Subgroups — #498636

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 165 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#13263	C₂	1	4	yes	maximal
3	#13159	C₂	1	4	yes	maximal
4	C₁₆×C₄⋊C₄	C₂	1	4	yes	maximal
5	C₄×C₄⋊C₁₆	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₂²	1	2	yes	—
7	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	4	yes	—
12	C₂×C₄×C₁₆	C₂²	1	4	yes	—
13	#1984	C₄	1	2	yes	—
14	#1939	C₄	1	4	yes	—
15	#1842	C₄	1	1	yes	characteristic
16	C₈×C₄⋊C₄	C₂²	1	4	yes	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1374	C₂²	1	8	yes	—
19	#1374	C₂²	1	4	yes	—
20	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
21	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
22	C₂²×C₁₆	C₂³	1	4	yes	—
23	C₂²×C₁₆	C₂³	1	4	yes	—
24	C₈×Q₈	—	4	2	no	—
25	C₈×Q₈	—	4	2	no	—
26	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂³	1	4	yes	—
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂³	1	2	yes	—
34	#391	C₈	1	2	yes	—
35	#389	C₈	1	4	yes	—
36	#384	C₈	1	1	yes	characteristic
37	C₄×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₄³	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₁₆	—	2	4	no	—
40	C₄⋊C₁₆	—	4	4	no	—
41	C₄⋊C₁₆	Q₈	1	8	yes	complemented
42	C₄×C₁₆	—	2	8	no	—
43	C₄×C₁₆	—	4	4	no	—
44	#336	C₂×C₄	1	8	yes	—
45	C₂²×Q₈	C₂×C₈	1	2	yes	—
46	C₂²×C₈	C₂²×C₄	1	4	yes	—
47	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂²×C₄	1	2	yes	—
49	C₂²×C₈	C₂⁴	1	1	yes	characteristic
50	C₄⋊Q₈	C₁₆	1	8	yes	complemented
51	C₄×Q₈	—	4	2	no	—
52	C₄×Q₈	—	4	2	no	—
53	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
54	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
55	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
56	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
57	C₂×C₄²	C₂×C₈	1	2	yes	—
58	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄²	C₂²×C₄	1	4	yes	—
62	C₂×C₄²	C₂²×C₄	1	2	yes	—
63	C₂×C₁₆	—	2	4	no	—
64	C₂×C₁₆	—	2	4	no	—
65	C₂×C₁₆	—	2	4	no	—
66	C₂×C₁₆	—	2	4	no	—
67	C₂×C₁₆	—	2	4	no	—
68	C₂×C₁₆	C₂×Q₈	1	8	yes	—
69	C₄⋊C₈	—	2	8	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	4	4	no	—
72	C₄×C₈	—	2	4	no	—
73	C₄×C₈	—	2	4	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	2	4	no	—
77	C₄×C₈	—	4	1	no	—
78	C₄×C₈	C₂×Q₈	1	4	yes	—
79	C₄×C₈	#41	1	4	yes	—
80	#95	C₂×C₈	1	8	yes	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	4	2	no	—
83	C₂×Q₈	C₂×C₁₆	1	8	yes	—
84	C₂²×C₄	C₂²×C₈	1	4	yes	—
85	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₈	1	2	yes	—
87	C₂²×C₄	C₂²×C₈	1	4	yes	—
88	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₈	1	2	yes	—
90	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	8	no	—
97	C₂×C₈	C₄×Q₈	1	4	yes	—
98	C₂×C₈	C₂²×Q₈	1	2	yes	—
99	C₂×C₈	C₂×#41	1	2	yes	—
100	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
102	C₄⋊C₄	—	2	8	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	C₂×C₁₆	1	16	yes	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	4	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	4	no	—
110	C₄²	—	2	4	no	—
111	C₄²	—	4	1	no	—
112	C₄²	C₂×C₁₆	1	4	yes	—
113	C₄²	C₄×Q₈	1	4	yes	—
114	C₄²	#131	1	4	yes	—
115	C₁₆	—	2	8	no	—
116	C₁₆	—	4	4	no	—
117	C₂³	C₂³×C₈	1	1	yes	characteristic
118	Q₈	—	4	8	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	8	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	C₈×Q₈	1	4	yes	—
130	C₂×C₄	C₈×Q₈	1	4	yes	—
131	C₂×C₄	C₂²×C₁₆	1	2	yes	—
132	C₂×C₄	C₂²×C₁₆	1	8	yes	—
133	C₂×C₄	C₂²×C₁₆	1	4	yes	—
134	C₂×C₄	#504	1	4	yes	—
135	C₂×C₄	C₂×C₄×Q₈	1	2	yes	—
136	C₂×C₄	#518	1	1	yes	characteristic
137	C₂×C₄	C₂×#131	1	2	yes	—
138	C₂×C₄	#568	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	4	1	no	—
141	C₈	—	4	4	no	—
142	C₈	#554	1	2	yes	—
143	C₈	#555	1	1	yes	characteristic
144	C₈	Q₈²	1	1	yes	characteristic
145	C₂²	C₂×C₈×Q₈	1	2	yes	—
146	C₂²	#2972	1	1	yes	characteristic
147	C₂²	C₂³×C₁₆	1	1	yes	characteristic
148	C₂²	C₂×#504	1	2	yes	—
149	C₂²	#3407	1	1	yes	characteristic
150	C₄	—	2	4	no	—
151	C₄	—	2	8	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	4	no	—
155	C₄	Q₈×C₁₆	1	8	yes	—
156	C₄	#2369	1	1	yes	characteristic
157	C₄	#2962	1	2	yes	—
158	C₄	#2979	1	1	yes	characteristic
159	C₂	#23247	1	1	yes	characteristic
160	C₂	C₂×Q₈×C₁₆	1	2	yes	—
161	C₂	#33476	1	2	yes	—
162	C₂	#33486	1	1	yes	characteristic
163	C₂	#33493	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₁₆
socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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
41. C₄⋊C₁₆	Q₈	semidirect
50. C₄⋊Q₈	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Q₈	2
#13263	2
#13159	2
C₁₆×C₄⋊C₄	2
C₄×C₄⋊C₁₆	2