Subgroups — #497928

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2183	C₂	1	2	yes	maximal
2	#20516	C₂	1	1	yes	maximal, characteristic
3	#13160	C₂	1	2	yes	maximal
4	#13160	C₂	1	2	yes	maximal
5	#13159	C₂	1	2	yes	maximal
6	#12899	C₂	1	2	yes	maximal
7	#12898	C₂	1	1	yes	maximal, characteristic
8	#12654	C₂	1	2	yes	maximal
9	#12653	C₂	1	1	yes	maximal, characteristic
10	C₂×#446	C₄	1	1	yes	characteristic
11	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
12	#2295	C₄	1	1	yes	characteristic
13	#2183	—	2	8	no	—
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×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
22	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
23	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₂×#346	C₂²	1	2	yes	—
25	C₂×#346	C₂²	1	2	yes	—
26	C₂×C₄×C₁₆	C₂²	1	2	yes	—
27	#1771	C₄	1	2	yes	—
28	#1770	C₄	1	1	yes	characteristic
29	C₈×C₄⋊C₄	C₂²	1	2	yes	—
30	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	#1768	C₄	1	2	yes	—
32	#1767	C₄	1	1	yes	characteristic
33	#1726	C₂²	1	2	yes	—
34	C₄²×C₈	C₂²	1	1	yes	characteristic
35	#1374	C₂²	1	2	yes	—
36	#1374	C₂²	1	2	yes	—
37	#1374	C₂²	1	2	yes	—
38	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₆	C₂³	1	2	yes	—
40	C₂²×C₁₆	C₂³	1	2	yes	—
41	C₂²×C₁₆	C₂³	1	2	yes	—
42	C₂²×C₁₆	C₂³	1	2	yes	—
43	#446	—	2	4	no	—
44	C₈×Q₈	D₄	1	4	yes	—
45	C₈×Q₈	Q₈	1	4	yes	—
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	1	yes	characteristic
49	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×#97	C₂×C₄	1	2	yes	—
51	C₂×#97	C₂×C₄	1	2	yes	—
52	C₂×#97	C₂×C₄	1	2	yes	—
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	2	yes	—
58	C₂×C₄×C₈	C₂³	1	2	yes	—
59	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	#391	C₈	1	1	yes	characteristic
62	#391	C₈	1	1	yes	characteristic
63	#389	C₈	1	1	yes	characteristic
64	#389	C₈	1	1	yes	characteristic
65	#387	C₈	1	2	yes	—
66	#387	C₈	1	2	yes	—
67	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
69	#376	C₂×C₄	1	2	yes	—
70	C₄³	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₁₆	—	2	4	no	—
72	C₄⋊C₁₆	—	2	4	no	—
73	C₄⋊C₁₆	—	2	4	no	—
74	C₄⋊C₁₆	—	2	4	no	—
75	C₄⋊C₁₆	—	2	2	no	—
76	C₄⋊C₁₆	—	4	1	no	—
77	C₄⋊C₁₆	—	4	1	no	—
78	C₄⋊C₁₆	—	4	2	no	—
79	C₄⋊C₁₆	—	4	2	no	—
80	C₄⋊C₁₆	Q₈	1	4	yes	complemented
81	#346	—	2	4	no	—
82	#346	—	2	4	no	—
83	C₄×C₁₆	—	2	4	no	—
84	#336	C₂×C₄	1	2	yes	—
85	#336	C₂×C₄	1	2	yes	—
86	#336	C₂×C₄	1	2	yes	—
87	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₄	1	2	yes	—
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₄	1	2	yes	—
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂⁴	1	1	yes	characteristic
94	C₄×Q₈	C₄⋊C₄	1	4	yes	—
95	C₄×Q₈	M₁₆	1	4	yes	—
96	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
97	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
98	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
101	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
102	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
103	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
104	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄²	C₂²×C₄	1	2	yes	—
108	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄²	C₂²×C₄	1	2	yes	—
