Subgroups — #128254

Order 512.

394 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	C₈×#131	C₂	1	1	yes	maximal, characteristic
2	C₄×#350	C₂	1	2	yes	maximal
3	#11402	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1312	C₂	1	1	yes	maximal, characteristic, complemented
5	#7141	C₂	1	2	yes	maximal, complemented
6	C₈×#41	C₄	1	1	yes	characteristic
7	C₈×#41	C₄	1	1	yes	characteristic
8	C₄×#131	C₂²	1	1	yes	characteristic
9	C₂×#350	C₄	1	4	yes	—
10	C₄×M₃₂	C₂²	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	#1450	—	2	1	no	—
13	#1450	C₂²	1	1	yes	characteristic
14	C₈×M₁₆	—	2	1	no	—
15	C₈×M₁₆	—	2	1	no	—
16	C₈×M₁₆	C₄	1	1	yes	characteristic
17	C₈×M₁₆	C₄	1	1	yes	characteristic
18	C₂×C₈²	—	2	1	no	—
19	C₂×C₈²	C₂²	1	1	yes	characteristic
20	#1379	C₄	1	4	yes	—
21	#1312	—	2	1	no	—
22	#1312	—	2	1	no	—
23	#1312	—	2	1	no	—
24	#1312	—	2	1	no	—
25	C₂×#131	C₂×C₄	1	2	yes	—
26	C₄×#41	C₂×C₄	1	1	yes	characteristic
27	C₂×M₃₂	C₂×C₄	1	4	yes	—
28	C₂²×C₁₆	C₂×C₄	1	4	yes	—
29	C₈×Q₈	C₈	1	2	yes	—
30	C₈×Q₈	C₈	1	2	yes	—
31	C₈×D₄	—	2	2	no	—
32	C₈×D₄	—	2	2	no	—
33	C₈×D₄	C₈	1	2	yes	—
34	C₈×D₄	C₈	1	2	yes	—
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	C₂×C₄	1	1	yes	characteristic
38	#431	C₂×C₄	1	1	yes	characteristic
39	#405	—	2	2	no	—
40	#405	C₂×C₄	1	2	yes	—
41	C₄×M₁₆	—	2	1	no	—
42	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	—	2	1	no	—
45	C₂×C₄×C₈	—	2	1	no	—
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	#350	—	2	8	no	—
50	#346	—	2	2	no	—
51	C₄×C₁₆	—	2	2	no	—
52	C₄×C₁₆	—	2	2	no	—
53	C₄×C₁₆	—	2	2	no	—
54	#322	—	2	1	no	—
55	#322	—	2	1	no	—
56	#322	—	2	1	no	—
57	C₈²	—	2	1	no	—
58	C₈²	—	2	1	no	—
59	C₈²	—	2	1	no	—
60	C₈²	D₄	1	1	yes	characteristic
61	C₈²	D₄	1	1	yes	characteristic
62	C₈²	D₄	1	1	yes	characteristic
63	C₈²	D₄	1	1	yes	characteristic
64	C₂×#41	C₄²	1	1	yes	characteristic
65	#131	—	2	4	no	—
66	#131	M₁₆	1	8	yes	—
67	C₂×M₁₆	—	2	2	no	—
68	C₂×M₁₆	C₄²	1	2	yes	—
69	C₂²×C₈	—	2	2	no	—
70	C₂²×C₈	—	2	2	no	—
71	C₂²×C₈	—	2	2	no	—
72	C₂²×C₈	M₁₆	1	2	yes	—
73	C₂²×C₈	M₁₆	1	2	yes	—
74	C₂²×C₈	C₂²×C₄	1	2	yes	—
75	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
77	C₄×D₄	—	2	1	no	—
78	C₄×D₄	—	2	1	no	—
79	C₄×D₄	C₂×C₈	1	1	yes	characteristic
80	C₄×D₄	C₂×C₈	1	1	yes	characteristic
81	#117	—	2	1	no	—
82	#117	C₄²	1	1	yes	characteristic
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
85	M₃₂	—	2	4	no	—
86	M₃₂	—	2	4	no	—
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	—	2	4	no	—
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	—	2	4	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	C₂×C₈	1	2	yes	—
94	C₄⋊C₈	C₂×C₈	1	2	yes	—
95	C₂²⋊C₈	—	2	1	no	—
96	C₂²⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	C₂²⋊C₈	—	2	2	no	—
99	#97	—	2	2	no	—
