Subgroups — #495439

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#13159	C₂	1	2	yes	maximal
4	#13105	C₂	1	4	yes	maximal
5	#12653	C₂	1	1	yes	maximal, characteristic
6	#12653	C₂	1	2	yes	maximal
7	#12591	C₂	1	2	yes	maximal
8	C₄×#346	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	4	yes	—
13	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₂×#346	C₂²	1	2	yes	—
15	C₂×#346	C₂²	1	2	yes	—
16	C₂×#346	C₂²	1	1	yes	characteristic
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	C₈×C₄⋊C₄	C₄	1	2	yes	—
20	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
21	C₈×C₄⋊C₄	C₂²	1	2	yes	—
22	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₈	C₄	1	2	yes	—
25	C₄²×C₈	C₂²	1	2	yes	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1374	C₂²	1	4	yes	—
28	#1374	C₂²	1	4	yes	—
29	#1374	C₂²	1	2	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	4	yes	—
35	C₂²×C₁₆	C₂³	1	2	yes	—
36	C₂²×C₁₆	C₂³	1	2	yes	—
37	C₈×Q₈	—	2	4	no	—
38	C₈×Q₈	—	2	4	no	—
39	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
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	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	C₂³	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	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
55	C₄×C₄⋊C₄	C₈	1	2	yes	—
56	C₄×C₄⋊C₄	C₈	1	2	yes	—
57	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
58	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
59	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₄³	C₂×C₄	1	2	yes	—
61	C₄³	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₁₆	—	2	8	no	—
63	C₄⋊C₁₆	—	2	4	no	—
64	#346	—	2	4	no	—
65	#346	—	2	2	no	—
66	#346	—	2	4	no	—
67	#346	Q₈	1	4	yes	complemented
68	C₄×C₁₆	—	2	4	no	—
69	#336	C₂×C₄	1	2	yes	—
70	#336	C₂×C₄	1	2	yes	—
71	#336	C₂×C₄	1	2	yes	—
72	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₄	1	2	yes	—
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	C₂⁴	1	1	yes	characteristic
79	C₄×Q₈	—	2	4	no	—
80	C₄×Q₈	—	2	4	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
84	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
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×C₄²	C₂×C₈	1	2	yes	—
90	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
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	1	yes	characteristic
96	C₂×C₄²	C₂×C₈	1	2	yes	—
97	C₂×C₄²	C₂²×C₄	1	2	yes	—
98	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
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×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	4	no	—
107	C₂×C₁₆	—	2	4	no	—
108	C₂×C₁₆	—	2	4	no	—
109	C₂×C₁₆	C₂×Q₈	1	4	yes	—
110	C₂×C₁₆	#41	1	4	yes	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	4	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄×C₈	—	2	4	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	—	2	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	4	no	—
120	C₄×C₈	—	2	4	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	4	no	—
123	C₄×C₈	M₁₆	1	4	yes	—
124	C₄×C₈	C₂×Q₈	1	2	yes	—
125	C₄×C₈	#41	1	4	yes	—
126	C₄×C₈	#41	1	2	yes	—
127	#95	C₂×C₈	1	2	yes	—
128	#95	C₂×C₈	1	2	yes	—
129	#95	C₂×C₈	1	2	yes	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	M₃₂	1	4	yes	—
133	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂²×C₄	C₂²×C₈	1	2	yes	—
135	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
136	C₂²×C₄	C₂²×C₈	1	2	yes	—
137	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	#117	1	2	yes	—
155	C₂×C₈	C₄×Q₈	1	2	yes	—
156	C₂×C₈	C₂×M₁₆	1	2	yes	—
157	C₂×C₈	C₂×M₁₆	1	2	yes	—
158	C₂×C₈	#131	1	2	yes	—
159	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₈	C₂×#41	1	1	yes	characteristic
161	C₂×C₈	C₂×#41	1	2	yes	—
162	C₂×C₈	C₂×#41	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	M₃₂	1	4	yes	—
169	C₄²	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	4	1	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	4	1	no	—
182	C₄²	—	4	2	no	—
183	C₄²	#117	1	2	yes	—
184	C₄²	C₄×Q₈	1	2	yes	—
185	C₄²	C₂×M₁₆	1	2	yes	—
186	C₄²	#131	1	2	yes	—
187	C₁₆	—	2	4	no	—
188	C₁₆	—	2	4	no	—
189	C₁₆	—	4	4	no	—
190	C₂³	C₂³×C₈	1	1	yes	characteristic
191	Q₈	—	2	8	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	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	#431	1	2	yes	—
215	C₂×C₄	#431	1	2	yes	—
216	C₂×C₄	C₈×Q₈	1	2	yes	—
217	C₂×C₄	C₈×Q₈	1	2	yes	—
218	C₂×C₄	C₂×M₃₂	1	4	yes	—
219	C₂×C₄	C₂×M₃₂	1	4	yes	—
220	C₂×C₄	C₂×M₃₂	1	2	yes	—
221	C₂×C₄	C₂×M₃₂	1	2	yes	—
222	C₂×C₄	C₂×#117	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
224	C₂×C₄	C₄×#41	1	1	yes	characteristic
225	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
226	C₂×C₄	C₂×#131	1	1	yes	characteristic
227	C₂×C₄	C₂×#131	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	2	no	—
231	C₈	—	4	2	no	—
232	C₈	#446	1	4	yes	—
233	C₈	#529	1	2	yes	—
234	C₈	#533	1	2	yes	—
235	C₂²	C₂×#431	1	1	yes	characteristic
236	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
237	C₂²	C₈×#41	1	1	yes	characteristic
238	C₂²	C₂²×M₃₂	1	1	yes	characteristic
239	C₂²	C₂²×M₃₂	1	1	yes	characteristic
240	C₂²	#3407	1	1	yes	characteristic
241	C₂²	#3407	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	1	no	—
251	C₄	#2163	1	4	yes	—
252	C₄	#2295	1	1	yes	characteristic
253	C₄	#2922	1	1	yes	characteristic
254	C₄	C₂×#446	1	1	yes	characteristic
255	C₄	#2965	1	1	yes	characteristic
256	C₂	#21339	1	1	yes	characteristic
257	C₂	C₂×#2163	1	1	yes	characteristic
258	C₂	#33421	1	1	yes	characteristic
259	C₂	Q₈×M₃₂	1	1	yes	characteristic
260	C₂	Q₈×M₃₂	1	1	yes	characteristic
261	C₂	#33484	1	1	yes	characteristic
262	C₂	#33485	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. #346	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
#13159	2
#13105	2
#12653	2
#12591	2
C₄×#346	2