Subgroups — #498516

Order 512.

460 conjugacy classes of subgroups.

The subgroup lattice has 241 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33476	C₂	1	2	yes	maximal
2	#21732	C₂	1	1	yes	maximal, characteristic
3	#13265	C₂	1	2	yes	maximal
4	#13162	C₂	1	2	yes	maximal
5	#13106	C₂	1	4	yes	maximal
6	#12665	C₂	1	2	yes	maximal
7	C₄×#364	C₂	1	1	yes	maximal, characteristic
8	#12653	C₂	1	1	yes	maximal, characteristic
9	#3160	C₄	1	1	yes	characteristic
10	#2960	C₂²	1	2	yes	—
11	#2183	—	2	4	no	—
12	#2183	—	2	2	no	—
13	Q₈×C₁₆	—	2	2	no	—
14	C₂×#364	C₂²	1	1	yes	characteristic
15	C₂×#364	C₂²	1	1	yes	characteristic
16	#2151	C₂²	1	4	yes	—
17	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
19	#2108	C₂²	1	4	yes	—
20	C₄×M₃₂	C₂²	1	2	yes	—
21	C₂×#346	C₂²	1	2	yes	—
22	#1994	C₄	1	2	yes	—
23	#1945	C₄	1	2	yes	—
24	#1945	C₄	1	2	yes	—
25	#1848	C₄	1	1	yes	characteristic
26	#1772	C₂²	1	2	yes	—
27	#1772	C₂²	1	2	yes	—
28	C₄²×C₈	C₂²	1	1	yes	characteristic
29	#1379	C₂²	1	4	yes	—
30	#1379	C₂²	1	4	yes	—
31	#1376	C₂²	1	2	yes	—
32	#1374	C₂²	1	2	yes	—
33	#533	C₂×C₄	1	2	yes	—
34	C₂×M₃₂	C₂³	1	2	yes	—
35	C₂×M₃₂	C₂³	1	2	yes	—
36	C₂²×C₁₆	C₂³	1	2	yes	—
37	C₂²×C₁₆	C₂³	1	2	yes	—
38	#500	C₈	1	1	yes	characteristic
39	#500	C₈	1	1	yes	characteristic
40	#499	C₈	1	2	yes	—
41	#499	C₈	1	2	yes	—
42	#498	C₈	1	2	yes	—
43	#446	—	2	2	no	—
44	#446	—	2	2	no	—
45	C₈×Q₈	—	2	2	no	—
46	C₈×Q₈	—	2	2	no	—
47	C₂×#108	C₂×C₄	1	2	yes	—
48	C₂×#108	C₂×C₄	1	2	yes	—
49	#427	C₂×C₄	1	2	yes	—
50	#427	C₂×C₄	1	2	yes	—
51	#424	C₂×C₄	1	4	yes	—
52	#423	C₂×C₄	1	2	yes	—
53	#405	C₂³	1	4	yes	—
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
60	C₄³	C₂×C₄	1	1	yes	characteristic
61	#364	—	2	2	no	—
62	#364	Q₈	1	4	yes	complemented
63	C₄⋊C₁₆	—	2	4	no	—
64	C₄⋊C₁₆	—	2	4	no	—
65	C₄⋊C₁₆	—	2	1	no	—
66	C₄⋊C₁₆	—	2	1	no	—
67	C₄⋊C₁₆	D₄	1	2	yes	—
68	C₄⋊C₁₆	D₄	1	2	yes	—
69	#346	—	2	4	no	—
70	#346	—	2	2	no	—
71	#346	—	2	4	no	—
72	C₄×C₁₆	—	2	4	no	—
73	C₄×C₁₆	—	2	2	no	—
74	#339	C₂×C₄	1	4	yes	—
75	#339	C₂×C₄	1	4	yes	—
76	C₂×M₁₆	C₂²×C₄	1	4	yes	—
77	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂⁴	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×C₈	1	2	yes	—
82	C₄⋊Q₈	C₂×C₈	1	2	yes	—
83	#125	C₂×C₈	1	4	yes	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	—	2	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	2	2	no	—
88	#117	C₂²×C₄	1	4	yes	—
89	C₂×C₄²	—	2	1	no	—
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
94	M₃₂	—	2	2	no	—
95	M₃₂	—	2	2	no	—
96	M₃₂	—	2	2	no	—
97	M₃₂	C₂×Q₈	1	4	yes	—
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	C₂×D₄	1	4	yes	—
106	#108	—	2	4	no	—
107	#108	—	4	2	no	—
108	#107	C₂×C₈	1	2	yes	—
109	#107	C₂×C₈	1	2	yes	—
110	#107	C₂×C₈	1	2	yes	—
111	#107	C₂×C₈	1	2	yes	—
112	#106	C₂×C₈	1	4	yes	—
113	#106	C₂×C₈	1	4	yes	—
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	#97	—	2	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	4	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	C₂×C₈	1	1	yes	characteristic
