Subgroups — #60666

Order 256.

881 conjugacy classes of subgroups.

The subgroup lattice has 139 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3504	C₂	1	2	yes	maximal
2	#3166	C₂	1	6	yes	maximal, complemented
3	#3164	C₂	1	6	yes	maximal, complemented
4	#3163	C₂	1	6	yes	maximal, complemented
5	#3160	C₂	1	3	yes	maximal, complemented
6	C₂×#499	C₂	1	3	yes	maximal, complemented
7	#2913	C₂	1	2	yes	maximal
8	#2912	C₂	1	2	yes	maximal
9	C₄×#131	C₂	1	1	yes	maximal, characteristic
10	C₂×#131	C₂²	1	1	yes	characteristic
11	C₂×#131	C₂²	1	1	yes	characteristic
12	#557	—	2	4	no	—
13	#554	—	2	12	no	—
14	#544	C₂²	1	12	yes	—
15	#533	C₂²	1	6	yes	—
16	C₂×#125	C₂²	1	6	yes	—
17	#520	C₂²	1	4	yes	—
18	C₄×#41	C₂²	1	1	yes	characteristic
19	#501	C₂²	1	24	yes	—
20	#500	C₂²	1	6	yes	—
21	#499	C₂²	1	3	yes	—
22	#499	C₂²	1	6	yes	—
23	#499	C₂²	1	6	yes	—
24	#499	C₂²	1	6	yes	—
25	#499	C₂²	1	1	yes	characteristic
26	#499	C₂²	1	6	yes	—
27	#498	C₂²	1	6	yes	—
28	#428	C₂²	1	12	yes	—
29	#428	C₂²	1	12	yes	—
30	#427	C₂²	1	6	yes	—
31	#427	C₂²	1	6	yes	—
32	C₂×#107	C₂²	1	6	yes	—
33	C₂×#106	C₂²	1	6	yes	—
34	#405	C₂²	1	3	yes	—
35	#405	C₂²	1	3	yes	—
36	C₄×M₁₆	C₂²	1	3	yes	—
37	C₂×C₄×C₈	C₂²	1	3	yes	—
38	C₂×#41	C₂³	1	1	yes	characteristic
39	#131	Q₈	1	8	yes	complemented
40	#131	Q₈	1	8	yes	complemented
41	C₂×M₁₆	C₂³	1	3	yes	—
42	C₂×M₁₆	C₂³	1	3	yes	—
43	C₂²×C₈	C₂³	1	3	yes	—
44	C₂²×C₈	C₂³	1	3	yes	—
45	C₄⋊Q₈	—	2	12	no	—
46	C₄⋊Q₈	C₂³	1	12	yes	—
47	#125	—	2	12	no	—
48	#125	—	2	12	no	—
49	#125	C₂³	1	6	yes	—
50	#125	C₂³	1	12	yes	—
51	#125	C₂³	1	2	yes	—
52	C₂²⋊Q₈	—	2	12	no	—
53	C₂²⋊Q₈	—	2	24	no	—
54	C₄×Q₈	—	2	4	no	—
55	C₄×Q₈	—	2	12	no	—
56	C₄×Q₈	D₄	1	2	yes	—
57	C₄×D₄	—	2	12	no	—
58	C₄×D₄	D₄	1	6	yes	—
59	#117	C₂³	1	12	yes	—
60	#117	C₂³	1	3	yes	—
61	C₂×C₄⋊C₄	—	2	12	no	—
62	C₂×C₄⋊C₄	C₂³	1	12	yes	—
63	C₂×C₄²	C₂³	1	3	yes	—
64	#107	C₂³	1	6	yes	—
65	#107	C₂³	1	12	yes	—
66	#107	C₂³	1	2	yes	—
67	#107	C₂³	1	12	yes	—
68	#106	C₂³	1	6	yes	—
69	#106	C₂³	1	12	yes	—
70	#106	C₂³	1	2	yes	—
71	#106	C₂³	1	12	yes	—
72	#97	C₂³	1	6	yes	—
