Subgroups — #491587

Order 512.

485 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33797	C₂	1	2	yes	maximal
2	#33734	C₂	1	1	yes	maximal, characteristic
3	#33726	C₂	1	1	yes	maximal, characteristic
4	#19648	C₂	1	2	yes	maximal
5	#12535	C₂	1	2	yes	maximal, complemented
6	#12533	C₂	1	2	yes	maximal, complemented
7	#12522	C₂	1	1	yes	maximal, characteristic, complemented
8	#12521	C₂	1	1	yes	maximal, characteristic, complemented
9	#12504	C₂	1	1	yes	maximal, characteristic, complemented
10	#12504	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic
12	#3231	—	2	2	no	—
13	#3185	—	2	2	no	—
14	#3170	C₂²	1	2	yes	—
15	#3160	C₂²	1	2	yes	—
16	#2254	C₂²	1	2	yes	—
17	#2254	C₂²	1	2	yes	—
18	#2253	—	2	2	no	—
19	#2252	C₂²	1	2	yes	—
20	#2252	C₂²	1	1	yes	characteristic
21	#2252	C₂²	1	1	yes	characteristic
22	#2252	C₂²	1	1	yes	characteristic
23	#2247	C₂²	1	1	yes	characteristic
24	#2241	C₂²	1	2	yes	—
25	#2241	C₂²	1	2	yes	—
26	#2241	C₂²	1	2	yes	—
27	#2237	—	2	4	no	—
28	#2229	—	2	4	no	—
29	#2216	—	2	2	no	—
30	C₄×Q₃₂	—	2	2	no	—
31	#2162	C₂²	1	1	yes	characteristic
32	#2158	—	2	2	no	—
33	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
34	#1718	—	2	2	no	—
35	#1716	—	2	2	no	—
36	#1713	C₂²	1	2	yes	—
37	#1710	C₂²	1	2	yes	—
38	#1707	C₂²	1	2	yes	—
39	#1707	C₂²	1	2	yes	—
40	C₈×M₁₆	C₂²	1	1	yes	characteristic
41	#1312	C₂²	1	1	yes	characteristic
42	#1312	C₂²	1	1	yes	characteristic
43	C₈×C₁₆	C₂²	1	1	yes	characteristic
44	C₈×C₁₆	C₂²	1	1	yes	characteristic
45	#533	—	2	2	no	—
46	#533	—	2	2	no	—
47	C₂×Q₃₂	—	2	2	no	—
48	C₂²×C₁₆	D₄	1	2	yes	—
49	#500	—	2	2	no	—
50	#500	C₂³	1	2	yes	—
51	#500	C₂³	1	2	yes	—
52	#500	C₂³	1	2	yes	—
53	#499	—	2	2	no	—
54	#499	—	2	2	no	—
55	#498	—	2	2	no	—
56	#498	—	2	2	no	—
57	#498	—	2	2	no	—
58	#494	—	2	2	no	—
59	#494	C₂³	1	2	yes	—
60	#487	—	2	2	no	—
61	#487	—	2	2	no	—
62	#487	—	2	2	no	—
63	#487	—	2	2	no	—
64	#487	—	2	2	no	—
65	#485	—	4	4	no	—
66	#479	—	4	4	no	—
67	#470	—	4	4	no	—
68	#467	—	2	2	no	—
69	#467	—	2	2	no	—
70	#441	—	4	4	no	—
71	C₄×Q₁₆	—	2	2	no	—
72	C₄×Q₁₆	—	2	2	no	—
73	#431	D₄	1	1	yes	characteristic
74	#427	—	2	2	no	—
75	#427	—	2	2	no	—
76	C₄×M₁₆	D₄	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
78	#367	—	2	4	no	—
79	#367	—	2	4	no	—
80	#366	—	2	2	no	—
81	#366	—	2	2	no	—
82	#366	—	2	2	no	—
83	#366	—	2	4	no	—
84	C₄⋊C₁₆	D₄	1	2	yes	—
85	#358	—	2	4	no	—
86	#358	—	2	4	no	—
87	#358	—	2	4	no	—
88	C₂²⋊C₁₆	D₄	1	2	yes	—
89	C₄×C₁₆	D₄	1	2	yes	—
90	C₄×C₁₆	C₂³	1	1	yes	characteristic
91	C₄×C₁₆	C₂³	1	1	yes	characteristic
92	C₄×C₁₆	C₂³	1	1	yes	characteristic
93	C₄×C₁₆	C₂³	1	1	yes	characteristic
94	#322	D₄	1	1	yes	characteristic
95	#322	D₄	1	1	yes	characteristic
96	C₈²	C₂³	1	1	yes	characteristic
97	C₈²	C₂³	1	1	yes	characteristic
98	C₂×Q₁₆	—	2	2	no	—
99	C₂×Q₁₆	—	2	2	no	—
100	C₂×Q₁₆	—	4	4	no	—
101	C₂×M₁₆	D₈	1	2	yes	—
102	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
104	C₄⋊Q₈	—	2	2	no	—
105	C₄⋊Q₈	—	2	2	no	—
106	C₄⋊Q₈	—	2	2	no	—
107	C₄⋊Q₈	—	2	2	no	—
108	#125	—	4	2	no	—
109	#125	—	4	2	no	—
110	C₂²⋊Q₈	—	4	2	no	—
111	C₂²⋊Q₈	—	4	2	no	—
