Subgroups — C₂×#13628 #512706

Order 512.

478 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	1	yes	maximal, characteristic
2	C₂³×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#2108	C₂	1	1	yes	maximal, characteristic
4	#13628	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
6	C₂²×M₃₂	C₄	1	1	yes	characteristic
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#405	C₄	1	1	yes	characteristic
9	C₂×M₆₄	—	2	4	no	—
10	C₂×M₆₄	C₂²	1	4	yes	—
11	C₂²×C₃₂	—	2	4	no	—
12	C₂²×C₃₂	C₂²	1	4	yes	—
13	C₂²×C₃₂	C₂²	1	2	yes	—
14	C₂²×C₃₂	C₂²	1	2	yes	—
15	#2108	C₄	1	4	yes	—
16	#2108	C₂²	1	4	yes	—
17	C₂×#346	C₄	1	2	yes	—
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	C₄⋊C₃₂	C₂²	1	16	yes	complemented
20	C₂²×M₁₆	C₈	1	1	yes	characteristic
21	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×#117	C₈	1	1	yes	characteristic
23	C₂×M₃₂	—	2	4	no	—
24	C₂×M₃₂	C₂×C₄	1	4	yes	—
25	C₂²×C₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₁₆	D₄	1	4	yes	—
29	C₂²×C₁₆	Q₈	1	4	yes	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	#405	C₈	1	4	yes	—
33	#405	C₂×C₄	1	4	yes	—
34	C₂×#97	C₈	1	2	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
36	M₆₄	—	2	8	no	—
37	C₂×C₃₂	—	2	4	no	—
38	C₂×C₃₂	—	2	8	no	—
39	C₂×C₃₂	—	2	4	no	—
40	C₂×C₃₂	C₂³	1	4	yes	—
41	C₂×C₃₂	C₂³	1	4	yes	—
42	#346	C₂×C₄	1	8	yes	—
43	C₄×C₁₆	C₂×C₄	1	4	yes	—
44	C₄×C₁₆	C₂³	1	4	yes	—
45	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
46	C₂×M₁₆	—	2	4	no	—
47	C₂×M₁₆	C₂×C₈	1	4	yes	—
48	C₂²×C₈	C₄⋊C₄	1	4	yes	—
49	C₂²×C₈	C₂×C₈	1	2	yes	—
50	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₈	C₂×C₈	1	2	yes	—
52	C₂²×C₈	M₁₆	1	4	yes	—
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²×C₄	1	2	yes	—
55	#117	C₁₆	1	4	yes	—
56	#117	C₂×C₈	1	4	yes	—
57	C₂×C₄⋊C₄	C₁₆	1	2	yes	—
58	C₂×C₂²⋊C₄	C₁₆	1	2	yes	—
59	C₂×C₄²	C₂×C₈	1	2	yes	—
60	M₃₂	—	2	8	no	—
61	C₂×C₁₆	—	2	4	no	—
62	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
63	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
64	C₂×C₁₆	C₂²×C₄	1	4	yes	—
65	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₁₆	C₂²×C₄	1	2	yes	—
67	C₂×C₁₆	C₂×D₄	1	2	yes	—
68	C₂×C₁₆	C₂×D₄	1	4	yes	—
69	C₂×C₁₆	C₂×Q₈	1	4	yes	—
70	C₂×C₁₆	C₂×Q₈	1	2	yes	—
71	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
72	#97	C₂×C₈	1	8	yes	—
73	C₄×C₈	C₂×C₈	1	4	yes	—
74	C₄×C₈	C₂²×C₄	1	4	yes	—
75	C₃₂	—	2	4	no	—
76	C₃₂	—	2	4	no	—
77	C₂⁴	C₂×C₁₆	1	1	yes	characteristic
78	C₂²×C₄	C₄⋊C₈	1	4	yes	—
79	C₂²×C₄	C₂×C₁₆	1	4	yes	—
80	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₁₆	1	2	yes	—
82	C₂²×C₄	M₃₂	1	4	yes	—
83	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
84	C₂²×C₄	C₂²×C₈	1	2	yes	—
85	M₁₆	—	2	8	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	C₄⋊C₈	1	4	yes	—
88	C₂×C₈	C₄⋊C₈	1	8	yes	—
89	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₈	C₂²×C₈	1	4	yes	—
92	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₈	C₂²×C₈	1	2	yes	—
94	C₂×C₈	C₂×M₁₆	1	2	yes	—
95	C₂×C₈	C₂×M₁₆	1	4	yes	—
96	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
97	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
98	C₂²⋊C₄	C₂×C₁₆	1	8	yes	—
99	C₄²	C₂×C₁₆	1	4	yes	—
100	C₄²	C₂²×C₈	1	4	yes	—
101	C₁₆	—	2	4	no	—
102	C₁₆	C₂×C₄⋊C₄	1	2	yes	—
103	C₁₆	C₂×C₄⋊C₄	1	4	yes	—
104	C₁₆	C₂²×D₄	1	1	yes	characteristic
105	C₁₆	C₂²×Q₈	1	1	yes	characteristic
106	C₂³	—	2	4	no	—
107	C₂³	C₄⋊C₁₆	1	4	yes	—
108	C₂³	C₂²×C₁₆	1	2	yes	—
109	C₂³	C₂²×C₁₆	1	1	yes	characteristic
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	C₄⋊C₁₆	1	8	yes	—
112	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
113	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
114	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
115	C₂×C₄	C₂²×C₁₆	1	8	yes	—
116	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₁₆	1	2	yes	—
118	C₂×C₄	C₂×M₃₂	1	2	yes	—
119	C₂×C₄	C₂×M₃₂	1	4	yes	—
120	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₈	C₂×C₄⋊C₈	1	2	yes	—
123	C₈	C₂×C₄⋊C₈	1	4	yes	—
124	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₈	C₂²×M₁₆	1	1	yes	characteristic
126	C₂²	—	2	8	no	—
127	C₂²	—	2	4	no	—
128	C₂²	C₂×C₄⋊C₁₆	1	4	yes	—
129	C₂²	C₂×C₄⋊C₁₆	1	2	yes	—
130	C₂²	#2258	1	4	yes	—
131	C₂²	C₂³×C₁₆	1	1	yes	characteristic
132	C₄	—	2	8	no	—
133	C₄	C₂×C₄⋊C₁₆	1	2	yes	—
134	C₄	C₂×C₄⋊C₁₆	1	4	yes	—
135	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
136	C₄	C₂²×M₃₂	1	1	yes	characteristic
137	C₂	—	2	4	no	—
138	C₂	#13628	1	4	yes	direct product factor
139	C₂	C₂²×C₄⋊C₁₆	1	1	yes	characteristic
140	C₂	C₂×#2258	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
4. #13628	C₂	direct (N × H)
5. C₂×C₄⋊C₃₂	C₂	semidirect
19. C₄⋊C₃₂	C₂²	semidirect
138. C₂	#13628	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
C₂³×C₃₂	2
C₂×#2108	2
#13628	2
C₂×C₄⋊C₃₂	2

Subgroups — C2×#13628 #512706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13628 #512706