Subgroups — #490495

Order 512.

451 conjugacy classes of subgroups.

The subgroup lattice has 265 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33628	C₂	1	2	yes	maximal
2	#19604	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	#12450	C₂	1	4	yes	maximal, complemented
5	#12448	C₂	1	2	yes	maximal, complemented
6	#12446	C₂	1	2	yes	maximal, complemented
7	#12004	C₂	1	1	yes	maximal, characteristic
8	#11942	C₂	1	2	yes	maximal
9	#3170	—	2	1	no	—
10	#3170	—	2	1	no	—
11	#3160	C₂²	1	2	yes	—
12	C₂×C₈×Q₈	—	2	1	no	—
13	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
14	#2229	—	2	4	no	—
15	#2227	C₂²	1	4	yes	—
16	#2225	C₂²	1	4	yes	—
17	#2159	—	2	2	no	—
18	#2150	C₂²	1	2	yes	—
19	#2141	—	2	2	no	—
20	#1713	C₂²	1	1	yes	characteristic
21	#1713	C₂²	1	1	yes	characteristic
22	#1710	C₂²	1	2	yes	—
23	#1707	—	2	1	no	—
24	#1707	—	2	1	no	—
25	C₈×Q₁₆	—	2	1	no	—
26	C₈×Q₁₆	—	2	1	no	—
27	C₈×Q₁₆	C₂²	1	2	yes	—
28	C₈×Q₁₆	C₂²	1	2	yes	—
29	C₂×#335	C₂²	1	1	yes	characteristic
30	C₂×Q₈⋊C₈	—	2	1	no	—
31	C₂×C₈²	C₂²	1	1	yes	characteristic
32	#1370	C₂²	1	2	yes	—
33	#1370	C₂²	1	2	yes	—
34	#1365	C₂²	1	4	yes	—
35	#1364	C₂²	1	4	yes	—
36	C₂²×Q₁₆	D₄	1	1	yes	characteristic
37	#533	—	2	2	no	—
38	C₂×C₄×Q₈	—	2	1	no	—
39	C₂×M₃₂	—	2	2	no	—
40	#500	—	2	1	no	—
41	#500	—	2	1	no	—
42	#500	C₂³	1	2	yes	—
43	#500	C₂³	1	2	yes	—
44	#499	—	2	2	no	—
45	#499	—	2	2	no	—
46	#498	—	2	2	no	—
47	#494	—	2	1	no	—
48	#494	—	2	1	no	—
49	#487	—	2	2	no	—
50	#487	—	2	2	no	—
51	#467	—	4	2	no	—
52	#467	—	4	1	no	—
53	#467	—	4	1	no	—
54	C₈×Q₈	—	2	2	no	—
55	C₈×Q₈	—	2	2	no	—
56	C₈×Q₈	—	4	1	no	—
57	C₈×Q₈	—	4	1	no	—
58	C₄×Q₁₆	—	2	1	no	—
59	C₄×Q₁₆	—	2	1	no	—
60	C₄×Q₁₆	—	4	2	no	—
61	C₄×Q₁₆	—	4	1	no	—
62	C₄×Q₁₆	—	4	1	no	—
63	C₄×Q₁₆	D₄	1	2	yes	—
64	C₄×Q₁₆	C₂³	1	2	yes	—
65	#427	—	2	2	no	—
66	#427	—	2	2	no	—
67	C₂×#107	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₈	—	2	1	no	—
69	C₂×C₄⋊C₈	—	4	1	no	—
70	C₂×#103	—	2	1	no	—
71	C₂×C₄×C₈	—	4	1	no	—
72	C₂×C₄×C₈	D₄	1	1	yes	characteristic
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	#367	—	2	4	no	—
76	#366	—	2	4	no	—
77	C₄⋊C₁₆	—	2	2	no	—
78	C₄⋊C₁₆	C₂³	1	4	yes	—
79	#358	—	2	4	no	—
80	#358	—	2	4	no	—
81	#335	—	2	1	no	—
82	#335	C₂³	1	2	yes	—
83	Q₈⋊C₈	—	2	2	no	—
84	Q₈⋊C₈	—	2	2	no	—
85	C₈²	D₄	1	1	yes	characteristic
86	C₈²	D₄	1	1	yes	characteristic
87	C₈²	C₂³	1	1	yes	characteristic
88	C₈²	C₂³	1	1	yes	characteristic
89	C₂²×Q₈	—	2	1	no	—
90	C₂×Q₁₆	—	2	1	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	4	2	no	—
93	C₂×Q₁₆	—	4	1	no	—
94	C₂×Q₁₆	—	4	1	no	—
95	C₂×Q₁₆	Q₁₆	1	4	yes	complemented
96	C₂×Q₁₆	C₂×D₄	1	2	yes	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	—	4	1	no	—
99	C₂²×C₈	D₈	1	1	yes	characteristic
100	C₂²×C₈	D₈	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₈	#41	1	1	yes	characteristic
103	C₄⋊Q₈	—	2	2	no	—
104	C₄⋊Q₈	—	2	2	no	—
105	#125	—	4	2	no	—
106	C₂²⋊Q₈	—	4	2	no	—
107	C₂²⋊Q₈	—	4	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	4	1	no	—
111	C₄×Q₈	—	4	1	no	—
112	C₄×Q₈	—	4	2	no	—
113	C₄×Q₈	—	4	2	no	—
114	#117	—	4	2	no	—
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄⋊C₄	—	4	1	no	—
117	C₂×C₄²	—	4	1	no	—
118	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
119	M₃₂	—	4	4	no	—
120	C₂×C₁₆	—	2	4	no	—
121	#107	—	2	1	no	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	4	2	no	—
127	#107	—	4	1	no	—
