Subgroups — C₂×#2916 #60112

Order 256.

1293 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568	C₂	1	2	yes	maximal
2	C₂²×#131	C₂	1	1	yes	maximal, characteristic
3	#2916	C₂	1	16	yes	maximal, direct product factor
4	C₂×#430	C₂	1	6	yes	maximal, complemented
5	C₂²×#108	C₂	1	6	yes	maximal, complemented
6	C₂²×#41	C₂²	1	1	yes	characteristic
7	#568	—	2	8	no	—
8	#568	C₂²	1	16	yes	—
9	C₂×#131	—	2	8	no	—
10	C₂×#131	C₄	1	8	yes	—
11	C₂×#131	C₄	1	8	yes	—
12	C₂×#131	C₂²	1	8	yes	—
13	C₂²×M₁₆	C₂²	1	12	yes	—
14	C₂²×M₁₆	C₂²	1	3	yes	—
15	C₂³×C₈	C₂²	1	3	yes	—
16	#430	C₂²	1	48	yes	complemented
17	C₂×#108	C₂²	1	4	yes	—
18	C₂×#108	C₂²	1	12	yes	—
19	C₂×#108	C₂²	1	48	yes	complemented
20	C₂×#41	C₂×C₄	1	4	yes	—
21	C₂×#41	C₂×C₄	1	4	yes	—
22	C₂×#41	D₄	1	4	yes	—
23	C₂×#41	D₄	1	4	yes	—
24	C₂×#41	Q₈	1	4	yes	—
25	C₂×#41	C₂³	1	4	yes	—
26	C₂²×Q₈	Q₈	1	1	yes	characteristic
27	C₂²×D₄	Q₈	1	3	yes	—
28	C₂³×C₄	C₂³	1	3	yes	—
29	#131	—	2	64	no	—
30	#131	C₂×C₄	1	64	yes	—
31	C₂×M₁₆	—	2	12	no	—
32	C₂×M₁₆	—	2	24	no	—
33	C₂×M₁₆	C₂×C₄	1	12	yes	—
34	C₂×M₁₆	C₂×C₄	1	12	yes	—
35	C₂×M₁₆	C₂³	1	8	yes	—
36	C₂×M₁₆	C₂³	1	48	yes	—
37	C₂×M₁₆	C₂³	1	12	yes	—
38	C₂²×C₈	—	2	12	no	—
39	C₂²×C₈	C₂×C₄	1	12	yes	—
40	C₂²×C₈	C₂×C₄	1	12	yes	—
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	3	yes	—
43	C₂²×C₈	C₂³	1	12	yes	—
44	#108	C₂³	1	16	yes	—
45	#108	C₂³	1	48	yes	—
46	C₂⁴	C₂×Q₈	1	3	yes	—
47	#41	C₄⋊C₄	1	16	yes	—
48	#41	C₄⋊C₄	1	16	yes	—
49	#41	C₂²×C₄	1	16	yes	—
50	#41	C₂×D₄	1	16	yes	—
51	C₂×Q₈	C₄⋊C₄	1	4	yes	—
52	C₂×Q₈	C₄⋊C₄	1	4	yes	—
53	C₂×Q₈	C₂×Q₈	1	4	yes	—
54	C₂×D₄	C₄⋊C₄	1	12	yes	—
55	C₂×D₄	C₄⋊C₄	1	12	yes	—
56	C₂×D₄	C₂×Q₈	1	12	yes	—
57	C₂²×C₄	C₂²×C₄	1	6	yes	—
58	C₂²×C₄	C₂²×C₄	1	6	yes	—
59	C₂²×C₄	C₂×D₄	1	6	yes	—
60	C₂²×C₄	C₂×D₄	1	6	yes	—
61	C₂²×C₄	C₂×Q₈	1	3	yes	—
62	C₂²×C₄	C₂×Q₈	1	6	yes	—
63	C₂²×C₄	C₂⁴	1	6	yes	—
64	C₂²×C₄	C₂⁴	1	1	yes	characteristic
65	M₁₆	—	2	8	no	—
66	M₁₆	—	2	48	no	—
67	M₁₆	C₂²×C₄	1	48	yes	—
68	M₁₆	C₂⁴	1	16	yes	—
69	C₂×C₈	—	2	48	no	—
70	C₂×C₈	—	2	8	no	—
71	C₂×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₈	C₂²×C₄	1	6	yes	—
74	C₂×C₈	C₂²×C₄	1	6	yes	—
75	C₂×C₈	C₂²×C₄	1	48	yes	—
76	C₂×C₈	C₂⁴	1	2	yes	—
77	C₂×C₈	C₂⁴	1	6	yes	—
78	C₂³	—	4	6	no	—
79	C₂³	C₂×C₄⋊C₄	1	6	yes	—
80	C₂³	C₂×C₄⋊C₄	1	6	yes	—
81	C₂³	C₂²×Q₈	1	1	yes	characteristic
82	C₂³	C₂²×Q₈	1	6	yes	—
83	Q₈	C₂×C₄⋊C₄	1	16	yes	—
84	D₄	C₂×C₄⋊C₄	1	48	yes	—
85	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
86	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
87	C₂×C₄	C₂×C₄⋊C₄	1	12	yes	—
88	C₂×C₄	C₂×C₄⋊C₄	1	12	yes	—
89	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄	C₂³×C₄	1	12	yes	—
92	C₂×C₄	C₂²×D₄	1	12	yes	—
93	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄	C₂²×Q₈	1	6	yes	—
97	C₂×C₄	C₂⁵	1	1	yes	characteristic
98	C₈	—	2	16	no	—
99	C₈	C₂³×C₄	1	4	yes	—
100	C₈	C₂³×C₄	1	12	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	—	2	12	no	—
103	C₂²	—	4	6	no	—
104	C₂²	—	4	6	no	—
105	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²	C₂²×C₄⋊C₄	1	12	yes	—
107	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²	C₂³×Q₈	1	1	yes	characteristic
109	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₄	C₂²×C₄⋊C₄	1	12	yes	—
112	C₄	C₂⁴×C₄	1	1	yes	characteristic
113	C₄	C₂³×D₄	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	—	2	12	no	—
117	C₂	#2916	1	2	yes	direct product factor
118	C₂	C₂³×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂²	C₂²×C₄⋊C₄
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #2916	C₂	direct (N × H)
4. C₂×#430	C₂	semidirect
5. C₂²×#108	C₂	semidirect
16. #430	C₂²	semidirect
19. C₂×#108	C₂²	semidirect
117. C₂	#2916	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#568	2
C₂²×#131	2
#2916	2
C₂×#430	2
C₂²×#108	2

Subgroups — C2×#2916 #60112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2916 #60112