Subgroups — C₄×#446 #20517

Order 256.

479 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446	C₂	1	2	yes	maximal
2	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
3	#1771	C₂	1	4	yes	maximal
4	#1770	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄×#97	C₂	1	2	yes	maximal
8	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈	C₄	1	2	yes	—
10	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
11	#446	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×#97	C₂²	1	4	yes	—
15	C₂×#97	C₂²	1	4	yes	—
16	C₂×C₄×C₈	C₂²	1	2	yes	—
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	1	yes	characteristic
21	C₄×C₄⋊C₄	C₄	1	4	yes	—
22	C₄×C₄⋊C₄	C₄	1	2	yes	—
23	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄³	C₂²	1	1	yes	characteristic
26	C₄³	C₂²	1	2	yes	—
27	#336	C₂²	1	4	yes	—
28	#336	C₂²	1	2	yes	—
29	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂³	1	2	yes	—
31	C₂²×C₈	C₂³	1	4	yes	—
32	C₂²×C₈	C₂³	1	2	yes	—
33	C₄×Q₈	—	2	8	no	—
34	C₄×Q₈	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²	C₂×C₄	1	4	yes	—
40	C₂×C₄²	C₂×C₄	1	4	yes	—
41	C₂×C₄²	C₂×C₄	1	2	yes	—
42	C₂×C₄²	C₂×C₄	1	2	yes	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	C₂×C₄²	C₂³	1	2	yes	—
45	C₂×C₄²	C₂³	1	1	yes	characteristic
46	C₂×C₄²	C₂³	1	2	yes	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	C₄⋊C₈	C₂×C₄	1	8	yes	—
49	C₄⋊C₈	C₂×C₄	1	16	yes	—
50	#97	—	2	8	no	—
51	#97	C₂×C₄	1	16	yes	—
52	C₄×C₈	—	2	4	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	2	4	no	—
55	C₄×C₈	C₂×C₄	1	8	yes	—
56	C₄×C₈	Q₈	1	4	yes	complemented
57	#95	C₂×C₄	1	2	yes	—
58	#95	C₂×C₄	1	4	yes	—
59	C₂×Q₈	—	2	4	no	—
60	C₂×Q₈	C₄²	1	4	yes	—
61	C₂²×C₄	C₂²×C₄	1	4	yes	—
62	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄	C₂²×C₄	1	2	yes	—
64	C₂²×C₄	C₂²×C₄	1	2	yes	—
65	C₂²×C₄	C₂²×C₄	1	2	yes	—
66	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄	C₂²×C₄	1	2	yes	—
68	C₂²×C₄	C₂⁴	1	1	yes	characteristic
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	C₂²×C₄	1	4	yes	—
74	C₂×C₈	C₂²×C₄	1	8	yes	—
75	C₂×C₈	C₂²×C₄	1	4	yes	—
76	C₂×C₈	C₂×Q₈	1	4	yes	—
77	C₂×C₈	#41	1	4	yes	—
78	C₄⋊C₄	—	2	8	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	C₄²	1	8	yes	—
81	C₄⋊C₄	C₄²	1	4	yes	—
82	C₄²	—	2	8	no	—
83	C₄²	—	2	4	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	4	no	—
87	C₄²	M₁₆	1	8	yes	—
88	C₄²	C₂²×C₄	1	4	yes	—
89	C₄²	C₂²×C₄	1	8	yes	—
90	C₄²	C₂×Q₈	1	2	yes	—
91	C₄²	#41	1	2	yes	—
92	C₂³	C₂³×C₄	1	1	yes	characteristic
93	Q₈	—	2	8	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₂×C₄²	1	2	yes	—
103	C₂×C₄	C₂×C₄²	1	4	yes	—
104	C₂×C₄	C₂×C₄²	1	4	yes	—
105	C₂×C₄	C₂×C₄²	1	2	yes	—
106	C₂×C₄	C₄×Q₈	1	4	yes	—
107	C₂×C₄	C₂×M₁₆	1	4	yes	—
108	C₂×C₄	C₂×M₁₆	1	4	yes	—
109	C₂×C₄	C₂×M₁₆	1	4	yes	—
110	C₂×C₄	#131	1	4	yes	—
111	C₂×C₄	C₂³×C₄	1	2	yes	—
112	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄	C₂×#41	1	2	yes	—
114	C₂×C₄	C₂×#41	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	—	2	8	no	—
117	C₈	C₄×Q₈	1	8	yes	complemented
118	C₂²	C₂²×C₄²	1	1	yes	characteristic
119	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₂²	C₄×#41	1	1	yes	characteristic
121	C₂²	C₂²×M₁₆	1	2	yes	—
122	C₂²	C₂×#131	1	2	yes	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	4	no	—
126	C₄	—	2	4	no	—
127	C₄	—	4	4	no	—
128	C₄	C₄×M₁₆	1	4	yes	—
129	C₄	#446	1	8	yes	direct product factor
130	C₄	C₂×C₄×Q₈	1	2	yes	—
131	C₄	C₄×#41	1	2	yes	—
132	C₂	C₄²×Q₈	1	1	yes	characteristic
133	C₂	C₂×C₄×M₁₆	1	1	yes	characteristic
134	C₂	C₄×#131	1	1	yes	characteristic
135	C₂	C₂×#446	1	2	yes	—
136	C₂	#2965	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂³
commutator	C₂²	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #446	C₄	direct (N × H)
56. C₄×C₈	Q₈	semidirect
117. C₈	C₄×Q₈	semidirect
129. C₄	#446	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×#446	2
C₄²×Q₈	2
#1771	2
#1770	2
C₄×C₄⋊C₈	2
C₄×#97	2
C₄²×C₈	2

Subgroups — C4×#446 #20517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#446 #20517