²⋊C₄	—	4	4	no	—
147	C₄²	—	4	4	no	—
148	C₄²	C₄×D₄	1	1	yes	characteristic
149	C₄²	C₂×D₈	1	1	yes	characteristic
150	C₄²	#135	1	1	yes	characteristic
151	C₄²	C₂²×D₄	1	1	yes	characteristic
152	C₁₆	—	2	1	no	—
153	C₁₆	—	2	1	no	—
154	C₁₆	—	2	1	no	—
155	C₁₆	—	4	2	no	—
156	C₁₆	#131	1	2	yes	—
157	C₂³	C₄×D₈	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	4	4	no	—
160	Q₈	—	16	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×D₈	1	1	yes	characteristic
167	C₂×C₄	C₄×D₈	1	1	yes	characteristic
168	C₂×C₄	C₄×D₈	1	1	yes	characteristic
169	C₂×C₄	C₂×Q₃₂	1	2	yes	—
170	C₂×C₄	#508	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
173	C₂×C₄	C₂×#135	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	2	no	—
179	C₈	—	8	2	no	—
180	C₈	—	8	2	no	—
181	C₈	C₂×Q₃₂	1	4	yes	—
182	C₈	C₂×#131	1	1	yes	characteristic
183	C₈	#568	1	1	yes	characteristic
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	C₄×Q₃₂	1	2	yes	—
187	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	8	4	no	—
192	C₄	—	8	4	no	—
193	C₄	C₄×Q₃₂	1	2	yes	—
194	C₄	#2935	1	1	yes	characteristic
195	C₄	#2957	1	1	yes	characteristic
196	C₄	C₂²×Q₃₂	1	1	yes	characteristic
197	C₄	C₂×#508	1	1	yes	characteristic
198	C₂	—	2	2	no	—
199	C₂	#18066	1	1	yes	characteristic
200	C₂	C₂×C₄×Q₃₂	1	1	yes	characteristic
201	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	C₂³	C₄×D₈
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
3. #12083	C₂	semidirect
4. #12072	C₂	semidirect
5. C₈×Q₃₂	C₂	semidirect
126. M₁₆	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#33734	2
#18068	2
#12083	2
#12072	2
C₈×Q₃₂	2
C₂×#1371	2
#11646	2
C₁₆×M₁₆	2