100	#97	—	2	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₈	C₂×C₈	1	2	yes	—
112	C₄×C₈	C₂×C₈	1	2	yes	—
113	C₄×C₈	C₂×D₄	1	1	yes	characteristic
114	C₄×C₈	C₂×D₄	1	1	yes	characteristic
115	C₄×C₈	#41	1	2	yes	—
116	#41	—	2	2	no	—
117	#41	#97	1	4	yes	—
118	C₂×Q₈	C₄×C₈	1	1	yes	characteristic
119	C₂×D₄	—	2	1	no	—
120	C₂×D₄	C₄×C₈	1	1	yes	characteristic
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
125	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	8	no	—
129	M₁₆	#97	1	4	yes	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	2	8	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	C₂²⋊C₈	1	2	yes	—
139	C₂×C₈	C₂²⋊C₈	1	2	yes	—
140	C₂×C₈	C₂²⋊C₈	1	2	yes	—
141	C₂×C₈	C₂²⋊C₈	1	2	yes	—
142	C₂×C₈	C₄×D₄	1	2	yes	—
143	C₂×C₈	C₄×D₄	1	2	yes	—
144	C₂×C₈	C₂×M₁₆	1	4	yes	—
145	C₂×C₈	#131	1	4	yes	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	C₄×C₈	1	1	yes	characteristic
149	C₄⋊C₄	C₄×C₈	1	1	yes	characteristic
150	C₂²⋊C₄	—	2	1	no	—
151	C₂²⋊C₄	—	2	1	no	—
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₄²	#117	1	1	yes	characteristic
158	C₄²	C₂²×C₈	1	1	yes	characteristic
159	C₄²	C₂²×C₈	1	1	yes	characteristic
160	C₁₆	—	4	4	no	—
161	C₁₆	—	4	4	no	—
162	C₂³	—	2	1	no	—
163	C₂³	C₄×M₁₆	1	1	yes	characteristic
164	Q₈	—	2	1	no	—
165	Q₈	#322	1	2	yes	—
166	D₄	—	2	4	no	—
167	D₄	—	2	1	no	—
168	D₄	#322	1	2	yes	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	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×#97	1	2	yes	—
180	C₂×C₄	#405	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
182	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
183	C₂×C₄	#431	1	1	yes	characteristic
184	C₂×C₄	#431	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	4	no	—
189	C₈	—	4	2	no	—
190	C₈	—	4	2	no	—
191	C₈	#350	1	4	yes	complemented
192	C₈	#435	1	4	yes	—
193	C₈	#435	1	4	yes	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	1	no	—
198	C₂²	—	4	1	no	—
199	C₂²	#1451	1	2	yes	—
200	C₂²	C₄×C₂²⋊C₈	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	1	no	—
205	C₄	—	4	1	no	—
206	C₄	C₂×#322	1	1	yes	characteristic
207	C₄	#1450	1	1	yes	characteristic
208	C₄	#1752	1	2	yes	—
209	C₄	C₂×#350	1	2	yes	—
210	C₄	#2117	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	—	4	2	no	—
213	C₂	#8144	1	1	yes	characteristic
214	C₂	C₄×#350	1	1	yes	characteristic
215	C₂	#12604	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×C₂²⋊C₈
prime radical	C₂×#41	C₄²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈² ◁ 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. #11402	C₂	semidirect
4. C₂×#1312	C₂	semidirect
5. #7141	C₂	semidirect
191. C₈	#350	semidirect

Maximal subgroups

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

subgroup	index
C₈×#131	2
C₄×#350	2
#11402	2
C₂×#1312	2
#7141	2