Subgroups — #497182

Order 512.

352 conjugacy classes of subgroups.

The subgroup lattice has 195 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21733	C₂	1	1	yes	maximal, characteristic
2	#13269	C₂	1	4	yes	maximal
3	#13164	C₂	1	2	yes	maximal
4	#13108	C₂	1	4	yes	maximal
5	#12881	C₂	1	1	yes	maximal, characteristic
6	#12667	C₂	1	2	yes	maximal
7	#12593	C₂	1	1	yes	maximal, characteristic
8	#3160	C₄	1	1	yes	characteristic
9	#2961	C₂²	1	2	yes	—
10	#2153	C₂²	1	4	yes	—
11	C₂×#364	C₂²	1	2	yes	—
12	#2110	C₂²	1	4	yes	—
13	C₄×M₃₂	C₂²	1	1	yes	characteristic
14	C₄×M₃₂	C₂²	1	1	yes	characteristic
15	#1995	C₄	1	2	yes	—
16	#1946	C₄	1	2	yes	—
17	#1945	C₄	1	2	yes	—
18	#1848	C₄	1	1	yes	characteristic
19	#1773	C₂²	1	2	yes	—
20	#1772	C₂²	1	2	yes	—
21	#1726	C₂²	1	1	yes	characteristic
22	#1378	C₂²	1	4	yes	—
23	#1378	C₂²	1	4	yes	—
24	#1377	C₂²	1	2	yes	—
25	#1375	C₂²	1	2	yes	—
26	#1375	C₂²	1	2	yes	—
27	#1375	C₂²	1	2	yes	—
28	#533	C₂×C₄	1	2	yes	—
29	C₂×M₃₂	C₂³	1	2	yes	—
30	C₂×M₃₂	C₂³	1	4	yes	—
31	C₂×M₃₂	C₂³	1	2	yes	—
32	#500	—	2	1	no	—
33	#499	—	2	1	no	—
34	#499	—	2	1	no	—
35	#498	—	2	1	no	—
36	#446	—	4	4	no	—
37	C₂×#108	C₂×C₄	1	2	yes	—
38	C₂×#108	C₂×C₄	1	2	yes	—
39	#427	C₂×C₄	1	2	yes	—
40	#427	C₂×C₄	1	2	yes	—
41	#424	C₂×C₄	1	4	yes	—
42	#423	C₂×C₄	1	2	yes	—
43	#405	C₂³	1	4	yes	—
44	C₂×#97	C₂×C₄	1	1	yes	characteristic
45	C₂×#97	C₂³	1	1	yes	characteristic
46	C₂×#97	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	#376	C₂×C₄	1	1	yes	characteristic
51	#364	—	2	2	no	—
52	#364	—	2	2	no	—
53	#364	—	4	4	no	—
54	#347	—	4	4	no	—
55	#346	—	2	1	no	—
56	#346	Q₈	1	2	yes	complemented
57	C₄×C₁₆	—	2	1	no	—
58	C₄×C₁₆	Q₈	1	2	yes	complemented
59	#339	C₂×C₄	1	4	yes	—
60	#339	C₂×C₄	1	4	yes	—
61	C₂×M₁₆	C₂²×C₄	1	4	yes	—
62	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂⁴	1	1	yes	characteristic
66	C₄⋊Q₈	—	2	2	no	—
67	#125	—	2	2	no	—
68	C₂²⋊Q₈	—	4	2	no	—
69	C₄×Q₈	—	4	2	no	—
70	#117	C₂²×C₄	1	4	yes	—
71	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
74	M₃₂	—	2	2	no	—
75	M₃₂	—	2	2	no	—
76	M₃₂	—	2	4	no	—
77	M₃₂	—	2	4	no	—
78	M₃₂	—	2	2	no	—
79	M₃₂	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	4	no	—
82	C₂×C₁₆	C₂×Q₈	1	4	yes	—
83	#108	—	2	2	no	—
84	#108	—	2	2	no	—
85	#108	—	4	2	no	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#106	—	2	2	no	—
89	#106	—	2	2	no	—
90	C₄⋊C₈	—	4	2	no	—
91	C₄⋊C₈	—	4	4	no	—
92	#97	—	2	4	no	—
93	#97	—	2	2	no	—
94	#97	—	2	2	no	—
95	#97	—	4	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	M₁₆	1	1	yes	characteristic
100	C₄×C₈	M₁₆	1	1	yes	characteristic
101	C₄×C₈	M₁₆	1	1	yes	characteristic
102	C₄×C₈	M₁₆	1	1	yes	characteristic
103	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
104	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₄×C₈	#41	1	1	yes	characteristic
106	C₄×C₈	#41	1	1	yes	characteristic
107	#95	—	4	1	no	—
108	C₂×Q₈	—	4	2	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	2	4	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
122	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
123	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₈	#131	1	1	yes	characteristic
128	C₂×C₈	#131	1	1	yes	characteristic
129	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₈	C₂×#41	1	1	yes	characteristic
131	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
132	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₄×Q₈	1	1	yes	characteristic
143	C₄²	C₄×Q₈	1	1	yes	characteristic
144	C₄²	C₂×M₁₆	1	1	yes	characteristic
145	C₄²	C₂×M₁₆	1	1	yes	characteristic
146	C₁₆	—	2	4	no	—
147	C₁₆	—	4	2	no	—
148	C₁₆	—	4	4	no	—
149	C₂³	#568	1	1	yes	characteristic
150	Q₈	—	16	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	8	1	no	—
158	C₂×C₄	—	8	1	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₂×C₄	#518	1	1	yes	characteristic
162	C₂×C₄	#519	1	1	yes	characteristic
163	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
164	C₂×C₄	C₂×#131	1	1	yes	characteristic
165	C₂×C₄	#568	1	1	yes	characteristic
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	4	1	no	—
169	C₈	—	4	4	no	—
170	C₈	#446	1	1	yes	characteristic
171	C₈	#446	1	1	yes	characteristic
172	C₈	#446	1	1	yes	characteristic
173	C₈	#446	1	1	yes	characteristic
174	C₈	#542	1	2	yes	—
175	C₈	#544	1	2	yes	—
176	C₂²	—	4	1	no	—
177	C₂²	#2961	1	1	yes	characteristic
178	C₂²	#2975	1	1	yes	characteristic
179	C₂²	#2985	1	1	yes	characteristic
180	C₄	—	4	1	no	—
181	C₄	—	4	1	no	—
182	C₄	—	4	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	8	1	no	—
185	C₄	—	8	4	no	—
186	C₄	#2331	1	1	yes	characteristic
187	C₄	C₂×#446	1	1	yes	characteristic
188	C₄	#2981	1	1	yes	characteristic
189	C₄	#2983	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#22311	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2331
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22311
prime radical	C₂³	#568
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
56. #346	Q₈	semidirect
58. C₄×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#21733	2
#13269	2
#13164	2
#13108	2
#12881	2
#12667	2
#12593	2