Subgroups — #109371

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 289 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1566	C₂	1	1	yes	maximal, characteristic
2	C₂²×#335	C₂	1	1	yes	maximal, characteristic
3	C₂×#1451	C₂	1	1	yes	maximal, characteristic
4	#10853	C₂	1	2	yes	maximal
5	C₂×#1301	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	1	yes	characteristic
7	C₂×#423	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#341	—	2	4	no	—
12	#1566	—	2	2	no	—
13	#1566	C₄	1	4	yes	—
14	C₂×#335	—	2	2	no	—
15	C₂×#335	—	2	2	no	—
16	C₂×#335	C₄	1	2	yes	—
17	C₂×#335	C₄	1	2	yes	—
18	C₂×#334	—	2	1	no	—
19	C₂×#334	—	2	1	no	—
20	#1451	—	2	2	no	—
21	#1451	C₄	1	4	yes	—
22	C₂×#322	—	2	1	no	—
23	C₂×#322	—	2	1	no	—
24	#1301	—	2	4	no	—
25	C₂²×M₁₆	—	2	2	no	—
26	C₂²×M₁₆	D₄	1	2	yes	—
27	C₂³×C₈	—	2	2	no	—
28	C₂³×C₈	Q₈	1	1	yes	characteristic
29	C₂³×C₈	Q₈	1	1	yes	characteristic
30	C₂²×C₄²	C₂³	1	1	yes	characteristic
31	#431	—	2	1	no	—
32	#431	—	2	1	no	—
33	#431	C₂×C₄	1	2	yes	—
34	#431	D₄	1	2	yes	—
35	#423	—	2	2	no	—
36	#423	—	2	2	no	—
37	#423	—	2	2	no	—
38	#423	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₈	—	2	1	no	—
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₈	—	2	2	no	—
42	C₂×C₄⋊C₈	—	2	2	no	—
43	C₂×C₄⋊C₈	—	2	2	no	—
44	C₂×C₄⋊C₈	—	2	2	no	—
45	C₂×C₄⋊C₈	—	2	2	no	—
46	C₂×C₄⋊C₈	—	2	2	no	—
47	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
49	C₄×M₁₆	—	2	1	no	—
50	C₄×M₁₆	—	2	1	no	—
51	C₄×M₁₆	C₂×C₄	1	2	yes	—
52	C₄×M₁₆	Q₈	1	2	yes	—
53	C₂×#97	—	2	1	no	—
54	C₂×C₄×C₈	—	2	1	no	—
55	C₂×C₄×C₈	—	2	1	no	—
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
59	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
60	C₂×C₄×C₈	D₄	1	1	yes	characteristic
61	C₂×C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	D₄	1	2	yes	—
63	C₂×C₄×C₈	D₄	1	2	yes	—
64	#341	—	4	8	no	—
65	#335	—	2	4	no	—
66	#335	—	2	4	no	—
67	#334	—	2	4	no	—
68	#334	—	2	4	no	—
69	#322	—	2	4	no	—
70	#322	—	2	4	no	—
71	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	2	4	no	—
76	C₂×M₁₆	—	2	4	no	—
77	C₂×M₁₆	—	2	4	no	—
78	C₂×M₁₆	—	4	2	no	—
79	C₂×M₁₆	—	4	4	no	—
80	C₂×M₁₆	—	4	4	no	—
81	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
82	C₂²×C₈	—	2	4	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	2	4	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	—	4	2	no	—
88	C₂²×C₈	—	4	2	no	—
89	C₂²×C₈	—	4	2	no	—
90	C₂²×C₈	—	4	4	no	—
91	C₂²×C₈	—	4	4	no	—
92	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₄⋊C₄	1	2	yes	—
96	C₂²×C₈	C₄⋊C₄	1	2	yes	—
97	C₂²×C₈	C₄⋊C₄	1	2	yes	—
98	C₂²×C₈	C₄⋊C₄	1	2	yes	—
99	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₄²	#41	1	1	yes	characteristic
109	C₂×C₄²	#41	1	1	yes	characteristic
110	C₂×C₄²	#41	1	1	yes	characteristic
111	C₂×C₄²	#41	1	1	yes	characteristic
112	C₄⋊C₈	—	2	4	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	4	2	no	—
119	C₄⋊C₈	—	4	2	no	—
120	C₄⋊C₈	—	4	2	no	—
121	C₄⋊C₈	—	4	2	no	—
122	C₂²⋊C₈	—	2	2	no	—
123	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
124	#97	—	2	2	no	—
125	#97	—	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	C₄²	1	2	yes	—
131	C₄×C₈	C₄²	1	2	yes	—
132	C₄×C₈	C₂²⋊C₄	1	2	yes	—
133	C₄×C₈	C₂²⋊C₄	1	2	yes	—
134	C₄×C₈	C₂²⋊C₄	1	2	yes	—
135	C₄×C₈	C₂²⋊C₄	1	2	yes	—
136	C₄×C₈	C₂²⋊C₄	1	2	yes	—
137	C₄×C₈	C₂²⋊C₄	1	2	yes	—
138	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
149	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
