Subgroups — #496164

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33728	C₂	1	1	yes	maximal, characteristic
2	#21733	C₂	1	2	yes	maximal
3	#13285	C₂	1	2	yes	maximal
4	#13170	C₂	1	2	yes	maximal
5	#13169	C₂	1	2	yes	maximal
6	#12910	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	#2255	C₄	1	8	yes	complemented
13	C₂×#368	C₂²	1	2	yes	—
14	C₂×#368	C₂²	1	2	yes	—
15	#2158	C₂²	1	2	yes	—
16	#2158	C₂²	1	2	yes	—
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×#346	C₂²	1	1	yes	characteristic
19	C₂×#346	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1995	—	2	2	no	—
24	#1946	—	2	2	no	—
25	#1945	C₂²	1	4	yes	—
26	#1848	C₂²	1	2	yes	—
27	#1773	—	2	2	no	—
28	#1772	C₂²	1	4	yes	—
29	#1726	C₂²	1	1	yes	characteristic
30	#1382	C₂²	1	4	yes	—
31	#1382	C₂²	1	4	yes	—
32	#533	—	2	2	no	—
33	C₂²×C₁₆	C₂³	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	1	1	yes	characteristic
35	C₂²×C₁₆	C₂³	1	1	yes	characteristic
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	#500	C₂×C₄	1	4	yes	—
38	#499	—	2	2	no	—
39	#499	C₂×C₄	1	4	yes	—
40	#498	—	2	2	no	—
41	#446	—	4	8	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	C₂³	1	4	yes	—
44	#427	—	2	2	no	—
45	#427	C₂³	1	4	yes	—
46	#424	—	2	4	no	—
47	#423	—	2	2	no	—
48	#405	—	2	4	no	—
49	C₂×#97	D₄	1	1	yes	characteristic
50	C₂×#97	D₄	1	1	yes	characteristic
51	C₂×#97	D₄	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	#376	D₄	1	1	yes	characteristic
56	#368	—	2	4	no	—
57	#368	—	2	4	no	—
58	#367	C₂×C₄	1	4	yes	—
59	#367	C₂×C₄	1	4	yes	—
60	#366	C₂×C₄	1	4	yes	—
61	#366	C₂×C₄	1	4	yes	—
62	#346	—	2	2	no	—
63	#346	C₂×C₄	1	4	yes	—
64	#346	Q₈	1	4	yes	complemented
65	C₄×C₁₆	—	2	2	no	—
66	C₄×C₁₆	—	2	2	no	—
67	C₄×C₁₆	Q₈	1	4	yes	complemented
68	#339	—	2	4	no	—
69	#339	—	2	4	no	—
70	C₂×M₁₆	—	2	4	no	—
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂⁴	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	4	no	—
76	#125	—	2	4	no	—
77	C₂²⋊Q₈	—	4	4	no	—
78	C₄×Q₈	—	4	4	no	—
79	#117	—	2	4	no	—
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	C₂²×C₄	1	2	yes	—
88	C₂×C₁₆	C₂²×C₄	1	2	yes	—
89	C₂×C₁₆	C₂×Q₈	1	2	yes	—
90	C₂×C₁₆	C₂×Q₈	1	2	yes	—
91	C₂×C₁₆	C₂×Q₈	1	2	yes	—
92	C₂×C₁₆	C₂×Q₈	1	2	yes	—
93	C₂×C₁₆	#41	1	2	yes	—
94	C₂×C₁₆	#41	1	2	yes	—
95	#108	—	2	4	no	—
96	#108	—	2	4	no	—
97	#108	—	4	4	no	—
98	#107	—	2	4	no	—
99	#107	C₂²×C₄	1	8	yes	—
100	#106	—	2	4	no	—
101	#106	—	2	4	no	—
102	C₄⋊C₈	—	4	8	no	—
103	C₄⋊C₈	—	4	4	no	—
104	#97	—	2	2	no	—
105	#97	—	4	4	no	—
106	#97	D₈	1	4	yes	—
107	#97	Q₁₆	1	4	yes	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	4	4	no	—
111	C₄×C₈	C₂²×C₄	1	2	yes	—
112	C₄×C₈	C₂×Q₈	1	2	yes	—
113	C₄×C₈	C₂×Q₈	1	2	yes	—
114	C₄×C₈	#41	1	2	yes	—
115	#95	—	2	2	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	#135	1	2	yes	—
120	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
121	M₁₆	—	4	4	no	—
122	M₁₆	—	4	4	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	4	no	—
126	C₂×C₈	C₄×D₄	1	2	yes	—
127	C₂×C₈	C₄⋊Q₈	1	2	yes	—
128	C₂×C₈	C₂×D₈	1	2	yes	—
129	C₂×C₈	C₂×D₈	1	2	yes	—
130	C₂×C₈	C₂×Q₁₆	1	2	yes	—
131	C₂×C₈	C₂×Q₁₆	1	2	yes	—
132	C₂×C₈	#135	1	2	yes	—
133	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₈	C₂×#41	1	1	yes	characteristic
137	C₂×C₈	C₂×#41	1	1	yes	characteristic
138	C₂×C₈	C₂×#41	1	1	yes	characteristic
139	C₄⋊C₄	—	2	8	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₂²⋊C₄	—	4	4	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	4	4	no	—
146	C₄²	C₄×D₄	1	2	yes	—
147	C₄²	C₄⋊Q₈	1	2	yes	—
148	C₄²	C₂×D₈	1	2	yes	—
149	C₄²	C₂×Q₁₆	1	2	yes	—
150	C₁₆	—	2	2	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	C₄×Q₈	1	4	yes	—
153	C₁₆	C₄×Q₈	1	4	yes	—
154	C₂³	C₂×#135	1	1	yes	characteristic
155	Q₈	—	16	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	C₄×D₈	1	4	yes	—
163	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₄	#532	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
167	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
168	C₂×C₄	C₂×#135	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	8	4	no	—
171	C₈	C₄×Q₁₆	1	4	yes	—
172	C₈	#500	1	4	yes	—
173	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
174	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₈	#533	1	1	yes	characteristic
176	C₈	#533	1	1	yes	characteristic
177	C₂²	—	2	2	no	—
178	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
179	C₂²	#3160	1	1	yes	characteristic
180	C₂²	#3170	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	8	8	no	—
185	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
186	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
187	C₄	C₂×#500	1	1	yes	characteristic
188	C₄	#3169	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	C₄×#500	1	1	yes	characteristic
192	C₂	#33464	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. #2255	C₄	semidirect
64. #346	Q₈	semidirect
67. 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
#33728	2
#21733	2
#13285	2
#13170	2
#13169	2
#12910	2
#12671	2
#12670	2
#12539	2