Subgroups — #496160

Order 512.

496 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33726	C₂	1	1	yes	maximal, characteristic
2	#21732	C₂	1	2	yes	maximal
3	#13285	C₂	1	2	yes	maximal
4	#13169	C₂	1	4	yes	maximal
5	#12909	C₂	1	1	yes	maximal, characteristic
6	#12670	C₂	1	4	yes	maximal
7	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	2	yes	—
9	#2960	—	2	2	no	—
10	#2254	C₄	1	2	yes	complemented
11	#2253	C₄	1	4	yes	complemented
12	#2252	C₄	1	2	yes	complemented
13	C₂×#368	C₂²	1	4	yes	—
14	#2158	C₂²	1	4	yes	—
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	#1994	—	2	2	no	—
20	#1945	—	2	2	no	—
21	#1945	C₂²	1	4	yes	—
22	#1848	C₂²	1	2	yes	—
23	#1772	—	2	2	no	—
24	#1772	C₂²	1	4	yes	—
25	C₄²×C₈	C₂²	1	1	yes	characteristic
26	#1382	C₂²	1	8	yes	—
27	#533	—	2	2	no	—
28	C₂²×C₁₆	C₂³	1	2	yes	—
29	C₂²×C₁₆	C₂³	1	2	yes	—
30	#500	C₂×C₄	1	4	yes	—
31	#499	—	2	2	no	—
32	#499	C₂×C₄	1	4	yes	—
33	#498	—	2	2	no	—
34	#446	—	4	4	no	—
35	C₈×Q₈	—	4	4	no	—
36	C₂×#108	—	2	2	no	—
37	C₂×#108	C₂³	1	4	yes	—
38	#427	—	2	2	no	—
39	#427	C₂³	1	4	yes	—
40	#424	—	2	4	no	—
41	#423	—	2	2	no	—
42	#405	—	2	4	no	—
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₄³	D₄	1	1	yes	characteristic
50	#368	—	2	8	no	—
51	#367	C₂×C₄	1	8	yes	—
52	#366	C₂×C₄	1	8	yes	—
53	C₄×C₁₆	—	2	4	no	—
54	C₄×C₁₆	—	2	1	no	—
55	C₄×C₁₆	—	2	1	no	—
56	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
57	C₄×C₁₆	C₂×C₄	1	2	yes	—
58	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
59	C₄×C₁₆	Q₈	1	8	yes	complemented
60	#339	—	2	4	no	—
61	#339	—	2	4	no	—
62	C₂×M₁₆	—	2	4	no	—
63	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₈	C₂⁴	1	1	yes	characteristic
67	C₄⋊Q₈	—	2	4	no	—
68	#125	—	2	4	no	—
69	C₂²⋊Q₈	—	4	4	no	—
70	C₄×Q₈	—	4	4	no	—
71	#117	—	2	4	no	—
72	C₂×C₄²	—	2	2	no	—
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	C₂²×C₄	1	2	yes	—
79	C₂×C₁₆	C₂²×C₄	1	2	yes	—
80	C₂×C₁₆	C₂×Q₈	1	4	yes	—
81	C₂×C₁₆	C₂×Q₈	1	4	yes	—
82	C₂×C₁₆	#41	1	4	yes	—
83	#108	—	2	8	no	—
84	#108	—	4	4	no	—
85	#107	—	2	4	no	—
86	#107	C₂²×C₄	1	8	yes	—
87	#106	—	2	4	no	—
88	#106	—	2	4	no	—
89	C₄⋊C₈	—	4	8	no	—
90	C₄⋊C₈	—	4	4	no	—
91	#97	—	4	4	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	—	4	4	no	—
95	C₄×C₈	D₈	1	2	yes	—
96	C₄×C₈	D₈	1	2	yes	—
97	C₄×C₈	Q₁₆	1	4	yes	—
98	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₄×C₈	C₂×Q₈	1	4	yes	—
101	C₄×C₈	#41	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	C₂×Q₈	—	4	4	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	#135	1	2	yes	—
107	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
108	M₁₆	—	4	4	no	—
109	M₁₆	—	4	4	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	4	4	no	—
113	C₂×C₈	C₄×D₄	1	1	yes	characteristic
114	C₂×C₈	C₄×D₄	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Q₈	1	2	yes	—
116	C₂×C₈	C₂×D₈	1	2	yes	—
117	C₂×C₈	C₂×D₈	1	1	yes	characteristic
118	C₂×C₈	C₂×D₈	1	1	yes	characteristic
119	C₂×C₈	C₂×Q₁₆	1	2	yes	—
120	C₂×C₈	C₂×Q₁₆	1	2	yes	—
121	C₂×C₈	#135	1	2	yes	—
122	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₈	C₂²×Q₈	1	2	yes	—
124	C₂×C₈	C₂×#41	1	2	yes	—
125	C₂×C₈	C₂×#41	1	1	yes	characteristic
126	C₄⋊C₄	—	2	8	no	—
127	C₄⋊C₄	—	4	8	no	—
128	C₄⋊C₄	—	4	4	no	—
129	C₂²⋊C₄	—	4	4	no	—
130	C₄²	—	2	8	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	4	4	no	—
134	C₄²	C₄×D₄	1	1	yes	characteristic
135	C₄²	C₄×D₄	1	1	yes	characteristic
136	C₄²	C₄⋊Q₈	1	2	yes	—
137	C₄²	C₂×D₈	1	2	yes	—
138	C₄²	C₂×Q₁₆	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₁₆	C₄×Q₈	1	4	yes	—
141	C₁₆	C₄×Q₈	1	4	yes	—
142	C₂³	C₂×#135	1	1	yes	characteristic
143	Q₈	—	8	8	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	8	no	—
150	C₂×C₄	C₄×D₈	1	2	yes	—
151	C₂×C₄	C₄×D₈	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₂×C₄	#532	1	1	yes	characteristic
155	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
157	C₂×C₄	C₂×#135	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	8	4	no	—
160	C₈	C₄×Q₁₆	1	2	yes	—
161	C₈	C₄×Q₁₆	1	2	yes	—
162	C₈	#500	1	2	yes	—
163	C₈	#500	1	2	yes	—
164	C₈	C₂×C₄×Q₈	1	2	yes	—
165	C₈	#533	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
168	C₂²	#3160	1	1	yes	characteristic
169	C₂²	#3170	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	8	8	no	—
174	C₄	#2178	1	4	yes	—
175	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
176	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
177	C₄	C₂×#500	1	1	yes	characteristic
178	C₄	#3169	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×#500	1	1	yes	characteristic
182	C₂	C₂×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×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×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×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
10. #2254	C₄	semidirect
11. #2253	C₄	semidirect
12. #2252	C₄	semidirect
59. 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
#33726	2
#21732	2
#13285	2
#13169	2
#12909	2
#12670	2
C₄²×C₁₆	2