Subgroups — #128248

Order 512.

398 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2108	C₂	1	1	yes	maximal, characteristic
2	#12602	C₂	1	1	yes	maximal, characteristic
3	#11399	C₂	1	2	yes	maximal, complemented
4	#8156	C₂	1	1	yes	maximal, characteristic
5	#7140	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂	C₄	1	1	yes	characteristic
7	C₂³×C₁₆	C₄	1	1	yes	characteristic
8	C₂×#405	C₂²	1	1	yes	characteristic
9	C₂×#349	C₄	1	2	yes	—
10	#2108	—	2	2	no	—
11	#2108	—	2	2	no	—
12	C₄×M₃₂	C₂²	1	2	yes	—
13	C₂×#346	C₄	1	1	yes	characteristic
14	C₂×#346	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1458	C₄	1	2	yes	—
18	#1450	C₂²	1	2	yes	—
19	#1377	C₄	1	2	yes	—
20	#1311	—	2	2	no	—
21	C₈×C₁₆	—	2	2	no	—
22	#1281	C₄	1	2	yes	—
23	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×#117	C₂×C₄	1	1	yes	characteristic
26	C₂×M₃₂	—	2	2	no	—
27	C₂×M₃₂	—	2	2	no	—
28	C₂×M₃₂	C₈	1	4	yes	complemented
29	C₂×M₃₂	C₂×C₄	1	4	yes	—
30	C₂²×C₁₆	—	2	1	no	—
31	C₂²×C₁₆	—	2	1	no	—
32	C₂²×C₁₆	—	2	2	no	—
33	C₂²×C₁₆	C₈	1	4	yes	complemented
34	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
38	#431	C₂×C₄	1	4	yes	—
39	#405	—	2	2	no	—
40	#405	—	2	2	no	—
41	C₂×#97	C₂×C₄	1	1	yes	characteristic
42	C₂×#97	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	#349	C₈	1	8	yes	complemented
46	#346	—	2	2	no	—
47	#346	—	2	2	no	—
48	#346	—	2	2	no	—
49	C₄×C₁₆	—	2	1	no	—
50	C₄×C₁₆	—	2	1	no	—
51	C₄×C₁₆	—	2	2	no	—
52	C₄×C₁₆	D₄	1	2	yes	—
53	C₄×C₁₆	D₄	1	2	yes	—
54	#324	C₈	1	4	yes	—
55	#323	C₈	1	4	yes	—
56	#322	—	2	2	no	—
57	C₈²	—	2	2	no	—
58	C₂³×C₄	C₄²	1	1	yes	characteristic
59	C₂×M₁₆	—	2	2	no	—
60	C₂×M₁₆	—	2	2	no	—
61	C₂×M₁₆	C₂×C₈	1	4	yes	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	C₄²	1	1	yes	characteristic
67	C₂²×C₈	C₄²	1	1	yes	characteristic
68	C₂²×C₈	C₂×C₈	1	2	yes	—
69	C₂²×C₈	C₂×C₈	1	2	yes	—
70	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	#117	—	2	2	no	—
73	#117	—	2	2	no	—
74	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
75	C₂×C₂²⋊C₄	C₂×C₈	1	2	yes	—
76	C₂×C₄²	C₄²	1	1	yes	characteristic
77	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
78	M₃₂	—	2	4	no	—
79	M₃₂	—	2	8	no	—
80	M₃₂	C₂×C₈	1	8	yes	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	4	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	4	no	—
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
90	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₁₆	C₂×C₈	1	2	yes	—
92	C₂×C₁₆	C₂×C₈	1	2	yes	—
93	C₂×C₁₆	C₂×C₈	1	2	yes	—
94	C₂×C₁₆	C₂×C₈	1	2	yes	—
95	C₂×C₁₆	M₁₆	1	2	yes	—
96	C₂×C₁₆	M₁₆	1	2	yes	—
97	C₄⋊C₈	—	2	4	no	—
98	C₂²⋊C₈	C₂×C₈	1	8	yes	—
99	#97	—	2	2	no	—
100	#97	—	2	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	2	4	no	—
104	C₄×C₈	C₂×D₄	1	2	yes	—
105	C₄×C₈	#41	1	2	yes	—
106	C₂⁴	C₄×C₈	1	1	yes	characteristic
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
112	C₂²×C₄	C₄×C₈	1	2	yes	—
113	C₂²×C₄	C₄×C₈	1	2	yes	—
114	C₂²×C₄	C₄×C₈	1	2	yes	—
115	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
116	C₂²×C₄	C₂²×C₈	1	2	yes	—
117	M₁₆	—	2	8	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₄×C₈	1	2	yes	—
128	C₂×C₈	C₄×C₈	1	2	yes	—
129	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₈	#117	1	1	yes	characteristic
131	C₂×C₈	C₄×D₄	1	2	yes	—
132	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₈	C₂²×C₈	1	2	yes	—
135	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₈	#131	1	2	yes	—
138	C₄⋊C₄	—	2	4	no	—
139	C₂²⋊C₄	—	2	4	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₄×D₄	1	2	yes	—
143	C₄²	#131	1	2	yes	—
144	C₁₆	—	2	2	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	2	8	no	—
147	C₁₆	C₂²⋊C₈	1	2	yes	—
148	C₁₆	C₂²⋊C₈	1	2	yes	—
149	C₁₆	C₂²⋊C₈	1	4	yes	—
150	C₂³	—	2	1	no	—
151	C₂³	—	2	1	no	—
152	C₂³	—	4	2	no	—
153	C₂³	C₈²	1	2	yes	—
154	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	C₈²	1	2	yes	—
165	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
168	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
169	C₂×C₄	#405	1	1	yes	characteristic
170	C₂×C₄	#405	1	1	yes	characteristic
171	C₂×C₄	C₈×D₄	1	4	yes	—
172	C₈	—	2	2	no	—
173	C₈	—	2	2	no	—
174	C₈	—	4	8	no	—
175	C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
177	C₈	#431	1	1	yes	characteristic
178	C₈	#431	1	1	yes	characteristic
179	C₈	C₈×D₄	1	4	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	C₂×C₈²	1	1	yes	characteristic
186	C₂²	C₈×M₁₆	1	1	yes	characteristic
187	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	C₈×M₁₆	1	1	yes	characteristic
192	C₄	#1450	1	1	yes	characteristic
193	C₄	C₄×C₂²⋊C₈	1	1	yes	characteristic
194	C₄	C₈×C₂²⋊C₄	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	C₈×C₂²⋊C₈	1	1	yes	characteristic
199	C₂	#12613	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂²	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
5. #7140	C₂	semidirect
28. C₂×M₃₂	C₈	semidirect
33. C₂²×C₁₆	C₈	semidirect
45. #349	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2108	2
#12602	2
#11399	2
#8156	2
#7140	2