Subgroups — #497174

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21715	C₂	1	1	yes	maximal, characteristic
2	#13264	C₂	1	2	yes	maximal
3	#13263	C₂	1	2	yes	maximal
4	#13160	C₂	1	1	yes	maximal, characteristic
5	#13160	C₂	1	1	yes	maximal, characteristic
6	#13105	C₂	1	2	yes	maximal
7	#13104	C₂	1	2	yes	maximal
8	#12880	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	#12592	C₂	1	1	yes	maximal, characteristic
12	C₂×C₈×Q₈	C₂²	1	2	yes	—
13	#2399	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	1	yes	characteristic
19	C₂×#346	C₂²	1	2	yes	—
20	C₂×C₄×C₁₆	C₂²	1	2	yes	—
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	#1985	C₄	1	2	yes	—
23	#1940	C₄	1	1	yes	characteristic
24	#1940	C₄	1	1	yes	characteristic
25	#1940	C₄	1	2	yes	—
26	#1843	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	#1726	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	C₂×C₄	1	2	yes	—
42	C₂²×C₁₆	C₂³	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	1	yes	characteristic
47	C₂²×C₁₆	C₂³	1	1	yes	characteristic
48	C₈×Q₈	—	4	2	no	—
49	C₈×Q₈	—	4	2	no	—
50	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
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×#97	C₂×C₄	1	1	yes	characteristic
57	C₂×#97	C₂×C₄	1	1	yes	characteristic
58	C₂×#97	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	2	yes	—
60	C₂×C₄×C₈	C₂³	1	2	yes	—
61	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	#398	C₈	1	2	yes	—
65	#398	C₈	1	2	yes	—
66	#395	C₈	1	2	yes	—
67	#395	C₈	1	2	yes	—
68	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
70	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
71	#376	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₁₆	—	2	2	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	C₄⋊C₁₆	—	4	2	no	—
75	C₄⋊C₁₆	—	4	2	no	—
76	#346	—	2	2	no	—
77	#346	—	4	2	no	—
78	C₄×C₁₆	—	4	2	no	—
79	C₄×C₁₆	Q₈	1	4	yes	complemented
80	#336	C₂×C₄	1	2	yes	—
81	#336	C₂×C₄	1	2	yes	—
82	#336	C₂×C₄	1	2	yes	—
83	#336	C₂×C₄	1	2	yes	—
84	C₂²×Q₈	C₂×C₈	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	2	yes	—
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂⁴	1	1	yes	characteristic
91	C₄×Q₈	—	4	2	no	—
92	C₄×Q₈	—	4	2	no	—
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	2	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	2	yes	—
99	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
100	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₄	1	2	yes	—
102	C₂×C₄²	C₂²×C₄	1	2	yes	—
103	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	2	1	no	—
110	C₂×C₁₆	—	2	1	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	1	no	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	C₂×Q₈	1	2	yes	—
122	C₂×C₁₆	C₂×Q₈	1	2	yes	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	4	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	4	2	no	—
129	#97	—	2	2	no	—
130	#97	—	4	1	no	—
131	#97	—	4	1	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	C₂×Q₈	1	2	yes	—
141	C₄×C₈	#41	1	2	yes	—
142	#95	C₂×C₈	1	2	yes	—
143	#95	C₂×C₈	1	2	yes	—
144	#95	C₂×C₈	1	2	yes	—
145	#95	C₂×C₈	1	1	yes	characteristic
146	#95	C₂×C₈	1	1	yes	characteristic
147	#95	C₂×C₈	1	2	yes	—
148	#95	C₂×C₈	1	1	yes	characteristic
149	#95	C₂×C₈	1	1	yes	characteristic
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂²×C₄	C₂²×C₈	1	2	yes	—
154	C₂²×C₄	C₂²×C₈	1	2	yes	—
155	C₂²×C₄	C₂²×C₈	1	2	yes	—
156	C₂²×C₄	C₂²×C₈	1	2	yes	—
157	C₂²×C₄	C₂²×C₈	1	2	yes	—
158	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₈	1	2	yes	—
161	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	C₄×Q₈	1	2	yes	—
177	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₈	C₂×#41	1	1	yes	characteristic
179	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
182	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₄×Q₈	1	2	yes	—
199	C₄²	#131	1	2	yes	—
200	C₁₆	—	2	2	no	—
201	C₁₆	—	2	2	no	—
202	C₁₆	—	4	1	no	—
203	C₁₆	—	4	1	no	—
204	C₁₆	—	4	2	no	—
205	C₁₆	—	4	2	no	—
206	C₂³	C₂³×C₈	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	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	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	C₈×Q₈	1	2	yes	—
228	C₂×C₄	C₈×Q₈	1	2	yes	—
229	C₂×C₄	#504	1	4	yes	—
230	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
231	C₂×C₄	#518	1	1	yes	characteristic
232	C₂×C₄	#519	1	1	yes	characteristic
233	C₂×C₄	C₂×#131	1	1	yes	characteristic
234	C₂×C₄	#568	1	1	yes	characteristic
235	C₂×C₄	#568	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	4	1	no	—
238	C₈	—	4	2	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	1	no	—
241	C₈	#542	1	1	yes	characteristic
242	C₈	#542	1	1	yes	characteristic
243	C₈	#544	1	1	yes	characteristic
244	C₈	#544	1	1	yes	characteristic
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	C₂×#504	1	2	yes	—
249	C₂²	#3407	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	4	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	#2331	1	1	yes	characteristic
259	C₄	#2961	1	1	yes	characteristic
260	C₄	#2981	1	1	yes	characteristic
261	C₄	#2985	1	1	yes	characteristic
262	C₂	#22307	1	1	yes	characteristic
263	C₂	#33474	1	1	yes	characteristic
264	C₂	#33475	1	1	yes	characteristic
265	C₂	#33490	1	2	yes	—
266	C₂	#33499	1	2	yes	—

	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
79. 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
#21715	2
#13264	2
#13263	2
#13160	2
#13105	2
#13104	2
#12880	2
#12662	2
C₁₆×C₄⋊C₄	2
#12592	2