125	C₄×C₈	C₂×C₈	1	1	yes	characteristic
126	C₄×C₈	C₂×C₈	1	1	yes	characteristic
127	C₄×C₈	C₂×C₈	1	1	yes	characteristic
128	C₄×C₈	C₂×D₄	1	1	yes	characteristic
129	C₄×C₈	C₂×D₄	1	1	yes	characteristic
130	C₄×C₈	C₂×Q₈	1	2	yes	—
131	C₄×C₈	#41	1	1	yes	characteristic
132	C₄×C₈	#41	1	1	yes	characteristic
133	C₄×C₈	#41	1	2	yes	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	#131	1	1	yes	characteristic
139	C₂²×C₄	#131	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
141	M₁₆	—	2	4	no	—
142	M₁₆	—	2	4	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	C₄×D₄	1	1	yes	characteristic
149	C₂×C₈	C₄×D₄	1	1	yes	characteristic
150	C₂×C₈	C₄×Q₈	1	2	yes	—
151	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
154	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₈	#131	1	2	yes	—
156	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₈	C₂×#41	1	1	yes	characteristic
159	C₂×C₈	C₂×#41	1	1	yes	characteristic
160	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
161	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	C₂²×C₈	1	4	yes	—
167	C₄⋊C₄	C₂²×C₈	1	4	yes	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	4	2	no	—
175	C₄²	C₄×D₄	1	1	yes	characteristic
176	C₄²	C₄×D₄	1	1	yes	characteristic
177	C₄²	C₄×Q₈	1	2	yes	—
178	C₄²	C₂²×C₈	1	1	yes	characteristic
179	C₄²	C₂²×C₈	1	1	yes	characteristic
180	C₁₆	—	2	4	no	—
181	C₁₆	—	2	4	no	—
182	C₁₆	—	4	2	no	—
183	C₁₆	—	4	2	no	—
184	C₂³	C₂×#131	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	8	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	C₈×D₄	1	1	yes	characteristic
198	C₂×C₄	C₈×D₄	1	1	yes	characteristic
199	C₂×C₄	C₈×D₄	1	1	yes	characteristic
200	C₂×C₄	C₈×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
203	C₂×C₄	#519	1	1	yes	characteristic
204	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
205	C₂×C₄	C₂×#131	1	1	yes	characteristic
206	C₂×C₄	#568	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	4	4	no	—
212	C₈	C₈×Q₈	1	2	yes	—
213	C₈	C₈×Q₈	1	2	yes	—
214	C₈	#547	1	1	yes	characteristic
215	C₈	D₄×Q₈	1	1	yes	characteristic
216	C₈	#556	1	1	yes	characteristic
217	C₈	#557	1	1	yes	characteristic
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
221	C₂²	#2962	1	1	yes	characteristic
222	C₂²	#2986	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	4	no	—
229	C₄	—	4	4	no	—
230	C₄	#2171	1	2	yes	—
231	C₄	#2365	1	1	yes	characteristic
232	C₄	#2935	1	1	yes	characteristic
233	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
234	C₄	#2982	1	1	yes	characteristic
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	#23149	1	1	yes	characteristic
238	C₂	C₂×#2171	1	1	yes	characteristic
239	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
62. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33476	2
#21732	2
#13265	2
#13162	2
#13106	2
#12665	2
C₄×#364	2
#12653	2