Subgroups — #72232012

Order 1024.

665 conjugacy classes of subgroups.

The subgroup lattice has 175 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512545	C₂	1	1	yes	maximal, characteristic
2	#483341	C₂	1	2	yes	maximal
3	#151038	C₂	1	6	yes	maximal, complemented
4	#151037	C₂	1	3	yes	maximal, complemented
5	C₂×#7463	C₂	1	3	yes	maximal, complemented
6	C₂×#2258	C₄	1	1	yes	characteristic
7	C₂×#2258	C₄	1	1	yes	characteristic
8	C₄×#504	C₂²	1	1	yes	characteristic
9	#17789	—	2	2	no	—
10	#13549	C₄	1	3	yes	—
11	#13549	C₄	1	3	yes	—
12	C₄×M₆₄	C₂²	1	3	yes	—
13	C₂×#1397	C₂²	1	3	yes	—
14	#12009	—	2	6	no	—
15	#12008	C₂²	1	6	yes	—
16	C₂×#1372	C₂²	1	6	yes	—
17	#7463	C₂²	1	3	yes	—
18	#7463	C₂²	1	6	yes	—
19	#7463	C₂²	1	1	yes	characteristic
20	#7463	C₂²	1	6	yes	—
21	C₂×#504	C₂×C₄	1	1	yes	characteristic
22	C₂×#504	Q₈	1	1	yes	characteristic
23	#2988	—	4	2	no	—
24	C₄×#131	D₄	1	1	yes	characteristic
25	#2258	—	2	4	no	—
26	#2258	C₈	1	8	yes	complemented
27	C₂×M₆₄	C₂×C₄	1	3	yes	—
28	C₂×M₆₄	C₂×C₄	1	3	yes	—
29	C₂²×C₃₂	C₂×C₄	1	3	yes	—
30	C₂²×C₃₂	C₂×C₄	1	3	yes	—
31	#2108	C₂×C₄	1	3	yes	—
32	#2108	Q₈	1	3	yes	—
33	C₄×M₃₂	D₄	1	3	yes	—
34	C₂×C₄×C₁₆	C₂³	1	3	yes	—
35	#1565	—	4	6	no	—
36	#1564	—	2	6	no	—
37	C₂×#335	—	2	6	no	—
38	#1397	C₂×C₄	1	3	yes	—
39	#1397	C₂×C₄	1	3	yes	—
40	#1397	C₂³	1	1	yes	characteristic
41	#1397	C₂³	1	3	yes	—
42	C₄×C₃₂	C₂×C₄	1	6	yes	—
43	#1372	—	2	6	no	—
44	#1372	C₂³	1	6	yes	—
45	#1372	C₂³	1	2	yes	—
46	#1371	—	2	6	no	—
47	C₂×#131	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#131	Q₁₆	1	1	yes	characteristic
49	C₄×#41	D₈	1	1	yes	characteristic
50	#504	—	2	4	no	—
51	#504	C₂×C₈	1	4	yes	—
52	#504	M₁₆	1	4	yes	—
53	C₂×M₃₂	C₄⋊C₄	1	3	yes	—
54	C₂×M₃₂	C₄⋊C₄	1	3	yes	—
55	C₂²×C₁₆	C₂²×C₄	1	3	yes	—
56	C₂²×C₁₆	C₂×Q₈	1	3	yes	—
57	C₈×Q₈	—	8	2	no	—
58	C₈×D₄	—	8	6	no	—
59	#431	—	4	6	no	—
60	C₂×C₄⋊C₈	—	4	6	no	—
61	#405	C₄⋊C₄	1	3	yes	—
62	#405	Q₁₆	1	3	yes	—
63	C₄×M₁₆	D₈	1	3	yes	—
64	C₂×C₄×C₈	C₂×D₄	1	3	yes	—
65	M₆₄	—	2	6	no	—
66	M₆₄	C₂×C₈	1	12	yes	complemented
67	C₂×C₃₂	—	2	6	no	—
68	C₂×C₃₂	C₂×C₈	1	12	yes	complemented
69	C₂×C₃₂	C₂²×C₄	1	3	yes	—
70	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃₂	C₂²×C₄	1	3	yes	—
72	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
73	#346	C₄⋊C₄	1	6	yes	—
74	C₄×C₁₆	C₂²×C₄	1	3	yes	—
75	C₄×C₁₆	C₂×D₄	1	3	yes	—
76	C₄×C₁₆	C₂×Q₈	1	3	yes	—
77	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
78	#335	—	2	6	no	—
79	#335	—	2	2	no	—
80	#335	—	4	6	no	—
81	#334	—	4	6	no	—
82	C₂×#41	#107	1	1	yes	characteristic
83	#131	—	2	4	no	—
84	#131	C₄⋊C₈	1	4	yes	—
85	#131	#108	1	4	yes	—
86	C₂×M₁₆	#107	1	3	yes	—
87	C₂×M₁₆	#107	1	3	yes	—
88	C₂²×C₈	—	8	6	no	—
89	C₂²×C₈	C₂×C₄⋊C₄	1	3	yes	—
90	C₂²×C₈	C₂×Q₁₆	1	3	yes	—
91	C₄×Q₈	SD₃₂	1	2	yes	—
92	C₄×D₄	SD₃₂	1	6	yes	—
