Subgroups — #497122

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33443	C₂	1	2	yes	maximal, complemented
2	#21707	C₂	1	1	yes	maximal, characteristic
3	#13197	C₂	1	2	yes	maximal, complemented
4	#12988	C₂	1	4	yes	maximal, complemented
5	#12986	C₂	1	4	yes	maximal, complemented
6	#12881	C₂	1	1	yes	maximal, characteristic, complemented
7	#12593	C₂	1	1	yes	maximal, characteristic, complemented
8	#3407	C₂²	1	4	yes	—
9	#3147	C₄	1	1	yes	characteristic
10	C₂×#444	C₂²	1	2	yes	—
11	C₂×#443	C₂²	1	2	yes	—
12	#2932	C₂²	1	2	yes	—
13	#2171	—	2	4	no	—
14	#2170	—	2	4	no	—
15	C₂×#364	C₂²	1	2	yes	—
16	#2117	C₂²	1	4	yes	—
17	#2117	C₂²	1	4	yes	—
18	#2115	C₂²	1	4	yes	—
19	C₄×M₃₂	C₂²	1	1	yes	characteristic
20	C₄×M₃₂	C₂²	1	1	yes	characteristic
21	#1975	C₄	1	2	yes	—
22	#1974	C₄	1	2	yes	—
23	#1963	C₄	1	2	yes	—
24	#1848	C₄	1	1	yes	characteristic
25	#1726	C₂²	1	1	yes	characteristic
26	#1377	C₂²	1	2	yes	—
27	#1375	C₂²	1	2	yes	—
28	#1375	C₂²	1	2	yes	—
29	#1375	C₂²	1	2	yes	—
30	C₂×#135	C₂×C₄	1	2	yes	—
31	C₂×#135	C₂×C₄	1	2	yes	—
32	C₂×#131	C₂³	1	4	yes	—
33	#532	C₂×C₄	1	2	yes	—
34	#504	—	2	8	no	—
35	#504	—	4	4	no	—
36	C₂×M₃₂	—	4	4	no	—
37	C₂×M₃₂	C₂³	1	2	yes	—
38	C₂×M₃₂	C₂³	1	4	yes	—
39	C₂×M₃₂	C₂³	1	2	yes	—
40	#495	C₈	1	2	yes	—
41	#495	C₈	1	2	yes	—
42	#494	C₈	1	1	yes	characteristic
43	#493	C₈	1	1	yes	characteristic
44	#492	C₈	1	2	yes	—
45	#444	—	4	4	no	—
46	#443	—	2	2	no	—
47	#443	—	2	2	no	—
48	#443	—	2	2	no	—
49	#443	—	2	2	no	—
50	#435	—	4	4	no	—
51	C₂×#108	C₂×C₄	1	2	yes	—
52	C₂×#108	C₂×C₄	1	2	yes	—
53	#423	C₂×C₄	1	2	yes	—
54	C₂×#104	C₂×C₄	1	4	yes	—
55	C₂×#104	C₂×C₄	1	4	yes	—
56	#408	C₂×C₄	1	4	yes	—
57	C₂×#97	C₂×C₄	1	1	yes	characteristic
58	C₂×#97	C₂³	1	1	yes	characteristic
59	C₂×#97	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	#376	C₂×C₄	1	1	yes	characteristic
64	#364	—	2	2	no	—
65	#364	—	2	2	no	—
66	#350	—	2	8	no	—
67	#350	—	4	4	no	—
68	#349	—	4	4	no	—
69	#346	—	2	1	no	—
70	#346	D₄	1	2	yes	complemented
71	C₄×C₁₆	—	2	1	no	—
72	C₄×C₁₆	D₄	1	2	yes	complemented
73	C₂×#41	C₂²×C₄	1	4	yes	—
74	#135	—	2	2	no	—
75	#135	—	2	2	no	—
76	#135	—	2	2	no	—
77	#135	—	2	2	no	—
78	#135	—	4	2	no	—
79	#135	—	4	2	no	—
80	C₂×Q₁₆	C₂×C₈	1	2	yes	—
81	C₂×Q₁₆	C₂×C₈	1	2	yes	—
82	C₂×SD₁₆	C₂×C₈	1	4	yes	—
83	C₂×SD₁₆	C₂×C₈	1	4	yes	—
84	C₂×D₈	C₂×C₈	1	2	yes	—
85	C₂×D₈	C₂×C₈	1	2	yes	—
86	#131	—	2	4	no	—
87	#131	—	2	4	no	—
88	#131	—	2	4	no	—
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	4	no	—
91	C₂×M₁₆	C₂²×C₄	1	4	yes	—
92	C₂²×C₈	—	2	4	no	—
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂⁴	1	1	yes	characteristic
97	C₄⋊Q₈	C₂×C₈	1	2	yes	—
98	#127	C₂×C₈	1	2	yes	—
99	#124	C₂×C₈	1	4	yes	—
100	#121	—	4	2	no	—
101	C₄×D₄	—	4	2	no	—
102	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
105	M₃₂	—	2	2	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	4	no	—
108	M₃₂	—	2	2	no	—
109	M₃₂	—	2	4	no	—
110	M₃₂	—	2	2	no	—
111	M₃₂	—	4	8	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	—	4	8	no	—
115	C₂×C₁₆	C₂×D₄	1	4	yes	—
116	#108	—	2	2	no	—
117	#108	—	2	2	no	—
118	#108	—	4	2	no	—
119	C₄⋊C₈	—	4	2	no	—
120	#104	—	2	4	no	—
121	#104	—	2	4	no	—
122	#104	—	4	4	no	—
123	C₂²⋊C₈	—	4	4	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	#97	—	4	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	C₂×C₈	1	1	yes	characteristic
