Subgroups — #497930

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 261 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₃₂	C₂	1	2	yes	maximal
2	#20516	C₂	1	1	yes	maximal, characteristic
3	#13163	C₂	1	2	yes	maximal
4	#13163	C₂	1	4	yes	maximal
5	#12901	C₂	1	2	yes	maximal
6	#12900	C₂	1	1	yes	maximal, characteristic
7	#12656	C₂	1	2	yes	maximal
8	C₄×#364	C₂	1	1	yes	maximal, characteristic
9	C₂×#446	C₄	1	1	yes	characteristic
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	#2295	C₄	1	1	yes	characteristic
12	#2183	—	2	2	no	—
13	#2183	—	2	4	no	—
14	Q₈×C₁₆	—	2	2	no	—
15	C₂×#364	C₂²	1	2	yes	—
16	C₂×#364	C₂²	1	1	yes	characteristic
17	C₂×#364	C₂²	1	2	yes	—
18	C₂×#364	C₂²	1	4	yes	—
19	C₂×#364	C₂²	1	1	yes	characteristic
20	#2150	C₂²	1	2	yes	—
21	#2150	C₂²	1	4	yes	—
22	C₄×M₃₂	C₂²	1	2	yes	—
23	C₄×M₃₂	C₂²	1	4	yes	—
24	#1771	C₄	1	2	yes	—
25	#1770	C₄	1	1	yes	characteristic
26	C₈×C₄⋊C₄	C₂²	1	2	yes	—
27	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	#1768	C₄	1	2	yes	—
29	#1767	C₄	1	1	yes	characteristic
30	#1726	C₂²	1	2	yes	—
31	C₄²×C₈	C₂²	1	1	yes	characteristic
32	#1376	C₂²	1	2	yes	—
33	#1376	C₂²	1	4	yes	—
34	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂×M₃₂	C₂³	1	2	yes	—
36	C₂×M₃₂	C₂³	1	4	yes	—
37	C₂×M₃₂	C₂³	1	2	yes	—
38	#446	C₈	1	4	yes	—
39	#446	C₈	1	4	yes	—
40	C₈×Q₈	D₄	1	2	yes	—
41	C₈×Q₈	D₄	1	2	yes	—
42	C₈×Q₈	Q₈	1	2	yes	—
43	C₈×Q₈	Q₈	1	2	yes	—
44	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₂×#97	C₂×C₄	1	2	yes	—
49	C₂×#97	C₂×C₄	1	2	yes	—
50	C₂×#97	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	2	yes	—
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	2	yes	—
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	#391	—	2	1	no	—
60	#389	—	2	1	no	—
61	#387	—	2	2	no	—
62	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
63	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
64	#376	C₂×C₄	1	2	yes	—
65	C₄³	C₂×C₄	1	1	yes	characteristic
66	#364	—	2	4	no	—
67	#364	—	2	2	no	—
68	#364	—	4	2	no	—
69	#364	—	4	4	no	—
70	#364	Q₈	1	4	yes	complemented
71	C₄⋊C₁₆	—	2	2	no	—
72	C₄⋊C₁₆	—	2	4	no	—
73	C₄⋊C₁₆	—	2	4	no	—
74	C₄⋊C₁₆	—	2	2	no	—
75	#346	—	2	4	no	—
76	#346	—	2	2	no	—
77	C₄×C₁₆	—	2	2	no	—
78	C₄×C₁₆	—	2	4	no	—
79	#336	C₂×C₄	1	2	yes	—
80	#336	C₂×C₄	1	2	yes	—
81	#336	C₂×C₄	1	2	yes	—
82	C₂²×Q₈	M₁₆	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₄	1	2	yes	—
85	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₄	1	2	yes	—
87	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂⁴	1	1	yes	characteristic
89	C₄×Q₈	C₄⋊C₄	1	2	yes	—
90	C₄×Q₈	C₄⋊C₄	1	2	yes	—
91	C₄×Q₈	C₂×C₈	1	2	yes	—
92	C₄×Q₈	C₂×C₈	1	2	yes	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	2	1	no	—
95	C₂×C₄⋊C₄	M₁₆	1	2	yes	—
96	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	—	2	1	no	—
99	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₄	1	2	yes	—
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂²×C₄	1	2	yes	—
104	M₃₂	—	2	2	no	—
105	M₃₂	—	2	4	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	4	no	—
108	M₃₂	—	2	2	no	—
109	M₃₂	C₂×Q₈	1	4	yes	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	4	no	—
113	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
115	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
