Subgroups — #102004

Order 512.

462 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#494	C₂	1	1	yes	maximal, characteristic
2	#9865	C₂	1	2	yes	maximal
3	C₄×#334	C₂	1	1	yes	maximal, characteristic
4	#8260	C₂	1	1	yes	maximal, characteristic
5	#7570	C₂	1	2	yes	maximal
6	C₂×#494	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₁₆	—	2	1	no	—
8	C₂×C₄×Q₁₆	—	2	1	no	—
9	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
10	#1969	—	2	1	no	—
11	#1969	—	2	1	no	—
12	#1956	C₄	1	2	yes	—
13	#1845	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₈	C₂²	1	2	yes	—
15	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×#334	—	2	1	no	—
19	C₂×#322	—	2	1	no	—
20	#1279	—	2	4	no	—
21	#1277	—	2	1	no	—
22	#1273	—	2	1	no	—
23	#1271	—	2	4	no	—
24	C₂²×Q₁₆	—	2	1	no	—
25	C₂²×Q₁₆	—	2	1	no	—
26	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
27	C₂×C₄×Q₈	—	2	2	no	—
28	#494	—	2	2	no	—
29	#494	C₈	1	4	yes	complemented
30	C₄×Q₁₆	—	4	2	no	—
31	C₄×Q₁₆	—	4	2	no	—
32	C₄×Q₁₆	—	4	2	no	—
33	C₄×Q₁₆	—	4	2	no	—
34	C₂×#107	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	2	no	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	D₄	1	2	yes	—
39	C₂×#103	—	2	2	no	—
40	C₂×#103	—	2	2	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	—	2	2	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	#391	—	2	2	no	—
51	#389	—	2	2	no	—
52	#389	—	2	2	no	—
53	#384	C₂×C₄	1	2	yes	—
54	C₄×C₄⋊C₄	—	2	2	no	—
55	C₄×C₄⋊C₄	—	2	2	no	—
56	C₄³	C₂³	1	1	yes	characteristic
57	#336	—	2	2	no	—
58	#336	—	2	1	no	—
59	#336	D₄	1	2	yes	—
60	#334	—	2	4	no	—
61	#322	—	4	2	no	—
62	C₂²×Q₈	—	2	2	no	—
63	C₂×Q₁₆	—	2	4	no	—
64	C₂×Q₁₆	—	2	4	no	—
65	C₂×Q₁₆	—	4	2	no	—
66	C₂×Q₁₆	—	4	2	no	—
67	C₂×Q₁₆	—	4	2	no	—
68	C₂×Q₁₆	—	4	2	no	—
69	C₂²×C₈	—	2	2	no	—
70	C₂²×C₈	—	2	2	no	—
71	C₂²×C₈	—	4	2	no	—
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	C₂×C₈	1	4	yes	—
80	C₄×Q₈	—	4	2	no	—
81	C₄×Q₈	—	4	2	no	—
82	C₄×Q₈	—	4	2	no	—
83	C₄×Q₈	—	4	2	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	2	2	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	—	4	2	no	—
90	C₂×C₄²	—	4	2	no	—
91	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄²	#41	1	1	yes	characteristic
97	C₂×C₄²	#41	1	1	yes	characteristic
98	#107	—	4	2	no	—
99	#107	—	4	2	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	2	4	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	4	2	no	—
104	#103	—	4	4	no	—
105	#103	—	4	4	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	4	1	no	—
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	—	4	1	no	—
115	C₄×C₈	—	4	1	no	—
116	C₄×C₈	—	4	4	no	—
117	C₄×C₈	C₂×C₈	1	2	yes	—
118	C₄×C₈	M₁₆	1	2	yes	—
119	C₄×C₈	D₈	1	2	yes	—
120	C₄×C₈	SD₁₆	1	2	yes	—
121	#95	—	4	2	no	—
122	#95	—	4	2	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	2	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
134	C₂²×C₄	#121	1	1	yes	characteristic