130	C₄×C₈	C₂×C₈	1	1	yes	characteristic
131	C₄×C₈	C₂×C₈	1	1	yes	characteristic
132	C₄×C₈	C₂×C₈	1	1	yes	characteristic
133	C₄×C₈	C₂×D₄	1	1	yes	characteristic
134	C₄×C₈	C₂×D₄	1	1	yes	characteristic
135	C₄×C₈	#41	1	1	yes	characteristic
136	C₄×C₈	#41	1	1	yes	characteristic
137	#95	—	4	1	no	—
138	#41	—	2	4	no	—
139	#41	—	2	4	no	—
140	#41	—	2	4	no	—
141	#41	—	2	4	no	—
142	C₂×Q₈	C₂²×C₈	1	4	yes	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	C₂²×C₈	1	4	yes	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
151	Q₁₆	—	2	2	no	—
152	Q₁₆	—	2	2	no	—
153	Q₁₆	—	4	2	no	—
154	SD₁₆	—	2	4	no	—
155	SD₁₆	—	2	4	no	—
156	SD₁₆	—	4	4	no	—
157	D₈	—	2	2	no	—
158	D₈	—	2	2	no	—
159	D₈	—	4	2	no	—
160	M₁₆	—	2	4	no	—
161	M₁₆	—	2	4	no	—
162	M₁₆	—	4	4	no	—
163	M₁₆	—	4	4	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	C₄×D₄	1	1	yes	characteristic
173	C₂×C₈	C₄×D₄	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
175	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
178	C₂×C₈	#131	1	1	yes	characteristic
179	C₂×C₈	#131	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₈	C₂×#41	1	1	yes	characteristic
182	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
183	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
184	C₄⋊C₄	—	4	2	no	—
185	C₂²⋊C₄	—	4	4	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	C₄×D₄	1	1	yes	characteristic
191	C₄²	C₄×D₄	1	1	yes	characteristic
192	C₄²	C₂²×C₈	1	1	yes	characteristic
193	C₄²	C₂²×C₈	1	1	yes	characteristic
194	C₁₆	—	2	4	no	—
195	C₁₆	—	4	4	no	—
196	C₁₆	—	4	2	no	—
197	C₂³	—	2	4	no	—
198	C₂³	#568	1	1	yes	characteristic
199	Q₈	—	2	4	no	—
200	Q₈	—	2	4	no	—
201	D₄	—	2	4	no	—
202	D₄	—	2	4	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	4	no	—
205	D₄	—	16	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×C₄	#518	1	1	yes	characteristic
219	C₂×C₄	#518	1	1	yes	characteristic
220	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
221	C₂×C₄	C₂×#131	1	1	yes	characteristic
222	C₂×C₄	#568	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	4	no	—
226	C₈	—	4	1	no	—
227	C₈	C₈×D₄	1	1	yes	characteristic
228	C₈	C₈×D₄	1	1	yes	characteristic
229	C₈	C₈×D₄	1	1	yes	characteristic
230	C₈	C₈×D₄	1	1	yes	characteristic
231	C₈	#535	1	2	yes	—
232	C₈	#540	1	2	yes	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	#2932	1	1	yes	characteristic
237	C₂²	#2972	1	1	yes	characteristic
238	C₂²	#2981	1	1	yes	characteristic
239	C₄	—	4	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	4	no	—
244	C₄	—	8	1	no	—
245	C₄	#2328	1	1	yes	characteristic
246	C₄	C₂×C₈×D₄	1	1	yes	characteristic
247	C₄	#2972	1	1	yes	characteristic
248	C₄	#2981	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	8	4	no	—
253	C₂	#22260	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2328
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22260
prime radical	#3147	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33443	C₂	semidirect
3. #13197	C₂	semidirect
4. #12988	C₂	semidirect
5. #12986	C₂	semidirect
6. #12881	C₂	semidirect
7. #12593	C₂	semidirect
70. #346	D₄	semidirect
72. C₄×C₁₆	D₄	semidirect

Maximal subgroups

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

subgroup	index
#33443	2
#21707	2
#13197	2
#12988	2
#12986	2
#12881	2
#12593	2