Subgroups — #483336

Order 512.

540 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1564	C₂	1	2	yes	maximal
3	#12009	C₂	1	8	yes	maximal, complemented
4	#12008	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	C₄	1	1	yes	characteristic
6	C₂²×M₃₂	C₄	1	1	yes	characteristic
7	C₂×#431	—	2	2	no	—
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄×M₃₂	C₄	1	4	yes	—
10	C₄×M₃₂	C₂²	1	4	yes	—
11	C₂×#346	C₄	1	2	yes	—
12	C₂×C₄×C₁₆	C₂²	1	2	yes	—
13	#1564	—	2	2	no	—
14	#1564	—	2	2	no	—
15	#1564	C₂²	1	4	yes	—
16	#1564	C₂²	1	4	yes	—
17	C₂×#335	C₂²	1	4	yes	—
18	C₂×#334	—	2	2	no	—
19	#1372	C₂²	1	8	yes	complemented
20	#1371	C₂²	1	8	yes	complemented
21	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₈	Q₈	1	1	yes	characteristic
23	C₂²×C₄²	D₄	1	1	yes	characteristic
24	C₂×M₃₂	—	2	4	no	—
25	C₂×M₃₂	C₈	1	8	yes	complemented
26	C₂×M₃₂	C₂×C₄	1	4	yes	—
27	C₂×M₃₂	C₂×C₄	1	4	yes	—
28	C₂²×C₁₆	C₂×C₄	1	2	yes	—
29	C₂²×C₁₆	C₂×C₄	1	2	yes	—
30	#431	—	2	4	no	—
31	#431	—	2	4	no	—
32	#431	—	4	2	no	—
33	#431	—	4	2	no	—
34	C₂×C₄⋊C₈	—	2	4	no	—
35	C₂×C₂²⋊C₈	—	4	2	no	—
36	C₂×C₄×C₈	—	4	2	no	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₈	D₄	1	2	yes	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	Q₈	1	2	yes	—
42	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	2	yes	—
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	#346	C₂×C₄	1	8	yes	—
46	C₄×C₁₆	C₂×C₄	1	4	yes	—
47	C₄×C₁₆	C₂³	1	4	yes	—
48	#335	—	2	4	no	—
49	#335	C₂³	1	8	yes	—
50	#334	—	2	4	no	—
51	#334	—	2	4	no	—
52	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₄	D₈	1	1	yes	characteristic
54	C₂³×C₄	Q₁₆	1	1	yes	characteristic
55	C₂²×C₈	—	2	4	no	—
56	C₂²×C₈	—	4	4	no	—
57	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₈	C₄⋊C₄	1	2	yes	—
59	C₂²×C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂×C₈	1	4	yes	—
62	C₂²×C₈	M₁₆	1	4	yes	—
63	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²×C₄	1	2	yes	—
65	C₂²×C₈	C₂×Q₈	1	2	yes	—
66	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₄²	C₄⋊C₄	1	2	yes	—
68	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄²	D₈	1	1	yes	characteristic
70	C₂×C₄²	D₈	1	2	yes	—
71	C₂×C₄²	Q₁₆	1	1	yes	characteristic
72	C₂×C₄²	Q₁₆	1	2	yes	—
73	C₂×C₄²	C₂×D₄	1	2	yes	—
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	M₃₂	—	2	8	no	—
76	M₃₂	C₂×C₈	1	16	yes	complemented
77	C₂×C₁₆	—	2	4	no	—
78	C₂×C₁₆	C₂×C₈	1	8	yes	complemented
79	C₂×C₁₆	C₂²×C₄	1	4	yes	—
80	C₂×C₁₆	C₂²×C₄	1	4	yes	—
81	C₄⋊C₈	—	2	8	no	—
82	C₄⋊C₈	—	4	4	no	—
83	C₂²⋊C₈	—	4	4	no	—
84	C₂²⋊C₈	—	4	4	no	—
85	C₄×C₈	—	4	4	no	—
86	C₄×C₈	—	4	4	no	—
87	C₄×C₈	C₄⋊C₄	1	2	yes	—
88	C₄×C₈	C₄⋊C₄	1	2	yes	—
89	C₄×C₈	C₄⋊C₄	1	2	yes	—
90	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₄×C₈	C₂²×C₄	1	2	yes	—
92	C₄×C₈	C₂×D₄	1	1	yes	characteristic
93	C₄×C₈	C₂×D₄	1	2	yes	—
94	C₄×C₈	C₂×Q₈	1	2	yes	—
95	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₄×C₈	C₂⁴	1	1	yes	characteristic
97	C₂⁴	#107	1	1	yes	characteristic
