Subgroups — C₂×#13627 #512705

Order 512.

510 conjugacy classes of subgroups.

The subgroup lattice has 142 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	#13627	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂³×C₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂×M₆₄	—	2	8	no	—
9	C₂×M₆₄	C₂²	1	8	yes	—
10	C₂²×C₃₂	C₂²	1	4	yes	—
11	C₂×C₄×C₁₆	C₄	1	2	yes	—
12	C₂×C₄×C₁₆	C₄	1	4	yes	—
13	C₂×C₄×C₁₆	C₂²	1	4	yes	—
14	C₂×C₄×C₁₆	C₂²	1	2	yes	—
15	C₄⋊C₃₂	C₂²	1	16	yes	complemented
16	C₂³×C₈	C₈	1	1	yes	characteristic
17	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄²	C₈	1	1	yes	characteristic
19	C₂²×C₁₆	—	2	4	no	—
20	C₂²×C₁₆	C₂×C₄	1	4	yes	—
21	C₂²×C₁₆	C₂×C₄	1	2	yes	—
22	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₁₆	D₄	1	4	yes	—
25	C₂²×C₁₆	Q₈	1	4	yes	—
26	C₂²×C₁₆	C₂³	1	2	yes	—
27	C₂²×C₁₆	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₈	C₈	1	4	yes	—
29	C₂×C₄×C₈	C₈	1	2	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
32	M₆₄	—	2	16	no	—
33	C₂×C₃₂	—	2	8	no	—
34	C₂×C₃₂	C₂³	1	8	yes	—
35	C₄×C₁₆	C₂×C₄	1	8	yes	—
36	C₄×C₁₆	C₂×C₄	1	4	yes	—
37	C₄×C₁₆	C₂³	1	4	yes	—
38	C₂³×C₄	C₁₆	1	2	yes	—
39	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₈	—	2	4	no	—
41	C₂²×C₈	C₄⋊C₄	1	4	yes	—
42	C₂²×C₈	C₂×C₈	1	4	yes	—
43	C₂²×C₈	C₂×C₈	1	2	yes	—
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×C₈	1	2	yes	—
46	C₂²×C₈	M₁₆	1	4	yes	—
47	C₂²×C₈	C₂²×C₄	1	2	yes	—
48	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄²	C₁₆	1	1	yes	characteristic
50	C₂×C₄²	C₁₆	1	4	yes	—
51	C₂×C₄²	C₁₆	1	1	yes	characteristic
52	C₂×C₄²	C₂×C₈	1	4	yes	—
53	C₂×C₄²	C₂×C₈	1	2	yes	—
54	C₂×C₁₆	—	2	8	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
57	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
58	C₂×C₁₆	C₂²×C₄	1	4	yes	—
59	C₂×C₁₆	C₂²×C₄	1	2	yes	—
60	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₁₆	C₂×D₄	1	2	yes	—
62	C₂×C₁₆	C₂×D₄	1	4	yes	—
63	C₂×C₁₆	C₂×Q₈	1	4	yes	—
64	C₂×C₁₆	C₂×Q₈	1	2	yes	—
65	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
66	C₄×C₈	C₂×C₈	1	8	yes	—
67	C₄×C₈	C₂×C₈	1	4	yes	—
68	C₄×C₈	C₂²×C₄	1	4	yes	—
69	C₃₂	—	2	8	no	—
70	C₂⁴	C₂×C₁₆	1	1	yes	characteristic
71	C₂²×C₄	—	2	8	no	—
72	C₂²×C₄	C₄⋊C₈	1	4	yes	—
73	C₂²×C₄	C₂×C₁₆	1	2	yes	—
74	C₂²×C₄	C₂×C₁₆	1	8	yes	—
75	C₂²×C₄	C₂×C₁₆	1	2	yes	—
76	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
77	C₂²×C₄	C₂×C₁₆	1	2	yes	—
78	C₂²×C₄	M₃₂	1	4	yes	—
79	C₂²×C₄	C₂²×C₈	1	2	yes	—
80	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₄⋊C₈	1	8	yes	—
84	C₂×C₈	C₄⋊C₈	1	4	yes	—
85	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₈	C₂²×C₈	1	4	yes	—
88	C₂×C₈	C₂²×C₈	1	2	yes	—
89	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₈	C₂×M₁₆	1	4	yes	—
91	C₂×C₈	C₂×M₁₆	1	2	yes	—
92	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
93	C₄²	C₂×C₁₆	1	4	yes	—
94	C₄²	C₂×C₁₆	1	4	yes	—
95	C₄²	C₂×C₁₆	1	4	yes	—
96	C₄²	C₂²×C₈	1	4	yes	—
97	C₁₆	—	2	4	no	—
98	C₁₆	C₂×C₄⋊C₄	1	4	yes	—
99	C₁₆	C₂×C₄⋊C₄	1	2	yes	—
100	C₁₆	C₂²×D₄	1	1	yes	characteristic
101	C₁₆	C₂²×Q₈	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	C₄⋊C₁₆	1	4	yes	—
104	C₂³	C₂²×C₁₆	1	2	yes	—
105	C₂³	C₂²×C₁₆	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	16	no	—
108	C₂×C₄	C₄⋊C₁₆	1	8	yes	—
109	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
110	C₂×C₄	M₆₄	1	8	yes	complemented
111	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
112	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
113	C₂×C₄	C₂²×C₁₆	1	4	yes	—
114	C₂×C₄	C₂²×C₁₆	1	2	yes	—
115	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₁₆	1	4	yes	—
117	C₂×C₄	C₂×M₃₂	1	4	yes	—
118	C₂×C₄	C₂×M₃₂	1	2	yes	—
119	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	C₂×C₄⋊C₈	1	2	yes	—
122	C₈	C₂×C₄⋊C₈	1	4	yes	—
123	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₈	C₂²×M₁₆	1	1	yes	characteristic
125	C₂²	—	2	8	no	—
126	C₂²	—	2	4	no	—
127	C₂²	C₂×C₄⋊C₁₆	1	4	yes	—
128	C₂²	C₂×C₄⋊C₁₆	1	2	yes	—
129	C₂²	C₂×M₆₄	1	4	yes	—
130	C₂²	C₂³×C₁₆	1	1	yes	characteristic
131	C₄	—	2	4	no	—
132	C₄	C₂×C₄⋊C₁₆	1	2	yes	—
133	C₄	C₂×C₄⋊C₁₆	1	4	yes	—
134	C₄	C₂×M₆₄	1	8	yes	complemented
135	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
136	C₄	C₂²×M₃₂	1	1	yes	characteristic
137	C₂	—	2	4	no	—
138	C₂	#13627	1	4	yes	direct product factor
139	C₂	C₂²×C₄⋊C₁₆	1	1	yes	characteristic
140	C₂	C₂²×M₆₄	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂³	C₂²×C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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. #13627	C₂	direct (N × H)
4. C₂×C₄⋊C₃₂	C₂	semidirect
15. C₄⋊C₃₂	C₂²	semidirect
110. C₂×C₄	M₆₄	semidirect
134. C₄	C₂×M₆₄	semidirect
138. C₂	#13627	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
#13627	2
C₂×C₄⋊C₃₂	2

Subgroups — C2×#13627 #512705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13627 #512705