Subgroups — #496151

Order 512.

512 conjugacy classes of subgroups.

The subgroup lattice has 195 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2253	C₂	1	1	yes	maximal, characteristic
2	#21723	C₂	1	2	yes	maximal
3	#13284	C₂	1	2	yes	maximal
4	#13168	C₂	1	2	yes	maximal
5	#13166	C₂	1	2	yes	maximal
6	#12906	C₂	1	1	yes	maximal, characteristic
7	#12674	C₂	1	2	yes	maximal
8	C₄×#367	C₂	1	2	yes	maximal
9	#12539	C₂	1	1	yes	maximal, characteristic
10	C₂×#499	C₂²	1	2	yes	—
11	#2330	—	2	2	no	—
12	#2253	C₄	1	8	yes	complemented
13	C₂×#367	C₂²	1	2	yes	—
14	C₂×#367	C₂²	1	2	yes	—
15	C₂×#367	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	#1992	—	2	2	no	—
24	#1944	—	2	2	no	—
25	#1942	C₂²	1	4	yes	—
26	#1845	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×#125	—	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	#499	—	2	4	no	—
38	#499	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	#389	—	2	4	no	—
50	#389	—	4	4	no	—
51	#387	—	4	4	no	—
52	#384	—	2	2	no	—
53	C₄×C₄⋊C₄	—	2	4	no	—
54	#376	D₄	1	1	yes	characteristic
55	#367	—	2	4	no	—
56	#367	—	2	4	no	—
57	#367	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	—
62	C₄×C₁₆	—	2	2	no	—
63	C₄×C₁₆	C₂×C₄	1	4	yes	—
64	C₄×C₁₆	Q₈	1	4	yes	—
65	#340	—	2	4	no	—
66	#340	—	2	4	no	—
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	C₂⁴	1	1	yes	characteristic
71	#125	—	2	8	no	—
72	#125	—	4	4	no	—
73	C₂×C₄⋊C₄	—	2	4	no	—
74	C₂×C₄⋊C₄	—	2	4	no	—
75	C₂×C₄⋊C₄	—	4	4	no	—
76	C₂×C₄⋊C₄	—	4	4	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	4	no	—
95	#107	—	2	4	no	—
96	#107	—	2	4	no	—
97	#107	—	4	4	no	—
98	#107	C₂²×C₄	1	8	yes	—
99	#106	—	2	8	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	4	no	—
111	#95	D₈	1	4	yes	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	—	4	4	no	—
114	C₂²×C₄	C₂×D₈	1	2	yes	—
115	C₂²×C₄	C₂×D₈	1	2	yes	—
116	C₂²×C₄	C₂×D₈	1	2	yes	—
117	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	C₄×D₄	1	2	yes	—
123	C₂×C₈	C₄⋊Q₈	1	2	yes	—
124	C₂×C₈	C₂×Q₁₆	1	2	yes	—
125	C₂×C₈	C₂×Q₁₆	1	2	yes	—
126	C₂×C₈	#135	1	2	yes	—
127	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₈	C₂×#41	1	1	yes	characteristic
131	C₂×C₈	C₂×#41	1	1	yes	characteristic
132	C₂×C₈	C₂×#41	1	1	yes	characteristic
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	16	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	16	4	no	—
139	C₄⋊C₄	—	16	4	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	16	4	no	—
143	C₄²	C₄×D₄	1	2	yes	—
144	C₄²	C₄⋊Q₈	1	2	yes	—
145	C₄²	C₂×Q₁₆	1	2	yes	—
146	C₄²	#135	1	2	yes	—
147	C₁₆	—	2	2	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₄×Q₈	1	4	yes	—
150	C₁₆	C₄×Q₈	1	4	yes	—
151	C₂³	C₂²×D₈	1	1	yes	characteristic
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	8	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	—	8	4	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	C₄×D₈	1	4	yes	—
163	C₂×C₄	#500	1	4	yes	—
164	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₄	#532	1	1	yes	characteristic
167	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
168	C₂×C₄	C₂×#135	1	1	yes	characteristic
169	C₂×C₄	C₂×#135	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	C₄×Q₁₆	1	4	yes	—
173	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
174	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₈	#533	1	1	yes	characteristic
176	C₈	#533	1	1	yes	characteristic
177	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
178	C₂²	C₂×#500	1	1	yes	characteristic
179	C₂²	#3169	1	1	yes	characteristic
180	C₂²	#3416	1	4	yes	—
181	C₄	—	2	4	no	—
182	C₄	—	2	4	no	—
183	C₄	—	4	2	no	—
184	C₄	—	8	8	no	—
185	C₄	—	16	4	no	—
186	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
187	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
188	C₄	#3160	1	1	yes	characteristic
189	C₄	#3170	1	1	yes	characteristic
190	C₂	C₄×#500	1	1	yes	characteristic
191	C₂	#33460	1	2	yes	—
192	C₂	#33730	1	2	yes	—
193	C₂	#33743	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. #2253	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2253	2
#21723	2
#13284	2
#13168	2
#13166	2
#12906	2
#12674	2
C₄×#367	2
#12539	2