150	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₄	#121	1	1	yes	characteristic
152	C₂²×C₄	#121	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂²×C₄	#125	1	1	yes	characteristic
156	C₂²×C₄	#125	1	1	yes	characteristic
157	C₂²×C₄	#127	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
160	C₂²×C₄	AGL(1,ℤ/8)	1	4	yes	—
161	C₂²×C₄	#137	1	4	yes	—
162	M₁₆	—	2	8	no	—
163	M₁₆	—	4	2	no	—
164	M₁₆	—	4	2	no	—
165	M₁₆	—	4	8	no	—
166	M₁₆	—	8	2	no	—
167	M₁₆	—	8	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	—	4	8	no	—
182	C₂×C₈	—	8	4	no	—
183	C₂×C₈	—	8	4	no	—
184	C₂×C₈	—	8	2	no	—
185	C₂×C₈	—	8	2	no	—
186	C₂×C₈	#95	1	2	yes	—
187	C₂×C₈	#95	1	2	yes	—
188	C₂×C₈	#95	1	2	yes	—
189	C₂×C₈	#95	1	2	yes	—
190	C₂×C₈	#108	1	4	yes	—
191	C₄²	C₂×C₄²	1	1	yes	characteristic
192	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₄²	#117	1	1	yes	characteristic
196	C₄²	#117	1	1	yes	characteristic
197	C₄²	C₄×D₄	1	1	yes	characteristic
198	C₄²	C₄×D₄	1	1	yes	characteristic
199	C₄²	C₄×Q₈	1	1	yes	characteristic
200	C₄²	C₄×Q₈	1	1	yes	characteristic
201	C₄²	#121	1	1	yes	characteristic
202	C₄²	#121	1	1	yes	characteristic
203	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
204	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
205	C₄²	#124	1	1	yes	characteristic
206	C₄²	#124	1	1	yes	characteristic
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	#384	1	1	yes	characteristic
210	C₂³	#389	1	1	yes	characteristic
211	C₂³	#389	1	1	yes	characteristic
212	C₂³	#501	1	4	yes	—
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	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	C₂×#95	1	1	yes	characteristic
227	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
229	C₂×C₄	#380	1	1	yes	characteristic
230	C₂×C₄	#380	1	1	yes	characteristic
231	C₂×C₄	#382	1	1	yes	characteristic
232	C₂×C₄	#389	1	1	yes	characteristic
233	C₂×C₄	#389	1	1	yes	characteristic
234	C₂×C₄	#390	1	1	yes	characteristic
235	C₂×C₄	#390	1	1	yes	characteristic
236	C₂×C₄	#391	1	1	yes	characteristic
237	C₂×C₄	#391	1	1	yes	characteristic
238	C₂×C₄	#428	1	4	yes	—
239	C₂×C₄	C₂×#108	1	2	yes	—
240	C₂×C₄	#430	1	2	yes	—
241	C₂×C₄	#440	1	4	yes	—
242	C₂×C₄	#441	1	2	yes	—
243	C₂×C₄	#442	1	2	yes	—
244	C₂×C₄	#468	1	4	yes	—
245	C₂×C₄	#469	1	2	yes	—
246	C₂×C₄	#470	1	2	yes	—
247	C₂×C₄	#496	1	2	yes	—
248	C₂×C₄	#497	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	4	4	no	—
253	C₈	—	4	4	no	—
254	C₈	—	8	4	no	—
255	C₈	—	8	4	no	—
256	C₈	#341	1	4	yes	complemented
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	8	no	—
262	C₂²	C₄⋊#95	1	1	yes	characteristic
263	C₂²	#1847	1	2	yes	—
264	C₂²	#1943	1	2	yes	—
265	C₂²	#1944	1	2	yes	—
266	C₂²	#1950	1	2	yes	—
267	C₂²	#1953	1	2	yes	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	C₂×#341	1	1	yes	characteristic
275	C₄	#1738	1	1	yes	characteristic
276	C₄	#1776	1	2	yes	—
277	C₄	#1809	1	2	yes	—
278	C₄	#1950	1	2	yes	—
279	C₄	#1971	1	1	yes	characteristic
280	C₄	#1972	1	1	yes	characteristic
281	C₄	#1973	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	#9625	1	4	yes	—
285	C₂	#10853	1	1	yes	characteristic
286	C₂	#10863	1	1	yes	characteristic
287	C₂	#10873	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#389
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
256. C₈	#341	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#1566	2
C₂²×#335	2
C₂×#1451	2
#10853	2
C₂×#1301	2