Subgroups — #127709

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 257 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×#1846	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#7275	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	—	2	1	no	—
6	C₂×#431	—	2	1	no	—
7	C₂²×#107	C₄	1	2	yes	—
8	C₂²×#106	—	2	1	no	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#382	—	2	1	no	—
11	#2150	—	2	2	no	—
12	#2150	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₁₆	—	2	1	no	—
14	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
15	#1846	—	2	1	no	—
16	#1846	—	2	1	no	—
17	#1846	C₄	1	2	yes	—
18	#1846	C₂²	1	2	yes	—
19	#1564	—	2	2	no	—
20	#1564	C₄	1	4	yes	complemented
21	C₂×#335	—	2	1	no	—
22	C₂×#334	C₂²	1	2	yes	—
23	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₄⋊C₄	—	2	2	no	—
26	C₂²×C₄²	D₄	1	1	yes	characteristic
27	C₂×M₃₂	—	2	4	no	—
28	C₂×M₃₂	—	4	2	no	—
29	C₂×M₃₂	—	4	2	no	—
30	C₂²×C₁₆	—	2	2	no	—
31	#431	—	2	4	no	—
32	#431	—	4	2	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	4	4	no	—
35	C₂×#107	C₂×C₄	1	4	yes	—
36	C₂×#106	—	2	2	no	—
37	C₂×#106	—	2	2	no	—
38	C₂×#106	—	4	2	no	—
39	C₂×#106	—	4	2	no	—
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₂²⋊C₈	—	4	1	no	—
42	C₂×C₄×C₈	—	4	1	no	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	2	yes	—
49	C₂×C₄×C₈	Q₈	1	2	yes	—
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	#382	—	2	2	no	—
52	#382	—	2	2	no	—
53	#382	—	4	1	no	—
54	#382	—	4	1	no	—
55	C₂×#95	—	4	1	no	—
56	C₂×#95	—	4	1	no	—
57	C₄⋊C₁₆	—	2	4	no	—
58	C₄⋊C₁₆	—	2	2	no	—
59	C₄⋊C₁₆	C₂×C₄	1	4	yes	complemented
60	#335	—	2	4	no	—
61	#334	—	2	2	no	—
62	#334	C₂×C₄	1	4	yes	complemented
63	C₂³×C₄	—	4	2	no	—
64	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂³×C₄	D₈	1	1	yes	characteristic
66	C₂³×C₄	SD₁₆	1	1	yes	characteristic
67	C₂²×C₈	—	2	2	no	—
68	C₂²×C₈	—	2	2	no	—
69	C₂²×C₈	—	4	2	no	—
70	C₂²×C₈	C₄²	1	2	yes	—
71	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₈	C₄⋊C₄	1	2	yes	—
75	C₂²×C₈	C₄⋊C₄	1	2	yes	—
76	C₂²×C₈	D₈	1	2	yes	—
77	C₂²×C₈	SD₁₆	1	2	yes	—
78	C₂²×C₈	SD₁₆	1	2	yes	—
79	C₂²×C₈	Q₁₆	1	2	yes	—
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	2	4	no	—
83	C₂×C₄⋊C₄	—	4	4	no	—
84	C₂×C₄⋊C₄	—	4	4	no	—
85	C₂×C₄⋊C₄	—	4	2	no	—
86	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
88	C₂×C₄²	C₄⋊C₄	1	2	yes	—
89	C₂×C₄²	D₈	1	1	yes	characteristic
90	C₂×C₄²	SD₁₆	1	2	yes	—
91	C₂×C₄²	SD₁₆	1	1	yes	characteristic
92	C₂×C₄²	Q₁₆	1	2	yes	—
93	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
94	M₃₂	—	4	8	no	—
95	C₂×C₁₆	—	2	4	no	—
96	C₂×C₁₆	—	4	2	no	—
97	C₂×C₁₆	—	4	2	no	—
98	#107	—	4	4	no	—
99	#107	—	4	4	no	—
100	#106	—	4	4	no	—
101	#106	—	4	2	no	—
102	#106	—	4	2	no	—
103	C₄⋊C₈	—	2	4	no	—
104	C₄⋊C₈	—	4	2	no	—
105	C₂²⋊C₈	—	4	4	no	—
106	C₄×C₈	—	4	4	no	—
107	C₄×C₈	C₄²	1	2	yes	—
108	C₄×C₈	C₂²⋊C₄	1	2	yes	—
109	C₄×C₈	C₂²⋊C₄	1	2	yes	—
110	C₄×C₈	C₄⋊C₄	1	2	yes	—
111	C₄×C₈	C₄⋊C₄	1	2	yes	—
112	C₄×C₈	C₄⋊C₄	1	2	yes	—
113	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₄×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
117	#95	—	4	2	no	—
118	#95	—	4	2	no	—
119	#95	—	4	2	no	—
120	#95	—	4	2	no	—
121	C₂⁴	#102	1	1	yes	characteristic
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	4	4	no	—
125	C₂²×C₄	—	8	4	no	—
126	C₂²×C₄	—	8	4	no	—
127	C₂²×C₄	—	8	4	no	—
128	C₂²×C₄	#95	1	2	yes	—
129	C₂²×C₄	#100	1	2	yes	—
