Subgroups — #490493

Order 512.

459 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33626	C₂	1	2	yes	maximal
2	#19604	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#135	C₂	1	1	yes	maximal, characteristic
4	#12450	C₂	1	2	yes	maximal, complemented
5	#12449	C₂	1	2	yes	maximal, complemented
6	#12448	C₂	1	1	yes	maximal, characteristic, complemented
7	#12447	C₂	1	1	yes	maximal, characteristic, complemented
8	#12446	C₂	1	1	yes	maximal, characteristic, complemented
9	#12445	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#1370	C₂	1	1	yes	maximal, characteristic, complemented
11	#11939	C₂	1	2	yes	maximal, complemented
12	#3170	—	2	1	no	—
13	#3170	—	2	1	no	—
14	#3160	C₂²	1	2	yes	—
15	C₈×#41	—	2	1	no	—
16	C₄×#135	C₂²	1	1	yes	characteristic
17	#2229	—	2	2	no	—
18	#2228	—	2	2	no	—
19	#2227	C₂²	1	2	yes	—
20	#2226	C₂²	1	2	yes	—
21	#2225	C₂²	1	2	yes	—
22	#2224	C₂²	1	2	yes	—
23	#2158	—	2	2	no	—
24	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
25	#2138	—	2	2	no	—
26	#1713	C₂²	1	1	yes	characteristic
27	#1713	C₂²	1	1	yes	characteristic
28	#1710	C₂²	1	2	yes	—
29	#1707	—	2	1	no	—
30	#1707	—	2	1	no	—
31	C₈×Q₁₆	C₂²	1	1	yes	characteristic
32	C₈×Q₁₆	C₂²	1	1	yes	characteristic
33	C₈×SD₁₆	—	2	1	no	—
34	C₈×SD₁₆	—	2	1	no	—
35	C₈×D₈	C₂²	1	1	yes	characteristic
36	C₈×D₈	C₂²	1	1	yes	characteristic
37	#1564	C₂²	1	1	yes	characteristic
38	#1476	—	2	1	no	—
39	C₂×C₈²	C₂²	1	1	yes	characteristic
40	#1370	C₂²	1	1	yes	characteristic
41	#1370	C₂²	1	1	yes	characteristic
42	#1370	C₂²	1	2	yes	—
43	#1365	C₂²	1	2	yes	—
44	#1365	C₂²	1	2	yes	—
45	#1364	C₂²	1	2	yes	—
46	#1364	C₂²	1	2	yes	—
47	C₂×#135	D₄	1	1	yes	characteristic
48	#533	—	2	2	no	—
49	C₄×#41	—	2	1	no	—
50	C₂²×C₁₆	—	2	2	no	—
51	#500	—	2	1	no	—
52	#500	—	2	1	no	—
53	#500	C₂³	1	2	yes	—
54	#500	C₂³	1	2	yes	—
55	#499	—	2	2	no	—
56	#499	—	2	2	no	—
57	#498	—	2	2	no	—
58	#494	—	2	1	no	—
59	#494	—	2	1	no	—
60	#487	—	2	2	no	—
61	#487	—	2	2	no	—
62	#467	—	4	2	no	—
63	#467	—	4	1	no	—
64	#467	—	4	1	no	—
65	C₈×Q₈	—	2	1	no	—
66	C₈×Q₈	—	2	1	no	—
67	C₄×Q₁₆	—	4	2	no	—
68	C₄×Q₁₆	—	4	1	no	—
69	C₄×Q₁₆	—	4	1	no	—
70	C₄×Q₁₆	D₄	1	1	yes	characteristic
71	C₄×Q₁₆	C₂³	1	1	yes	characteristic
72	C₄×SD₁₆	—	2	1	no	—
73	C₄×SD₁₆	—	2	1	no	—
74	C₄×D₈	D₄	1	1	yes	characteristic
75	C₄×D₈	C₂³	1	1	yes	characteristic
76	C₈×D₄	—	2	1	no	—
77	C₈×D₄	—	2	1	no	—
78	C₈×D₄	—	4	1	no	—
79	C₈×D₄	—	4	1	no	—
80	#431	—	2	1	no	—
81	#431	—	4	1	no	—
82	#427	—	2	2	no	—
83	#427	—	2	2	no	—
84	#427	D₄	1	1	yes	characteristic
85	#416	—	2	1	no	—
86	C₂×C₄×C₈	—	4	1	no	—
87	C₂×C₄×C₈	D₄	1	1	yes	characteristic
88	C₂×C₄×C₈	D₄	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
