Subgroups — #512476

Order 512.

426 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33261	C₂	1	2	yes	maximal
2	#13168	C₂	1	2	yes	maximal
3	#13167	C₂	1	2	yes	maximal
4	#13165	C₂	1	2	yes	maximal
5	#13165	C₂	1	2	yes	maximal
6	#12907	C₂	1	1	yes	maximal, characteristic
7	#12905	C₂	1	1	yes	maximal, characteristic
8	#12890	C₂	1	2	yes	maximal
9	#12539	C₂	1	1	yes	maximal, characteristic
10	#2647	—	2	2	no	—
11	C₂×#367	C₂²	1	2	yes	—
12	C₂×#366	C₂²	1	2	yes	—
13	C₂×#366	C₂²	1	2	yes	—
14	C₂×#366	C₂²	1	2	yes	—
15	C₂×#346	C₂²	1	1	yes	characteristic
16	C₂×#346	C₂²	1	1	yes	characteristic
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	#1944	—	2	2	no	—
22	#1943	—	2	2	no	—
23	#1942	C₂²	1	4	yes	—
24	#1942	C₂²	1	4	yes	—
25	#1846	C₂²	1	2	yes	—
26	#1845	C₂²	1	2	yes	—
27	#1836	—	2	2	no	—
28	#1726	C₂²	1	1	yes	characteristic
29	#1380	C₂²	1	4	yes	—
30	#1380	C₂²	1	4	yes	—
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	C₂²×C₁₆	C₂³	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	1	1	yes	characteristic
35	C₂×#107	—	2	2	no	—
36	C₂×#107	C₂³	1	4	yes	—
37	C₂×#107	C₂³	1	4	yes	—
38	C₂×#106	—	2	2	no	—
39	C₂×#97	D₄	1	1	yes	characteristic
40	C₂×#97	D₄	1	1	yes	characteristic
41	C₂×#97	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	#400	—	4	4	no	—
46	#398	—	4	4	no	—
47	#389	—	2	4	no	—
48	#389	—	2	4	no	—
49	#384	—	2	2	no	—
50	#382	—	2	2	no	—
51	#376	D₄	1	1	yes	characteristic
52	#367	—	2	4	no	—
53	#366	—	2	4	no	—
54	#366	—	2	4	no	—
55	#366	—	2	4	no	—
56	#346	—	2	2	no	—
57	#346	—	2	2	no	—
58	#346	Q₈	1	4	yes	—
59	C₄×C₁₆	—	2	2	no	—
60	C₄×C₁₆	—	2	2	no	—
61	C₄×C₁₆	Q₈	1	4	yes	—
62	#340	—	2	4	no	—
63	#340	—	2	4	no	—
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₈	C₂⁴	1	1	yes	characteristic
68	C₂×C₄⋊C₄	—	2	4	no	—
69	C₂×C₄⋊C₄	—	2	4	no	—
70	C₂×C₄⋊C₄	—	4	4	no	—
71	C₂×C₄⋊C₄	—	4	4	no	—
72	C₂×C₄⋊C₄	—	4	4	no	—
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	C₂×Q₈	1	2	yes	—
81	C₂×C₁₆	C₂×Q₈	1	2	yes	—
82	C₂×C₁₆	C₂×Q₈	1	2	yes	—
83	C₂×C₁₆	C₂×Q₈	1	2	yes	—
84	C₂×C₁₆	#41	1	2	yes	—
85	C₂×C₁₆	#41	1	2	yes	—
86	C₂×C₁₆	#41	1	2	yes	—
87	C₂×C₁₆	#41	1	2	yes	—
88	#107	—	2	4	no	—
89	#107	—	2	4	no	—
90	#107	—	2	4	no	—
91	#107	—	2	4	no	—
92	#107	—	4	4	no	—
93	#106	—	4	4	no	—
94	#97	—	2	2	no	—
95	#97	—	2	2	no	—
96	#97	Q₁₆	1	4	yes	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₂×Q₈	1	2	yes	—
102	C₄×C₈	C₂×Q₈	1	2	yes	—
103	#95	—	4	4	no	—
104	#95	—	4	4	no	—
105	#95	—	4	4	no	—
106	#95	D₈	1	2	yes	—
107	#95	D₈	1	2	yes	—
108	C₂²×C₄	—	4	4	no	—
109	C₂²×C₄	—	4	4	no	—
110	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₈	1	2	yes	—
112	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
113	C₂²×C₄	C₂×D₈	1	2	yes	—
114	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	C₄⋊Q₈	1	2	yes	—
120	C₂×C₈	C₂×Q₁₆	1	2	yes	—
121	C₂×C₈	C₂×Q₁₆	1	2	yes	—
122	C₂×C₈	#135	1	2	yes	—
123	C₂×C₈	#135	1	2	yes	—
124	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₈	C₂×#41	1	1	yes	characteristic
127	C₂×C₈	C₂×#41	1	1	yes	characteristic
128	C₂×C₈	C₂×#41	1	1	yes	characteristic
129	C₂×C₈	C₂×#41	1	1	yes	characteristic
130	C₄⋊C₄	—	4	4	no	—
131	C₄⋊C₄	—	4	4	no	—
132	C₄⋊C₄	—	4	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	16	4	no	—
135	C₄⋊C₄	—	16	4	no	—
136	C₄⋊C₄	—	16	4	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	C₄⋊Q₈	1	2	yes	—
142	C₄²	C₂×Q₁₆	1	2	yes	—
143	C₁₆	—	2	2	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	#125	1	4	yes	—
146	C₁₆	#125	1	4	yes	—
147	C₂³	C₂²×D₈	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	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	—	8	4	no	—
163	C₂×C₄	#500	1	4	yes	—
164	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₄	#532	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	#499	1	4	yes	—
173	C₈	C₂×#125	1	1	yes	characteristic
174	C₈	C₂×#125	1	1	yes	characteristic
175	C₈	#533	1	1	yes	characteristic
176	C₈	#533	1	1	yes	characteristic
177	C₂²	C₂×#500	1	1	yes	characteristic
178	C₂²	#3169	1	1	yes	characteristic
179	C₂²	#3169	1	1	yes	characteristic
180	C₂²	#3415	1	2	yes	—
181	C₂²	#3417	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	16	4	no	—
187	C₄	—	16	4	no	—
188	C₄	#2859	1	1	yes	characteristic
189	C₄	C₂×#499	1	1	yes	characteristic
190	C₄	#3170	1	1	yes	characteristic
191	C₄	#3170	1	1	yes	characteristic
192	C₂	#33258	1	1	yes	characteristic
193	C₂	#33729	1	2	yes	—
194	C₂	#33739	1	2	yes	—
195	C₂	#33740	1	1	yes	characteristic
196	C₂	#33741	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
#33261	2
#13168	2
#13167	2
#13165	2
#12907	2
#12905	2
#12890	2
#12539	2