110	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	C₂×Q₈	1	4	yes	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
126	C₄⋊C₈	C₄⋊C₄	1	8	yes	—
127	#97	—	2	4	no	—
128	#97	—	2	4	no	—
129	#97	—	4	2	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	4	1	no	—
137	C₄×C₈	C₂×D₄	1	4	yes	—
138	C₄×C₈	C₂×D₄	1	2	yes	—
139	C₄×C₈	C₂×Q₈	1	4	yes	—
140	C₄×C₈	C₂×Q₈	1	2	yes	—
141	C₄×C₈	C₂×Q₈	1	2	yes	—
142	C₄×C₈	#41	1	2	yes	—
143	#95	C₂×C₈	1	2	yes	—
144	#95	C₂×C₈	1	2	yes	—
145	#95	C₂×C₈	1	2	yes	—
146	#95	C₂×C₈	1	2	yes	—
147	#95	C₂×C₈	1	2	yes	—
148	#95	C₂×C₈	1	2	yes	—
149	#95	C₂×C₈	1	2	yes	—
150	C₂×Q₈	—	4	1	no	—
151	C₂×Q₈	—	4	1	no	—
152	C₂×Q₈	C₄⋊C₈	1	4	yes	—
153	C₂²×C₄	C₂²×C₈	1	2	yes	—
154	C₂²×C₄	C₂²×C₈	1	2	yes	—
155	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²×C₈	1	2	yes	—
157	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₈	1	2	yes	—
159	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
177	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
178	C₂×C₈	C₄×Q₈	1	2	yes	—
179	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₈	C₂×#41	1	1	yes	characteristic
183	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	C₄⋊C₈	1	8	yes	—
190	C₄⋊C₄	C₄⋊C₈	1	4	yes	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	—	4	1	no	—
199	C₄²	C₂×C₄⋊C₄	1	4	yes	—
200	C₄²	C₂×C₄⋊C₄	1	2	yes	—
201	C₄²	C₄×Q₈	1	2	yes	—
202	C₄²	C₂×M₁₆	1	2	yes	—
203	C₄²	C₂×M₁₆	1	4	yes	—
204	C₄²	#131	1	2	yes	—
205	C₁₆	—	2	4	no	—
206	C₁₆	—	4	2	no	—
207	C₁₆	—	4	2	no	—
208	C₁₆	—	4	2	no	—
209	C₂³	C₂³×C₈	1	1	yes	characteristic
210	Q₈	—	4	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
235	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
236	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
238	C₂×C₄	C₈×Q₈	1	2	yes	—
239	C₂×C₄	C₈×Q₈	1	2	yes	—
240	C₂×C₄	#504	1	2	yes	—
241	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
243	C₂×C₄	#520	1	1	yes	characteristic
244	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
245	C₂×C₄	C₂×#131	1	1	yes	characteristic
246	C₂×C₄	#568	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	4	no	—
249	C₈	—	2	2	no	—
250	C₈	—	4	1	no	—
251	C₈	—	4	2	no	—
252	C₈	D₄×Q₈	1	1	yes	characteristic
253	C₈	#550	1	1	yes	characteristic
254	C₈	#557	1	1	yes	characteristic
255	C₈	Q₈²	1	1	yes	characteristic
256	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
257	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
258	C₂²	#2988	1	1	yes	characteristic
259	C₂²	C₂×#504	1	1	yes	characteristic
260	C₂²	#3407	1	1	yes	characteristic
261	C₂²	#3407	1	1	yes	characteristic
262	C₂²	#3407	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	2	no	—
273	C₄	Q₈×C₄⋊C₄	1	1	yes	characteristic
274	C₄	#2906	1	1	yes	characteristic
275	C₄	Q₈×M₁₆	1	1	yes	characteristic
276	C₄	#2991	1	1	yes	characteristic
277	C₂	Q₈×C₄⋊C₈	1	1	yes	characteristic
278	C₂	#33397	1	1	yes	characteristic
279	C₂	#33398	1	1	yes	characteristic
280	C₂	#33475	1	1	yes	characteristic
281	C₂	#33476	1	1	yes	characteristic
282	C₂	#33506	1	1	yes	characteristic
283	C₂	#33509	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
80. 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
C₂×#2183	2
#20516	2
#13160	2
#13159	2
#12899	2
#12898	2
#12654	2
#12653	2