Subgroups — #498514

Order 512.

528 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2183	C₂	1	2	yes	maximal
2	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	#13263	C₂	1	2	yes	maximal
4	#13159	C₂	1	2	yes	maximal
5	#13105	C₂	1	4	yes	maximal
6	#12662	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂×C₈×Q₈	C₂²	1	2	yes	—
10	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	#2183	—	2	8	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
13	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×#346	C₂²	1	4	yes	—
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₁₆	C₂²	1	2	yes	—
20	#1984	C₄	1	2	yes	—
21	#1939	C₄	1	2	yes	—
22	#1939	C₄	1	2	yes	—
23	#1842	C₄	1	1	yes	characteristic
24	C₈×C₄⋊C₄	C₂²	1	2	yes	—
25	C₈×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1374	C₂²	1	2	yes	—
28	#1374	C₂²	1	2	yes	—
29	#1374	C₂²	1	4	yes	—
30	#1374	C₂²	1	4	yes	—
31	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
32	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
33	C₂²×C₁₆	C₂³	1	2	yes	—
34	C₂²×C₁₆	C₂³	1	2	yes	—
35	C₂²×C₁₆	C₂³	1	2	yes	—
36	C₂²×C₁₆	C₂³	1	2	yes	—
37	C₈×Q₈	—	2	4	no	—
38	C₈×Q₈	—	2	4	no	—
39	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	4	yes	—
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	#391	C₈	1	2	yes	—
51	#389	C₈	1	2	yes	—
52	#389	C₈	1	2	yes	—
53	#384	C₈	1	1	yes	characteristic
54	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
56	C₄³	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₁₆	—	2	8	no	—
58	C₄⋊C₁₆	—	2	2	no	—
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	D₄	1	4	yes	—
61	C₄⋊C₁₆	Q₈	1	4	yes	complemented
62	#346	—	2	8	no	—
63	C₄×C₁₆	—	2	4	no	—
64	C₄×C₁₆	—	2	4	no	—
65	#336	C₂×C₄	1	4	yes	—
66	#336	C₂×C₄	1	4	yes	—
67	C₂²×Q₈	C₂×C₈	1	2	yes	—
68	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₈	C₂²×C₄	1	4	yes	—
70	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂⁴	1	1	yes	characteristic
73	C₄⋊Q₈	—	2	4	no	—
74	C₄×Q₈	—	2	4	no	—
75	C₄×Q₈	—	2	4	no	—
76	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
77	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
78	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
81	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
82	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄²	C₂²×C₄	1	4	yes	—
87	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	C₂×D₄	1	4	yes	—
101	C₂×C₁₆	C₂×Q₈	1	4	yes	—
102	C₄⋊C₈	—	2	8	no	—
103	C₄⋊C₈	—	2	4	no	—
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	4	2	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	4	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	C₂×D₄	1	2	yes	—
114	C₄×C₈	C₂×Q₈	1	2	yes	—
115	C₄×C₈	#41	1	2	yes	—
116	C₄×C₈	#41	1	2	yes	—
117	#95	C₂×C₈	1	4	yes	—
118	#95	C₂×C₈	1	4	yes	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	2	4	no	—
121	C₂×Q₈	—	2	4	no	—
122	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₈	1	4	yes	—
124	C₂²×C₄	C₂²×C₈	1	4	yes	—
125	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
126	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
127	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
128	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
129	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
130	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	C₄×D₄	1	2	yes	—
142	C₂×C₈	C₄×Q₈	1	2	yes	—
143	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₈	C₂×#41	1	1	yes	characteristic
146	C₂×C₈	C₂×#41	1	1	yes	characteristic
147	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	8	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	4	1	no	—
162	C₄²	M₃₂	1	4	yes	—
163	C₄²	C₄×D₄	1	2	yes	—
164	C₄²	C₄×Q₈	1	2	yes	—
165	C₄²	#131	1	2	yes	—
166	C₄²	#131	1	2	yes	—
167	C₁₆	—	2	4	no	—
168	C₁₆	—	2	4	no	—
169	C₁₆	—	4	2	no	—
170	C₁₆	—	4	2	no	—
171	C₂³	C₂³×C₈	1	1	yes	characteristic
172	Q₈	—	4	8	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	C₈×D₄	1	2	yes	—
189	C₂×C₄	C₈×D₄	1	2	yes	—
190	C₂×C₄	C₈×Q₈	1	2	yes	—
191	C₂×C₄	C₈×Q₈	1	2	yes	—
192	C₂×C₄	C₂×M₃₂	1	2	yes	—
193	C₂×C₄	C₂×M₃₂	1	2	yes	—
194	C₂×C₄	C₂×M₃₂	1	2	yes	—
195	C₂×C₄	#504	1	2	yes	—
196	C₂×C₄	#504	1	2	yes	—
197	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
199	C₂×C₄	#519	1	1	yes	characteristic
200	C₂×C₄	C₂×#131	1	1	yes	characteristic
201	C₂×C₄	C₂×#131	1	1	yes	characteristic
202	C₂×C₄	#568	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	4	1	no	—
207	C₈	#547	1	1	yes	characteristic
208	C₈	D₄×Q₈	1	1	yes	characteristic
209	C₈	#556	1	1	yes	characteristic
210	C₈	#557	1	1	yes	characteristic
211	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
212	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
213	C₂²	#2982	1	1	yes	characteristic
214	C₂²	C₂²×M₃₂	1	1	yes	characteristic
215	C₂²	C₂×#504	1	1	yes	characteristic
216	C₂²	C₂×#504	1	1	yes	characteristic
217	C₂²	#3407	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	4	no	—
222	C₄	—	4	4	no	—
223	C₄	—	4	1	no	—
224	C₄	#2169	1	4	yes	—
225	C₄	#2183	1	4	yes	complemented
226	C₄	#2365	1	1	yes	characteristic
227	C₄	#2935	1	1	yes	characteristic
228	C₄	#2962	1	1	yes	characteristic
229	C₄	#2986	1	1	yes	characteristic
230	C₂	#23149	1	1	yes	characteristic
231	C₂	C₂×#2169	1	1	yes	characteristic
232	C₂	#33438	1	1	yes	characteristic
233	C₂	C₂×#2183	1	1	yes	characteristic
234	C₂	#33476	1	1	yes	characteristic
235	C₂	#33497	1	1	yes	characteristic
236	C₂	#33500	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
61. C₄⋊C₁₆	Q₈	semidirect
225. C₄	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2183	2
C₈×C₄⋊Q₈	2
#13263	2
#13159	2
#13105	2
#12662	2
C₄×C₄⋊C₁₆	2