Subgroups — #497177

Order 512.

460 conjugacy classes of subgroups.

The subgroup lattice has 274 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	#13160	C₂	1	2	yes	maximal
5	#13105	C₂	1	2	yes	maximal
6	#13104	C₂	1	2	yes	maximal
7	#12879	C₂	1	1	yes	maximal, characteristic
8	#12663	C₂	1	2	yes	maximal
9	#12591	C₂	1	1	yes	maximal, characteristic
10	C₂×C₈×Q₈	C₂²	1	2	yes	—
11	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
12	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₂×#346	C₂²	1	1	yes	characteristic
16	C₂×#346	C₂²	1	2	yes	—
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	#1984	C₄	1	2	yes	—
20	#1939	C₄	1	2	yes	—
21	#1939	C₄	1	1	yes	characteristic
22	#1939	C₄	1	1	yes	characteristic
23	#1842	C₄	1	1	yes	characteristic
24	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₈×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄²×C₈	C₂²	1	1	yes	characteristic
28	#1374	C₂²	1	2	yes	—
29	#1374	C₂²	1	2	yes	—
30	#1374	C₂²	1	2	yes	—
31	#1374	C₂²	1	2	yes	—
32	#1374	C₂²	1	2	yes	—
33	#1374	C₂²	1	2	yes	—
34	#1374	C₂²	1	2	yes	—
35	#1374	C₂²	1	1	yes	characteristic
36	#1374	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
38	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
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	C₈×Q₈	—	4	2	no	—
44	C₈×Q₈	—	4	2	no	—
45	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₈	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	2	yes	—
55	C₂×C₄×C₈	C₂³	1	2	yes	—
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	#391	C₈	1	2	yes	—
60	#389	C₈	1	2	yes	—
61	#389	C₈	1	1	yes	characteristic
62	#389	C₈	1	1	yes	characteristic
63	#384	C₈	1	1	yes	characteristic
64	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
67	C₄³	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₁₆	—	2	4	no	—
69	C₄⋊C₁₆	—	4	2	no	—
70	C₄⋊C₁₆	—	4	2	no	—
71	#346	—	2	2	no	—
72	#346	—	4	2	no	—
73	#346	Q₈	1	4	yes	complemented
74	C₄×C₁₆	—	4	2	no	—
75	#336	C₂×C₄	1	2	yes	—
76	#336	C₂×C₄	1	2	yes	—
77	#336	C₂×C₄	1	2	yes	—
78	#336	C₂×C₄	1	2	yes	—
79	C₂²×Q₈	C₂×C₈	1	2	yes	—
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₄	1	2	yes	—
82	C₂²×C₈	C₂²×C₄	1	2	yes	—
83	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₈	C₂⁴	1	1	yes	characteristic
86	C₄⋊Q₈	—	2	4	no	—
87	C₄×Q₈	—	4	2	no	—
88	C₄×Q₈	—	4	2	no	—
89	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
91	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
92	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
93	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
94	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
99	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
101	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂²×C₄	1	2	yes	—
104	C₂×C₄²	C₂²×C₄	1	2	yes	—
105	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
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	C₂×Q₈	1	4	yes	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	4	1	no	—
121	C₄⋊C₈	—	4	1	no	—
122	C₄⋊C₈	—	4	2	no	—
123	C₄⋊C₈	—	4	2	no	—
124	C₄⋊C₈	—	4	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	4	1	no	—
133	C₄×C₈	—	4	1	no	—
134	C₄×C₈	C₂×Q₈	1	2	yes	—
135	C₄×C₈	#41	1	2	yes	—
136	#95	C₂×C₈	1	2	yes	—
137	#95	C₂×C₈	1	2	yes	—
138	#95	C₂×C₈	1	2	yes	—
139	#95	C₂×C₈	1	2	yes	—
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₈	1	2	yes	—
145	C₂²×C₄	C₂²×C₈	1	2	yes	—
146	C₂²×C₄	C₂²×C₈	1	2	yes	—
147	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
148	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₈	1	2	yes	—
151	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₄×Q₈	1	2	yes	—
169	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
170	C₂×C₈	C₂×#41	1	1	yes	characteristic
171	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	M₃₂	1	4	yes	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	C₄×Q₈	1	2	yes	—
195	C₄²	#131	1	2	yes	—
196	C₁₆	—	2	4	no	—
197	C₁₆	—	4	2	no	—
198	C₁₆	—	4	2	no	—
199	C₁₆	—	4	2	no	—
200	C₂³	C₂³×C₈	1	1	yes	characteristic
201	Q₈	—	4	8	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	1	no	—
208	C₂×C₄	—	2	1	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	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	C₈×Q₈	1	2	yes	—
228	C₂×C₄	C₈×Q₈	1	2	yes	—
229	C₂×C₄	C₂×M₃₂	1	2	yes	—
230	C₂×C₄	C₂×M₃₂	1	4	yes	—
231	C₂×C₄	#504	1	2	yes	—
232	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
233	C₂×C₄	#518	1	1	yes	characteristic
234	C₂×C₄	#519	1	1	yes	characteristic
235	C₂×C₄	C₂×#131	1	1	yes	characteristic
236	C₂×C₄	#568	1	1	yes	characteristic
237	C₂×C₄	#568	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	4	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	4	1	no	—
243	C₈	#542	1	2	yes	—
244	C₈	#544	1	2	yes	—
245	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
246	C₂²	#2972	1	1	yes	characteristic
247	C₂²	#2982	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	C₂²	#3407	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	1	no	—
262	C₄	#2331	1	1	yes	characteristic
263	C₄	#2961	1	1	yes	characteristic
264	C₄	#2981	1	1	yes	characteristic
265	C₄	#2985	1	1	yes	characteristic
266	C₂	#22307	1	1	yes	characteristic
267	C₂	#33476	1	1	yes	characteristic
268	C₂	Q₈×M₃₂	1	1	yes	characteristic
269	C₂	#33488	1	1	yes	characteristic
270	C₂	#33494	1	1	yes	characteristic
271	C₂	#33497	1	1	yes	characteristic
272	C₂	#33499	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
73. #346	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
#13160	2
#13105	2
#13104	2
#12879	2
#12663	2
#12591	2