130	C₂²×C₄	#101	1	2	yes	—
131	C₂²×C₄	#102	1	1	yes	characteristic
132	C₂²×C₄	#102	1	1	yes	characteristic
133	C₂²×C₄	#102	1	2	yes	—
134	C₂²×C₄	#102	1	1	yes	characteristic
135	C₂²×C₄	#102	1	2	yes	—
136	C₂²×C₄	#103	1	2	yes	—
137	C₂²×C₄	#103	1	2	yes	—
138	C₂²×C₄	#103	1	2	yes	—
139	C₂²×C₄	#106	1	2	yes	—
140	C₂²×C₄	#107	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	4	4	no	—
151	C₂×C₈	—	8	4	no	—
152	C₂×C₈	#95	1	2	yes	—
153	C₂×C₈	#95	1	2	yes	—
154	C₂×C₈	#102	1	2	yes	—
155	C₂×C₈	#102	1	4	yes	—
156	C₂×C₈	#103	1	2	yes	—
157	C₂×C₈	#103	1	4	yes	—
158	C₂×C₈	#106	1	4	yes	—
159	C₂×C₈	#107	1	4	yes	—
160	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
161	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₈	C₂×D₈	1	1	yes	characteristic
165	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
166	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
167	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	8	4	no	—
171	C₄²	#95	1	2	yes	—
172	C₄²	#102	1	2	yes	—
173	C₄²	#103	1	2	yes	—
174	C₄²	#103	1	2	yes	—
175	C₄²	#106	1	2	yes	—
176	C₄²	#107	1	2	yes	—
177	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₄²	C₂×SD₁₆	1	1	yes	characteristic
180	C₄²	C₂×Q₁₆	1	1	yes	characteristic
181	C₁₆	—	4	4	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	#331	1	2	yes	—
185	C₂³	#332	1	2	yes	—
186	C₂³	#340	1	2	yes	—
187	C₂³	C₂×#102	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	8	4	no	—
197	C₂×C₄	—	8	4	no	—
198	C₂×C₄	—	8	4	no	—
199	C₂×C₄	—	16	8	no	—
200	C₂×C₄	#331	1	2	yes	—
201	C₂×C₄	#332	1	4	yes	—
202	C₂×C₄	#332	1	2	yes	—
203	C₂×C₄	#333	1	4	yes	—
204	C₂×C₄	#340	1	2	yes	—
205	C₂×C₄	#340	1	2	yes	—
206	C₂×C₄	#340	1	2	yes	—
207	C₂×C₄	#340	1	4	yes	—
208	C₂×C₄	#343	1	4	yes	—
209	C₂×C₄	C₂×#95	1	1	yes	characteristic
210	C₂×C₄	C₂×#100	1	1	yes	characteristic
211	C₂×C₄	C₂×#101	1	1	yes	characteristic
212	C₂×C₄	C₂×#102	1	1	yes	characteristic
213	C₂×C₄	C₂×#102	1	1	yes	characteristic
214	C₂×C₄	C₂×#103	1	1	yes	characteristic
215	C₂×C₄	C₂×#103	1	1	yes	characteristic
216	C₂×C₄	C₂×#103	1	1	yes	characteristic
217	C₂×C₄	C₂×#106	1	1	yes	characteristic
218	C₂×C₄	C₂×#107	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	8	4	no	—
223	C₈	#340	1	4	yes	—
224	C₈	C₂×#102	1	1	yes	characteristic
225	C₈	C₂×#103	1	1	yes	characteristic
226	C₈	C₂×#106	1	1	yes	characteristic
227	C₈	C₂×#107	1	1	yes	characteristic
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	#1296	1	4	yes	—
233	C₂²	C₂×#331	1	1	yes	characteristic
234	C₂²	C₂×#332	1	1	yes	characteristic
235	C₂²	C₂×#340	1	1	yes	characteristic
236	C₂²	#2143	1	2	yes	—
237	C₂²	#2145	1	2	yes	—
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	1	no	—
244	C₄	—	16	4	no	—
245	C₄	#1296	1	4	yes	—
246	C₄	C₂×#332	1	1	yes	characteristic
247	C₄	C₂×#333	1	1	yes	characteristic
248	C₄	C₂×#340	1	1	yes	characteristic
249	C₄	C₂×#343	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	C₂×#1296	1	1	yes	characteristic
252	C₂	#11926	1	2	yes	—
253	C₂	#11935	1	2	yes	—
254	C₂	#12561	1	1	yes	characteristic
255	C₂	#12567	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×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
4. #7275	C₂	semidirect
12. #2150	C₄	semidirect
20. #1564	C₄	semidirect
59. C₄⋊C₁₆	C₂×C₄	semidirect
62. #334	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2150	2
C₂×#1846	2
C₂×#1564	2
#7275	2