Subgroups — #129169

Order 512.

284 conjugacy classes of subgroups.

The subgroup lattice has 199 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12662	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#1370	C₂	1	1	yes	maximal, characteristic
4	C₂×#1312	C₂	1	1	yes	maximal, characteristic
5	C₈×#106	C₂	1	1	yes	maximal, characteristic
6	#7167	C₂	1	1	yes	maximal, characteristic
7	#7167	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#346	—	2	1	no	—
11	C₂×C₄×C₁₆	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×#106	C₄	1	1	yes	characteristic
15	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×#334	C₄	1	1	yes	characteristic
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1374	—	2	1	no	—
20	#1374	—	2	1	no	—
21	#1370	C₄	1	4	yes	complemented
22	#1312	C₄	1	4	yes	complemented
23	#1279	C₄	1	1	yes	characteristic
24	#1279	C₄	1	1	yes	characteristic
25	C₂²×C₁₆	—	2	1	no	—
26	C₂²×C₁₆	—	2	1	no	—
27	C₂²×C₁₆	—	2	1	no	—
28	C₂²×C₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₁₆	D₄	1	1	yes	characteristic
30	C₂×#106	C₈	1	2	yes	—
31	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	—	2	1	no	—
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₁₆	—	2	1	no	—
41	C₄⋊C₁₆	—	2	1	no	—
42	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
43	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
44	#346	—	4	2	no	—
45	C₄×C₁₆	—	2	1	no	—
46	C₄×C₁₆	—	4	2	no	—
47	C₄×C₁₆	C₂×C₄	1	2	yes	—
48	C₄×C₁₆	C₂×C₄	1	2	yes	—
49	C₄×C₁₆	Q₈	1	2	yes	—
50	#340	C₈	1	2	yes	—
51	#336	—	2	1	no	—
52	#336	—	2	1	no	—
53	#334	C₈	1	4	yes	—
54	C₈²	C₂×C₄	1	2	yes	—
55	C₈²	C₂×C₄	1	2	yes	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	—	2	1	no	—
59	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₈	#41	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
64	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
65	C₂×C₄²	—	2	1	no	—
66	C₂×C₄²	—	2	1	no	—
67	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₁₆	—	2	1	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	4	1	no	—
76	C₂×C₁₆	—	4	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	4	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
82	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
83	#106	—	2	4	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	C₂×C₈	1	2	yes	—
87	C₄⋊C₈	C₂×C₈	1	2	yes	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	4	1	no	—
90	C₄×C₈	—	4	1	no	—
91	C₄×C₈	—	4	1	no	—
92	C₄×C₈	—	4	1	no	—
93	C₄×C₈	C₄²	1	2	yes	—
94	C₄×C₈	C₄²	1	2	yes	—
95	C₄×C₈	C₂×C₈	1	2	yes	—
96	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₄×C₈	#41	1	1	yes	characteristic
100	#95	—	2	1	no	—
101	#95	—	2	1	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
105	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
106	C₂²×C₄	#131	1	1	yes	characteristic
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	C₄×C₈	1	2	yes	—
120	C₂×C₈	C₄×C₈	1	2	yes	—
121	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₈	#117	1	1	yes	characteristic
123	C₂×C₈	C₄×D₄	1	1	yes	characteristic
124	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
125	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₈	#137	1	1	yes	characteristic
128	C₂×C₈	#137	1	1	yes	characteristic
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	M₃₂	1	4	yes	—
133	C₄²	—	4	1	no	—
134	C₄²	—	4	1	no	—
135	C₄²	—	4	1	no	—
136	C₄²	—	4	1	no	—
137	C₄²	C₂×C₄²	1	1	yes	characteristic
138	C₄²	C₄×Q₈	1	1	yes	characteristic
139	C₄²	C₂²×C₈	1	1	yes	characteristic
140	C₄²	#131	1	1	yes	characteristic
141	C₁₆	—	2	2	no	—
142	C₁₆	—	2	2	no	—
143	C₁₆	—	4	2	no	—
144	C₁₆	—	4	2	no	—
145	C₁₆	—	4	2	no	—
146	C₂³	C₈×D₄	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
156	C₂×C₄	#405	1	1	yes	characteristic
157	C₂×C₄	C₈×D₄	1	1	yes	characteristic
158	C₂×C₄	#440	1	1	yes	characteristic
159	C₂×C₄	#440	1	1	yes	characteristic
160	C₂×C₄	#444	1	1	yes	characteristic
161	C₂×C₄	#444	1	1	yes	characteristic
162	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
163	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
164	C₂×C₄	C₂×M₃₂	1	2	yes	—
165	C₂×C₄	#504	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	2	2	no	—
168	C₈	—	2	2	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	4	2	no	—
172	C₈	—	4	2	no	—
173	C₈	#428	1	1	yes	characteristic
174	C₈	#428	1	1	yes	characteristic
175	C₈	#441	1	1	yes	characteristic
176	C₈	#442	1	1	yes	characteristic
177	C₂²	SD₁₆⋊C₈	1	1	yes	characteristic
178	C₂²	SD₁₆⋊C₈	1	1	yes	characteristic
179	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
180	C₂²	#2168	1	2	yes	—
181	C₂²	#2170	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	4	no	—
185	C₄	—	8	2	no	—
186	C₄	SD₁₆⋊C₈	1	1	yes	characteristic
187	C₄	SD₁₆⋊C₈	1	1	yes	characteristic
188	C₄	#1773	1	1	yes	characteristic
189	C₄	#1776	1	1	yes	characteristic
190	C₄	C₄×M₃₂	1	1	yes	characteristic
191	C₄	#2108	1	1	yes	characteristic
192	C₄	#2183	1	2	yes	—
193	C₂	#8285	1	1	yes	characteristic
194	C₂	#12056	1	2	yes	—
195	C₂	#12057	1	2	yes	—
196	C₂	#12663	1	1	yes	characteristic
197	C₂	#12664	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
21. #1370	C₄	semidirect
22. #1312	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12662	2
C₁₆×C₄⋊C₄	2
C₂×#1370	2
C₂×#1312	2
C₈×#106	2
#7167	2