Subgroups — C₂×#2108 #33323

Order 256.

427 conjugacy classes of subgroups.

The subgroup lattice has 112 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#405	C₂	1	1	yes	maximal, characteristic
4	#2108	C₂	1	8	yes	maximal, direct product factor
5	C₂×#346	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆	C₄	1	1	yes	characteristic
8	C₂³×C₈	C₂²	1	1	yes	characteristic
9	C₂×#117	C₄	1	1	yes	characteristic
10	C₂×M₃₂	C₄	1	8	yes	complemented
11	C₂×M₃₂	C₂²	1	4	yes	—
12	C₂²×C₁₆	C₄	1	8	yes	complemented
13	C₂²×C₁₆	C₂²	1	2	yes	—
14	C₂²×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆	C₂²	1	4	yes	—
16	C₂²×C₁₆	C₂²	1	1	yes	characteristic
17	#405	C₄	1	4	yes	—
18	#405	C₂²	1	4	yes	—
19	C₂×#97	C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	2	yes	—
21	#346	C₂²	1	8	yes	complemented
22	C₄×C₁₆	C₂²	1	8	yes	complemented
23	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆	C₈	1	8	yes	—
25	C₂×M₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₈	C₂³	1	2	yes	—
33	#117	C₈	1	4	yes	—
34	#117	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄	C₈	1	2	yes	—
36	C₂×C₂²⋊C₄	C₈	1	2	yes	—
37	C₂×C₄²	C₂×C₄	1	2	yes	—
38	M₃₂	C₂×C₄	1	16	yes	complemented
39	C₂×C₁₆	C₂×C₄	1	4	yes	—
40	C₂×C₁₆	C₂×C₄	1	4	yes	—
41	C₂×C₁₆	C₂×C₄	1	8	yes	—
42	C₂×C₁₆	C₂×C₄	1	16	yes	complemented
43	C₂×C₁₆	C₂³	1	2	yes	—
44	C₂×C₁₆	C₂³	1	4	yes	—
45	C₂×C₁₆	C₂³	1	2	yes	—
46	#97	C₂×C₄	1	8	yes	—
47	C₄×C₈	C₂×C₄	1	4	yes	—
48	C₄×C₈	C₂³	1	4	yes	—
49	C₂⁴	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄	C₄²	1	4	yes	—
51	C₂²×C₄	C₂×C₈	1	4	yes	—
52	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄	C₂×C₈	1	2	yes	—
54	C₂²×C₄	C₂×C₈	1	4	yes	—
55	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄	C₂²×C₄	1	2	yes	—
57	M₁₆	C₂×C₈	1	16	yes	—
58	C₂×C₈	C₄²	1	4	yes	—
59	C₂×C₈	C₄²	1	8	yes	—
60	C₂×C₈	C₂×C₈	1	8	yes	—
61	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈	C₂²×C₄	1	2	yes	—
63	C₂×C₈	C₂²×C₄	1	2	yes	—
64	C₂×C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₈	C₂²×C₄	1	4	yes	—
66	C₂×C₈	C₂²×C₄	1	4	yes	—
67	C₂×C₈	C₂²×C₄	1	4	yes	—
68	C₂×C₈	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄	C₂×C₈	1	8	yes	—
70	C₂²⋊C₄	C₂×C₈	1	8	yes	—
71	C₄²	C₂×C₈	1	4	yes	—
72	C₄²	C₂²×C₄	1	4	yes	—
73	C₁₆	C₂²×C₄	1	4	yes	—
74	C₁₆	C₂²×C₄	1	4	yes	—
75	C₁₆	C₂²×C₄	1	8	yes	—
76	C₂³	—	2	4	no	—
77	C₂³	C₄×C₈	1	4	yes	—
78	C₂³	C₂²×C₈	1	2	yes	—
79	C₂³	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄	—	2	8	no	—
81	C₂×C₄	C₄×C₈	1	8	yes	—
82	C₂×C₄	C₄×C₈	1	4	yes	—
83	C₂×C₄	C₂×C₄²	1	4	yes	—
84	C₂×C₄	C₂×C₄²	1	2	yes	—
85	C₂×C₄	C₂²×C₈	1	8	yes	—
86	C₂×C₄	C₂²×C₈	1	2	yes	—
87	C₂×C₄	C₂²×C₈	1	4	yes	—
88	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×C₈	1	2	yes	—
90	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
91	C₈	C₂×C₄²	1	2	yes	—
92	C₈	C₂×C₄²	1	4	yes	—
93	C₈	C₂²×C₈	1	8	yes	—
94	C₈	C₂³×C₄	1	1	yes	characteristic
95	C₈	C₂³×C₄	1	1	yes	characteristic
96	C₂²	—	2	8	no	—
97	C₂²	—	2	4	no	—
98	C₂²	C₂×C₄×C₈	1	4	yes	—
99	C₂²	C₂×C₄×C₈	1	2	yes	—
100	C₂²	#504	1	4	yes	—
101	C₂²	C₂³×C₈	1	1	yes	characteristic
102	C₄	—	2	8	no	—
103	C₄	C₂×C₄×C₈	1	2	yes	—
104	C₄	C₂×C₄×C₈	1	4	yes	—
105	C₄	C₂²×C₄²	1	1	yes	characteristic
106	C₄	C₂³×C₈	1	1	yes	characteristic
107	C₂	—	2	4	no	—
108	C₂	#2108	1	4	yes	direct product factor
109	C₂	C₂²×C₄×C₈	1	1	yes	characteristic
110	C₂	C₂×#504	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂²
commutator	C₂	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	C₂²×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #2108	C₂	direct (N × H)
5. C₂×#346	C₂	semidirect
6. C₂×C₄×C₁₆	C₂	semidirect
10. C₂×M₃₂	C₄	semidirect
12. C₂²×C₁₆	C₄	semidirect
21. #346	C₂²	semidirect
22. C₄×C₁₆	C₂²	semidirect
38. M₃₂	C₂×C₄	semidirect
42. C₂×C₁₆	C₂×C₄	semidirect
108. C₂	#2108	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×M₃₂	2
C₂³×C₁₆	2
C₂×#405	2
#2108	2
C₂×#346	2
C₂×C₄×C₁₆	2

Subgroups — C2×#2108 #33323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2108 #33323