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	C₄⋊C₈	1	4	yes	—
101	C₂²×C₄	#107	1	2	yes	—
102	C₂²×C₄	#107	1	1	yes	characteristic
103	C₂²×C₄	#107	1	2	yes	—
104	C₂²×C₄	#107	1	1	yes	characteristic
105	C₂²×C₄	#107	1	1	yes	characteristic
106	C₂²×C₄	#107	1	2	yes	—
107	C₂²×C₄	#108	1	4	yes	—
108	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
109	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₈	1	2	yes	—
112	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
113	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	8	no	—
119	C₂×C₈	—	8	8	no	—
120	C₂×C₈	C₄⋊C₈	1	8	yes	—
121	C₂×C₈	C₄⋊C₈	1	4	yes	—
122	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₈	C₂²×C₈	1	2	yes	—
127	C₂×C₈	C₂²×C₈	1	4	yes	—
128	C₂×C₈	C₂×M₁₆	1	2	yes	—
129	C₂×C₈	C₂×M₁₆	1	4	yes	—
130	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
132	C₄²	#107	1	2	yes	—
133	C₄²	#107	1	2	yes	—
134	C₄²	#107	1	2	yes	—
135	C₄²	C₂×C₄⋊C₄	1	2	yes	—
136	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₄²	C₂×D₈	1	2	yes	—
138	C₄²	C₂×D₈	1	1	yes	characteristic
139	C₄²	C₂×Q₁₆	1	2	yes	—
140	C₄²	C₂×Q₁₆	1	1	yes	characteristic
141	C₄²	C₂²×D₄	1	1	yes	characteristic
142	C₁₆	—	2	4	no	—
143	C₁₆	C₂²×C₈	1	8	yes	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	1	no	—
146	C₂³	#335	1	4	yes	—
147	C₂³	C₂×#107	1	2	yes	—
148	C₂³	C₂×#107	1	1	yes	characteristic
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	#335	1	4	yes	—
158	C₂×C₄	#335	1	8	yes	—
159	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
160	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
161	C₂×C₄	C₂×#107	1	2	yes	—
162	C₂×C₄	C₂×#107	1	1	yes	characteristic
163	C₂×C₄	C₂×#107	1	2	yes	—
164	C₂×C₄	C₂×#107	1	2	yes	—
165	C₂×C₄	C₂×#107	1	1	yes	characteristic
166	C₂×C₄	C₂×#107	1	1	yes	characteristic
167	C₂×C₄	C₂×#108	1	4	yes	—
168	C₂×C₄	C₂×#108	1	2	yes	—
169	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
171	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
172	C₈	—	2	1	no	—
173	C₈	—	2	2	no	—
174	C₈	—	2	1	no	—
175	C₈	—	8	8	no	—
176	C₈	C₂×C₄⋊C₈	1	2	yes	—
177	C₈	C₂×C₄⋊C₈	1	4	yes	—
178	C₈	C₂³×C₈	1	1	yes	characteristic
179	C₈	C₂²×M₁₆	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	4	4	no	—
185	C₂²	C₂×#335	1	2	yes	—
186	C₂²	C₂×#335	1	4	yes	—
187	C₂²	C₂²×#107	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	2	no	—
192	C₄	C₂×#335	1	4	yes	—
193	C₄	C₂×#335	1	2	yes	—
194	C₄	#2161	1	2	yes	—
195	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
196	C₄	C₂²×#108	1	1	yes	characteristic
197	C₄	#3414	1	2	yes	—
198	C₂	—	2	4	no	—
199	C₂	—	2	2	no	—
200	C₂	C₂²×#335	1	1	yes	characteristic
201	C₂	#33409	1	1	yes	characteristic
202	C₂	C₂×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ 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. #12009	C₂	semidirect
4. #12008	C₂	semidirect
19. #1372	C₂²	semidirect
20. #1371	C₂²	semidirect
25. C₂×M₃₂	C₈	semidirect
76. M₃₂	C₂×C₈	semidirect
78. C₂×C₁₆	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
C₂×#1564	2
#12009	2
#12008	2