Subgroups — #203912

Order 512.

664 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21177	C₂	1	2	yes	maximal
2	C₂×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×#1566	C₂	1	1	yes	maximal, characteristic
4	C₂×#1451	C₂	1	1	yes	maximal, characteristic
5	#9625	C₂	1	8	yes	maximal, complemented
6	#8079	C₂	1	2	yes	maximal
7	C₂×#431	C₂²	1	1	yes	characteristic
8	C₂×#430	—	2	2	no	—
9	C₂×#423	C₂²	1	2	yes	—
10	C₂×#423	C₂²	1	2	yes	—
11	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#1950	—	2	4	no	—
14	#1950	C₂²	1	8	yes	—
15	#1738	—	2	2	no	—
16	#1567	C₄	1	2	yes	—
17	#1567	C₄	1	4	yes	—
18	#1567	C₄	1	2	yes	—
19	#1566	—	2	1	no	—
20	#1566	—	2	1	no	—
21	#1566	C₂²	1	4	yes	—
22	C₂×#335	C₂²	1	4	yes	—
23	C₂×#334	—	2	2	no	—
24	#1451	—	2	2	no	—
25	#1451	C₂²	1	4	yes	—
26	C₂×#322	—	2	1	no	—
27	C₂×#322	—	2	1	no	—
28	#1301	C₂²	1	8	yes	—
29	C₂²×M₁₆	—	2	2	no	—
30	C₂²×M₁₆	—	2	2	no	—
31	C₂²×M₁₆	D₄	1	2	yes	—
32	C₂³×C₈	Q₈	1	2	yes	—
33	C₂²×C₄²	C₂³	1	1	yes	characteristic
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	Q₈	1	2	yes	—
37	#431	C₂³	1	2	yes	—
38	#430	—	4	8	no	—
39	C₂×#108	—	2	8	no	—
40	#423	—	2	2	no	—
41	#423	—	2	2	no	—
42	#423	—	2	2	no	—
43	#423	—	2	2	no	—
44	#423	—	2	2	no	—
45	#423	C₂×C₄	1	4	yes	—
46	#423	C₂×C₄	1	4	yes	—
47	#423	C₂³	1	4	yes	—
48	C₂×C₄⋊C₈	—	2	2	no	—
49	C₂×C₄⋊C₈	—	2	1	no	—
50	C₂×C₄⋊C₈	—	2	2	no	—
51	C₂×C₄⋊C₈	—	2	2	no	—
52	C₂×C₄⋊C₈	C₂³	1	4	yes	—
53	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
54	C₄×M₁₆	—	2	1	no	—
55	C₄×M₁₆	—	2	1	no	—
56	C₄×M₁₆	D₄	1	2	yes	—
57	C₄×M₁₆	C₂³	1	2	yes	—
58	C₂×#97	—	2	1	no	—
59	C₂×C₄×C₈	—	2	1	no	—
60	C₂×C₄×C₈	—	2	1	no	—
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	D₄	1	2	yes	—
65	C₂×C₄×C₈	D₄	1	1	yes	characteristic
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	2	yes	—
70	#341	—	2	8	no	—
71	#335	C₂×C₄	1	8	yes	—
72	#335	C₂×C₄	1	8	yes	—
73	#334	—	2	4	no	—
74	#334	—	2	4	no	—
75	#322	—	2	4	no	—
76	#322	—	2	4	no	—
77	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂×M₁₆	—	2	4	no	—
81	C₂×M₁₆	—	2	4	no	—
82	C₂×M₁₆	—	2	2	no	—
83	C₂×M₁₆	—	2	4	no	—
84	C₂×M₁₆	—	2	4	no	—
85	C₂×M₁₆	—	2	4	no	—
86	C₂×M₁₆	—	4	4	no	—
87	C₂×M₁₆	—	4	4	no	—
88	C₂×M₁₆	—	4	2	no	—
89	C₂×M₁₆	—	4	4	no	—
90	C₂×M₁₆	—	4	4	no	—
91	C₂×M₁₆	C₂×D₄	1	4	yes	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	2	4	no	—
95	C₂²×C₈	—	4	4	no	—
96	C₂²×C₈	—	4	2	no	—
97	C₂²×C₈	C₄⋊C₄	1	2	yes	—
98	C₂²×C₈	C₄⋊C₄	1	2	yes	—
99	C₂²×C₈	C₄⋊C₄	1	2	yes	—
100	C₂²×C₈	C₄⋊C₄	1	2	yes	—
101	C₂²×C₈	C₂×Q₈	1	2	yes	—
102	C₂²×C₈	C₂×Q₈	1	2	yes	—
103	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₄	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×D₄	1	1	yes	characteristic
108	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₄²	#41	1	1	yes	characteristic
111	C₂×C₄²	#41	1	1	yes	characteristic
112	C₂×C₄²	#41	1	1	yes	characteristic
113	C₂×C₄²	#41	1	1	yes	characteristic
114	C₂×C₄²	C₂⁴	1	1	yes	characteristic
115	#108	—	4	16	no	—
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	4	no	—
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	4	2	no	—
124	C₄⋊C₈	—	4	2	no	—
125	C₄⋊C₈	C₂²×C₄	1	8	yes	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	C₂×Q₈	1	4	yes	—
128	#97	—	2	2	no	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	C₄⋊C₄	1	2	yes	—
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈	C₄⋊C₄	1	2	yes	—
139	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₄	1	2	yes	—
141	C₄×C₈	C₂²×C₄	1	2	yes	—
142	C₄×C₈	C₂×D₄	1	2	yes	—
143	C₄×C₈	C₂×D₄	1	2	yes	—
144	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
150	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
