Subgroups — #72231876

Order 1024.

885 conjugacy classes of subgroups.

The subgroup lattice has 165 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512545	C₂	1	2	yes	maximal
2	#479899	C₂	1	1	yes	maximal, characteristic
3	#149502	C₂	1	3	yes	maximal, complemented
4	#149501	C₂	1	6	yes	maximal, complemented
5	C₂×#7332	C₂	1	3	yes	maximal, complemented
6	C₂×#2258	C₄	1	4	yes	—
7	C₄×#504	C₄	1	1	yes	characteristic
8	C₄×#504	C₂²	1	1	yes	characteristic
9	C₈×#131	C₄	1	1	yes	characteristic
10	#13549	C₄	1	12	yes	—
11	C₄×M₆₄	C₂²	1	6	yes	—
12	C₂×#1397	C₂²	1	6	yes	—
13	#11399	C₄	1	3	yes	—
14	#11399	C₂²	1	3	yes	—
15	C₈×M₃₂	C₄	1	3	yes	—
16	C₂×#1311	C₂²	1	3	yes	—
17	#7332	C₂²	1	1	yes	characteristic
18	#7332	C₂²	1	6	yes	—
19	#7332	C₂²	1	3	yes	—
20	#7332	C₂²	1	6	yes	—
21	C₂×#504	C₈	1	2	yes	—
22	C₂×#504	C₂×C₄	1	2	yes	—
23	C₈×#41	C₈	1	2	yes	—
24	C₄×#131	C₂×C₄	1	1	yes	characteristic
25	#2258	—	2	16	no	—
26	C₂×M₆₄	C₂×C₄	1	12	yes	—
27	C₂²×C₃₂	C₂×C₄	1	12	yes	—
28	#2108	C₈	1	6	yes	—
29	#2108	C₂×C₄	1	6	yes	—
30	C₄×M₃₂	C₂×C₄	1	3	yes	—
31	C₄×M₃₂	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₁₆	C₂×C₄	1	3	yes	—
33	C₂×C₄×C₁₆	C₂³	1	3	yes	—
34	#1450	C₂×C₄	1	3	yes	—
35	C₈×M₁₆	C₈	1	6	yes	—
36	C₂×C₈²	C₂×C₄	1	3	yes	—
37	#1397	C₂×C₄	1	12	yes	—
38	#1397	C₂³	1	2	yes	—
39	#1397	C₂³	1	6	yes	—
40	C₄×C₃₂	C₂×C₄	1	12	yes	—
41	#1311	C₂×C₄	1	3	yes	—
42	#1311	C₂×C₄	1	3	yes	—
43	#1311	C₂³	1	1	yes	characteristic
44	#1311	C₂³	1	3	yes	—
45	C₈×C₁₆	C₂×C₄	1	6	yes	—
46	C₂×#131	C₄²	1	1	yes	characteristic
47	C₂×#131	C₂×C₈	1	1	yes	characteristic
48	C₄×#41	C₂×C₈	1	1	yes	characteristic
49	#504	C₁₆	1	16	yes	—
50	#504	M₁₆	1	16	yes	—
51	C₂×M₃₂	C₄²	1	6	yes	—
52	C₂×M₃₂	C₂×C₈	1	6	yes	—
53	C₂²×C₁₆	C₂×C₈	1	6	yes	—
54	C₂²×C₁₆	C₂²×C₄	1	6	yes	—
55	C₈×Q₈	—	2	2	no	—
56	C₈×D₄	—	2	6	no	—
57	#431	C₂×C₈	1	6	yes	—
58	#405	C₄²	1	3	yes	—
59	#405	C₂×C₈	1	3	yes	—
60	C₄×M₁₆	C₂×C₈	1	3	yes	—
61	C₂×C₄×C₈	C₂×C₈	1	6	yes	—
62	C₂×C₄×C₈	C₂²×C₄	1	3	yes	—
63	M₆₄	—	2	24	no	—
64	C₂×C₃₂	—	2	24	no	—
65	C₂×C₃₂	C₂²×C₄	1	12	yes	—
66	C₂×C₃₂	C₂²×C₄	1	4	yes	—
67	#346	C₄²	1	6	yes	—
68	#346	C₂×C₈	1	6	yes	—
69	C₄×C₁₆	C₂×C₈	1	6	yes	—
70	C₄×C₁₆	C₂²×C₄	1	6	yes	—
71	C₄×C₁₆	C₂²×C₄	1	3	yes	—
72	C₄×C₁₆	C₂²×C₄	1	3	yes	—
73	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
74	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
75	#322	C₂×C₈	1	6	yes	—
76	C₈²	C₂×C₈	1	6	yes	—
77	C₈²	C₂²×C₄	1	3	yes	—
78	C₈²	C₂²×C₄	1	1	yes	characteristic
79	C₂×#41	C₄×C₈	1	1	yes	characteristic
80	#131	#97	1	8	yes	—
81	#131	C₂×C₁₆	1	8	yes	—
82	C₂×M₁₆	C₄×C₈	1	3	yes	—
83	C₂×M₁₆	C₄×C₈	1	3	yes	—
84	C₂²×C₈	—	2	6	no	—
