Subgroups — #496152

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2252	C₂	1	1	yes	maximal, characteristic
2	#21725	C₂	1	2	yes	maximal
3	#13283	C₂	1	2	yes	maximal
4	#13168	C₂	1	2	yes	maximal
5	#13165	C₂	1	2	yes	maximal
6	#12907	C₂	1	1	yes	maximal, characteristic
7	#12674	C₂	1	2	yes	maximal
8	C₄×#366	C₂	1	2	yes	maximal
9	#12539	C₂	1	1	yes	maximal, characteristic
10	C₂×#500	C₂²	1	2	yes	—
11	#2331	—	2	2	no	—
12	#2252	C₄	1	8	yes	complemented
13	C₂×#367	C₂²	1	2	yes	—
14	C₂×#366	C₂²	1	2	yes	—
15	C₂×#366	C₂²	1	2	yes	—
16	C₂×#366	C₂²	1	2	yes	—
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×#346	C₂²	1	1	yes	characteristic
19	C₂×#346	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1993	—	2	2	no	—
24	#1943	—	2	2	no	—
25	#1942	C₂²	1	4	yes	—
26	#1846	C₂²	1	2	yes	—
27	#1776	—	2	2	no	—
28	C₄×#107	C₂²	1	4	yes	—
29	#1726	C₂²	1	1	yes	characteristic
30	#1380	C₂²	1	4	yes	—
31	#1380	C₂²	1	4	yes	—
32	C₂×C₄⋊Q₈	—	2	2	no	—
33	C₂²×C₁₆	C₂³	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	1	1	yes	characteristic
35	C₂²×C₁₆	C₂³	1	1	yes	characteristic
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	#500	—	2	4	no	—
38	#500	C₂×C₄	1	8	yes	—
39	C₂×#107	—	2	2	no	—
40	C₂×#107	C₂³	1	4	yes	—
41	C₂×#107	C₂³	1	4	yes	—
42	C₂×#106	—	2	2	no	—
43	C₂×#97	D₄	1	1	yes	characteristic
44	C₂×#97	D₄	1	1	yes	characteristic
45	C₂×#97	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	#391	—	4	4	no	—
50	#389	—	2	4	no	—
51	#387	—	4	4	no	—
52	#382	—	2	2	no	—
53	C₄×C₄⋊C₄	—	2	4	no	—
54	#376	D₄	1	1	yes	characteristic
55	#367	—	2	4	no	—
56	#366	—	2	4	no	—
57	#366	C₂×C₄	1	8	yes	—
58	#366	C₂×C₄	1	8	yes	—
59	#346	—	2	2	no	—
60	#346	—	2	2	no	—
61	#346	Q₈	1	4	yes	complemented
62	C₄×C₁₆	—	2	2	no	—
63	C₄×C₁₆	C₂×C₄	1	4	yes	—
64	C₄×C₁₆	Q₈	1	4	yes	complemented
65	#340	—	2	4	no	—
66	#340	—	2	4	no	—
67	C₂²×Q₈	—	4	2	no	—
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₈	C₂⁴	1	1	yes	characteristic
72	C₄⋊Q₈	—	2	8	no	—
73	C₄⋊Q₈	—	4	4	no	—
74	C₂×C₄⋊C₄	—	2	4	no	—
75	C₂×C₄⋊C₄	—	2	4	no	—
76	C₂×C₄⋊C₄	—	4	2	no	—
77	C₂×C₄⋊C₄	—	4	4	no	—
78	C₂×C₄²	—	4	4	no	—
79	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	C₂²×C₄	1	2	yes	—
87	C₂×C₁₆	C₂²×C₄	1	2	yes	—
88	C₂×C₁₆	C₂×Q₈	1	2	yes	—
89	C₂×C₁₆	C₂×Q₈	1	2	yes	—
90	C₂×C₁₆	C₂×Q₈	1	2	yes	—
91	C₂×C₁₆	C₂×Q₈	1	2	yes	—
92	C₂×C₁₆	#41	1	2	yes	—
93	C₂×C₁₆	#41	1	2	yes	—
94	#107	—	2	8	no	—
95	#107	—	2	4	no	—
96	#107	—	2	4	no	—
97	#107	—	2	4	no	—
98	#107	C₂²×C₄	1	8	yes	—
99	#106	—	4	4	no	—
100	#97	—	2	2	no	—
101	#97	—	2	2	no	—
102	#97	Q₁₆	1	4	yes	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×C₄	1	2	yes	—
106	C₄×C₈	C₂×Q₈	1	2	yes	—
107	C₄×C₈	C₂×Q₈	1	2	yes	—
108	C₄×C₈	#41	1	2	yes	—
