Subgroups — #141846

Order 512.

300 conjugacy classes of subgroups.

The subgroup lattice has 211 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21732	C₂	1	1	yes	maximal, characteristic
2	C₄×#364	C₂	1	1	yes	maximal, characteristic
3	#8266	C₂	1	1	yes	maximal, characteristic
4	#7181	C₂	1	2	yes	maximal
5	#7180	C₂	1	2	yes	maximal
6	#3160	C₄	1	1	yes	characteristic
7	#2960	—	2	1	no	—
8	C₂×#364	C₂²	1	1	yes	characteristic
9	C₂×#364	C₂²	1	1	yes	characteristic
10	C₄×M₃₂	—	2	1	no	—
11	#1994	—	2	1	no	—
12	#1945	—	2	1	no	—
13	#1945	C₄	1	2	yes	—
14	#1848	C₄	1	1	yes	characteristic
15	#1772	—	2	1	no	—
16	#1772	C₂²	1	2	yes	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1564	—	2	1	no	—
19	#1450	C₂²	1	2	yes	—
20	#1376	—	2	1	no	—
21	#1276	—	2	1	no	—
22	#533	—	2	1	no	—
23	C₂×M₃₂	—	2	1	no	—
24	C₂×M₃₂	—	2	1	no	—
25	#500	—	2	1	no	—
26	#499	—	2	1	no	—
27	#499	C₈	1	2	yes	complemented
28	#498	C₈	1	2	yes	complemented
29	#446	—	4	2	no	—
30	C₈×Q₈	—	4	2	no	—
31	#431	—	2	2	no	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	C₂×C₄	1	2	yes	—
34	#427	—	2	1	no	—
35	#427	C₂×C₄	1	2	yes	—
36	#424	—	2	2	no	—
37	#423	—	2	1	no	—
38	#405	—	2	2	no	—
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	C₄³	D₄	1	1	yes	characteristic
46	#364	—	2	1	no	—
47	#364	—	2	1	no	—
48	#364	Q₈	1	2	yes	complemented
49	#364	Q₈	1	2	yes	complemented
50	#346	—	4	1	no	—
51	C₄×C₁₆	—	4	1	no	—
52	#339	—	2	2	no	—
53	#339	—	2	2	no	—
54	#335	—	2	2	no	—
55	#334	—	2	2	no	—
56	#322	—	2	2	no	—
57	C₈²	—	2	2	no	—
58	C₂×M₁₆	—	2	2	no	—
59	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₄⋊Q₈	—	2	2	no	—
64	#125	—	2	2	no	—
65	C₂²⋊Q₈	—	4	2	no	—
66	C₄×Q₈	—	4	2	no	—
67	#117	—	2	2	no	—
68	C₂×C₄²	—	4	1	no	—
69	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
72	M₃₂	—	2	2	no	—
73	M₃₂	—	2	2	no	—
74	M₃₂	—	4	1	no	—
75	M₃₂	—	4	1	no	—
76	C₂×C₁₆	—	4	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	#108	—	2	2	no	—
79	#108	—	2	2	no	—
80	#108	—	4	1	no	—
81	#108	—	4	1	no	—
82	#107	—	2	2	no	—
83	#107	—	2	2	no	—
84	#106	—	2	2	no	—
85	#106	C₂×C₈	1	4	yes	—
86	C₄⋊C₈	—	2	4	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₄⋊C₈	—	4	4	no	—
89	C₂²⋊C₈	—	4	2	no	—
90	#97	—	4	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	4	2	no	—
98	C₄×C₈	—	4	2	no	—
99	C₄×C₈	C₂×C₈	1	1	yes	characteristic
100	C₄×C₈	C₂×C₈	1	1	yes	characteristic
101	C₄×C₈	M₁₆	1	1	yes	characteristic
102	C₄×C₈	M₁₆	1	1	yes	characteristic
103	C₄×C₈	SD₁₆	1	1	yes	characteristic
104	C₄×C₈	SD₁₆	1	1	yes	characteristic
105	C₄×C₈	Q₁₆	1	1	yes	characteristic
106	C₄×C₈	Q₁₆	1	1	yes	characteristic
107	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₄×C₈	#41	1	1	yes	characteristic
110	C₄×C₈	#41	1	1	yes	characteristic
111	C₂×Q₈	—	4	2	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	M₁₆	—	4	2	no	—
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	4	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
125	C₂×C₈	#103	1	1	yes	characteristic
126	C₂×C₈	#103	1	1	yes	characteristic
127	C₂×C₈	#104	1	1	yes	characteristic
128	C₂×C₈	#104	1	1	yes	characteristic
129	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
130	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
131	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₈	#131	1	1	yes	characteristic
136	C₂×C₈	#131	1	1	yes	characteristic
137	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
139	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	4	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	C₂²⋊C₈	1	1	yes	characteristic
151	C₄²	C₂²⋊C₈	1	1	yes	characteristic
152	C₄²	#103	1	1	yes	characteristic
153	C₄²	#103	1	1	yes	characteristic
154	C₄²	#124	1	1	yes	characteristic
155	C₄²	C₄⋊Q₈	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₁₆	—	4	2	no	—
158	C₂³	#421	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	#391	1	1	yes	characteristic
171	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₂×C₄	#408	1	1	yes	characteristic
173	C₂×C₄	C₂×#103	1	1	yes	characteristic
174	C₂×C₄	#418	1	1	yes	characteristic
175	C₂×C₄	C₂×#104	1	1	yes	characteristic
176	C₈	—	4	1	no	—
177	C₈	—	8	4	no	—
178	C₈	—	8	2	no	—
179	C₈	Q₈⋊C₈	1	1	yes	characteristic
180	C₈	Q₈⋊C₈	1	1	yes	characteristic
181	C₈	Q₈⋊C₈	1	1	yes	characteristic
182	C₈	Q₈⋊C₈	1	1	yes	characteristic
183	C₈	C₈×Q₈	1	1	yes	characteristic
184	C₈	C₈×Q₈	1	1	yes	characteristic
185	C₈	#446	1	1	yes	characteristic
186	C₈	#446	1	1	yes	characteristic
187	C₈	#476	1	1	yes	characteristic
188	C₈	#476	1	1	yes	characteristic
189	C₈	#477	1	1	yes	characteristic
190	C₈	#477	1	1	yes	characteristic
191	C₂²	—	4	1	no	—
192	C₂²	#1478	1	1	yes	characteristic
193	C₂²	#1481	1	1	yes	characteristic
194	C₂²	#1986	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	1	no	—
199	C₄	—	4	1	no	—
200	C₄	—	4	1	no	—
201	C₄	—	8	4	no	—
202	C₄	C₂×Q₈⋊C₈	1	1	yes	characteristic
203	C₄	#1484	1	1	yes	characteristic
204	C₄	#1984	1	1	yes	characteristic
205	C₄	#1987	1	1	yes	characteristic
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#9864	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#476
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9864
prime radical	C₂³	#421
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ 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
27. #499	C₈	semidirect
28. #498	C₈	semidirect
48. #364	Q₈	semidirect
49. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#21732	2
C₄×#364	2
#8266	2
#7181	2
#7180	2