85	C₂²×C₈	C₂×C₄²	1	3	yes	—
86	C₂²×C₈	C₂²×C₈	1	3	yes	—
87	C₄×Q₈	M₃₂	1	2	yes	—
88	C₄×D₄	M₃₂	1	6	yes	—
89	#117	C₄×C₈	1	3	yes	—
90	C₂×C₄²	C₂²×C₈	1	3	yes	—
91	M₃₂	#97	1	24	yes	—
92	M₃₂	C₂×C₁₆	1	24	yes	—
93	C₂×C₁₆	C₂×C₁₆	1	24	yes	—
94	C₂×C₁₆	C₂×C₄²	1	6	yes	—
95	C₂×C₁₆	C₂²×C₈	1	2	yes	—
96	C₂×C₁₆	C₂²×C₈	1	6	yes	—
97	C₂×C₁₆	C₂×M₁₆	1	24	yes	—
98	C₂×C₁₆	C₂³×C₄	1	2	yes	—
99	C₄⋊C₈	—	2	6	no	—
100	C₂²⋊C₈	—	2	6	no	—
101	#97	C₄×C₈	1	6	yes	—
102	C₄×C₈	—	2	6	no	—
103	C₄×C₈	C₂×C₄²	1	3	yes	—
104	C₄×C₈	C₂²×C₈	1	2	yes	—
105	C₄×C₈	C₂²×C₈	1	6	yes	—
106	C₄×C₈	C₂²×C₈	1	3	yes	—
107	C₄×C₈	C₂²×C₈	1	3	yes	—
108	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
109	C₃₂	—	2	4	no	—
110	C₃₂	—	2	12	no	—
111	#41	#322	1	4	yes	—
112	#41	C₄×C₁₆	1	4	yes	—
113	C₂×Q₈	#346	1	1	yes	characteristic
114	C₂×D₄	#346	1	3	yes	—
115	C₂²×C₄	C₂×C₄×C₈	1	3	yes	—
116	C₂²×C₄	C₂×M₃₂	1	6	yes	—
117	M₁₆	#322	1	12	yes	—
118	M₁₆	C₄×C₁₆	1	12	yes	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	6	no	—
121	C₂×C₈	—	8	6	no	—
122	C₂×C₈	C₂×C₄×C₈	1	3	yes	—
123	C₂×C₈	C₂×C₄×C₈	1	3	yes	—
124	C₂×C₈	C₂×#97	1	12	yes	—
125	C₂×C₈	C₂²×C₁₆	1	12	yes	—
126	C₂×C₈	C₂²×C₄²	1	1	yes	characteristic
127	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
128	C₄⋊C₄	#346	1	6	yes	—
129	C₂²⋊C₄	#346	1	6	yes	—
130	C₄²	C₂×C₄×C₈	1	3	yes	—
131	C₄²	C₂×M₃₂	1	6	yes	—
132	C₄²	C₂³×C₈	1	1	yes	characteristic
133	C₁₆	C₂×#97	1	12	yes	—
134	C₁₆	C₂²×C₁₆	1	4	yes	—
135	C₁₆	C₂²×C₁₆	1	12	yes	—
136	C₁₆	C₂²×M₁₆	1	4	yes	—
137	C₂³	C₂×#346	1	3	yes	—
138	Q₈	#1312	1	4	yes	—
139	D₄	#1312	1	12	yes	—
140	C₂×C₄	—	4	6	no	—
141	C₂×C₄	C₂×#322	1	6	yes	—
142	C₂×C₄	C₂×C₄×C₁₆	1	6	yes	—
143	C₂×C₄	C₂×#346	1	6	yes	—
144	C₂×C₄	C₂×#346	1	3	yes	—
145	C₂×C₄	C₂²×C₄×C₈	1	1	yes	characteristic
146	C₂×C₄	C₂²×M₃₂	1	2	yes	—
147	C₈	—	4	6	no	—
148	C₈	—	4	2	no	—
149	C₈	C₂×#322	1	6	yes	—
150	C₈	C₂×C₄×C₁₆	1	6	yes	—
151	C₈	C₂²×#97	1	2	yes	—
152	C₈	C₂³×C₁₆	1	2	yes	—
153	C₂²	—	2	6	no	—
154	C₂²	C₂×#1312	1	6	yes	—
155	C₂²	C₂²×#346	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	2	6	no	—
158	C₄	C₂×#1312	1	6	yes	—
159	C₄	C₂²×#322	1	1	yes	characteristic
160	C₄	C₂²×C₄×C₁₆	1	1	yes	characteristic
161	C₂	—	2	6	no	—
162	C₂	C₂²×#1312	1	1	yes	characteristic
163	C₂	#512548	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₄	C₂²×C₄×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂×#41	C₄×C₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #149502	C₂	semidirect
4. #149501	C₂	semidirect
5. C₂×#7332	C₂	semidirect

Maximal subgroups

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

subgroup	index
#512545	2
#479899	2
#149502	2
#149501	2
C₂×#7332	2