Subgroups — #128249

Order 512.

410 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#504	C₂	1	1	yes	maximal, characteristic
2	C₄×#350	C₂	1	1	yes	maximal, characteristic
3	#11399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#8184	C₂	1	1	yes	maximal, characteristic
6	#7141	C₂	1	1	yes	maximal, characteristic, complemented
7	#7138	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#504	C₄	1	2	yes	—
9	C₄×#131	C₂²	1	1	yes	characteristic
10	C₂×#350	C₄	1	2	yes	—
11	#2108	—	2	2	no	—
12	#2108	C₄	1	2	yes	—
13	C₄×M₃₂	—	2	1	no	—
14	C₄×M₃₂	C₂²	1	1	yes	characteristic
15	C₄×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₆	—	2	1	no	—
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#1476	C₄	1	2	yes	—
20	#1450	C₂²	1	1	yes	characteristic
21	C₂×C₈²	C₂²	1	1	yes	characteristic
22	#1379	C₄	1	2	yes	—
23	#1311	—	2	1	no	—
24	C₈×C₁₆	—	2	2	no	—
25	C₈×C₁₆	—	2	1	no	—
26	#1278	C₄	1	2	yes	—
27	C₂×#131	C₂×C₄	1	1	yes	characteristic
28	C₂×#131	C₂×C₄	1	1	yes	characteristic
29	C₄×#41	C₂×C₄	1	1	yes	characteristic
30	#504	—	2	4	no	—
31	#504	C₈	1	8	yes	complemented
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	C₂×C₄	1	2	yes	—
34	C₂×M₃₂	C₂×C₄	1	2	yes	—
35	C₂²×C₁₆	—	2	2	no	—
36	C₂²×C₁₆	C₂×C₄	1	2	yes	—
37	C₂²×C₁₆	C₂×C₄	1	2	yes	—
38	#431	C₂×C₄	1	2	yes	—
39	#405	—	2	1	no	—
40	#405	—	2	1	no	—
41	#405	C₂×C₄	1	1	yes	characteristic
42	#405	C₂×C₄	1	1	yes	characteristic
43	C₄×M₁₆	—	2	1	no	—
44	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	—	2	1	no	—
46	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	#350	C₈	1	8	yes	complemented
49	#346	—	2	2	no	—
50	#346	—	2	1	no	—
51	#346	—	2	1	no	—
52	C₄×C₁₆	—	2	2	no	—
53	C₄×C₁₆	—	2	1	no	—
54	C₄×C₁₆	—	2	1	no	—
55	C₄×C₁₆	—	2	1	no	—
56	C₄×C₁₆	—	2	1	no	—
57	C₄×C₁₆	D₄	1	2	yes	—
58	C₄×C₁₆	D₄	1	1	yes	characteristic
59	C₄×C₁₆	D₄	1	1	yes	characteristic
60	Q₈⋊C₈	C₈	1	4	yes	—
61	D₄⋊C₈	C₈	1	4	yes	—
62	#322	—	2	1	no	—
63	C₈²	—	2	1	no	—
64	C₈²	—	2	1	no	—
65	C₈²	—	2	1	no	—
66	C₂×#41	C₄²	1	1	yes	characteristic
67	#131	—	2	2	no	—
68	#131	—	2	2	no	—
69	#131	C₂×C₈	1	4	yes	—
70	#131	C₂×C₈	1	4	yes	—
71	C₂×M₁₆	—	2	1	no	—
72	C₂×M₁₆	—	2	1	no	—
73	C₂×M₁₆	C₄²	1	1	yes	characteristic
74	C₂×M₁₆	C₄²	1	1	yes	characteristic
75	C₂²×C₈	—	2	2	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₄×Q₈	C₂×C₈	1	2	yes	—
81	C₄×D₄	—	2	2	no	—
82	C₄×D₄	C₂×C₈	1	2	yes	—
83	#117	—	2	1	no	—
84	#117	C₄²	1	1	yes	characteristic
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
87	M₃₂	—	2	2	no	—
88	M₃₂	—	2	2	no	—
89	M₃₂	—	2	8	no	—
90	M₃₂	C₂×C₈	1	4	yes	—
91	M₃₂	C₂×C₈	1	4	yes	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	8	no	—
97	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
98	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
99	C₂×C₁₆	C₂×C₈	1	4	yes	—
100	C₂×C₁₆	C₂×C₈	1	4	yes	—
101	C₂×C₁₆	M₁₆	1	4	yes	—
102	C₄⋊C₈	C₂×C₈	1	4	yes	—