90	#367	—	2	2	no	—
91	#367	—	2	2	no	—
92	#366	—	2	2	no	—
93	#366	—	2	2	no	—
94	C₄⋊C₁₆	—	2	2	no	—
95	C₄⋊C₁₆	C₂³	1	2	yes	—
96	C₄⋊C₁₆	C₂³	1	2	yes	—
97	#358	—	2	2	no	—
98	#358	—	2	2	no	—
99	#357	—	2	2	no	—
100	#357	—	2	2	no	—
101	#335	C₂³	1	1	yes	characteristic
102	#335	C₂³	1	1	yes	characteristic
103	#334	—	2	1	no	—
104	Q₈⋊C₈	—	2	1	no	—
105	Q₈⋊C₈	—	2	1	no	—
106	D₄⋊C₈	—	2	1	no	—
107	D₄⋊C₈	—	2	1	no	—
108	C₈²	D₄	1	1	yes	characteristic
109	C₈²	D₄	1	1	yes	characteristic
110	C₈²	C₂³	1	1	yes	characteristic
111	C₈²	C₂³	1	1	yes	characteristic
112	C₂×#41	—	2	1	no	—
113	#135	—	2	2	no	—
114	#135	Q₁₆	1	4	yes	complemented
115	C₂×Q₁₆	—	4	2	no	—
116	C₂×Q₁₆	—	4	1	no	—
117	C₂×Q₁₆	—	4	1	no	—
118	C₂×Q₁₆	C₂×D₄	1	1	yes	characteristic
119	C₂×SD₁₆	—	2	1	no	—
120	C₂×D₈	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	4	1	no	—
123	C₂²×C₈	—	4	1	no	—
124	C₂²×C₈	D₈	1	1	yes	characteristic
125	C₂²×C₈	D₈	1	1	yes	characteristic
126	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	#125	—	4	2	no	—
131	C₂²⋊Q₈	—	4	2	no	—
132	C₂²⋊Q₈	—	4	2	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×Q₈	—	4	2	no	—
136	C₄×Q₈	—	4	2	no	—
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	—	2	1	no	—
139	C₄×D₄	—	4	1	no	—
140	C₄×D₄	—	4	1	no	—
141	#117	—	2	1	no	—
142	#117	—	4	1	no	—
143	#117	—	4	2	no	—
144	C₂×C₄²	—	4	1	no	—
145	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	4	4	no	—
149	#107	—	2	2	no	—
150	#107	—	2	2	no	—
151	#107	—	2	2	no	—
152	#107	—	2	2	no	—
153	#107	—	4	2	no	—
154	#107	—	4	1	no	—
155	#107	—	4	1	no	—
156	#107	C₂×D₄	1	1	yes	characteristic
157	#107	C₂×D₄	1	1	yes	characteristic
158	#106	—	2	1	no	—
159	#106	—	4	2	no	—
160	#106	—	4	2	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₄⋊C₈	—	4	1	no	—
164	C₄⋊C₈	—	4	1	no	—
165	#103	—	2	1	no	—
166	#103	—	2	1	no	—
167	#103	—	4	4	no	—
168	#103	—	4	2	no	—
169	#103	—	4	2	no	—
170	#102	—	2	1	no	—
171	#102	—	2	1	no	—
172	C₂²⋊C₈	—	4	1	no	—
173	C₂²⋊C₈	—	4	1	no	—
174	C₂²⋊C₈	—	4	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	—	4	1	no	—
178	C₄×C₈	—	4	1	no	—
179	C₄×C₈	D₈	1	1	yes	characteristic
180	C₄×C₈	D₈	1	1	yes	characteristic
181	C₄×C₈	C₂×D₄	1	1	yes	characteristic
182	C₄×C₈	C₂×D₄	1	1	yes	characteristic
183	C₄×C₈	C₂×D₄	1	1	yes	characteristic
184	C₄×C₈	C₂×D₄	1	1	yes	characteristic
185	C₄×C₈	C₂×D₄	1	1	yes	characteristic
186	C₄×C₈	C₂×D₄	1	1	yes	characteristic
187	C₄×C₈	#41	1	1	yes	characteristic
188	C₄×C₈	C₂⁴	1	1	yes	characteristic
189	#41	—	2	4	no	—
190	#41	—	4	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	#121	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
201	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
