Subgroups — #102116

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#500	C₂	1	1	yes	maximal, characteristic
2	#9870	C₂	1	2	yes	maximal
3	#9637	C₂	1	2	yes	maximal
4	#8805	C₂	1	1	yes	maximal, characteristic
5	C₄×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₄	1	1	yes	characteristic
7	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
8	#1990	C₄	1	2	yes	—
9	#1942	—	2	2	no	—
10	#1845	C₄	1	1	yes	characteristic
11	#1842	C₂²	1	2	yes	—
12	C₄×#107	—	2	2	no	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#334	—	2	2	no	—
17	C₂×Q₈⋊C₈	—	2	4	no	—
18	C₂×#322	—	2	2	no	—
19	#1279	—	2	4	no	—
20	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×Q₈	—	2	2	no	—
22	#500	—	2	2	no	—
23	#500	C₈	1	4	yes	complemented
24	C₂×#107	—	2	2	no	—
25	C₂×#107	—	2	2	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄×C₈	—	2	1	no	—
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	—	2	1	no	—
32	C₂×C₄×C₈	—	2	1	no	—
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	2	yes	—
36	C₂×C₄×C₈	D₄	1	2	yes	—
37	C₂×C₄×C₈	D₄	1	2	yes	—
38	C₂×C₄×C₈	D₄	1	2	yes	—
39	#391	—	2	2	no	—
40	#389	—	2	2	no	—
41	#389	—	2	2	no	—
42	#384	C₂×C₄	1	2	yes	—
43	C₄×C₄⋊C₄	—	2	2	no	—
44	C₄×C₄⋊C₄	—	2	2	no	—
45	C₄³	C₂³	1	1	yes	characteristic
46	#340	—	2	4	no	—
47	#334	—	4	4	no	—
48	Q₈⋊C₈	—	4	8	no	—
49	#322	—	2	8	no	—
50	C₂²×Q₈	—	2	2	no	—
51	C₂²×C₈	—	2	2	no	—
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	—	4	2	no	—
54	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
56	C₄⋊Q₈	—	2	2	no	—
57	C₄⋊Q₈	—	2	2	no	—
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	C₂×C₈	1	4	yes	—
60	C₄×Q₈	—	4	4	no	—
61	C₄×Q₈	—	4	4	no	—
62	C₂×C₄⋊C₄	—	2	2	no	—
63	C₂×C₄⋊C₄	—	2	2	no	—
64	C₂×C₄⋊C₄	—	2	2	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄²	—	4	2	no	—
69	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	#41	1	1	yes	characteristic
75	C₂×C₄²	#41	1	1	yes	characteristic
76	#107	—	2	8	no	—
77	#107	—	4	4	no	—
78	C₄⋊C₈	—	4	4	no	—
79	C₄⋊C₈	—	4	4	no	—
80	C₄⋊C₈	—	4	4	no	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	2	4	no	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	4	no	—
87	C₄×C₈	—	2	4	no	—
88	C₄×C₈	—	4	2	no	—
89	C₄×C₈	—	4	2	no	—
90	C₄×C₈	C₂×C₈	1	2	yes	—
91	C₄×C₈	M₁₆	1	2	yes	—
92	C₄×C₈	SD₁₆	1	4	yes	—
93	C₄×C₈	Q₁₆	1	4	yes	—
94	#95	—	4	2	no	—
95	#95	—	4	2	no	—
96	C₂×Q₈	—	2	4	no	—
97	C₂×Q₈	—	4	2	no	—
98	C₂×Q₈	—	4	4	no	—
99	C₂×Q₈	—	4	4	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
105	C₂²×C₄	#121	1	1	yes	characteristic
106	C₂²×C₄	#121	1	1	yes	characteristic
107	C₂²×C₄	#124	1	1	yes	characteristic
108	C₂²×C₄	#127	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	—	8	2	no	—
119	C₂×C₈	—	8	2	no	—
120	C₂×C₈	C₂²⋊C₈	1	4	yes	—
121	C₂×C₈	#103	1	4	yes	—
122	C₂×C₈	#103	1	4	yes	—
123	C₂×C₈	#104	1	4	yes	—
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	4	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₄⋊C₄	—	4	4	no	—
131	C₄⋊C₄	—	8	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	8	2	no	—
136	C₄²	—	8	2	no	—
137	C₄²	C₂²×C₈	1	1	yes	characteristic
138	C₄²	C₂×M₁₆	1	1	yes	characteristic
139	C₄²	#131	1	1	yes	characteristic
140	C₄²	#131	1	1	yes	characteristic
141	C₄²	C₂×SD₁₆	1	2	yes	—
142	C₄²	C₂×Q₁₆	1	2	yes	—
143	C₄²	AGL(1,ℤ/8)	1	4	yes	—
144	C₄²	#137	1	4	yes	—
145	C₂³	#390	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	Q₈	—	8	4	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂×C₄	#408	1	1	yes	characteristic
162	C₂×C₄	C₂×#103	1	2	yes	—
163	C₂×C₄	C₂×#104	1	2	yes	—
164	C₂×C₄	C₈×D₄	1	1	yes	characteristic
165	C₂×C₄	C₈×D₄	1	1	yes	characteristic
166	C₂×C₄	#436	1	1	yes	characteristic
167	C₂×C₄	#436	1	1	yes	characteristic
168	C₂×C₄	#441	1	2	yes	—
169	C₂×C₄	#442	1	2	yes	—
170	C₂×C₄	#444	1	2	yes	—
171	C₂×C₄	#444	1	2	yes	—
172	C₂×C₄	#469	1	2	yes	—
173	C₂×C₄	#470	1	2	yes	—
174	C₂×C₄	#487	1	2	yes	—
175	C₂×C₄	#492	1	1	yes	characteristic
176	C₂×C₄	#494	1	1	yes	characteristic
177	C₂×C₄	#496	1	2	yes	—
178	C₂×C₄	#497	1	2	yes	—
179	C₈	—	2	4	no	—
180	C₈	—	4	4	no	—
181	C₈	—	4	4	no	—
182	C₈	—	16	2	no	—
183	C₈	Q₈⋊C₈	1	8	yes	complemented
184	C₂²	#1954	1	1	yes	characteristic
185	C₂²	#1956	1	1	yes	characteristic
186	C₂²	#1962	1	1	yes	characteristic
187	C₂²	#1971	1	1	yes	characteristic
188	C₂²	#1972	1	1	yes	characteristic
189	C₂²	#1975	1	1	yes	characteristic
190	C₂²	#1975	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	8	4	no	—
195	C₄	—	16	2	no	—
196	C₄	C₂×Q₈⋊C₈	1	2	yes	—
197	C₄	#1493	1	2	yes	—
198	C₄	#1579	1	2	yes	—
199	C₄	#1580	1	2	yes	—
200	C₄	#1593	1	2	yes	—
201	C₄	#1594	1	2	yes	—
202	C₄	#1714	1	2	yes	—
203	C₄	#1723	1	2	yes	—
204	C₂	#9637	1	1	yes	characteristic
205	C₂	#9658	1	1	yes	characteristic
206	C₂	#9659	1	1	yes	characteristic
207	C₂	#9836	1	1	yes	characteristic
208	C₂	#9849	1	1	yes	characteristic
209	C₂	#9850	1	1	yes	characteristic
210	C₂	#9861	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#127
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
23. #500	C₈	semidirect
183. C₈	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#500	2
#9870	2
#9637	2
#8805	2
C₄×#322	2