Subgroups — #72231965

Order 1024.

653 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512789	C₂	1	2	yes	maximal
2	#479943	C₂	1	1	yes	maximal, characteristic
3	#149513	C₂	1	3	yes	maximal, complemented
4	#149509	C₂	1	6	yes	maximal, complemented
5	#149502	C₂	1	3	yes	maximal, complemented
6	C₁₆×#41	C₄	1	1	yes	characteristic
7	C₁₆×#41	C₂²	1	1	yes	characteristic
8	C₈×#131	C₄	1	1	yes	characteristic
9	Q₈×C₃₂	C₄	1	4	yes	—
10	#13646	C₄	1	12	yes	—
11	#13640	C₂²	1	6	yes	—
12	C₄×M₆₄	C₂²	1	6	yes	—
13	#11411	C₄	1	3	yes	—
14	#11408	C₄	1	3	yes	—
15	#11408	C₂²	1	3	yes	—
16	C₂×#1311	C₂²	1	3	yes	—
17	#7332	C₂²	1	6	yes	—
18	#7332	C₂²	1	3	yes	—
19	#7332	C₂²	1	1	yes	characteristic
20	#7330	C₂²	1	6	yes	—
21	C₈×#41	C₈	1	1	yes	characteristic
22	C₈×#41	C₂×C₄	1	1	yes	characteristic
23	C₄×#131	C₈	1	1	yes	characteristic
24	C₂×M₆₄	C₂×C₄	1	12	yes	—
25	Q₈×C₁₆	C₈	1	2	yes	—
26	Q₈×C₁₆	C₂×C₄	1	2	yes	—
27	D₄×C₁₆	C₈	1	6	yes	—
28	D₄×C₁₆	C₂×C₄	1	6	yes	—
29	#2162	C₂×C₄	1	3	yes	—
30	#2162	C₂×C₄	1	3	yes	—
31	C₂×C₄×C₁₆	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₁₆	C₂³	1	3	yes	—
33	#1450	C₈	1	3	yes	—
34	C₈×M₁₆	C₈	1	3	yes	—
35	C₈×M₁₆	C₂×C₄	1	3	yes	—
36	C₂×C₈²	C₂×C₄	1	3	yes	—
37	C₄⋊C₃₂	C₂×C₄	1	12	yes	—
38	C₂²⋊C₃₂	C₂×C₄	1	12	yes	—
39	#1397	C₂³	1	6	yes	—
40	#1397	C₂³	1	2	yes	—
41	C₄×C₃₂	C₂×C₄	1	12	yes	—
42	#1312	C₂×C₄	1	6	yes	—
43	#1311	C₂×C₄	1	3	yes	—
44	#1311	C₂×C₄	1	3	yes	—
45	#1311	C₂³	1	1	yes	characteristic
46	#1311	C₂³	1	3	yes	—
47	C₂×#131	—	2	1	no	—
48	C₄×#41	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₁₆	C₂×C₈	1	6	yes	—
50	C₂²×C₁₆	C₂²×C₄	1	6	yes	—
51	C₈×Q₈	—	2	1	no	—
52	C₈×Q₈	C₄²	1	1	yes	characteristic
53	C₈×Q₈	C₂×C₈	1	1	yes	characteristic
54	C₈×D₄	—	2	3	no	—
55	C₈×D₄	C₄²	1	3	yes	—
56	C₈×D₄	C₂×C₈	1	3	yes	—
57	#431	C₂×C₈	1	3	yes	—
58	#431	C₂×C₈	1	3	yes	—
59	#405	—	2	3	no	—
60	C₄×M₁₆	C₂×C₈	1	3	yes	—
61	C₂×C₄×C₈	C₂×C₈	1	3	yes	—
62	C₂×C₄×C₈	C₂×C₈	1	3	yes	—
63	C₂×C₄×C₈	C₂²×C₄	1	3	yes	—
64	M₆₄	—	4	12	no	—
65	C₂×C₃₂	C₂²×C₄	1	4	yes	—
66	C₂×C₃₂	C₂²×C₄	1	12	yes	—
67	C₄⋊C₁₆	C₄²	1	6	yes	—
68	C₄⋊C₁₆	C₂×C₈	1	6	yes	—
69	C₂²⋊C₁₆	C₄²	1	6	yes	—
70	C₂²⋊C₁₆	C₂×C₈	1	6	yes	—
71	C₄×C₁₆	C₂×C₈	1	6	yes	—
72	C₄×C₁₆	C₂²×C₄	1	6	yes	—
73	C₄×C₁₆	C₂²×C₄	1	3	yes	—
74	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
75	C₄×C₁₆	C₂²×C₄	1	3	yes	—
76	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
77	#322	C₂×C₈	1	6	yes	—
78	C₈²	C₂×C₈	1	3	yes	—
79	C₈²	C₂×C₈	1	3	yes	—
80	C₈²	C₂²×C₄	1	1	yes	characteristic
81	C₈²	C₂²×C₄	1	3	yes	—
82	C₂×#41	M₃₂	1	1	yes	characteristic
83	#131	—	4	4	no	—
84	C₂×M₁₆	—	2	3	no	—
85	C₂²×C₈	—	2	3	no	—
86	C₂²×C₈	—	2	3	no	—
87	C₂²×C₈	C₂×C₄²	1	3	yes	—
88	C₂²×C₈	C₂²×C₈	1	3	yes	—
89	C₄×Q₈	C₄×C₈	1	1	yes	characteristic
90	C₄×Q₈	M₃₂	1	1	yes	characteristic
91	C₄×D₄	C₄×C₈	1	3	yes	—
92	C₄×D₄	M₃₂	1	3	yes	—
93	#117	M₃₂	1	3	yes	—