128	#107	—	4	1	no	—
129	#107	C₂×D₄	1	2	yes	—
130	#106	—	4	2	no	—
131	#106	—	4	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	4	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄⋊C₈	—	4	1	no	—
136	#103	—	2	2	no	—
137	#103	—	2	2	no	—
138	#103	—	4	4	no	—
139	#103	—	4	2	no	—
140	#103	—	4	2	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	4	2	no	—
143	C₄×C₈	—	4	2	no	—
144	C₄×C₈	D₈	1	1	yes	characteristic
145	C₄×C₈	D₈	1	1	yes	characteristic
146	C₄×C₈	C₂×D₄	1	1	yes	characteristic
147	C₄×C₈	C₂×D₄	1	1	yes	characteristic
148	C₄×C₈	C₂×D₄	1	1	yes	characteristic
149	C₄×C₈	C₂×D₄	1	1	yes	characteristic
150	C₄×C₈	C₂×D₄	1	1	yes	characteristic
151	C₄×C₈	C₂×D₄	1	1	yes	characteristic
152	C₄×C₈	#41	1	1	yes	characteristic
153	C₄×C₈	C₂⁴	1	1	yes	characteristic
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂²×C₄	—	4	1	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	#121	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
164	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
165	Q₁₆	—	2	2	no	—
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	8	4	no	—
168	Q₁₆	—	8	2	no	—
169	Q₁₆	—	8	2	no	—
170	Q₁₆	C₂×Q₁₆	1	4	yes	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	8	2	no	—
179	C₂×C₈	#121	1	1	yes	characteristic
180	C₂×C₈	#121	1	1	yes	characteristic
181	C₂×C₈	C₂×D₈	1	1	yes	characteristic
182	C₂×C₈	C₂×D₈	1	1	yes	characteristic
183	C₂×C₈	C₂×D₈	1	1	yes	characteristic
184	C₂×C₈	C₂×D₈	1	1	yes	characteristic
185	C₂×C₈	C₂×Q₁₆	1	2	yes	—
186	C₂×C₈	#135	1	2	yes	—
187	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₈	C₂×#41	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	8	2	no	—
198	C₄⋊C₄	—	8	2	no	—
199	C₂²⋊C₄	—	8	2	no	—
200	C₄²	—	4	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	8	2	no	—
203	C₄²	#121	1	1	yes	characteristic
204	C₄²	C₂×D₈	1	1	yes	characteristic
205	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₄²	C₂²×D₄	1	1	yes	characteristic
207	C₁₆	—	4	4	no	—
208	C₂³	#459	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	4	4	no	—
211	Q₈	—	4	2	no	—
212	Q₈	—	8	4	no	—
213	Q₈	—	8	2	no	—
214	Q₈	—	8	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	#459	1	1	yes	characteristic
225	C₂×C₄	#459	1	1	yes	characteristic
226	C₂×C₄	#459	1	1	yes	characteristic
227	C₂×C₄	#467	1	2	yes	—
228	C₂×C₄	#473	1	2	yes	—
229	C₂×C₄	C₂×#121	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
231	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	1	no	—
239	C₈	—	8	2	no	—
240	C₈	#467	1	2	yes	—
241	C₈	#510	1	2	yes	—
242	C₈	C₂²×Q₁₆	1	1	yes	characteristic
243	C₈	C₂×#135	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	#1669	1	2	yes	—
247	C₂²	C₂×#459	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	4	no	—
252	C₄	—	8	2	no	—
253	C₄	—	16	2	no	—
254	C₄	—	16	2	no	—
255	C₄	#1669	1	2	yes	—
256	C₄	C₂×#467	1	1	yes	characteristic
257	C₄	C₂×#473	1	1	yes	characteristic
258	C₄	C₂×#510	1	1	yes	characteristic
259	C₄	#3414	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	C₂×#1669	1	1	yes	characteristic
262	C₂	#33581	1	1	yes	characteristic
263	C₂	#33594	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
4. #12450	C₂	semidirect
5. #12448	C₂	semidirect
6. #12446	C₂	semidirect
95. C₂×Q₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#33628	2
#19604	2
C₂×C₈×Q₁₆	2
#12450	2
#12448	2
#12446	2
#12004	2
#11942	2