Subgroups — #498071

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#13263	C₂	1	2	yes	maximal
3	#13105	C₂	1	4	yes	maximal
4	#12898	C₂	1	1	yes	maximal, characteristic
5	#12879	C₂	1	2	yes	maximal
6	#12662	C₂	1	2	yes	maximal
7	#12653	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	C₄²×Q₈	C₄	1	1	yes	characteristic
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×#346	C₂²	1	4	yes	—
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
20	C₈×C₄⋊C₄	C₄	1	2	yes	—
21	C₈×C₄⋊C₄	C₂²	1	2	yes	—
22	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₈	C₄	1	2	yes	—
24	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
25	C₄²×C₈	C₂²	1	2	yes	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1374	C₂²	1	2	yes	—
28	#1374	C₂²	1	4	yes	—
29	#1374	C₂²	1	4	yes	—
30	#1374	C₂²	1	4	yes	—
31	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
32	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
33	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	1	2	yes	—
35	C₂²×C₁₆	C₂³	1	4	yes	—
36	C₂²×C₁₆	C₂³	1	2	yes	—
37	C₈×Q₈	—	2	4	no	—
38	C₈×Q₈	—	2	2	no	—
39	C₈×Q₈	—	2	2	no	—
40	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
48	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	2	yes	—
51	C₂×C₄×C₈	C₂³	1	2	yes	—
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₄⋊C₄	C₈	1	2	yes	—
56	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
57	C₄×C₄⋊C₄	C₈	1	2	yes	—
58	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
59	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₄³	C₂×C₄	1	2	yes	—
62	C₄³	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₁₆	—	2	4	no	—
64	C₄⋊C₁₆	—	2	2	no	—
65	C₄⋊C₁₆	—	2	4	no	—
66	C₄⋊C₁₆	—	4	2	no	—
67	C₄⋊C₁₆	Q₈	1	4	yes	complemented
68	#346	—	4	4	no	—
69	C₄×C₁₆	—	2	4	no	—
70	#336	C₂×C₄	1	2	yes	—
71	#336	C₂×C₄	1	2	yes	—
72	#336	C₂×C₄	1	2	yes	—
73	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₄	1	2	yes	—
75	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂²×C₄	1	2	yes	—
79	C₂²×C₈	C₂⁴	1	1	yes	characteristic
80	C₄×Q₈	—	2	4	no	—
81	C₄×Q₈	—	2	4	no	—
82	C₄×Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
85	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
88	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
90	C₂×C₄²	C₂×C₈	1	2	yes	—
91	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄²	C₂×C₈	1	2	yes	—
93	C₂×C₄²	C₂×C₈	1	2	yes	—
94	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
95	C₂×C₄²	C₂×C₈	1	2	yes	—
96	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄²	C₂²×C₄	1	2	yes	—
99	C₂×C₄²	C₂²×C₄	1	2	yes	—
100	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	2	4	no	—
108	C₂×C₁₆	—	2	4	no	—
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	C₂×Q₈	1	4	yes	—
112	C₄⋊C₈	—	2	4	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	M₁₆	1	4	yes	—
116	C₄×C₈	—	2	4	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	4	no	—
126	C₄×C₈	—	4	1	no	—
127	C₄×C₈	—	4	2	no	—
128	C₄×C₈	C₂×Q₈	1	2	yes	—
129	C₄×C₈	#41	1	2	yes	—
130	C₄×C₈	#41	1	2	yes	—
131	C₄×C₈	#41	1	2	yes	—
132	#95	C₂×C₈	1	2	yes	—
133	#95	C₂×C₈	1	2	yes	—
134	#95	C₂×C₈	1	2	yes	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	2	2	no	—
137	C₂×Q₈	M₃₂	1	4	yes	—
138	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₈	1	2	yes	—
140	C₂²×C₄	C₂²×C₈	1	2	yes	—
141	C₂²×C₄	C₂²×C₈	1	2	yes	—
142	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₈	1	2	yes	—
146	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	#117	1	2	yes	—
162	C₂×C₈	C₄×Q₈	1	2	yes	—
163	C₂×C₈	C₂×M₁₆	1	4	yes	—
164	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₈	C₂×#41	1	1	yes	characteristic
166	C₂×C₈	C₂×#41	1	1	yes	characteristic
167	C₂×C₈	C₂×#41	1	1	yes	characteristic
168	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	4	2	no	—
189	C₄²	M₃₂	1	4	yes	—
190	C₄²	#117	1	2	yes	—
191	C₄²	C₄×Q₈	1	2	yes	—
192	C₄²	C₂×M₁₆	1	2	yes	—
193	C₄²	#131	1	2	yes	—
194	C₁₆	—	2	4	no	—
195	C₁₆	—	4	2	no	—
196	C₁₆	—	4	4	no	—
197	C₂³	C₂³×C₈	1	1	yes	characteristic
198	Q₈	—	2	8	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	#431	1	2	yes	—
220	C₂×C₄	#431	1	2	yes	—
221	C₂×C₄	C₈×Q₈	1	2	yes	—
222	C₂×C₄	C₈×Q₈	1	2	yes	—
223	C₂×C₄	C₂×M₃₂	1	4	yes	—
224	C₂×C₄	C₂×M₃₂	1	2	yes	—
225	C₂×C₄	C₂×M₃₂	1	2	yes	—
226	C₂×C₄	C₂×M₃₂	1	2	yes	—
227	C₂×C₄	C₂×M₃₂	1	2	yes	—
228	C₂×C₄	#504	1	2	yes	—
229	C₂×C₄	C₂×#117	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
231	C₂×C₄	#520	1	1	yes	characteristic
232	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
233	C₂×C₄	C₂×#131	1	1	yes	characteristic
234	C₂×C₄	#568	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	2	no	—
237	C₈	—	2	4	no	—
238	C₈	—	4	2	no	—
239	C₈	—	4	1	no	—
240	C₈	#554	1	1	yes	characteristic
241	C₈	#556	1	1	yes	characteristic
242	C₈	#557	1	1	yes	characteristic
243	C₈	#559	1	1	yes	characteristic
244	C₂²	C₂×#431	1	1	yes	characteristic
245	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
246	C₂²	#2988	1	1	yes	characteristic
247	C₂²	C₂²×M₃₂	1	1	yes	characteristic
248	C₂²	C₂²×M₃₂	1	1	yes	characteristic
249	C₂²	C₂×#504	1	1	yes	characteristic
250	C₂²	#3407	1	1	yes	characteristic
251	C₄	—	2	4	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	#2163	1	4	yes	—
260	C₄	#2183	1	4	yes	—
261	C₄	#2355	1	1	yes	characteristic
262	C₄	#2925	1	1	yes	characteristic
263	C₄	Q₈×M₁₆	1	1	yes	characteristic
264	C₄	#2990	1	1	yes	characteristic
265	C₂	#22821	1	1	yes	characteristic
266	C₂	C₂×#2163	1	1	yes	characteristic
267	C₂	#33423	1	1	yes	characteristic
268	C₂	C₂×#2183	1	1	yes	characteristic
269	C₂	Q₈×M₃₂	1	1	yes	characteristic
270	C₂	#33503	1	1	yes	characteristic
271	C₂	#33505	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ 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
67. 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
C₄×C₈×Q₈	2
#13263	2
#13105	2
#12898	2
#12879	2
#12662	2
#12653	2
C₄×C₄⋊C₁₆	2