Subgroups — #60660

Order 256.

885 conjugacy classes of subgroups.

The subgroup lattice has 143 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3458	C₂	1	2	yes	maximal
2	#3164	C₂	1	8	yes	maximal, complemented
3	#3163	C₂	1	8	yes	maximal, complemented
4	#3162	C₂	1	4	yes	maximal, complemented
5	#3160	C₂	1	4	yes	maximal, complemented
6	#2911	C₂	1	2	yes	maximal
7	C₂×#427	C₂	1	2	yes	maximal
8	C₂×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈	C₂²	1	1	yes	characteristic
11	#544	—	2	4	no	—
12	#544	C₂²	1	16	yes	—
13	#543	—	2	4	no	—
14	#533	—	2	8	no	—
15	#533	C₂²	1	8	yes	—
16	C₂×#117	C₂²	1	4	yes	—
17	C₂×#117	C₂²	1	1	yes	characteristic
18	#501	C₂²	1	32	yes	complemented
19	#500	C₂²	1	8	yes	complemented
20	#499	C₂²	1	8	yes	complemented
21	#499	C₂²	1	8	yes	complemented
22	#498	C₂²	1	8	yes	complemented
23	#428	C₂²	1	16	yes	—
24	#427	C₂²	1	4	yes	—
25	#427	C₂²	1	8	yes	—
26	#427	C₂²	1	4	yes	—
27	#427	C₂²	1	4	yes	—
28	#427	C₂²	1	4	yes	—
29	C₂×#107	C₂²	1	4	yes	—
30	C₂×#106	C₂²	1	4	yes	—
31	#405	C₂²	1	4	yes	—
32	#405	C₂²	1	4	yes	—
33	C₂×#97	C₂²	1	2	yes	—
34	C₂×C₄×C₈	C₂²	1	2	yes	—
35	C₂³×C₄	C₂³	1	1	yes	characteristic
36	C₂×M₁₆	Q₈	1	8	yes	complemented
37	C₂×M₁₆	C₂³	1	4	yes	—
38	C₂²×C₈	Q₈	1	8	yes	complemented
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	2	yes	—
42	C₂²×C₈	C₂³	1	2	yes	—
43	C₂²×C₈	C₂³	1	2	yes	—
44	C₄⋊Q₈	—	2	16	no	—
45	C₄⋊Q₈	C₂³	1	16	yes	—
46	#125	—	2	16	no	—
47	#125	C₂³	1	16	yes	—
48	C₂²⋊Q₈	—	2	16	no	—
49	C₂²⋊Q₈	—	2	32	no	—
50	C₄×Q₈	—	2	16	no	—
51	#117	—	2	4	no	—
52	#117	—	2	4	no	—
53	#117	D₄	1	4	yes	—
54	#117	C₂³	1	8	yes	—
55	#117	C₂³	1	8	yes	—
56	#117	C₂³	1	4	yes	—
57	C₂×C₄⋊C₄	D₄	1	2	yes	—
58	C₂×C₄⋊C₄	C₂³	1	8	yes	—
59	C₂×C₂²⋊C₄	—	2	4	no	—
60	C₂×C₂²⋊C₄	D₄	1	2	yes	—
61	C₂×C₄²	—	2	4	no	—
62	C₂×C₄²	C₂³	1	2	yes	—
63	#107	C₂³	1	8	yes	—
64	#107	C₂³	1	8	yes	—
65	#107	C₂³	1	16	yes	—
66	#106	C₂³	1	8	yes	—
67	#106	C₂³	1	8	yes	—
68	#106	C₂³	1	16	yes	—
69	#97	C₂³	1	8	yes	—
70	C₄×C₈	C₂³	1	4	yes	—
71	C₄×C₈	C₂³	1	4	yes	—
72	C₂⁴	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈	—	2	16	no	—
74	C₂²×C₄	—	2	8	no	—
75	C₂²×C₄	C₂×D₄	1	4	yes	—
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄	1	2	yes	—
