Subgroups — #231293

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23153	C₂	1	2	yes	maximal
2	C₄×#500	C₂	1	1	yes	maximal, characteristic
3	#9865	C₂	1	2	yes	maximal
4	#9595	C₂	1	2	yes	maximal
5	#9540	C₂	1	4	yes	maximal
6	#8287	C₂	1	2	yes	maximal
7	C₄×#334	C₂	1	1	yes	maximal, characteristic
8	#8260	C₂	1	1	yes	maximal, characteristic
9	C₂×#500	C₄	1	1	yes	characteristic
10	C₂×#446	—	2	2	no	—
11	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
12	#1990	C₄	1	2	yes	—
13	#1984	—	2	2	no	—
14	#1942	C₄	1	2	yes	—
15	#1942	C₄	1	2	yes	—
16	#1939	C₂²	1	4	yes	—
17	#1917	—	2	4	no	—
18	#1845	C₄	1	1	yes	characteristic
19	C₄×#107	C₂²	1	2	yes	—
20	C₄×#107	C₂²	1	2	yes	—
21	#1770	C₂²	1	4	yes	—
22	C₄×C₄⋊C₈	C₂²	1	2	yes	—
23	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₄²×C₈	C₂²	1	1	yes	characteristic
26	C₂×#334	C₂²	1	1	yes	characteristic
27	C₂×#334	C₂²	1	1	yes	characteristic
28	C₂×#322	C₂²	1	2	yes	—
29	#1279	C₂²	1	4	yes	—
30	#1279	C₂²	1	4	yes	—
31	#1277	C₂²	1	2	yes	—
32	#1273	C₂²	1	2	yes	—
33	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₂×C₄×Q₈	—	2	2	no	—
35	#500	—	2	4	no	—
36	#446	—	4	8	no	—
37	C₂×#107	C₂×C₄	1	2	yes	—
38	C₂×#107	C₂×C₄	1	2	yes	—
39	C₂×#107	C₂×C₄	1	2	yes	—
40	C₂×#107	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₈	—	2	4	no	—
42	C₂×C₄⋊C₈	—	2	1	no	—
43	C₂×C₄⋊C₈	D₄	1	2	yes	—
44	C₂×C₄⋊C₈	C₂³	1	2	yes	—
45	C₂×C₄⋊C₈	C₂³	1	2	yes	—
46	C₂×#97	—	2	4	no	—
47	C₂×C₄×C₈	—	2	2	no	—
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	2	yes	—
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	2	yes	—
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	#391	—	2	2	no	—
57	#389	—	2	2	no	—
58	#389	C₂×C₄	1	4	yes	—
59	#384	C₂×C₄	1	2	yes	—
60	C₄×C₄⋊C₄	—	2	2	no	—
61	C₄×C₄⋊C₄	C₂³	1	4	yes	—
62	C₄³	C₂³	1	1	yes	characteristic
63	#340	C₂×C₄	1	4	yes	—
64	#340	C₂×C₄	1	4	yes	—
65	#336	—	2	4	no	—
66	#336	—	2	4	no	—
67	#336	—	2	2	no	—
68	#336	—	2	1	no	—
69	#336	D₄	1	2	yes	—
70	#334	—	2	2	no	—
71	#334	Q₈	1	4	yes	complemented
72	#322	—	2	4	no	—
73	C₂²×Q₈	—	2	2	no	—
74	C₂²×C₈	—	2	2	no	—
75	C₂²×C₈	—	2	2	no	—
76	C₂²×C₈	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₄	1	4	yes	—
82	C₄⋊Q₈	—	2	4	no	—
83	C₄⋊Q₈	—	2	4	no	—
84	C₄×Q₈	—	4	4	no	—
85	C₄×Q₈	—	4	4	no	—
86	C₂×C₄⋊C₄	—	2	2	no	—
87	C₂×C₄⋊C₄	—	2	4	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄⋊C₄	C₂²×C₄	1	4	yes	—
91	C₂×C₄⋊C₄	C₂²×C₄	1	4	yes	—
92	C₂×C₄²	—	2	4	no	—
93	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄²	#41	1	1	yes	characteristic
98	C₂×C₄²	#41	1	1	yes	characteristic
99	C₂×C₄²	C₂⁴	1	1	yes	characteristic
100	#107	—	2	4	no	—
101	#107	—	2	4	no	—
102	#107	—	2	4	no	—
103	#107	—	4	2	no	—
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	4	2	no	—
109	C₄⋊C₈	—	4	8	no	—
110	C₄⋊C₈	C₂×Q₈	1	4	yes	—
111	#97	—	4	8	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	1	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	1	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	4	1	no	—