93	#117	#107	1	3	yes	—
94	C₂×C₄²	C₂×D₈	1	3	yes	—
95	M₃₂	—	2	6	no	—
96	M₃₂	C₄⋊C₈	1	12	yes	—
97	C₂×C₁₆	—	2	6	no	—
98	C₂×C₁₆	C₂×C₄⋊C₄	1	3	yes	—
99	C₂×C₁₆	C₂×C₄⋊C₄	1	3	yes	—
100	C₂×C₁₆	C₂²×C₈	1	6	yes	—
101	C₂×C₁₆	C₂×M₁₆	1	6	yes	—
102	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₁₆	C₂²×Q₈	1	1	yes	characteristic
104	C₄⋊C₈	—	4	2	no	—
105	C₄⋊C₈	—	4	6	no	—
106	C₄⋊C₈	—	8	6	no	—
107	C₂²⋊C₈	—	8	6	no	—
108	#97	#107	1	6	yes	—
109	C₄×C₈	—	8	6	no	—
110	C₄×C₈	C₂×C₄⋊C₄	1	3	yes	—
111	C₄×C₈	C₂×D₈	1	3	yes	—
112	C₄×C₈	C₂×Q₁₆	1	3	yes	—
113	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
114	C₃₂	—	2	1	no	—
115	C₃₂	—	2	3	no	—
116	C₃₂	C₂²×C₈	1	2	yes	—
117	C₃₂	C₂²×C₈	1	6	yes	—
118	#41	#335	1	4	yes	—
119	#41	#368	1	4	yes	—
120	C₂×Q₈	#367	1	1	yes	characteristic
121	C₂×D₄	#367	1	3	yes	—
122	C₂²×C₄	C₂×#107	1	3	yes	—
123	C₂²×C₄	C₂×SD₃₂	1	6	yes	—
124	M₁₆	—	2	6	no	—
125	M₁₆	#335	1	12	yes	—
126	C₂×C₈	—	2	6	no	—
127	C₂×C₈	—	8	2	no	—
128	C₂×C₈	—	8	6	no	—
129	C₂×C₈	—	32	6	no	—
130	C₂×C₈	C₂×C₄⋊C₈	1	6	yes	—
131	C₂×C₈	C₂×#107	1	3	yes	—
132	C₂×C₈	C₂×#107	1	3	yes	—
133	C₂×C₈	C₂×#108	1	6	yes	—
134	C₂×C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₈	C₂²×Q₁₆	1	1	yes	characteristic
136	C₄⋊C₄	#367	1	6	yes	—
137	C₂²⋊C₄	#367	1	6	yes	—
138	C₄²	C₂×#107	1	3	yes	—
139	C₄²	C₂×SD₃₂	1	6	yes	—
140	C₄²	C₂²×D₈	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	2	3	no	—
143	C₁₆	C₂×C₄⋊C₈	1	6	yes	—
144	C₁₆	C₂³×C₈	1	1	yes	characteristic
145	C₁₆	C₂²×M₁₆	1	1	yes	characteristic
146	C₂³	C₂×#367	1	3	yes	—
147	Q₈	#1372	1	4	yes	—
148	D₄	#1372	1	12	yes	—
149	C₂×C₄	—	4	6	no	—
150	C₂×C₄	C₂×#335	1	6	yes	—
151	C₂×C₄	C₂×#367	1	6	yes	—
152	C₂×C₄	C₂×#367	1	3	yes	—
153	C₂×C₄	C₂×#368	1	6	yes	—
154	C₂×C₄	C₂²×#107	1	1	yes	characteristic
155	C₂×C₄	C₂²×SD₃₂	1	2	yes	—
156	C₈	—	2	1	no	—
157	C₈	—	2	3	no	—
158	C₈	—	16	2	no	—
159	C₈	—	16	6	no	—
160	C₈	C₂×#335	1	6	yes	—
161	C₈	C₂²×C₄⋊C₈	1	1	yes	characteristic
162	C₈	C₂²×#108	1	1	yes	characteristic
163	C₂²	—	2	6	no	—
164	C₂²	C₂×#1372	1	6	yes	—
165	C₂²	C₂²×#367	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	6	no	—
168	C₄	C₂×#1372	1	6	yes	—
169	C₄	C₂²×#335	1	1	yes	characteristic
170	C₄	C₂²×#368	1	1	yes	characteristic
171	C₂	—	2	6	no	—
172	C₂	C₂²×#1372	1	1	yes	characteristic
173	C₂	#512724	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×#367
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂×#41	#107
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ C₄×#504 ◁ 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. #151038	C₂	semidirect
4. #151037	C₂	semidirect
5. C₂×#7463	C₂	semidirect
26. #2258	C₈	semidirect
66. M₆₄	C₂×C₈	semidirect
68. C₂×C₃₂	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
#512545	2
#483341	2
#151038	2
#151037	2
C₂×#7463	2