Subgroups — #17335

Order 256.

407 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2965	C₂	1	2	yes	maximal
2	C₄×#131	C₂	1	1	yes	maximal, characteristic
3	#1455	C₂	1	2	yes	maximal, complemented
4	#1453	C₂	1	2	yes	maximal, complemented
5	#1451	C₂	1	2	yes	maximal, complemented
6	#1450	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×M₁₆	C₂	1	4	yes	maximal, complemented
8	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131	C₄	1	2	yes	—
10	C₄×#41	C₂²	1	1	yes	characteristic
11	#446	C₄	1	4	yes	—
12	#436	C₄	1	4	yes	—
13	#435	C₄	1	8	yes	—
14	#431	C₂²	1	4	yes	—
15	#431	C₂²	1	2	yes	—
16	#405	C₄	1	4	yes	—
17	#405	C₄	1	2	yes	—
18	C₄×M₁₆	C₂²	1	1	yes	characteristic
19	C₄×M₁₆	C₂²	1	2	yes	—
20	C₄×M₁₆	C₂²	1	4	yes	—
21	C₂×C₄×C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	2	yes	—
24	#322	C₂²	1	4	yes	—
25	#322	C₂²	1	4	yes	—
26	#322	C₂²	1	2	yes	—
27	C₈²	C₂²	1	1	yes	characteristic
28	C₈²	C₂²	1	1	yes	characteristic
29	C₈²	C₂²	1	2	yes	—
30	C₈²	C₂²	1	2	yes	—
31	C₂×#41	C₂×C₄	1	1	yes	characteristic
32	#131	—	2	8	no	—
33	C₂×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×M₁₆	C₂×C₄	1	4	yes	—
35	C₂×M₁₆	C₂×C₄	1	8	yes	—
36	C₂²×C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₈	C₂×C₄	1	2	yes	—
39	C₄×Q₈	C₂×C₄	1	2	yes	—
40	C₄×D₄	C₂×C₄	1	4	yes	—
41	C₄×D₄	C₂×C₄	1	2	yes	—
42	#117	C₂×C₄	1	2	yes	—
43	#117	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₄²	C₂³	1	2	yes	—
46	C₄⋊C₈	C₂×C₄	1	8	yes	—
47	C₄⋊C₈	C₂×C₄	1	4	yes	—
48	C₂²⋊C₈	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈	C₂×C₄	1	8	yes	—
50	#97	C₂×C₄	1	2	yes	—
51	#97	C₂×C₄	1	4	yes	—
52	#97	C₂×C₄	1	8	yes	—
53	C₄×C₈	C₂×C₄	1	4	yes	—
54	C₄×C₈	C₂×C₄	1	2	yes	—
55	C₄×C₈	C₂×C₄	1	4	yes	—
56	C₄×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₈	C₂³	1	2	yes	—
58	C₄×C₈	C₂³	1	2	yes	—
59	C₄×C₈	C₂³	1	4	yes	—
60	C₄×C₈	C₂³	1	2	yes	—
61	C₄×C₈	C₂³	1	1	yes	characteristic
62	#41	—	2	4	no	—
63	C₂×Q₈	C₄²	1	1	yes	characteristic
64	C₂×D₄	C₄²	1	2	yes	—
65	C₂×D₄	C₄²	1	1	yes	characteristic
66	C₂²×C₄	C₂²×C₄	1	2	yes	—
67	C₂²×C₄	C₂²×C₄	1	4	yes	—
68	C₂²×C₄	C₂²×C₄	1	2	yes	—
69	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
70	M₁₆	—	2	4	no	—
71	M₁₆	—	2	8	no	—
72	M₁₆	—	2	16	no	—
73	C₂×C₈	—	2	8	no	—
74	C₂×C₈	—	2	8	no	—
75	C₂×C₈	M₁₆	1	8	yes	complemented
76	C₂×C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₈	C₂²×C₄	1	4	yes	—
78	C₂×C₈	C₂²×C₄	1	8	yes	—
79	C₂×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₈	C₂²×C₄	1	4	yes	—
82	C₄⋊C₄	C₄²	1	4	yes	—
83	C₄⋊C₄	C₄²	1	2	yes	—
84	C₂²⋊C₄	C₄²	1	4	yes	—
85	C₂²⋊C₄	C₄²	1	2	yes	—
86	C₄²	C₂²×C₄	1	1	yes	characteristic
87	C₄²	C₂²×C₄	1	2	yes	—
88	C₄²	C₂²×C₄	1	2	yes	—
89	C₄²	C₂²×C₄	1	4	yes	—
90	C₄²	C₂⁴	1	1	yes	characteristic
91	C₂³	C₂×C₄²	1	2	yes	—
92	C₂³	C₂×C₄²	1	1	yes	characteristic
93	Q₈	—	2	2	no	—
94	D₄	—	2	4	no	—
95	D₄	—	2	2	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	C₂×C₄²	1	2	yes	—
100	C₂×C₄	C₂×C₄²	1	4	yes	—
101	C₂×C₄	C₂×C₄²	1	2	yes	—
102	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄	C₂×M₁₆	1	4	yes	—
104	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄	C₂³×C₄	1	2	yes	—
106	C₈	—	2	4	no	—
107	C₈	—	2	4	no	—
108	C₈	—	2	8	no	—
109	C₈	C₂×M₁₆	1	4	yes	—
110	C₈	C₂×M₁₆	1	4	yes	—
111	C₈	#131	1	8	yes	complemented
112	C₂²	—	2	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	4	no	—
115	C₂²	C₄×M₁₆	1	2	yes	—
116	C₂²	C₂²×C₄²	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	4	no	—
120	C₄	C₄×M₁₆	1	2	yes	—
121	C₄	C₂²×M₁₆	1	2	yes	—
122	C₄	C₂×#131	1	4	yes	—
123	C₂	—	2	2	no	—
124	C₂	—	4	2	no	—
125	C₂	C₂×C₄×M₁₆	1	1	yes	characteristic
126	C₂	C₄×#131	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂³
commutator, socle	C₂²	C₂²×C₄²
Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1455	C₂	semidirect
4. #1453	C₂	semidirect
5. #1451	C₂	semidirect
6. #1450	C₂	semidirect
7. C₈×M₁₆	C₂	semidirect
8. C₂×C₈²	C₂	semidirect
75. C₂×C₈	M₁₆	semidirect
111. C₈	#131	semidirect

Maximal subgroups

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

subgroup	index
#2965	2
C₄×#131	2
#1455	2
#1453	2
#1451	2
#1450	2
C₈×M₁₆	2
C₂×C₈²	2