Subgroups — #60193

Order 256.

905 conjugacy classes of subgroups.

The subgroup lattice has 145 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3458	C₂	1	1	yes	maximal, characteristic
2	#2962	C₂	1	16	yes	maximal, complemented
3	#2961	C₂	1	4	yes	maximal, complemented
4	#2960	C₂	1	4	yes	maximal, complemented
5	#2905	C₂	1	1	yes	maximal, characteristic
6	C₂×#424	C₂	1	2	yes	maximal
7	#2873	C₂	1	2	yes	maximal
8	C₂×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆	C₂²	1	2	yes	—
10	C₂²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈	C₂²	1	1	yes	characteristic
12	#544	C₄	1	4	yes	—
13	#544	C₄	1	8	yes	—
14	#543	C₄	1	4	yes	—
15	#533	C₂²	1	4	yes	—
16	#533	C₂²	1	8	yes	—
17	C₂×#117	C₂²	1	1	yes	characteristic
18	C₂×#117	C₂²	1	2	yes	—
19	#446	C₂²	1	16	yes	complemented
20	#446	C₂²	1	32	yes	complemented
21	C₈×Q₈	C₂²	1	16	yes	complemented
22	#424	C₂²	1	4	yes	—
23	#424	C₂²	1	4	yes	—
24	#424	C₂²	1	8	yes	—
25	#424	C₂²	1	8	yes	—
26	#423	C₂²	1	4	yes	—
27	C₂×C₄⋊C₈	C₂²	1	8	yes	—
28	#405	C₂²	1	8	yes	—
29	#405	C₂²	1	8	yes	—
30	#405	C₂²	1	8	yes	—
31	C₄×M₁₆	C₂²	1	8	yes	—
32	C₂×#97	C₂²	1	2	yes	—
33	C₂×C₄×C₈	C₂²	1	2	yes	—
34	C₂³×C₄	C₂³	1	1	yes	characteristic
35	C₂×M₁₆	—	2	8	no	—
36	C₂×M₁₆	Q₈	1	8	yes	complemented
37	C₂×M₁₆	C₂³	1	4	yes	—
38	C₂×M₁₆	C₂³	1	4	yes	—
39	C₂×M₁₆	C₂³	1	4	yes	—
40	C₂²×C₈	—	2	4	no	—
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	4	yes	—
44	C₂²×C₈	C₂³	1	4	yes	—
45	C₂²×C₈	C₂³	1	2	yes	—
46	C₄⋊Q₈	C₂×C₄	1	8	yes	—
47	C₄⋊Q₈	C₂×C₄	1	16	yes	—
48	#125	C₂×C₄	1	8	yes	—
49	#125	C₂×C₄	1	16	yes	—
50	C₂²⋊Q₈	C₂×C₄	1	16	yes	—
51	C₂²⋊Q₈	C₂×C₄	1	32	yes	—
52	C₄×Q₈	C₂³	1	16	yes	—
53	#117	C₂×C₄	1	4	yes	—
54	#117	C₂×C₄	1	4	yes	—
55	#117	C₂×C₄	1	4	yes	—
56	#117	C₂³	1	4	yes	—
57	#117	C₂³	1	4	yes	—
58	#117	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
61	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄	C₂×C₄	1	4	yes	—
63	C₂×C₄²	C₂³	1	2	yes	—
64	C₂×C₄²	C₂³	1	4	yes	—
65	C₄⋊C₈	C₂³	1	16	yes	—
66	C₄⋊C₈	C₂³	1	16	yes	—
67	C₄⋊C₈	C₂³	1	16	yes	—
68	#97	C₂³	1	8	yes	—
69	#97	C₂³	1	16	yes	—
70	C₄×C₈	C₂³	1	8	yes	—
71	C₄×C₈	C₂³	1	16	yes	—
72	C₂⁴	C₂²×C₄	1	1	yes	characteristic
73	C₂×Q₈	C₂²×C₄	1	16	yes	—
