Subgroups — #497171

Order 512.

484 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#13263	C₂	1	4	yes	maximal
3	#13159	C₂	1	2	yes	maximal
4	#13104	C₂	1	4	yes	maximal
5	#12879	C₂	1	1	yes	maximal, characteristic
6	C₁₆×C₄⋊C₄	C₂	1	2	yes	maximal
7	#12590	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₂²	1	2	yes	—
9	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	4	yes	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#1984	C₄	1	2	yes	—
16	#1939	C₄	1	2	yes	—
17	#1939	C₄	1	2	yes	—
18	#1842	C₄	1	1	yes	characteristic
19	C₈×C₄⋊C₄	C₂²	1	2	yes	—
20	C₈×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄²×C₈	C₂²	1	1	yes	characteristic
22	#1374	C₂²	1	2	yes	—
23	#1374	C₂²	1	2	yes	—
24	#1374	C₂²	1	2	yes	—
25	#1374	C₂²	1	4	yes	—
26	#1374	C₂²	1	4	yes	—
27	#1374	C₂²	1	2	yes	—
28	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
29	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	4	yes	—
32	C₂²×C₁₆	C₂³	1	2	yes	—
33	C₈×Q₈	—	4	4	no	—
34	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	4	yes	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	#391	C₈	1	2	yes	—
46	#389	C₈	1	2	yes	—
47	#389	C₈	1	2	yes	—
48	#384	C₈	1	1	yes	characteristic
49	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₄³	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₁₆	—	2	4	no	—
53	C₄⋊C₁₆	—	4	4	no	—
54	C₄×C₁₆	—	2	2	no	—
55	C₄×C₁₆	—	4	4	no	—
56	C₄×C₁₆	Q₈	1	4	yes	complemented
57	#336	C₂×C₄	1	4	yes	—
58	#336	C₂×C₄	1	4	yes	—
59	C₂²×Q₈	C₂×C₈	1	2	yes	—
60	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₈	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	C₂⁴	1	1	yes	characteristic
65	C₄⋊Q₈	C₁₆	1	8	yes	complemented
66	C₄×Q₈	—	4	2	no	—
67	C₄×Q₈	—	4	2	no	—
68	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
69	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
70	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
71	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
72	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
73	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
74	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄²	C₂²×C₄	1	4	yes	—
80	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	—	2	4	no	—
86	C₂×C₁₆	—	2	4	no	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	C₂×Q₈	1	4	yes	—
91	C₄⋊C₈	—	2	4	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	4	2	no	—
94	C₄⋊C₈	—	4	4	no	—
95	C₄×C₈	—	2	4	no	—
96	C₄×C₈	—	2	4	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	—	4	1	no	—
102	C₄×C₈	C₂×Q₈	1	2	yes	—
103	C₄×C₈	#41	1	2	yes	—
104	#95	C₂×C₈	1	4	yes	—
105	#95	C₂×C₈	1	4	yes	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	C₂×C₁₆	1	8	yes	—
109	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
110	C₂²×C₄	C₂²×C₈	1	4	yes	—
111	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
112	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
113	C₂²×C₄	C₂²×C₈	1	4	yes	—
114	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
115	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
116	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
117	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	C₄×Q₈	1	2	yes	—
130	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₈	C₂×#41	1	1	yes	characteristic
132	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
133	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
139	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	4	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	1	no	—
149	C₄²	C₂×C₁₆	1	4	yes	—
150	C₄²	C₄×Q₈	1	2	yes	—
151	C₄²	#131	1	2	yes	—
152	C₁₆	—	2	4	no	—
153	C₁₆	—	4	2	no	—
154	C₁₆	—	4	4	no	—
155	C₂³	C₂³×C₈	1	1	yes	characteristic
156	Q₈	—	4	8	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	C₈×Q₈	1	2	yes	—
175	C₂×C₄	C₈×Q₈	1	2	yes	—
176	C₂×C₄	C₂²×C₁₆	1	2	yes	—
177	C₂×C₄	C₂²×C₁₆	1	2	yes	—
178	C₂×C₄	C₂²×C₁₆	1	8	yes	—
179	C₂×C₄	C₂²×C₁₆	1	2	yes	—
180	C₂×C₄	#504	1	2	yes	—
181	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
182	C₂×C₄	#518	1	1	yes	characteristic
183	C₂×C₄	#519	1	1	yes	characteristic
184	C₂×C₄	C₂×#131	1	1	yes	characteristic
185	C₂×C₄	#568	1	1	yes	characteristic
186	C₂×C₄	#568	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	4	1	no	—
189	C₈	—	4	4	no	—
190	C₈	—	4	1	no	—
191	C₈	#542	1	2	yes	—
192	C₈	#544	1	2	yes	—
193	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
194	C₂²	#2972	1	1	yes	characteristic
195	C₂²	#2982	1	1	yes	characteristic
196	C₂²	C₂³×C₁₆	1	1	yes	characteristic
197	C₂²	C₂×#504	1	1	yes	characteristic
198	C₂²	#3407	1	1	yes	characteristic
199	C₂²	#3407	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	8	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	1	no	—
205	C₄	—	4	4	no	—
206	C₄	—	4	1	no	—
207	C₄	Q₈×C₁₆	1	4	yes	—
208	C₄	#2331	1	1	yes	characteristic
209	C₄	#2961	1	1	yes	characteristic
210	C₄	#2981	1	1	yes	characteristic
211	C₄	#2985	1	1	yes	characteristic
212	C₂	#22307	1	1	yes	characteristic
213	C₂	C₂×Q₈×C₁₆	1	1	yes	characteristic
214	C₂	#33475	1	1	yes	characteristic
215	C₂	#33486	1	1	yes	characteristic
216	C₂	#33495	1	1	yes	characteristic
217	C₂	#33496	1	1	yes	characteristic
218	C₂	#33499	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₁₆
socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ 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. C₄×C₁₆	Q₈	semidirect
65. C₄⋊Q₈	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Q₈	2
#13263	2
#13159	2
#13104	2
#12879	2
C₁₆×C₄⋊C₄	2
#12590	2