135	C₂²×C₄	#121	1	1	yes	characteristic
136	C₂²×C₄	#124	1	1	yes	characteristic
137	C₂²×C₄	#127	1	1	yes	characteristic
138	Q₁₆	—	4	8	no	—
139	Q₁₆	—	4	8	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	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	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	—	8	2	no	—
155	C₂×C₈	—	8	2	no	—
156	C₂×C₈	C₂²⋊C₈	1	2	yes	—
157	C₂×C₈	C₂²⋊C₈	1	2	yes	—
158	C₂×C₈	#102	1	2	yes	—
159	C₂×C₈	#102	1	2	yes	—
160	C₂×C₈	#104	1	2	yes	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	8	2	no	—
169	C₄⋊C₄	—	8	2	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	—	8	2	no	—
177	C₄²	—	8	2	no	—
178	C₄²	C₂²×C₈	1	1	yes	characteristic
179	C₄²	C₂×M₁₆	1	1	yes	characteristic
180	C₄²	#131	1	1	yes	characteristic
181	C₄²	#131	1	1	yes	characteristic
182	C₄²	C₂×D₈	1	1	yes	characteristic
183	C₄²	C₂×SD₁₆	1	1	yes	characteristic
184	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄²	#137	1	4	yes	—
187	C₂³	#390	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	8	2	no	—
191	Q₈	—	8	2	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	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
208	C₂×C₄	#408	1	1	yes	characteristic
209	C₂×C₄	C₂×#102	1	1	yes	characteristic
210	C₂×C₄	#418	1	1	yes	characteristic
211	C₂×C₄	C₂×#104	1	1	yes	characteristic
212	C₂×C₄	#421	1	1	yes	characteristic
213	C₂×C₄	C₈×D₄	1	1	yes	characteristic
214	C₂×C₄	C₈×D₄	1	1	yes	characteristic
215	C₂×C₄	#436	1	1	yes	characteristic
216	C₂×C₄	#436	1	1	yes	characteristic
217	C₂×C₄	#441	1	2	yes	—
218	C₂×C₄	#444	1	2	yes	—
219	C₂×C₄	#459	1	1	yes	characteristic
220	C₂×C₄	#459	1	1	yes	characteristic
221	C₂×C₄	#460	1	1	yes	characteristic
222	C₂×C₄	#460	1	1	yes	characteristic
223	C₂×C₄	#470	1	2	yes	—
224	C₂×C₄	#473	1	2	yes	—
225	C₂×C₄	#497	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	4	1	no	—
229	C₈	—	8	2	no	—
230	C₈	—	8	4	no	—
231	C₈	—	8	2	no	—
232	C₈	D₄⋊C₈	1	4	yes	—
233	C₂²	#1954	1	1	yes	characteristic
234	C₂²	#1971	1	1	yes	characteristic
235	C₂²	#1975	1	1	yes	characteristic
236	C₂²	#1976	1	1	yes	characteristic
237	C₂²	#1979	1	1	yes	characteristic
238	C₂²	#1980	1	1	yes	characteristic
239	C₂²	#1982	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	8	4	no	—
247	C₄	—	16	2	no	—
248	C₄	C₂×D₄⋊C₈	1	1	yes	characteristic
249	C₄	#1478	1	1	yes	characteristic
250	C₄	#1480	1	1	yes	characteristic
251	C₄	#1484	1	1	yes	characteristic
252	C₄	#1579	1	2	yes	—
253	C₄	#1593	1	2	yes	—
254	C₄	#1689	1	2	yes	—
255	C₄	#1690	1	2	yes	—
256	C₂	#9658	1	1	yes	characteristic
257	C₂	#9669	1	1	yes	characteristic
258	C₂	#9674	1	1	yes	characteristic
259	C₂	#9720	1	1	yes	characteristic
260	C₂	#9727	1	1	yes	characteristic
261	C₂	#9742	1	1	yes	characteristic
262	C₂	#9753	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
29. #494	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#494	2
#9865	2
C₄×#334	2
#8260	2
#7570	2