Subgroups — C₄×#405 #20330

Order 256.

763 conjugacy classes of subgroups.

The subgroup lattice has 119 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405	C₂	1	4	yes	maximal
2	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×#117	C₂	1	1	yes	maximal, characteristic
5	#1726	C₂	1	4	yes	maximal, complemented
6	C₄×#97	C₂	1	2	yes	maximal, complemented
7	C₄²×C₈	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₈	C₂²	1	2	yes	—
10	C₂×#117	C₂²	1	2	yes	—
11	C₂²×C₄²	C₂²	1	1	yes	characteristic
12	#405	C₄	1	32	yes	direct product factor
13	C₄×M₁₆	C₄	1	8	yes	complemented
14	C₂×#97	C₂²	1	2	yes	—
15	C₂×#97	C₂²	1	8	yes	—
16	C₂×C₄×C₈	C₄	1	8	yes	complemented
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	8	yes	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄	C₄	1	4	yes	—
23	C₄×C₂²⋊C₄	C₄	1	4	yes	—
24	#376	C₂²	1	2	yes	—
25	C₄³	C₂²	1	2	yes	—
26	C₂³×C₄	C₂×C₄	1	2	yes	—
27	C₂³×C₄	C₂³	1	1	yes	characteristic
28	C₂×M₁₆	C₂×C₄	1	8	yes	—
29	C₂×M₁₆	C₂×C₄	1	16	yes	—
30	C₂²×C₈	C₂×C₄	1	8	yes	—
31	C₂²×C₈	C₂×C₄	1	16	yes	—
32	C₂²×C₈	C₂³	1	2	yes	—
33	C₂²×C₈	C₂³	1	2	yes	—
34	C₂²×C₈	C₂³	1	4	yes	—
35	#117	C₂×C₄	1	16	yes	—
36	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₄²	C₂×C₄	1	8	yes	—
39	C₂×C₄²	C₂×C₄	1	2	yes	—
40	C₂×C₄²	C₂×C₄	1	4	yes	—
41	C₂×C₄²	C₂×C₄	1	4	yes	—
42	C₂×C₄²	C₂³	1	4	yes	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	C₂×C₄²	C₂³	1	1	yes	characteristic
45	#97	C₂×C₄	1	8	yes	—
46	#97	C₂×C₄	1	32	yes	complemented
47	C₄×C₈	C₂×C₄	1	4	yes	—
48	C₄×C₈	C₂×C₄	1	8	yes	—
49	C₄×C₈	C₂×C₄	1	4	yes	—
50	C₄×C₈	C₂×C₄	1	8	yes	—
51	C₄×C₈	C₂×C₄	1	32	yes	complemented
52	#95	C₂×C₄	1	8	yes	—
53	C₂⁴	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄	—	2	8	no	—
55	C₂²×C₄	C₄²	1	8	yes	—
56	C₂²×C₄	C₂²×C₄	1	2	yes	—
57	C₂²×C₄	C₂²×C₄	1	8	yes	—
58	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄	C₂²×C₄	1	2	yes	—
60	C₂²×C₄	C₂²×C₄	1	2	yes	—
61	C₂²×C₄	C₂²×C₄	1	8	yes	—
62	C₂²×C₄	C₂²×C₄	1	2	yes	—
63	C₂²×C₄	C₂⁴	1	1	yes	characteristic
64	M₁₆	C₄²	1	32	yes	complemented
65	C₂×C₈	C₄²	1	32	yes	complemented
66	C₂×C₈	C₂²×C₄	1	4	yes	—
67	C₂×C₈	C₂²×C₄	1	4	yes	—
68	C₂×C₈	C₂²×C₄	1	8	yes	—
69	C₂×C₈	C₂²×C₄	1	8	yes	—
70	C₂×C₈	C₂²×C₄	1	16	yes	—
71	C₂×C₈	C₂²×C₄	1	8	yes	—
72	C₄⋊C₄	C₄²	1	16	yes	—
73	C₂²⋊C₄	C₄²	1	16	yes	—
74	C₄²	—	2	16	no	—
75	C₄²	C₄²	1	8	yes	—
76	C₄²	C₂²×C₄	1	2	yes	—
77	C₄²	C₂²×C₄	1	2	yes	—
78	C₄²	C₂²×C₄	1	2	yes	—
79	C₄²	C₂²×C₄	1	16	yes	—
80	C₄²	C₂²×C₄	1	2	yes	—
81	C₂³	—	2	4	no	—
82	C₂³	C₂×C₄²	1	4	yes	—
83	C₂³	C₂×C₄²	1	2	yes	—
84	C₂³	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	16	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	16	no	—
88	C₂×C₄	C₂×C₄²	1	4	yes	—
89	C₂×C₄	C₂×C₄²	1	16	yes	—
90	C₂×C₄	C₂×C₄²	1	2	yes	—
91	C₂×C₄	C₂×C₄²	1	4	yes	—
92	C₂×C₄	C₂×C₄²	1	8	yes	—
93	C₂×C₄	C₂×C₄²	1	8	yes	—
94	C₂×C₄	C₂×C₄²	1	4	yes	—
95	C₂×C₄	#131	1	8	yes	—
96	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄	C₂³×C₄	1	4	yes	—
99	C₈	C₂×C₄²	1	8	yes	—
100	C₈	C₂×C₄²	1	8	yes	—
101	C₈	C₂×C₄²	1	16	yes	—
102	C₂²	—	2	4	no	—
103	C₂²	—	2	8	no	—
104	C₂²	C₄³	1	4	yes	—
105	C₂²	C₂²×C₄²	1	1	yes	characteristic
106	C₂²	C₂²×C₄²	1	2	yes	—
107	C₂²	C₂×#131	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	2	16	no	—
110	C₄	C₄³	1	4	yes	—
111	C₄	#405	1	8	yes	direct product factor
112	C₄	C₂²×C₄²	1	2	yes	—
113	C₄	C₂²×C₄²	1	2	yes	—
114	C₂	—	2	4	no	—
115	C₂	C₂×C₄³	1	1	yes	characteristic
116	C₂	C₂×#405	1	2	yes	—
117	C₂	C₄×#131	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂×C₄³
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #1726	C₂	semidirect
6. C₄×#97	C₂	semidirect
7. C₄²×C₈	C₂	semidirect
12. #405	C₄	direct (N × H)
13. C₄×M₁₆	C₄	semidirect
16. C₂×C₄×C₈	C₄	semidirect
46. #97	C₂×C₄	semidirect
51. C₄×C₈	C₂×C₄	semidirect
64. M₁₆	C₄²	semidirect
65. C₂×C₈	C₄²	semidirect
111. C₄	#405	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×#405	2
C₂×C₄×M₁₆	2
C₂²×C₄×C₈	2
C₄×#117	2
#1726	2
C₄×#97	2
C₄²×C₈	2

Subgroups — C4×#405 #20330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#405 #20330