Subgroups — #101936

Order 512.

306 conjugacy classes of subgroups.

The subgroup lattice has 217 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#9601	C₂	1	2	yes	maximal
2	#8806	C₂	1	1	yes	maximal, characteristic
3	C₄×#334	C₂	1	1	yes	maximal, characteristic
4	#8260	C₂	1	1	yes	maximal, characteristic
5	#7031	C₂	1	2	yes	maximal
6	#1842	C₂²	1	2	yes	—
7	C₄×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	—	2	2	no	—
12	C₂×#335	—	2	1	no	—
13	C₂×#335	—	2	1	no	—
14	C₂×#334	C₄	1	1	yes	characteristic
15	C₂×#334	C₄	1	1	yes	characteristic
16	C₂×#322	—	2	1	no	—
17	#1277	—	2	2	no	—
18	#1277	C₄	1	2	yes	—
19	#1273	—	2	1	no	—
20	C₂×C₄⋊C₈	—	2	2	no	—
21	C₂×C₄⋊C₈	—	2	2	no	—
22	C₂×C₄⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	—	2	2	no	—
29	C₂×C₄×C₈	—	2	1	no	—
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
36	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
37	C₄³	C₂³	1	1	yes	characteristic
38	#336	—	2	2	no	—
39	#336	—	2	1	no	—
40	#336	Q₈	1	2	yes	—
41	#335	—	4	2	no	—
42	#335	—	4	4	no	—
43	#335	—	4	2	no	—
44	#334	—	2	2	no	—
45	#334	C₈	1	4	yes	complemented
46	#322	—	4	2	no	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	—	2	1	no	—
50	C₂²×C₈	—	2	1	no	—
51	C₂²×C₈	—	2	2	no	—
52	C₂²×C₈	—	4	2	no	—
53	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄²	#41	1	1	yes	characteristic
63	C₂×C₄²	#41	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	4	2	no	—
72	C₄⋊C₈	—	4	2	no	—
73	C₄⋊C₈	—	4	2	no	—
74	C₄⋊C₈	—	8	2	no	—
75	C₄⋊C₈	C₂×C₈	1	4	yes	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	4	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	—	4	2	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	—	4	1	no	—
87	C₄×C₈	—	4	2	no	—
88	C₄×C₈	C₂×C₈	1	2	yes	—
89	C₄×C₈	M₁₆	1	2	yes	—
90	C₄×C₈	D₈	1	2	yes	—
91	C₄×C₈	Q₁₆	1	2	yes	—
92	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
95	C₂²×C₄	#121	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂²×C₄	#125	1	1	yes	characteristic
98	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	4	1	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	—	4	1	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	8	2	no	—
122	C₂×C₈	—	8	2	no	—
123	C₂×C₈	—	8	2	no	—
124	C₂×C₈	C₄⋊C₈	1	2	yes	—
125	C₂×C₈	C₄⋊C₈	1	2	yes	—
126	C₂×C₈	#107	1	2	yes	—
127	C₂×C₈	#107	1	2	yes	—
128	C₂×C₈	#108	1	2	yes	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	4	1	no	—
135	C₄²	C₂²×C₈	1	1	yes	characteristic
136	C₄²	C₂×M₁₆	1	1	yes	characteristic
137	C₄²	#131	1	1	yes	characteristic
138	C₄²	#131	1	1	yes	characteristic
139	C₄²	C₂×D₈	1	1	yes	characteristic
140	C₄²	C₂×Q₁₆	1	1	yes	characteristic
141	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₄²	AGL(1,ℤ/8)	1	2	yes	—
143	C₄²	#137	1	1	yes	characteristic
144	C₄²	#137	1	2	yes	—
145	C₂³	#389	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
156	C₂×C₄	#424	1	1	yes	characteristic
157	C₂×C₄	C₂×#107	1	1	yes	characteristic
158	C₂×C₄	#428	1	1	yes	characteristic
159	C₂×C₄	C₂×#108	1	1	yes	characteristic
160	C₂×C₄	#430	1	1	yes	characteristic
161	C₂×C₄	C₈×D₄	1	1	yes	characteristic
162	C₂×C₄	#436	1	1	yes	characteristic
163	C₂×C₄	#440	1	2	yes	—
164	C₂×C₄	#444	1	2	yes	—
165	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
166	C₂×C₄	#446	1	1	yes	characteristic
167	C₂×C₄	#459	1	1	yes	characteristic
168	C₂×C₄	#462	1	1	yes	characteristic
169	C₂×C₄	#469	1	1	yes	characteristic
170	C₂×C₄	#470	1	1	yes	characteristic
171	C₂×C₄	#472	1	1	yes	characteristic
172	C₂×C₄	#473	1	1	yes	characteristic
173	C₂×C₄	#474	1	1	yes	characteristic
174	C₂×C₄	#477	1	1	yes	characteristic
175	C₂×C₄	#501	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	4	1	no	—
179	C₈	—	4	4	no	—
180	C₈	—	8	2	no	—
181	C₈	—	8	2	no	—
182	C₈	—	8	1	no	—
183	C₈	—	8	1	no	—
184	C₈	—	16	2	no	—
185	C₈	#335	1	4	yes	complemented
186	C₂²	#1939	1	1	yes	characteristic
187	C₂²	#1944	1	1	yes	characteristic
188	C₂²	#1946	1	1	yes	characteristic
189	C₂²	#1947	1	1	yes	characteristic
190	C₂²	#1950	1	1	yes	characteristic
191	C₂²	#1952	1	1	yes	characteristic
192	C₂²	#1953	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	1	no	—
199	C₄	C₂×#335	1	1	yes	characteristic
200	C₄	#1565	1	1	yes	characteristic
201	C₄	#1571	1	1	yes	characteristic
202	C₄	#1574	1	1	yes	characteristic
203	C₄	SD₁₆⋊C₈	1	2	yes	—
204	C₄	#1592	1	2	yes	—
205	C₄	#1687	1	1	yes	characteristic
206	C₄	#1689	1	1	yes	characteristic
207	C₄	#1694	1	1	yes	characteristic
208	C₄	#1696	1	1	yes	characteristic
209	C₂	#9596	1	1	yes	characteristic
210	C₂	#9601	1	1	yes	characteristic
211	C₂	#9608	1	1	yes	characteristic
212	C₂	#9620	1	1	yes	characteristic
213	C₂	#9626	1	1	yes	characteristic
214	C₂	#9630	1	1	yes	characteristic
215	C₂	#9636	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#389
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
45. #334	C₈	semidirect
185. C₈	#335	semidirect

Maximal subgroups

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

subgroup	index
#9601	2
#8806	2
C₄×#334	2
#8260	2
#7031	2