121	C₄×C₈	—	4	4	no	—
122	C₄×C₈	M₁₆	1	4	yes	—
123	C₄×C₈	C₂×Q₈	1	2	yes	—
124	C₄×C₈	#41	1	2	yes	—
125	#95	—	2	4	no	—
126	#95	—	2	4	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂×Q₈	—	4	4	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
134	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
135	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₄	C₂×#41	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	C₄×Q₈	1	2	yes	—
159	C₂×C₈	C₄×Q₈	1	2	yes	—
160	C₂×C₈	C₂×M₁₆	1	2	yes	—
161	C₂×C₈	C₂×M₁₆	1	2	yes	—
162	C₂×C₈	#131	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	8	no	—
170	C₄⋊C₄	—	8	2	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	4	no	—
177	C₄²	—	4	4	no	—
178	C₄²	C₂×M₁₆	1	2	yes	—
179	C₄²	#131	1	2	yes	—
180	C₄²	AGL(1,ℤ/8)	1	4	yes	—
181	C₄²	C₂²×Q₈	1	1	yes	characteristic
182	C₄²	C₂×#41	1	1	yes	characteristic
183	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
184	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂³	C₂×C₄×D₄	1	1	yes	characteristic
186	Q₈	—	8	8	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	#436	1	2	yes	—
203	C₂×C₄	#436	1	2	yes	—
204	C₂×C₄	#442	1	2	yes	—
205	C₂×C₄	#442	1	2	yes	—
206	C₂×C₄	#444	1	2	yes	—
207	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
208	C₂×C₄	#519	1	1	yes	characteristic
209	C₂×C₄	#547	1	1	yes	characteristic
210	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
211	C₂×C₄	#556	1	1	yes	characteristic
212	C₂×C₄	#557	1	1	yes	characteristic
213	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
214	C₂×C₄	C₂×#131	1	1	yes	characteristic
215	C₂×C₄	C₂×#131	1	1	yes	characteristic
216	C₂×C₄	#568	1	1	yes	characteristic
217	C₂×C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
218	C₂×C₄	#578	1	2	yes	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	4	4	no	—
223	C₈	—	4	1	no	—
224	C₈	—	8	2	no	—
225	C₈	—	8	2	no	—
226	C₈	#446	1	4	yes	complemented
227	C₂²	#2365	1	1	yes	characteristic
228	C₂²	C₂×#436	1	1	yes	characteristic
229	C₂²	#2935	1	1	yes	characteristic
230	C₂²	C₂×#442	1	1	yes	characteristic
231	C₂²	#2950	1	1	yes	characteristic
232	C₂²	C₂×#444	1	1	yes	characteristic
233	C₂²	#2957	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	8	4	no	—
240	C₄	—	8	4	no	—
241	C₄	#1594	1	4	yes	—
242	C₄	C₂×#446	1	1	yes	characteristic
243	C₄	#2962	1	1	yes	characteristic
244	C₄	#2983	1	1	yes	characteristic
245	C₄	#2986	1	1	yes	characteristic
246	C₄	#3353	1	2	yes	—
247	C₄	#3386	1	2	yes	—
248	C₂	C₂×#1594	1	1	yes	characteristic
249	C₂	#17897	1	1	yes	characteristic
250	C₂	#18073	1	1	yes	characteristic
251	C₂	#18090	1	1	yes	characteristic
252	C₂	#23153	1	1	yes	characteristic
253	C₂	#23161	1	1	yes	characteristic
254	C₂	#23171	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
71. #334	Q₈	semidirect
226. C₈	#446	semidirect

Maximal subgroups

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

subgroup	index
#23153	2
C₄×#500	2
#9865	2
#9595	2
#9540	2
#8287	2
C₄×#334	2
#8260	2