Subgroups — #498528

Order 512.

496 conjugacy classes of subgroups.

The subgroup lattice has 192 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	#23158	C₂	1	2	yes	maximal
3	#13276	C₂	1	2	yes	maximal
4	#13168	C₂	1	2	yes	maximal
5	#13109	C₂	1	4	yes	maximal
6	#12674	C₂	1	2	yes	maximal
7	C₄×#367	C₂	1	1	yes	maximal, characteristic
8	#12662	C₂	1	1	yes	maximal, characteristic
9	C₂×#500	C₂²	1	2	yes	—
10	#2369	—	2	2	no	—
11	#2252	C₄	1	8	yes	complemented
12	C₂×#367	C₂²	1	1	yes	characteristic
13	C₂×#367	C₂²	1	1	yes	characteristic
14	C₂×#366	C₂²	1	2	yes	—
15	C₂×#366	C₂²	1	2	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×#346	C₂²	1	2	yes	—
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#1987	—	2	2	no	—
20	#1942	C₂²	1	4	yes	—
21	#1921	—	2	4	no	—
22	C₄×#107	C₂²	1	4	yes	—
23	C₄×#107	C₂²	1	2	yes	—
24	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	#1380	C₂²	1	2	yes	—
26	#1380	C₂²	1	4	yes	—
27	#1380	C₂²	1	4	yes	—
28	#1374	C₂²	1	2	yes	—
29	C₂×C₄⋊Q₈	—	2	2	no	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	#500	—	2	4	no	—
34	#500	C₂×C₄	1	8	yes	—
35	C₂×#107	—	2	2	no	—
36	C₂×#107	C₂³	1	2	yes	—
37	C₂×#107	C₂³	1	4	yes	—
38	C₂×#107	C₂³	1	2	yes	—
39	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	2	yes	—
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	#391	—	4	4	no	—
43	#389	—	2	4	no	—
44	#389	—	4	2	no	—
45	C₄×C₄⋊C₄	—	2	4	no	—
46	C₄×C₄⋊C₄	—	2	2	no	—
47	C₄×C₄⋊C₄	—	4	2	no	—
48	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
49	#367	—	2	2	no	—
50	#367	Q₈	1	4	yes	complemented
51	#366	C₂×C₄	1	8	yes	—
52	#366	C₂×C₄	1	8	yes	—
53	C₄⋊C₁₆	—	2	2	no	—
54	#346	—	2	4	no	—
55	C₄×C₁₆	C₂×C₄	1	4	yes	—
56	#340	—	2	2	no	—
57	#340	—	2	4	no	—
58	#340	—	2	4	no	—
59	#336	D₄	1	2	yes	—
60	C₂²×Q₈	—	4	2	no	—
61	C₂²×C₈	C₂×D₄	1	2	yes	—
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	C₂⁴	1	1	yes	characteristic
64	C₄⋊Q₈	—	2	8	no	—
65	C₄⋊Q₈	—	4	4	no	—
66	C₂×C₄⋊C₄	—	2	4	no	—
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄⋊C₄	—	4	4	no	—
71	C₂×C₄⋊C₄	D₈	1	2	yes	—
72	C₂×C₄²	—	4	2	no	—
73	C₂×C₄²	—	4	4	no	—
74	C₂×C₄²	C₂×D₄	1	2	yes	—
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	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	C₂²×C₄	1	2	yes	—
82	C₂×C₁₆	C₂²×C₄	1	2	yes	—
83	C₂×C₁₆	C₂×Q₈	1	2	yes	—
84	C₂×C₁₆	#41	1	2	yes	—
85	#107	—	2	8	no	—
86	#107	—	2	4	no	—
87	#107	—	2	2	no	—
88	#107	—	2	2	no	—
89	#107	—	2	2	no	—
90	#107	C₂²×C₄	1	8	yes	—
91	#107	C₂×Q₈	1	4	yes	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₂²×C₄	1	2	yes	—
96	C₄×C₈	#41	1	2	yes	—
97	#95	—	4	4	no	—
98	#95	—	4	2	no	—
99	#95	—	4	4	no	—
100	#95	D₈	1	2	yes	—
101	C₂×Q₈	—	4	8	no	—
102	C₂×Q₈	—	16	2	no	—
103	C₂×Q₈	—	16	2	no	—
104	C₂²×C₄	—	4	2	no	—
105	C₂²×C₄	—	4	4	no	—
106	C₂²×C₄	—	4	2	no	—
107	C₂²×C₄	C₂×D₈	1	4	yes	—
108	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
109	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	C₄×D₄	1	2	yes	—
117	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₈	C₂×#41	1	1	yes	characteristic
120	C₂×C₈	C₂×#41	1	1	yes	characteristic
121	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	2	8	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	8	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	16	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	8	4	no	—
135	C₄²	—	16	4	no	—
136	C₄²	C₄×D₄	1	2	yes	—
137	C₄²	#135	1	2	yes	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	4	2	no	—
140	C₁₆	C₄×Q₈	1	4	yes	—
141	C₂³	C₂²×D₈	1	1	yes	characteristic
142	Q₈	—	16	8	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	C₄×D₈	1	2	yes	—
157	C₂×C₄	C₄×D₈	1	2	yes	—
158	C₂×C₄	#510	1	4	yes	—
159	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂×C₄	#547	1	1	yes	characteristic
161	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
162	C₂×C₄	C₂×#135	1	1	yes	characteristic
163	C₂×C₄	#578	1	1	yes	characteristic
164	C₂×C₄	#578	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	4	2	no	—
167	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₈	#519	1	1	yes	characteristic
169	C₈	#556	1	1	yes	characteristic
170	C₈	#557	1	1	yes	characteristic
171	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
172	C₂²	#3334	1	1	yes	characteristic
173	C₂²	Q₈×D₈	1	1	yes	characteristic
174	C₂²	C₂×#510	1	2	yes	—
175	C₂²	#3415	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	4	4	no	—
178	C₄	—	8	4	no	—
179	C₄	—	8	8	no	—
180	C₄	—	16	2	no	—
181	C₄	#2176	1	4	yes	—
182	C₄	#2365	1	1	yes	characteristic
183	C₄	#2950	1	1	yes	characteristic
184	C₄	#3342	1	1	yes	characteristic
185	C₄	#3383	1	1	yes	characteristic
186	C₂	#23157	1	1	yes	characteristic
187	C₂	C₂×#2176	1	1	yes	characteristic
188	C₂	#33456	1	1	yes	characteristic
189	C₂	#33924	1	2	yes	—
190	C₂	#33952	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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #2252	C₄	semidirect
50. #367	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
#23158	2
#13276	2
#13168	2
#13109	2
#12674	2
C₄×#367	2
#12662	2