112	C₂²⋊Q₈	—	4	2	no	—
113	C₂²⋊Q₈	—	4	2	no	—
114	C₄×Q₈	—	4	2	no	—
115	C₄×Q₈	—	4	2	no	—
116	C₄×Q₈	—	4	2	no	—
117	C₄×Q₈	—	4	2	no	—
118	#117	—	4	2	no	—
119	#117	—	4	2	no	—
120	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
121	Q₃₂	—	4	4	no	—
122	C₂×C₁₆	—	2	4	no	—
123	C₂×C₁₆	C₂×D₄	1	2	yes	—
124	C₂×C₁₆	C₂×D₄	1	2	yes	—
125	C₂×C₁₆	C₂×D₄	1	2	yes	—
126	C₂×C₁₆	C₂×D₄	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	—	2	2	no	—
130	#107	—	2	2	no	—
131	#107	—	2	2	no	—
132	#107	—	2	2	no	—
133	#107	—	4	4	no	—
134	#106	—	4	2	no	—
135	#106	—	4	2	no	—
136	#106	—	4	4	no	—
137	#106	—	4	4	no	—
138	C₄⋊C₈	D₈	1	2	yes	—
139	#103	—	4	2	no	—
140	#103	—	4	2	no	—
141	#103	—	4	4	no	—
142	#103	—	4	4	no	—
143	#103	—	4	4	no	—
144	#103	—	4	4	no	—
145	C₂²⋊C₈	D₈	1	2	yes	—
146	#97	D₈	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
153	C₄×C₈	C₂×D₄	1	1	yes	characteristic
154	C₄×C₈	C₂⁴	1	1	yes	characteristic
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	#127	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
162	Q₁₆	—	4	4	no	—
163	Q₁₆	—	4	4	no	—
164	Q₁₆	—	8	8	no	—
165	M₁₆	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	#127	1	1	yes	characteristic
169	C₂×C₈	#127	1	1	yes	characteristic
170	C₂×C₈	C₂×D₈	1	2	yes	—
171	C₂×C₈	C₂×D₈	1	2	yes	—
172	C₂×C₈	C₂×D₈	1	2	yes	—
173	C₂×C₈	C₂×D₈	1	2	yes	—
174	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	8	2	no	—
185	C₄⋊C₄	—	8	2	no	—
186	C₄⋊C₄	—	8	2	no	—
187	C₄⋊C₄	—	8	2	no	—
188	C₂²⋊C₄	—	8	2	no	—
189	C₂²⋊C₄	—	8	2	no	—
190	C₄²	—	8	2	no	—
191	C₄²	—	8	2	no	—
192	C₄²	#127	1	1	yes	characteristic
193	C₄²	C₂×D₈	1	1	yes	characteristic
194	C₄²	C₂×D₈	1	1	yes	characteristic
195	C₄²	C₂²×D₄	1	1	yes	characteristic
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	2	no	—
198	C₁₆	—	2	2	no	—
199	C₁₆	#137	1	4	yes	—
200	C₂³	#493	1	1	yes	characteristic
201	Q₈	—	8	2	no	—
202	Q₈	—	8	4	no	—
203	Q₈	—	8	4	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	4	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	#493	1	1	yes	characteristic
215	C₂×C₄	#493	1	1	yes	characteristic
216	C₂×C₄	#493	1	1	yes	characteristic
217	C₂×C₄	C₂×#127	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	#508	1	4	yes	—
226	C₈	C₂×#137	1	2	yes	—
227	C₈	#578	1	2	yes	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂×#493	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	16	2	no	—
236	C₄	—	16	2	no	—
237	C₄	—	16	2	no	—
238	C₄	—	16	2	no	—
239	C₄	#3152	1	1	yes	characteristic
240	C₄	#3156	1	1	yes	characteristic
241	C₄	C₂×#508	1	2	yes	—
242	C₄	#3417	1	2	yes	—
243	C₂	—	4	1	no	—
244	C₂	#19630	1	1	yes	characteristic
245	C₂	#33713	1	1	yes	characteristic
246	C₂	#33721	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ 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
5. #12535	C₂	semidirect
6. #12533	C₂	semidirect
7. #12522	C₂	semidirect
8. #12521	C₂	semidirect
9. #12504	C₂	semidirect
10. #12504	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33797	2
#33734	2
#33726	2
#19648	2
#12535	2
#12533	2
#12522	2
#12521	2
#12504	2
C₁₆×M₁₆	2