Subgroups — #497173

Order 512.

470 conjugacy classes of subgroups.

The subgroup lattice has 295 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	#13264	C₂	1	2	yes	maximal
3	#13263	C₂	1	2	yes	maximal
4	#13159	C₂	1	1	yes	maximal, characteristic
5	#13159	C₂	1	1	yes	maximal, characteristic
6	#13105	C₂	1	2	yes	maximal
7	#13104	C₂	1	2	yes	maximal
8	#12879	C₂	1	1	yes	maximal, characteristic
9	#12662	C₂	1	1	yes	maximal, characteristic
10	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#12590	C₂	1	1	yes	maximal, characteristic
12	C₂×C₈×Q₈	C₂²	1	2	yes	—
13	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₂×#346	C₂²	1	2	yes	—
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	2	yes	—
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	#1984	C₄	1	2	yes	—
23	#1939	C₄	1	1	yes	characteristic
24	#1939	C₄	1	1	yes	characteristic
25	#1939	C₄	1	2	yes	—
26	#1842	C₄	1	1	yes	characteristic
27	C₈×C₄⋊C₄	C₂²	1	2	yes	—
28	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₄²×C₈	C₂²	1	1	yes	characteristic
31	#1374	C₂²	1	1	yes	characteristic
32	#1374	C₂²	1	1	yes	characteristic
33	#1374	C₂²	1	1	yes	characteristic
34	#1374	C₂²	1	1	yes	characteristic
35	#1374	C₂²	1	2	yes	—
36	#1374	C₂²	1	2	yes	—
37	#1374	C₂²	1	2	yes	—
38	#1374	C₂²	1	2	yes	—
39	#1374	C₂²	1	2	yes	—
40	#1374	C₂²	1	2	yes	—
41	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
42	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	C₂²×C₁₆	C₂³	1	2	yes	—
46	C₂²×C₁₆	C₂³	1	2	yes	—
47	C₂²×C₁₆	C₂³	1	1	yes	characteristic
48	C₂²×C₁₆	C₂³	1	1	yes	characteristic
49	C₈×Q₈	—	4	2	no	—
50	C₈×Q₈	—	4	2	no	—
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	2	yes	—
54	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂³	1	2	yes	—
61	C₂×C₄×C₈	C₂³	1	2	yes	—
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	#391	C₈	1	2	yes	—
66	#389	C₈	1	1	yes	characteristic
67	#389	C₈	1	2	yes	—
68	#389	C₈	1	1	yes	characteristic
69	#384	C₈	1	1	yes	characteristic
70	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
71	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
72	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₄³	C₂×C₄	1	1	yes	characteristic
74	C₄⋊C₁₆	—	2	2	no	—
75	C₄⋊C₁₆	—	2	2	no	—
76	C₄⋊C₁₆	—	4	2	no	—
77	C₄⋊C₁₆	—	4	2	no	—
78	#346	—	4	2	no	—
79	C₄×C₁₆	—	2	2	no	—
80	C₄×C₁₆	—	4	2	no	—
81	C₄×C₁₆	Q₈	1	4	yes	complemented
82	#336	C₂×C₄	1	2	yes	—
83	#336	C₂×C₄	1	2	yes	—
84	#336	C₂×C₄	1	2	yes	—
85	#336	C₂×C₄	1	2	yes	—
86	C₂²×Q₈	C₂×C₈	1	2	yes	—
87	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	2	yes	—
89	C₂²×C₈	C₂²×C₄	1	2	yes	—
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂⁴	1	1	yes	characteristic
93	C₄⋊Q₈	—	2	4	no	—
94	C₄×Q₈	—	4	2	no	—
95	C₄×Q₈	—	4	2	no	—
96	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
98	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
99	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
103	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
104	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
109	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄²	C₂²×C₄	1	2	yes	—
111	C₂×C₄²	C₂²×C₄	1	2	yes	—
112	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	C₂×Q₈	1	2	yes	—
131	C₂×C₁₆	C₂×Q₈	1	2	yes	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	C₄⋊C₈	—	4	2	no	—
137	C₄⋊C₈	—	4	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	4	1	no	—
148	C₄×C₈	—	4	1	no	—
149	C₄×C₈	C₂×Q₈	1	2	yes	—
150	C₄×C₈	#41	1	2	yes	—
151	#95	C₂×C₈	1	2	yes	—
152	#95	C₂×C₈	1	2	yes	—
153	#95	C₂×C₈	1	2	yes	—
154	#95	C₂×C₈	1	2	yes	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₈	1	2	yes	—
160	C₂²×C₄	C₂²×C₈	1	2	yes	—
161	C₂²×C₄	C₂²×C₈	1	2	yes	—
162	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₈	1	2	yes	—
165	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	C₄×Q₈	1	2	yes	—
184	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
185	C₂×C₈	C₂×#41	1	1	yes	characteristic
186	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	M₃₂	1	4	yes	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₄×Q₈	1	2	yes	—
212	C₄²	#131	1	2	yes	—
213	C₁₆	—	2	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	—	4	1	no	—
216	C₁₆	—	4	1	no	—
217	C₁₆	—	4	2	no	—
218	C₁₆	—	4	2	no	—
219	C₂³	C₂³×C₈	1	1	yes	characteristic
220	Q₈	—	4	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	C₈×Q₈	1	2	yes	—
245	C₂×C₄	C₈×Q₈	1	2	yes	—
246	C₂×C₄	C₂×M₃₂	1	4	yes	—
247	C₂×C₄	C₂×M₃₂	1	2	yes	—
248	C₂×C₄	#504	1	2	yes	—
249	C₂×C₄	#504	1	2	yes	—
250	C₂×C₄	#504	1	2	yes	—
251	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
252	C₂×C₄	#518	1	1	yes	characteristic
253	C₂×C₄	#519	1	1	yes	characteristic
254	C₂×C₄	C₂×#131	1	1	yes	characteristic
255	C₂×C₄	#568	1	1	yes	characteristic
256	C₂×C₄	#568	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	4	1	no	—
262	C₈	#542	1	1	yes	characteristic
263	C₈	#542	1	1	yes	characteristic
264	C₈	#544	1	1	yes	characteristic
265	C₈	#544	1	1	yes	characteristic
266	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
267	C₂²	#2972	1	1	yes	characteristic
268	C₂²	#2982	1	1	yes	characteristic
269	C₂²	C₂²×M₃₂	1	1	yes	characteristic
270	C₂²	C₂×#504	1	1	yes	characteristic
271	C₂²	C₂×#504	1	1	yes	characteristic
272	C₂²	C₂×#504	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	1	no	—
282	C₄	#2183	1	4	yes	—
283	C₄	#2331	1	1	yes	characteristic
284	C₄	#2961	1	1	yes	characteristic
285	C₄	#2981	1	1	yes	characteristic
286	C₄	#2985	1	1	yes	characteristic
287	C₂	#22307	1	1	yes	characteristic
288	C₂	C₂×#2183	1	1	yes	characteristic
289	C₂	#33474	1	1	yes	characteristic
290	C₂	#33488	1	1	yes	characteristic
291	C₂	#33493	1	1	yes	characteristic
292	C₂	#33498	1	1	yes	characteristic
293	C₂	#33500	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
81. 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
C₈×C₄⋊Q₈	2
#13264	2
#13263	2
#13159	2
#13105	2
#13104	2
#12879	2
#12662	2
C₁₆×C₄⋊C₄	2
#12590	2