78	C₂²×C₄	C₂×Q₈	1	4	yes	—
79	C₂²×C₄	C₂×Q₈	1	4	yes	—
80	C₂²×C₄	C₂⁴	1	2	yes	—
81	C₂²×C₄	C₂⁴	1	1	yes	characteristic
82	M₁₆	C₂×Q₈	1	16	yes	—
83	C₂×C₈	C₂×Q₈	1	4	yes	—
84	C₂×C₈	C₂×Q₈	1	4	yes	—
85	C₂×C₈	C₂×Q₈	1	8	yes	—
86	C₂×C₈	C₂×Q₈	1	8	yes	—
87	C₂×C₈	C₂×Q₈	1	8	yes	—
88	C₂×C₈	C₂⁴	1	1	yes	characteristic
89	C₂×C₈	C₂⁴	1	1	yes	characteristic
90	C₂×C₈	C₂⁴	1	4	yes	—
91	C₂×C₈	C₂⁴	1	2	yes	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	2	16	no	—
94	C₄⋊C₄	—	2	32	no	—
95	C₄⋊C₄	C₂×D₄	1	8	yes	—
96	C₄⋊C₄	C₂⁴	1	16	yes	—
97	C₂²⋊C₄	—	2	8	no	—
98	C₂²⋊C₄	—	2	8	no	—
99	C₂²⋊C₄	C₂×D₄	1	8	yes	—
100	C₄²	—	2	8	no	—
101	C₄²	—	2	8	no	—
102	C₄²	C₂×D₄	1	4	yes	—
103	C₄²	C₂⁴	1	4	yes	—
104	C₂³	—	4	2	no	—
105	C₂³	C₄⋊Q₈	1	4	yes	—
106	C₂³	C₂²×D₄	1	1	yes	characteristic
107	C₂³	C₂²×D₄	1	2	yes	—
108	Q₈	—	8	16	no	—
109	C₂×C₄	—	2	16	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	8	8	no	—
112	C₂×C₄	C₄⋊Q₈	1	4	yes	—
113	C₂×C₄	C₄⋊Q₈	1	8	yes	—
114	C₂×C₄	C₂²×D₄	1	8	yes	—
115	C₂×C₄	C₂²×D₄	1	2	yes	—
116	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄	C₂²×Q₈	1	4	yes	—
118	C₂×C₄	C₂²×Q₈	1	2	yes	—
119	C₂×C₄	C₂²×Q₈	1	2	yes	—
120	C₂×C₄	C₂²×Q₈	1	4	yes	—
121	C₂×C₄	C₂⁵	1	1	yes	characteristic
122	C₈	C₂²×Q₈	1	2	yes	—
123	C₈	C₂²×Q₈	1	2	yes	—
124	C₈	C₂²×Q₈	1	4	yes	—
125	C₈	C₂²×Q₈	1	8	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	4	4	no	—
129	C₂²	C₂×C₄⋊Q₈	1	2	yes	—
130	C₂²	C₂×C₄⋊Q₈	1	4	yes	—
131	C₂²	C₂³×D₄	1	1	yes	characteristic
132	C₄	—	2	8	no	—
133	C₄	—	4	16	no	—
134	C₄	C₂×C₄⋊Q₈	1	4	yes	—
135	C₄	C₂×C₄⋊Q₈	1	2	yes	—
136	C₄	C₂³×Q₈	1	1	yes	characteristic
137	C₄	C₂³×Q₈	1	1	yes	characteristic
138	C₂	—	2	2	no	—
139	C₂	—	2	4	no	—
140	C₂	C₂²×C₄⋊Q₈	1	1	yes	characteristic
141	C₂	#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂²	C₂³×D₄
prime radical	C₂⁴	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #3164	C₂	semidirect
3. #3163	C₂	semidirect
4. #3162	C₂	semidirect
5. #3160	C₂	semidirect
18. #501	C₂²	semidirect
19. #500	C₂²	semidirect
20. #499	C₂²	semidirect
21. #499	C₂²	semidirect
22. #498	C₂²	semidirect
36. C₂×M₁₆	Q₈	semidirect
38. C₂²×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3458	2
#3164	2
#3163	2
#3162	2
#3160	2
#2911	2
C₂×#427	2
C₂×#405	2