Subgroups — #72234397

Order 2048.

450 conjugacy classes of subgroups.

The subgroup lattice has 134 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234395	C₂	1	4	yes	maximal, complemented
2	C₈×M₁₂₈	C₂	1	2	yes	maximal
3	C₂×#93664	C₂	1	1	yes	maximal, characteristic
4	C₄×M₁₂₈	C₄	1	4	yes	—
5	C₂×C₈×C₃₂	C₄	1	1	yes	characteristic
6	C₂×C₈×C₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₆²	C₄	1	1	yes	characteristic
8	#94504	C₂²	1	4	yes	—
9	C₈×C₆₄	C₄	1	4	yes	—
10	#93664	C₄	1	2	yes	—
11	#93664	C₂²	1	2	yes	—
12	C₂×M₁₂₈	C₈	1	8	yes	—
13	C₂×C₄×C₃₂	C₈	1	2	yes	—
14	C₂×C₄×C₃₂	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₁₆	C₈	1	2	yes	—
16	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
17	#7510	C₈	1	8	yes	—
18	C₄×C₆₄	C₂×C₄	1	8	yes	—
19	C₈×C₃₂	C₈	1	2	yes	—
20	C₈×C₃₂	C₂×C₄	1	2	yes	—
21	C₈×C₃₂	C₂×C₄	1	1	yes	characteristic
22	C₈×C₃₂	C₂×C₄	1	2	yes	—
23	C₈×C₃₂	C₂³	1	1	yes	characteristic
24	C₁₆²	C₈	1	2	yes	—
25	C₁₆²	C₂×C₄	1	2	yes	—
26	C₂²×C₃₂	C₁₆	1	4	yes	—
27	C₂²×C₃₂	C₂×C₈	1	4	yes	—
28	C₂×C₄×C₁₆	C₁₆	1	4	yes	—
29	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈²	C₂×C₈	1	1	yes	characteristic
32	M₁₂₈	—	2	16	no	—
33	C₂×C₆₄	C₂×C₈	1	16	yes	—
34	C₄×C₃₂	C₁₆	1	4	yes	—
35	C₄×C₃₂	C₄²	1	2	yes	—
36	C₄×C₃₂	C₂×C₈	1	4	yes	—
37	C₄×C₃₂	C₂×C₈	1	4	yes	—
38	C₄×C₃₂	C₂²×C₄	1	2	yes	—
39	C₈×C₁₆	C₁₆	1	4	yes	—
40	C₈×C₁₆	C₄²	1	1	yes	characteristic
41	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
42	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
43	C₈×C₁₆	C₂×C₈	1	4	yes	—
44	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₁₆	—	2	4	no	—
46	C₂²×C₁₆	C₄×C₈	1	2	yes	—
47	C₂²×C₁₆	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₈	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₁₆	1	2	yes	—
50	C₂×C₃₂	C₃₂	1	16	yes	—
51	C₂×C₃₂	C₄×C₈	1	4	yes	—
52	C₂×C₃₂	C₂×C₁₆	1	8	yes	—
53	C₂×C₃₂	M₃₂	1	16	yes	—
54	C₂×C₃₂	C₂²×C₈	1	4	yes	—
55	C₄×C₁₆	—	2	4	no	—
56	C₄×C₁₆	C₄×C₈	1	2	yes	—
57	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
58	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
59	C₄×C₁₆	C₂×C₁₆	1	2	yes	—
60	C₄×C₁₆	C₂×C₁₆	1	8	yes	—
61	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
63	C₈²	C₄×C₈	1	1	yes	characteristic
64	C₈²	C₂×C₁₆	1	2	yes	—
65	C₈²	C₂²×C₈	1	1	yes	characteristic
66	C₆₄	—	2	16	no	—
67	C₂²×C₈	C₈²	1	1	yes	characteristic