74	C₂²×C₄	C₂²×C₄	1	4	yes	—
75	C₂²×C₄	C₂²×C₄	1	8	yes	—
76	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄	C₂²×C₄	1	2	yes	—
78	C₂²×C₄	C₂×Q₈	1	4	yes	—
79	C₂²×C₄	#41	1	4	yes	—
80	C₂²×C₄	C₂⁴	1	1	yes	characteristic
81	C₂²×C₄	C₂⁴	1	2	yes	—
82	M₁₆	—	2	8	no	—
83	M₁₆	—	2	8	no	—
84	M₁₆	C₂×Q₈	1	16	yes	—
85	C₂×C₈	—	2	8	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	8	no	—
89	C₂×C₈	C₂×Q₈	1	8	yes	—
90	C₂×C₈	C₂⁴	1	2	yes	—
91	C₂×C₈	C₂⁴	1	8	yes	—
92	C₂×C₈	C₂⁴	1	2	yes	—
93	C₂×C₈	C₂⁴	1	4	yes	—
94	C₄⋊C₄	C₂²×C₄	1	8	yes	—
95	C₄⋊C₄	C₂²×C₄	1	8	yes	—
96	C₄⋊C₄	C₂²×C₄	1	8	yes	—
97	C₄⋊C₄	C₂²×C₄	1	32	yes	—
98	C₄⋊C₄	C₂²×C₄	1	16	yes	—
99	C₂²⋊C₄	C₂²×C₄	1	8	yes	—
100	C₂²⋊C₄	C₂²×C₄	1	8	yes	—
101	C₂²⋊C₄	C₂²×C₄	1	8	yes	—
102	C₄²	C₂²×C₄	1	4	yes	—
103	C₄²	C₂²×C₄	1	8	yes	—
104	C₄²	C₂⁴	1	4	yes	—
105	C₄²	C₂⁴	1	8	yes	—
106	C₂³	—	4	2	no	—
107	C₂³	C₄×Q₈	1	4	yes	—
108	C₂³	C₂³×C₄	1	1	yes	characteristic
109	C₂³	C₂³×C₄	1	2	yes	—
110	Q₈	—	4	16	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	8	no	—
113	C₂×C₄	C₄×Q₈	1	8	yes	—
114	C₂×C₄	C₄×Q₈	1	4	yes	—
115	C₂×C₄	C₂³×C₄	1	8	yes	—
116	C₂×C₄	C₂³×C₄	1	16	yes	—
117	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄	C₂³×C₄	1	2	yes	—
119	C₂×C₄	C₂²×Q₈	1	2	yes	—
120	C₂×C₄	C₂²×Q₈	1	4	yes	—
121	C₂×C₄	C₂×#41	1	4	yes	—
122	C₂×C₄	C₂×#41	1	2	yes	—
123	C₂×C₄	C₂⁵	1	1	yes	characteristic
124	C₈	—	2	2	no	—
125	C₈	—	2	2	no	—
126	C₈	—	2	8	no	—
127	C₈	C₂²×Q₈	1	8	yes	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	4	4	no	—
131	C₂²	C₂×C₄×Q₈	1	2	yes	—
132	C₂²	C₂×C₄×Q₈	1	4	yes	—
133	C₂²	C₂⁴×C₄	1	1	yes	characteristic
134	C₄	—	2	8	no	—
135	C₄	—	2	16	no	—
136	C₄	C₂×C₄×Q₈	1	2	yes	—
137	C₄	C₂×C₄×Q₈	1	4	yes	—
138	C₄	C₂³×Q₈	1	1	yes	characteristic
139	C₄	C₂²×#41	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	—	2	4	no	—
142	C₂	C₂²×C₄×Q₈	1	1	yes	characteristic
143	C₂	#3573	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #2962	C₂	semidirect
3. #2961	C₂	semidirect
4. #2960	C₂	semidirect
19. #446	C₂²	semidirect
20. #446	C₂²	semidirect
21. C₈×Q₈	C₂²	semidirect
36. C₂×M₁₆	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
#2962	2
#2961	2
#2960	2
#2905	2
C₂×#424	2
#2873	2
C₂×#405	2