Subgroups — #141017

Order 512.

360 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33311	C₂	1	1	yes	maximal, characteristic
2	#12898	C₂	1	1	yes	maximal, characteristic
3	C₂×#1363	C₂	1	4	yes	maximal
4	C₄×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×#499	—	2	1	no	—
6	C₂×#499	C₂²	1	2	yes	—
7	#2859	—	2	1	no	—
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	#1846	—	2	1	no	—
12	#1846	C₄	1	2	yes	—
13	#1845	C₄	1	1	yes	characteristic
14	#1845	C₄	1	1	yes	characteristic
15	#1844	—	2	1	no	—
16	C₄×C₄⋊C₈	—	2	1	no	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×#334	C₂²	1	2	yes	—
19	#1363	—	2	8	no	—
20	#1277	—	2	1	no	—
21	C₂×#125	—	2	2	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	#499	—	2	4	no	—
24	#499	—	2	4	no	—
25	#499	—	4	4	no	—
26	C₂×#107	—	2	1	no	—
27	C₂×#107	—	2	1	no	—
28	C₂×#107	C₂×C₄	1	2	yes	—
29	C₂×#107	C₂×C₄	1	2	yes	—
30	C₂×#106	—	2	2	no	—
31	C₂×#106	—	2	2	no	—
32	C₂×C₄⋊C₈	—	2	2	no	—
33	C₂×C₄×C₈	—	4	1	no	—
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#389	—	4	1	no	—
41	#389	—	4	1	no	—
42	#389	—	4	1	no	—
43	#389	—	4	1	no	—
44	#384	—	2	1	no	—
45	#384	—	2	1	no	—
46	#382	—	2	2	no	—
47	C₄³	D₄	1	1	yes	characteristic
48	C₄⋊C₁₆	—	2	4	no	—
49	C₄⋊C₁₆	—	4	2	no	—
50	C₄⋊C₁₆	—	4	2	no	—
51	#336	—	4	1	no	—
52	#334	—	2	2	no	—
53	#334	D₄	1	4	yes	—
54	C₂²×C₈	—	4	2	no	—
55	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	#125	—	4	4	no	—
60	#125	—	4	4	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	2	2	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄⋊C₄	—	4	1	no	—
66	C₂×C₄⋊C₄	—	4	1	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄²	D₈	1	1	yes	characteristic
72	C₂×C₄²	D₈	1	1	yes	characteristic
73	C₂×C₄²	SD₁₆	1	1	yes	characteristic
74	C₂×C₄²	SD₁₆	1	1	yes	characteristic
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₆	—	4	2	no	—
77	C₂×C₁₆	—	4	2	no	—
78	C₂×C₁₆	—	4	2	no	—
79	#107	—	2	4	no	—
80	#107	—	2	4	no	—
81	#107	—	4	2	no	—
82	#107	—	4	2	no	—
83	#106	—	4	4	no	—
84	#106	—	4	2	no	—
85	#106	—	4	2	no	—
86	C₄⋊C₈	—	2	4	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	8	2	no	—
92	C₄×C₈	SD₁₆	1	2	yes	—
93	C₄×C₈	SD₁₆	1	2	yes	—
94	C₄×C₈	Q₁₆	1	2	yes	—
95	C₄×C₈	Q₁₆	1	2	yes	—
96	C₄×C₈	C₂×D₄	1	2	yes	—
97	C₄×C₈	#41	1	2	yes	—
98	#95	—	4	2	no	—
99	#95	—	4	2	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	#102	1	1	yes	characteristic
103	C₂²×C₄	#102	1	1	yes	characteristic
104	C₂²×C₄	#102	1	1	yes	characteristic
105	C₂²×C₄	#102	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
108	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	4	4	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	#103	1	2	yes	—
119	C₂×C₈	#103	1	2	yes	—
120	C₂×C₈	C₄×D₄	1	2	yes	—
121	C₂×C₈	#124	1	1	yes	characteristic
122	C₂×C₈	#127	1	1	yes	characteristic
123	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	8	2	no	—
132	C₄⋊C₄	—	8	2	no	—
133	C₄⋊C₄	—	8	2	no	—
134	C₄⋊C₄	—	8	2	no	—
135	C₄⋊C₄	—	8	4	no	—
136	C₄⋊C₄	—	8	4	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	4	1	no	—
140	C₄²	—	4	1	no	—
141	C₄²	#101	1	2	yes	—
142	C₄²	#101	1	2	yes	—
143	C₄²	#103	1	2	yes	—
144	C₄²	#103	1	2	yes	—
145	C₄²	#121	1	2	yes	—
146	C₄²	AGL(1,ℤ/8)	1	2	yes	—
147	C₁₆	—	8	2	no	—
148	C₂³	C₂×#102	1	1	yes	characteristic
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	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	#332	1	2	yes	—
166	C₂×C₄	#332	1	2	yes	—
167	C₂×C₄	#332	1	2	yes	—
168	C₂×C₄	#332	1	2	yes	—
169	C₂×C₄	#359	1	2	yes	—
170	C₂×C₄	#360	1	2	yes	—
171	C₂×C₄	#390	1	1	yes	characteristic
172	C₂×C₄	C₂×#101	1	1	yes	characteristic
173	C₂×C₄	C₂×#101	1	1	yes	characteristic
174	C₂×C₄	C₂×#103	1	1	yes	characteristic
175	C₂×C₄	C₂×#103	1	1	yes	characteristic
176	C₂×C₄	#418	1	1	yes	characteristic
177	C₂×C₄	#459	1	2	yes	—
178	C₂×C₄	#460	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	8	4	no	—
183	C₈	#487	1	1	yes	characteristic
184	C₈	#487	1	1	yes	characteristic
185	C₈	#492	1	1	yes	characteristic
186	C₈	#494	1	1	yes	characteristic
187	C₂²	C₂×#332	1	1	yes	characteristic
188	C₂²	C₂×#332	1	1	yes	characteristic
189	C₂²	#1982	1	1	yes	characteristic
190	C₂²	C₂×#359	1	1	yes	characteristic
191	C₂²	#2143	1	1	yes	characteristic
192	C₂²	C₂×#360	1	1	yes	characteristic
193	C₂²	#2145	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	16	2	no	—
200	C₄	—	16	2	no	—
201	C₄	#1363	1	4	yes	complemented
202	C₄	#1549	1	1	yes	characteristic
203	C₄	#1549	1	1	yes	characteristic
204	C₄	#1956	1	1	yes	characteristic
205	C₄	#1966	1	1	yes	characteristic
206	C₂	#9694	1	1	yes	characteristic
207	C₂	C₂×#1363	1	1	yes	characteristic
208	C₂	#11934	1	1	yes	characteristic
209	C₂	#11935	1	1	yes	characteristic
210	C₂	#11936	1	1	yes	characteristic
211	C₂	#13251	1	1	yes	characteristic
212	C₂	#13255	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
201. C₄	#1363	semidirect

Maximal subgroups

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

subgroup	index
#33311	2
#12898	2
C₂×#1363	2
C₄×#334	2