109	#95	—	4	4	no	—
110	#95	—	4	2	no	—
111	#95	—	4	2	no	—
112	#95	—	4	4	no	—
113	#95	D₈	1	2	yes	—
114	#95	D₈	1	2	yes	—
115	C₂×Q₈	—	4	8	no	—
116	C₂×Q₈	—	16	4	no	—
117	C₂²×C₄	—	4	4	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	C₂×D₈	1	2	yes	—
120	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
121	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
122	C₂²×C₄	C₂×D₈	1	2	yes	—
123	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	C₄×D₄	1	2	yes	—
129	C₂×C₈	C₄⋊Q₈	1	2	yes	—
130	C₂×C₈	C₂×Q₁₆	1	2	yes	—
131	C₂×C₈	C₂×Q₁₆	1	2	yes	—
132	C₂×C₈	#135	1	2	yes	—
133	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₈	C₂×#41	1	1	yes	characteristic
137	C₂×C₈	C₂×#41	1	1	yes	characteristic
138	C₂×C₈	C₂×#41	1	1	yes	characteristic
139	C₄⋊C₄	—	2	8	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	16	4	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	16	4	no	—
148	C₄²	C₄×D₄	1	2	yes	—
149	C₄²	C₄⋊Q₈	1	2	yes	—
150	C₄²	C₂×Q₁₆	1	2	yes	—
151	C₄²	#135	1	2	yes	—
152	C₁₆	—	2	2	no	—
153	C₁₆	—	2	2	no	—
154	C₁₆	C₄×Q₈	1	4	yes	—
155	C₁₆	C₄×Q₈	1	4	yes	—
156	C₂³	C₂²×D₈	1	1	yes	characteristic
157	Q₈	—	16	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	8	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	C₄×D₈	1	2	yes	—
172	C₂×C₄	C₄×D₈	1	2	yes	—
173	C₂×C₄	#500	1	4	yes	—
174	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	#532	1	1	yes	characteristic
177	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
178	C₂×C₄	C₂×#135	1	1	yes	characteristic
179	C₂×C₄	C₂×#135	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	C₄×Q₁₆	1	4	yes	—
183	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
184	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
185	C₈	#533	1	1	yes	characteristic
186	C₈	#533	1	1	yes	characteristic
187	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
188	C₂²	C₂×#500	1	1	yes	characteristic
189	C₂²	#3169	1	1	yes	characteristic
190	C₂²	#3415	1	2	yes	—
191	C₂²	#3417	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	4	no	—
195	C₄	—	4	2	no	—
196	C₄	—	8	8	no	—
197	C₄	—	16	4	no	—
198	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
199	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
200	C₄	#3160	1	1	yes	characteristic
201	C₄	#3170	1	1	yes	characteristic
202	C₂	C₄×#500	1	1	yes	characteristic
203	C₂	#33456	1	1	yes	characteristic
204	C₂	#33458	1	1	yes	characteristic
205	C₂	#33729	1	2	yes	—
206	C₂	#33739	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #2252	C₄	semidirect
61. #346	Q₈	semidirect
64. C₄×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2252	2
#21725	2
#13283	2
#13168	2
#13165	2
#12907	2
#12674	2
C₄×#366	2
#12539	2