94	C₂×C₄²	C₂²×C₈	1	3	yes	—
95	C₂×C₁₆	—	2	12	no	—
96	C₂×C₁₆	—	2	12	no	—
97	C₂×C₁₆	C₂×C₄²	1	6	yes	—
98	C₂×C₁₆	C₂²×C₈	1	6	yes	—
99	C₂×C₁₆	C₂²×C₈	1	2	yes	—
100	C₂×C₁₆	C₂³×C₄	1	2	yes	—
101	C₄⋊C₈	—	2	3	no	—
102	C₄⋊C₈	C₄×C₈	1	3	yes	—
103	C₄⋊C₈	C₄×C₈	1	3	yes	—
104	C₂²⋊C₈	—	2	3	no	—
105	C₂²⋊C₈	C₄×C₈	1	6	yes	—
106	#97	—	2	3	no	—
107	C₄×C₈	—	2	3	no	—
108	C₄×C₈	—	2	3	no	—
109	C₄×C₈	C₂×C₄²	1	3	yes	—
110	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
111	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₈	1	3	yes	—
113	C₄×C₈	C₂²×C₈	1	3	yes	—
114	C₄×C₈	C₂²×C₈	1	3	yes	—
115	C₄×C₈	C₂²×C₈	1	3	yes	—
116	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
117	C₃₂	—	2	12	no	—
118	C₃₂	—	2	4	no	—
119	#41	—	2	4	no	—
120	C₂×Q₈	#346	1	1	yes	characteristic
121	C₂×D₄	#346	1	3	yes	—
122	C₂²×C₄	C₂×C₄×C₈	1	3	yes	—
123	C₂²×C₄	C₂×M₃₂	1	3	yes	—
124	C₂²×C₄	C₂×M₃₂	1	3	yes	—
125	M₁₆	—	4	6	no	—
126	C₂×C₈	—	2	6	no	—
127	C₂×C₈	—	2	6	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	3	no	—
131	C₂×C₈	—	2	3	no	—
132	C₂×C₈	—	4	6	no	—
133	C₂×C₈	—	8	3	no	—
134	C₂×C₈	C₂×C₄×C₈	1	3	yes	—
135	C₂×C₈	C₂×C₄×C₈	1	3	yes	—
136	C₂×C₈	C₂²×C₄²	1	1	yes	characteristic
137	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
138	C₄⋊C₄	#346	1	3	yes	—
139	C₄⋊C₄	#346	1	3	yes	—
140	C₂²⋊C₄	#346	1	6	yes	—
141	C₄²	C₂×C₄×C₈	1	3	yes	—
142	C₄²	C₂×M₃₂	1	3	yes	—
143	C₄²	C₂×M₃₂	1	3	yes	—
144	C₄²	C₂³×C₈	1	1	yes	characteristic
145	C₁₆	—	2	6	no	—
146	C₁₆	—	2	6	no	—
147	C₁₆	#504	1	4	yes	—
148	C₁₆	#568	1	4	yes	—
149	C₂³	C₂×#346	1	3	yes	—
150	Q₈	#1398	1	4	yes	—
151	D₄	—	2	6	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	2	3	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	4	3	no	—
156	C₂×C₄	C₂×#346	1	3	yes	—
157	C₂×C₄	C₂×#346	1	3	yes	—
158	C₂×C₄	C₂×#346	1	3	yes	—
159	C₂×C₄	C₂²×C₄×C₈	1	1	yes	characteristic
160	C₂×C₄	C₂²×M₃₂	1	1	yes	characteristic
161	C₂×C₄	C₂²×M₃₂	1	1	yes	characteristic
162	C₈	—	2	3	no	—
163	C₈	—	2	3	no	—
164	C₈	—	4	1	no	—
165	C₈	—	4	3	no	—
166	C₈	—	4	3	no	—
167	C₈	—	4	1	no	—
168	C₈	#2875	1	2	yes	—
169	C₈	C₂×#504	1	2	yes	—
170	C₂²	—	2	3	no	—
171	C₂²	—	2	3	no	—
172	C₂²	—	2	3	no	—
173	C₂²	C₂²×#346	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	3	no	—
177	C₄	—	2	3	no	—
178	C₄	C₂×#1398	1	6	yes	—
179	C₄	#17316	1	1	yes	characteristic
180	C₄	C₄×#504	1	1	yes	characteristic
181	C₂	—	4	3	no	—
182	C₂	#479949	1	1	yes	characteristic
183	C₂	C₂²×#1398	1	1	yes	characteristic
184	C₂	#512548	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂×#41	M₃₂
LCS	G ▷ C₂×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. #149513	C₂	semidirect
4. #149509	C₂	semidirect
5. #149502	C₂	semidirect

Maximal subgroups

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

subgroup	index
#512789	2
#479943	2
#149513	2
#149509	2
#149502	2