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₈	C₂³	1	3	yes	—
75	C₄×C₈	C₂³	1	3	yes	—
76	C₄×C₈	C₂³	1	3	yes	—
77	#41	C₂×Q₈	1	4	yes	—
78	#41	C₂×Q₈	1	4	yes	—
79	C₂×Q₈	—	2	12	no	—
80	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
81	C₂×D₄	C₂×D₄	1	3	yes	—
82	C₂²×C₄	—	2	12	no	—
83	C₂²×C₄	C₂×D₄	1	6	yes	—
84	C₂²×C₄	C₂⁴	1	3	yes	—
85	M₁₆	C₂×Q₈	1	12	yes	—
86	M₁₆	C₂×Q₈	1	12	yes	—
87	C₂×C₈	C₂×Q₈	1	12	yes	—
88	C₂×C₈	C₂×Q₈	1	12	yes	—
89	C₂×C₈	C₂⁴	1	1	yes	characteristic
90	C₂×C₈	C₂⁴	1	1	yes	characteristic
91	C₂×C₈	C₂⁴	1	3	yes	—
92	C₂×C₈	C₂⁴	1	3	yes	—
93	C₄⋊C₄	—	2	12	no	—
94	C₄⋊C₄	—	2	24	no	—
95	C₄⋊C₄	—	2	12	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	12	no	—
98	C₄⋊C₄	C₂×D₄	1	6	yes	—
99	C₄⋊C₄	C₂⁴	1	12	yes	—
100	C₄⋊C₄	C₂⁴	1	4	yes	—
101	C₂²⋊C₄	—	2	12	no	—
102	C₂²⋊C₄	C₂×D₄	1	6	yes	—
103	C₄²	—	2	12	no	—
104	C₄²	C₂×D₄	1	6	yes	—
105	C₄²	C₂⁴	1	1	yes	characteristic
106	C₄²	C₂⁴	1	3	yes	—
107	C₂³	C₂²×D₄	1	3	yes	—
108	Q₈	—	8	12	no	—
109	Q₈	C₄⋊Q₈	1	4	yes	—
110	D₄	C₄⋊Q₈	1	12	yes	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	12	no	—
113	C₂×C₄	—	4	6	no	—
114	C₂×C₄	—	8	12	no	—
115	C₂×C₄	C₂²×D₄	1	2	yes	—
116	C₂×C₄	C₂²×D₄	1	6	yes	—
117	C₂×C₄	C₂²×D₄	1	3	yes	—
118	C₂×C₄	C₂²×Q₈	1	6	yes	—
119	C₂×C₄	C₂²×Q₈	1	6	yes	—
120	C₂×C₄	C₂⁵	1	1	yes	characteristic
121	C₈	C₂²×Q₈	1	2	yes	—
122	C₈	C₂²×Q₈	1	2	yes	—
123	C₈	C₂²×Q₈	1	6	yes	—
124	C₈	C₂²×Q₈	1	6	yes	—
125	C₂²	—	2	6	no	—
126	C₂²	C₂×C₄⋊Q₈	1	6	yes	—
127	C₂²	C₂³×D₄	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	6	no	—
130	C₄	—	4	12	no	—
131	C₄	—	4	4	no	—
132	C₄	C₂×C₄⋊Q₈	1	6	yes	—
133	C₄	C₂³×Q₈	1	1	yes	characteristic
134	C₄	C₂³×Q₈	1	1	yes	characteristic
135	C₂	—	2	6	no	—
136	C₂	C₂²×C₄⋊Q₈	1	1	yes	characteristic
137	C₂	#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #3166	C₂	semidirect
3. #3164	C₂	semidirect
4. #3163	C₂	semidirect
5. #3160	C₂	semidirect
6. C₂×#499	C₂	semidirect
39. #131	Q₈	semidirect
40. #131	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3504	2
#3166	2
#3164	2
#3163	2
#3160	2
C₂×#499	2
#2913	2
#2912	2
C₄×#131	2