202	Q₁₆	—	2	1	no	—
203	Q₁₆	—	8	4	no	—
204	Q₁₆	—	8	2	no	—
205	Q₁₆	—	8	2	no	—
206	Q₁₆	C₂×Q₁₆	1	2	yes	—
207	SD₁₆	—	2	4	no	—
208	D₈	—	2	1	no	—
209	D₈	C₂×Q₁₆	1	2	yes	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	8	2	no	—
219	C₂×C₈	—	8	2	no	—
220	C₂×C₈	#121	1	1	yes	characteristic
221	C₂×C₈	#121	1	1	yes	characteristic
222	C₂×C₈	C₂×D₈	1	1	yes	characteristic
223	C₂×C₈	C₂×D₈	1	1	yes	characteristic
224	C₂×C₈	C₂×D₈	1	1	yes	characteristic
225	C₂×C₈	C₂×D₈	1	1	yes	characteristic
226	C₂×C₈	C₂×Q₁₆	1	2	yes	—
227	C₂×C₈	#135	1	2	yes	—
228	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂×#41	1	1	yes	characteristic
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	8	2	no	—
239	C₄⋊C₄	—	8	2	no	—
240	C₂²⋊C₄	—	4	1	no	—
241	C₂²⋊C₄	—	4	1	no	—
242	C₂²⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	8	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	8	2	no	—
248	C₄²	#121	1	1	yes	characteristic
249	C₄²	C₂×D₈	1	1	yes	characteristic
250	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₄²	C₂²×D₄	1	1	yes	characteristic
252	C₁₆	—	4	2	no	—
253	C₁₆	—	4	2	no	—
254	C₂³	—	4	1	no	—
255	C₂³	#459	1	1	yes	characteristic
256	Q₈	—	2	2	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	8	4	no	—
259	Q₈	—	8	2	no	—
260	Q₈	—	8	2	no	—
261	D₄	—	2	2	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	#459	1	1	yes	characteristic
277	C₂×C₄	#459	1	1	yes	characteristic
278	C₂×C₄	#459	1	1	yes	characteristic
279	C₂×C₄	#467	1	2	yes	—
280	C₂×C₄	#473	1	2	yes	—
281	C₂×C₄	C₂×#121	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	4	1	no	—
291	C₈	—	8	1	no	—
292	C₈	—	8	1	no	—
293	C₈	#467	1	2	yes	—
294	C₈	#508	1	2	yes	—
295	C₈	C₂²×Q₁₆	1	1	yes	characteristic
296	C₈	C₂×#135	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	1	no	—
302	C₂²	#1669	1	2	yes	—
303	C₂²	C₂×#459	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	1	no	—
310	C₄	—	16	2	no	—
311	C₄	—	16	2	no	—
312	C₄	#1669	1	2	yes	—
313	C₄	C₂×#467	1	1	yes	characteristic
314	C₄	C₂×#473	1	1	yes	characteristic
315	C₄	C₂×#508	1	1	yes	characteristic
316	C₄	#3414	1	1	yes	characteristic
317	C₂	—	2	2	no	—
318	C₂	—	8	2	no	—
319	C₂	C₂×#1669	1	1	yes	characteristic
320	C₂	#33579	1	1	yes	characteristic
321	C₂	#33594	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #19604	C₂	semidirect
4. #12450	C₂	semidirect
5. #12449	C₂	semidirect
6. #12448	C₂	semidirect
7. #12447	C₂	semidirect
8. #12446	C₂	semidirect
9. #12445	C₂	semidirect
10. C₂×#1370	C₂	semidirect
11. #11939	C₂	semidirect
114. #135	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#33626	2
#19604	2
C₈×#135	2
#12450	2
#12449	2
#12448	2
#12447	2
#12446	2
#12445	2
C₂×#1370	2
#11939	2