Subgroups — #512802

Order 512.

363 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33432	C₂	1	1	yes	maximal, characteristic
2	#13646	C₂	1	4	yes	maximal, complemented
3	#13645	C₂	1	4	yes	maximal, complemented
4	#13641	C₂	1	2	yes	maximal, complemented
5	#13554	C₂	1	2	yes	maximal, complemented
6	#13553	C₂	1	2	yes	maximal, complemented
7	C₂×#504	C₂²	1	1	yes	characteristic
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	#2929	C₄	1	1	yes	characteristic
10	C₂×M₆₄	C₂²	1	4	yes	—
11	C₂×M₆₄	C₂²	1	2	yes	complemented
12	C₂²×C₃₂	C₂²	1	2	yes	complemented
13	#2168	C₄	1	4	yes	—
14	D₄×C₁₆	C₂²	1	4	yes	—
15	#2151	C₄	1	1	yes	characteristic
16	#2113	C₄	1	1	yes	characteristic
17	#2112	C₄	1	1	yes	characteristic
18	#2108	C₂²	1	1	yes	characteristic
19	C₄⋊C₃₂	C₂²	1	4	yes	complemented
20	C₂²⋊C₃₂	C₂²	1	4	yes	—
21	C₂²⋊C₃₂	C₂²	1	2	yes	—
22	C₂²⋊C₃₂	C₂²	1	2	yes	—
23	C₂²⋊C₃₂	C₂²	1	4	yes	complemented
24	#1397	C₂²	1	2	yes	complemented
25	C₄×C₃₂	C₂²	1	2	yes	complemented
26	C₂×#131	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	#525	C₈	1	1	yes	characteristic
29	#504	—	4	2	no	—
30	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
31	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
32	C₂×M₃₂	C₂×C₄	1	2	yes	—
33	C₂²×C₁₆	—	4	2	no	—
34	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	C₂²×C₁₆	C₂³	1	2	yes	—
38	C₂²×C₁₆	C₂³	1	2	yes	—
39	#435	C₈	1	4	yes	—
40	C₈×D₄	C₂×C₄	1	4	yes	—
41	#424	C₈	1	1	yes	characteristic
42	#408	C₈	1	1	yes	characteristic
43	#407	C₈	1	1	yes	characteristic
44	#405	C₂×C₄	1	1	yes	characteristic
45	M₆₄	—	4	4	no	—
46	M₆₄	—	4	2	no	—
47	C₂×C₃₂	—	4	2	no	—
48	C₂×C₃₂	C₂³	1	4	yes	—
49	C₂×C₃₂	C₂³	1	4	yes	—
50	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
51	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
52	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
53	#346	C₂×C₄	1	2	yes	—
54	C₄×C₁₆	C₂³	1	2	yes	—
55	C₂×#41	C₂×C₈	1	1	yes	characteristic
56	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
57	#131	—	4	1	no	—
58	#131	—	4	1	no	—
59	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
60	C₂×M₁₆	C₂×C₈	1	2	yes	—
61	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₈	—	4	1	no	—
63	C₂²×C₈	—	4	1	no	—
64	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₈	C₂×C₈	1	2	yes	—
66	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₈	C₂²×C₄	1	2	yes	—
68	C₂²×C₈	C₂²×C₄	1	2	yes	—
69	#123	C₁₆	1	2	yes	—
70	C₂²⋊Q₈	C₁₆	1	2	yes	—
71	#121	C₁₆	1	2	yes	—
72	#120	C₁₆	1	2	yes	—
73	C₄×D₄	C₂×C₈	1	4	yes	—
74	#117	C₂×C₈	1	1	yes	characteristic
75	M₃₂	—	4	1	no	—
76	M₃₂	—	4	2	no	—
77	M₃₂	—	4	1	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	4	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	4	1	no	—
83	C₂×C₁₆	—	4	2	no	—
84	C₂×C₁₆	C₂²×C₄	1	2	yes	—
