Subgroups — C₂²×#1422 #512708

Order 512.

502 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₆₄	C₂	1	2	yes	maximal
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1422	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆	C₄	1	1	yes	characteristic
5	C₂³×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×M₆₄	—	2	6	no	—
8	C₂×M₆₄	C₂²	1	12	yes	—
9	C₂²×C₃₂	—	2	2	no	—
10	C₂×C₄×C₁₆	C₄	1	6	yes	—
11	C₂×C₄×C₁₆	C₂²	1	6	yes	—
12	#1422	C₂²	1	16	yes	direct product factor
13	C₂³×C₈	C₈	1	1	yes	characteristic
14	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄²	C₈	1	1	yes	characteristic
16	C₂²×C₁₆	—	2	1	no	—
17	C₂²×C₁₆	—	2	3	no	—
18	C₂²×C₁₆	C₂×C₄	1	6	yes	—
19	C₂²×C₁₆	C₂×C₄	1	3	yes	—
20	C₂²×C₁₆	D₄	1	3	yes	—
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	Q₈	1	1	yes	characteristic
23	C₂²×C₁₆	Q₈	1	3	yes	—
24	C₂²×C₁₆	C₂³	1	3	yes	—
25	C₂×C₄×C₈	C₈	1	6	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
27	M₆₄	—	2	12	no	—
28	M₆₄	C₂³	1	8	yes	—
29	C₂×C₃₂	—	2	12	no	—
30	C₄×C₁₆	C₂×C₄	1	12	yes	—
31	C₄×C₁₆	C₂³	1	4	yes	—
32	C₂³×C₄	—	2	1	no	—
33	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₈	C₁₆	1	6	yes	—
35	C₂²×C₈	C₁₆	1	2	yes	—
36	C₂²×C₈	C₄⋊C₄	1	3	yes	—
37	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₈	1	3	yes	—
39	C₂²×C₈	C₂×C₈	1	6	yes	—
40	C₂²×C₈	M₁₆	1	1	yes	characteristic
41	C₂²×C₈	M₁₆	1	3	yes	—
42	C₂²×C₈	C₂²×C₄	1	3	yes	—
43	C₂×C₄²	—	2	3	no	—
44	C₂×C₄²	C₂×C₈	1	6	yes	—
45	C₂×C₁₆	—	2	6	no	—
46	C₂×C₁₆	—	2	6	no	—
47	C₂×C₁₆	C₄⋊C₄	1	3	yes	—
48	C₂×C₁₆	C₄⋊C₄	1	6	yes	—
49	C₂×C₁₆	C₄⋊C₄	1	3	yes	—
50	C₂×C₁₆	C₂²×C₄	1	4	yes	—
51	C₂×C₁₆	C₂²×C₄	1	3	yes	—
52	C₂×C₁₆	C₂×D₄	1	3	yes	—
53	C₂×C₁₆	C₂×D₄	1	3	yes	—
54	C₂×C₁₆	C₂×Q₈	1	3	yes	—
55	C₂×C₁₆	C₂×Q₈	1	3	yes	—
56	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
57	C₄×C₈	C₂×C₈	1	12	yes	—
58	C₄×C₈	C₂²×C₄	1	4	yes	—
59	C₃₂	—	2	8	no	—
60	C₂⁴	M₃₂	1	1	yes	characteristic
61	C₂²×C₄	—	2	8	no	—
62	C₂²×C₄	—	2	6	no	—
63	C₂²×C₄	C₄⋊C₈	1	3	yes	—
64	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
65	C₂²×C₄	C₂×C₁₆	1	3	yes	—
66	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
67	C₂²×C₄	M₃₂	1	3	yes	—
68	C₂²×C₄	C₂²×C₈	1	3	yes	—
69	C₂×C₈	C₄⋊C₈	1	6	yes	—
70	C₂×C₈	C₄⋊C₈	1	3	yes	—
71	C₂×C₈	C₄⋊C₈	1	3	yes	—
72	C₂×C₈	C₂×C₁₆	1	12	yes	—
73	C₂×C₈	C₂×C₁₆	1	12	yes	—
74	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₈	C₂²×C₈	1	3	yes	—
77	C₂×C₈	C₂²×C₈	1	4	yes	—
78	C₂×C₈	C₂×M₁₆	1	3	yes	—
79	C₂×C₈	C₂×M₁₆	1	3	yes	—
80	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
81	C₄²	—	2	6	no	—
82	C₄²	C₂²×C₈	1	4	yes	—
83	C₁₆	—	2	4	no	—
84	C₁₆	C₂×C₄⋊C₄	1	3	yes	—
85	C₁₆	C₂×C₄⋊C₄	1	3	yes	—
86	C₁₆	C₂²×D₄	1	1	yes	characteristic
87	C₁₆	C₂²×Q₈	1	1	yes	characteristic
88	C₂³	—	2	4	no	—
89	C₂³	C₄⋊C₁₆	1	3	yes	—
90	C₂³	C₄⋊C₁₆	1	1	yes	characteristic
91	C₂³	C₂×M₃₂	1	3	yes	—
92	C₂×C₄	—	2	24	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄⋊C₁₆	1	6	yes	—
95	C₂×C₄	C₄⋊C₁₆	1	3	yes	—
96	C₂×C₄	C₄⋊C₁₆	1	3	yes	—
97	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
98	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
99	C₂×C₄	C₂²×C₁₆	1	3	yes	—
100	C₂×C₄	C₂²×C₁₆	1	3	yes	—
101	C₂×C₄	C₂×M₃₂	1	3	yes	—
102	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
103	C₈	C₂×C₄⋊C₈	1	3	yes	—
104	C₈	C₂×C₄⋊C₈	1	3	yes	—
105	C₈	C₂²×C₁₆	1	8	yes	—
106	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₈	C₂²×M₁₆	1	1	yes	characteristic
108	C₂²	—	2	6	no	—
109	C₂²	—	2	6	no	—
110	C₂²	#1422	1	4	yes	direct product factor
111	C₂²	C₂×C₄⋊C₁₆	1	3	yes	—
112	C₂²	C₂×C₄⋊C₁₆	1	3	yes	—
113	C₂²	C₂²×M₃₂	1	1	yes	characteristic
114	C₄	—	2	8	no	—
115	C₄	C₂×C₄⋊C₁₆	1	3	yes	—
116	C₄	C₂×C₄⋊C₁₆	1	3	yes	—
117	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
118	C₄	C₂³×C₁₆	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	3	no	—
121	C₂	C₂×#1422	1	6	yes	direct product factor
122	C₂	C₂²×C₄⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂³	C₄⋊C₁₆
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×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
3. C₂×#1422	C₂	direct (N × H)
12. #1422	C₂²	direct (N × H)
110. C₂²	#1422	direct (N × H)
121. C₂	C₂×#1422	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₆₄	2
C₂²×C₄×C₁₆	2
C₂×#1422	2

Subgroups — C22×#1422 #512708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1422 #512708