68	C₂²×C₈	C₄×C₁₆	1	1	yes	characteristic
69	C₂²×C₈	M₆₄	1	4	yes	—
70	C₂×C₄²	C₄×C₁₆	1	1	yes	characteristic
71	C₂×C₁₆	—	2	8	no	—
72	C₂×C₁₆	—	4	8	no	—
73	C₂×C₁₆	C₈²	1	2	yes	—
74	C₂×C₁₆	C₄×C₁₆	1	2	yes	—
75	C₂×C₁₆	#346	1	8	yes	—
76	C₂×C₁₆	C₂×C₃₂	1	8	yes	—
77	C₂×C₁₆	C₂×C₄×C₈	1	2	yes	—
78	C₂×C₁₆	C₂²×C₁₆	1	2	yes	—
79	C₄×C₈	C₈²	1	1	yes	characteristic
80	C₄×C₈	C₄×C₁₆	1	1	yes	characteristic
81	C₄×C₈	C₄×C₁₆	1	1	yes	characteristic
82	C₄×C₈	C₄×C₁₆	1	2	yes	—
83	C₄×C₈	M₆₄	1	4	yes	—
84	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
85	C₄×C₈	C₂²×C₁₆	1	2	yes	—
86	C₃₂	#346	1	8	yes	—
87	C₃₂	C₂×C₃₂	1	16	yes	—
88	C₃₂	C₂×M₃₂	1	8	yes	—
89	C₂²×C₄	C₈×C₁₆	1	1	yes	characteristic
90	C₂²×C₄	#1397	1	2	yes	—
91	C₂×C₈	—	2	8	no	—
92	C₂×C₈	C₈×C₁₆	1	1	yes	characteristic
93	C₂×C₈	C₈×C₁₆	1	1	yes	characteristic
94	C₂×C₈	#1311	1	4	yes	—
95	C₂×C₈	C₄×C₃₂	1	4	yes	—
96	C₂×C₈	#1397	1	4	yes	—
97	C₂×C₈	C₂×C₈²	1	1	yes	characteristic
98	C₂×C₈	C₂×C₄×C₁₆	1	1	yes	characteristic
99	C₂×C₈	C₂×M₆₄	1	4	yes	—
100	C₄²	C₈×C₁₆	1	1	yes	characteristic
101	C₄²	#1397	1	2	yes	—
102	C₄²	C₂×C₄×C₁₆	1	1	yes	characteristic
103	C₁₆	—	4	8	no	—
104	C₁₆	#1311	1	4	yes	—
105	C₁₆	C₄×C₃₂	1	4	yes	—
106	C₁₆	C₂×#346	1	4	yes	—
107	C₁₆	C₂²×C₃₂	1	4	yes	—
108	C₂³	#7329	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	#7052	1	2	yes	—
111	C₂×C₄	C₈×C₃₂	1	2	yes	—
112	C₂×C₄	#7329	1	2	yes	—
113	C₂×C₄	#7329	1	1	yes	characteristic
114	C₂×C₄	C₂×C₈×C₁₆	1	1	yes	characteristic
115	C₂×C₄	C₂×#1397	1	2	yes	—
116	C₈	—	2	4	no	—
117	C₈	#7052	1	2	yes	—
118	C₈	C₈×C₃₂	1	2	yes	—
119	C₈	#7511	1	8	yes	—
120	C₈	C₂×#1311	1	2	yes	—
121	C₈	C₂×C₄×C₃₂	1	2	yes	—
122	C₂²	—	2	2	no	—
123	C₂²	#93665	1	2	yes	—
124	C₂²	C₂×#7329	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	#93665	1	2	yes	—
127	C₄	C₂×#7052	1	1	yes	characteristic
128	C₄	C₂×C₈×C₃₂	1	1	yes	characteristic
129	C₄	C₂×#7511	1	4	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#93665	1	1	yes	characteristic
132	C₂	#72234396	1	2	yes	—

	subgroup	G/H
center	C₈×C₁₆	C₄²
commutator	C₄	C₂×C₈×C₃₂
Frattini, cosocle	C₈×C₃₂	C₂³
socle	C₂²	C₂×#7329
prime radical	C₂³	#7329
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72234395	C₂	semidirect

Maximal subgroups

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

subgroup	index
#72234395	2
C₈×M₁₂₈	2
C₂×#93664	2