151	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
152	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
153	C₂²×C₄	#121	1	1	yes	characteristic
154	C₂²×C₄	#121	1	1	yes	characteristic
155	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄	#125	1	1	yes	characteristic
158	C₂²×C₄	#125	1	1	yes	characteristic
159	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
160	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	AGL(1,ℤ/8)	1	4	yes	—
163	C₂²×C₄	#137	1	4	yes	—
164	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
166	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
167	M₁₆	—	2	4	no	—
168	M₁₆	—	2	4	no	—
169	M₁₆	—	4	8	no	—
170	M₁₆	—	4	8	no	—
171	M₁₆	—	4	4	no	—
172	M₁₆	—	4	4	no	—
173	M₁₆	—	4	4	no	—
174	M₁₆	—	8	4	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	8	4	no	—
190	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
191	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
192	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
193	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
194	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₄²	C₄×D₄	1	1	yes	characteristic
197	C₄²	C₄×D₄	1	1	yes	characteristic
198	C₄²	C₄×Q₈	1	1	yes	characteristic
199	C₄²	C₄×Q₈	1	1	yes	characteristic
200	C₄²	#121	1	1	yes	characteristic
201	C₄²	#121	1	1	yes	characteristic
202	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
203	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
204	C₄²	#125	1	1	yes	characteristic
205	C₄²	#125	1	1	yes	characteristic
206	C₄²	C₂³×C₄	1	1	yes	characteristic
207	C₄²	C₂²×D₄	1	1	yes	characteristic
208	C₄²	C₂×#41	1	1	yes	characteristic
209	C₄²	C₂×#41	1	1	yes	characteristic
210	C₂³	—	4	1	no	—
211	C₂³	—	4	1	no	—
212	C₂³	#389	1	1	yes	characteristic
213	C₂³	#389	1	1	yes	characteristic
214	C₂³	#501	1	4	yes	—
215	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
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	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	#389	1	1	yes	characteristic
229	C₂×C₄	#389	1	1	yes	characteristic
230	C₂×C₄	#389	1	1	yes	characteristic
231	C₂×C₄	#389	1	1	yes	characteristic
232	C₂×C₄	#389	1	1	yes	characteristic
233	C₂×C₄	#389	1	1	yes	characteristic
234	C₂×C₄	#441	1	2	yes	—
235	C₂×C₄	#441	1	2	yes	—
236	C₂×C₄	#442	1	2	yes	—
237	C₂×C₄	#442	1	2	yes	—
238	C₂×C₄	#468	1	4	yes	—
239	C₂×C₄	#468	1	4	yes	—
240	C₂×C₄	#501	1	4	yes	—
241	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
244	C₂×C₄	C₂×#121	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₂×#125	1	1	yes	characteristic
247	C₂×C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
248	C₂×C₄	C₂×#137	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	8	no	—
254	C₈	—	4	4	no	—
255	C₈	—	8	4	no	—
256	C₈	#430	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	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	4	no	—
264	C₂²	#1943	1	2	yes	—
265	C₂²	#1943	1	2	yes	—
266	C₂²	C₂×#389	1	1	yes	characteristic
267	C₂²	C₂×#501	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	#1943	1	2	yes	—
275	C₄	#1943	1	2	yes	—
276	C₄	#1953	1	2	yes	—
277	C₄	C₂×#430	1	2	yes	—
278	C₄	C₂×#441	1	1	yes	characteristic
279	C₄	C₂×#442	1	1	yes	characteristic
280	C₄	C₂×#468	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	C₂×#1943	1	1	yes	characteristic
285	C₂	#21177	1	1	yes	characteristic
286	C₂	C₂×#1953	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
5. #9625	C₂	semidirect
256. C₈	#430	semidirect

Maximal subgroups

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

subgroup	index
#21177	2
C₂×#1567	2
C₂×#1566	2
C₂×#1451	2
#9625	2
#8079	2