85	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×C₄	1	2	yes	—
87	C₂×C₁₆	C₂²×C₄	1	2	yes	—
88	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₈	C₂×C₈	1	4	yes	—
90	C₂²⋊C₈	C₂×C₈	1	4	yes	—
91	C₂²⋊C₈	C₂×C₈	1	4	yes	—
92	#97	C₂×C₈	1	2	yes	—
93	C₄×C₈	C₂²×C₄	1	2	yes	—
94	C₃₂	—	2	4	no	—
95	C₃₂	—	2	4	no	—
96	C₂⁴	C₂×C₁₆	1	1	yes	characteristic
97	#41	—	4	1	no	—
98	#41	—	4	1	no	—
99	C₂×Q₈	C₂×C₁₆	1	1	yes	characteristic
100	C₂×D₄	C₂×C₁₆	1	4	yes	—
101	C₂×D₄	C₂×C₁₆	1	2	yes	—
102	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
103	C₂²×C₄	—	4	1	no	—
104	C₂²×C₄	—	4	1	no	—
105	C₂²×C₄	C₂×C₁₆	1	2	yes	—
106	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
107	C₂²×C₄	C₂²×C₈	1	2	yes	—
108	C₂²×C₄	C₂²×C₈	1	2	yes	—
109	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
110	M₁₆	—	4	1	no	—
111	M₁₆	—	4	2	no	—
112	M₁₆	—	4	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	C₂²×C₈	1	2	yes	—
122	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₈	C₂²×C₈	1	2	yes	—
124	C₂×C₈	C₂²×C₈	1	2	yes	—
125	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
126	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
127	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
128	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
129	C₂²⋊C₄	C₂×C₁₆	1	4	yes	—
130	C₂²⋊C₄	C₂×C₁₆	1	4	yes	—
131	C₄²	C₂²×C₈	1	2	yes	—
132	C₁₆	—	2	2	no	—
133	C₁₆	—	2	1	no	—
134	C₁₆	—	2	2	no	—
135	C₁₆	—	2	2	no	—
136	C₁₆	H(5,𝔽₂)	1	2	yes	—
137	C₂³	—	4	1	no	—
138	C₂³	—	4	2	no	—
139	C₂³	C₂²×C₁₆	1	2	yes	—
140	C₂³	C₂²×C₁₆	1	1	yes	characteristic
141	C₂³	C₂²×C₁₆	1	2	yes	—
142	Q₈	—	4	1	no	—
143	D₄	—	4	4	no	—
144	D₄	—	4	1	no	—
145	D₄	—	4	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	C₂²×C₁₆	1	4	yes	—
155	C₂×C₄	C₂²×C₁₆	1	2	yes	—
156	C₂×C₄	C₂²×C₁₆	1	2	yes	—
157	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
158	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	2	no	—
163	C₈	#518	1	1	yes	characteristic
164	C₈	#568	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	4	4	no	—
173	C₂²	C₂³×C₁₆	1	1	yes	characteristic
174	C₄	—	2	4	no	—
175	C₄	—	2	2	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	2	no	—
178	C₄	#2972	1	1	yes	characteristic
179	C₄	#3407	1	1	yes	characteristic
180	C₂	—	4	2	no	—
181	C₂	—	4	2	no	—
182	C₂	—	4	1	no	—
183	C₂	#33486	1	1	yes	characteristic
184	C₂	#33974	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	#525	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13646	C₂	semidirect
3. #13645	C₂	semidirect
4. #13641	C₂	semidirect
5. #13554	C₂	semidirect
6. #13553	C₂	semidirect
11. C₂×M₆₄	C₂²	semidirect
12. C₂²×C₃₂	C₂²	semidirect
19. C₄⋊C₃₂	C₂²	semidirect
23. C₂²⋊C₃₂	C₂²	semidirect
24. #1397	C₂²	semidirect
25. C₄×C₃₂	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#33432	2
#13646	2
#13645	2
#13641	2
#13554	2
#13553	2