103	C₂²⋊C₈	—	2	2	no	—
104	#97	—	2	1	no	—
105	#97	—	2	1	no	—
106	#97	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	C₂×C₈	1	4	yes	—
113	C₄×C₈	C₂×D₄	1	1	yes	characteristic
114	C₄×C₈	C₂×D₄	1	1	yes	characteristic
115	C₄×C₈	#41	1	1	yes	characteristic
116	C₄×C₈	#41	1	1	yes	characteristic
117	#41	—	2	1	no	—
118	#41	—	2	1	no	—
119	#41	C₄×C₈	1	2	yes	—
120	#41	C₄×C₈	1	2	yes	—
121	C₂×Q₈	C₄×C₈	1	1	yes	characteristic
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	C₄×C₈	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
127	C₂²×C₄	#131	1	2	yes	—
128	M₁₆	—	2	1	no	—
129	M₁₆	—	2	1	no	—
130	M₁₆	—	2	4	no	—
131	M₁₆	—	2	4	no	—
132	M₁₆	C₄×C₈	1	2	yes	—
133	M₁₆	C₄×C₈	1	2	yes	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₈	#117	1	1	yes	characteristic
143	C₂×C₈	C₄×D₄	1	1	yes	characteristic
144	C₂×C₈	C₄×D₄	1	1	yes	characteristic
145	C₂×C₈	C₂²×C₈	1	2	yes	—
146	C₂×C₈	C₂²×C₈	1	2	yes	—
147	C₂×C₈	C₂×M₁₆	1	2	yes	—
148	C₂×C₈	#131	1	2	yes	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	C₄×C₈	1	2	yes	—
151	C₂²⋊C₄	—	2	1	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	2	no	—
155	C₄²	C₄×D₄	1	1	yes	characteristic
156	C₄²	C₄×D₄	1	1	yes	characteristic
157	C₄²	C₂²×C₈	1	2	yes	—
158	C₁₆	—	2	4	no	—
159	C₁₆	—	2	4	no	—
160	C₁₆	—	2	4	no	—
161	C₁₆	C₂²⋊C₈	1	4	yes	—
162	C₁₆	C₂²⋊C₈	1	4	yes	—
163	C₂³	—	2	1	no	—
164	C₂³	#405	1	1	yes	characteristic
165	Q₈	—	2	1	no	—
166	Q₈	C₈²	1	2	yes	—
167	D₄	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	2	1	no	—
170	D₄	C₈²	1	2	yes	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
181	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
182	C₂×C₄	#405	1	1	yes	characteristic
183	C₂×C₄	C₈×D₄	1	2	yes	—
184	C₂×C₄	C₈×D₄	1	2	yes	—
185	C₈	—	2	8	no	—
186	C₈	—	2	2	no	—
187	C₈	—	2	2	no	—
188	C₈	—	4	4	no	—
189	C₈	C₂×C₂²⋊C₈	1	2	yes	—
190	C₈	#431	1	2	yes	—
191	C₈	C₈×D₄	1	2	yes	—
192	C₈	C₈×D₄	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	C₈×M₁₆	1	1	yes	characteristic
197	C₂²	#1450	1	1	yes	characteristic
198	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	C₂×C₈²	1	1	yes	characteristic
203	C₄	C₈×M₁₆	1	1	yes	characteristic
204	C₄	C₄×C₂²⋊C₈	1	1	yes	characteristic
205	C₄	C₈×C₂²⋊C₄	1	1	yes	characteristic
206	C₄	#2170	1	4	yes	—
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	4	2	no	—
210	C₂	C₈×C₂²⋊C₈	1	1	yes	characteristic
211	C₂	#12614	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	D₄
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	C₂×#41	C₄²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ 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
3. #11399	C₂	semidirect
4. C₂×C₈×C₁₆	C₂	semidirect
6. #7141	C₂	semidirect
7. #7138	C₂	semidirect
31. #504	C₈	semidirect
48. #350	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#504	2
C₄×#350	2
#11399	2
C₂×C₈×C₁₆	2
#8184	2
#7141	2
#7138	2