116	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
117	C₄⋊C₈	C₂×C₈	1	2	yes	—
118	C₄⋊C₈	C₂×C₈	1	2	yes	—
119	C₄⋊C₈	C₂×C₈	1	8	yes	—
120	#97	—	2	4	no	—
121	#97	—	4	2	no	—
122	#97	C₂×C₈	1	8	yes	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	C₂×C₈	1	2	yes	—
130	C₄×C₈	C₂×C₈	1	2	yes	—
131	C₄×C₈	C₂×D₄	1	1	yes	characteristic
132	C₄×C₈	C₂×D₄	1	4	yes	—
133	C₄×C₈	C₂×D₄	1	1	yes	characteristic
134	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
135	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
136	C₄×C₈	C₂×Q₈	1	4	yes	—
137	C₄×C₈	C₂×Q₈	1	2	yes	—
138	C₄×C₈	#41	1	2	yes	—
139	#95	—	2	2	no	—
140	#95	—	2	2	no	—
141	#95	—	2	1	no	—
142	#95	—	2	2	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	C₄⋊C₈	1	2	yes	—
145	C₂×Q₈	C₄⋊C₈	1	2	yes	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×M₁₆	1	2	yes	—
151	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
152	C₂²×C₄	C₂×M₁₆	1	2	yes	—
153	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
164	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₈	C₄×Q₈	1	2	yes	—
166	C₂×C₈	C₂²×C₈	1	2	yes	—
167	C₂×C₈	C₂²×C₈	1	4	yes	—
168	C₂×C₈	C₂²×C₈	1	2	yes	—
169	C₂×C₈	#131	1	2	yes	—
170	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
172	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₈	C₂×#41	1	1	yes	characteristic
174	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	C₄⋊C₈	1	4	yes	—
179	C₄⋊C₄	C₄⋊C₈	1	4	yes	—
180	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
181	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	4	2	no	—
189	C₄²	C₂×C₄⋊C₄	1	4	yes	—
190	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₄²	C₄×Q₈	1	2	yes	—
193	C₄²	C₂²×C₈	1	1	yes	characteristic
194	C₄²	C₂²×C₈	1	1	yes	characteristic
195	C₄²	C₂²×C₈	1	4	yes	—
196	C₁₆	—	2	4	no	—
197	C₁₆	—	4	2	no	—
198	C₁₆	—	4	4	no	—
199	C₂³	C₂²×M₁₆	1	1	yes	characteristic
200	Q₈	—	2	4	no	—
201	Q₈	—	4	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
218	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
220	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
223	C₂×C₄	#520	1	1	yes	characteristic
224	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
225	C₂×C₄	C₂×#131	1	1	yes	characteristic
226	C₂×C₄	#568	1	1	yes	characteristic
227	C₈	—	2	1	no	—
228	C₈	—	2	1	no	—
229	C₈	—	2	4	no	—
230	C₈	—	2	2	no	—
231	C₈	—	2	4	no	—
232	C₈	C₈×Q₈	1	4	yes	—
233	C₈	D₄×Q₈	1	1	yes	characteristic
234	C₈	#550	1	1	yes	characteristic
235	C₈	#557	1	1	yes	characteristic
236	C₈	Q₈²	1	1	yes	characteristic
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
240	C₂²	Q₈×M₁₆	1	1	yes	characteristic
241	C₂²	#2991	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	8	2	no	—
250	C₄	#2153	1	2	yes	—
251	C₄	Q₈×C₄⋊C₄	1	1	yes	characteristic
252	C₄	#2906	1	1	yes	characteristic
253	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
254	C₄	#2988	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	Q₈×C₄⋊C₈	1	1	yes	characteristic
258	C₂	C₂×#2153	1	1	yes	characteristic
259	C₂	#33401	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂³	C₂²×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
70. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₃₂	2
#20516	2
#13163	2
#12901	2
#12900	2
#12656	2
C₄×#364	2