Subgroups — #495318

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 271 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33451	C₂	1	1	yes	maximal, characteristic
2	#21294	C₂	1	2	yes	maximal, complemented
3	#13242	C₂	1	1	yes	maximal, characteristic
4	#13241	C₂	1	1	yes	maximal, characteristic
5	#13170	C₂	1	1	yes	maximal, characteristic, complemented
6	#13169	C₂	1	1	yes	maximal, characteristic, complemented
7	#13152	C₂	1	2	yes	maximal, complemented
8	#13151	C₂	1	2	yes	maximal, complemented
9	#12676	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#359	C₂	1	2	yes	maximal, complemented
11	#12539	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#508	C₂²	1	1	yes	characteristic
13	C₄×#135	C₂²	1	2	yes	—
14	#2923	—	2	2	no	—
15	#2177	C₄	1	2	yes	complemented
16	#2176	C₄	1	2	yes	complemented
17	#2175	C₄	1	4	yes	complemented
18	C₂×#368	C₂²	1	1	yes	characteristic
19	C₂×#368	C₂²	1	1	yes	characteristic
20	#2158	C₂²	1	1	yes	characteristic
21	C₂×#359	C₂²	1	2	yes	—
22	C₂×#359	C₂²	1	2	yes	—
23	#2138	C₂²	1	2	yes	—
24	C₂×#346	C₂²	1	1	yes	characteristic
25	C₂×#346	C₂²	1	1	yes	characteristic
26	C₂×#346	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
30	#1974	C₂²	1	2	yes	—
31	#1974	C₂²	1	2	yes	—
32	#1945	C₂²	1	2	yes	—
33	#1945	C₂²	1	2	yes	—
34	#1938	—	2	2	no	—
35	#1938	—	2	2	no	—
36	C₄×#108	C₂²	1	2	yes	—
37	#1759	—	2	2	no	—
38	#1726	C₂²	1	1	yes	characteristic
39	#1382	C₂²	1	2	yes	—
40	#1382	C₂²	1	2	yes	—
41	#1381	C₂²	1	2	yes	—
42	C₂×#135	C₂³	1	2	yes	—
43	C₄×#41	—	2	2	no	—
44	#508	—	2	2	no	—
45	#508	—	2	2	no	—
46	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₃₂	C₂×C₄	1	2	yes	—
48	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₁₆	C₂³	1	1	yes	characteristic
50	C₂²×C₁₆	C₂³	1	1	yes	characteristic
51	C₂²×C₁₆	C₂³	1	1	yes	characteristic
52	C₂²×C₁₆	C₂³	1	1	yes	characteristic
53	C₄×Q₁₆	C₂×C₄	1	4	yes	—
54	C₄×SD₁₆	—	2	4	no	—
55	C₄×D₈	C₂×C₄	1	4	yes	—
56	#433	—	4	4	no	—
57	#432	—	4	4	no	—
58	C₂×#108	C₂³	1	2	yes	—
59	C₂×#108	C₂³	1	2	yes	—
60	#427	C₂³	1	2	yes	—
61	#424	—	2	2	no	—
62	#424	—	2	2	no	—
63	C₂×#104	—	2	2	no	—
64	C₂×#104	—	2	2	no	—
65	#416	—	2	2	no	—
66	#408	—	2	2	no	—
67	#408	—	2	2	no	—
68	C₄×M₁₆	—	2	2	no	—
69	C₂×#97	D₄	1	1	yes	characteristic
70	C₂×#97	D₄	1	1	yes	characteristic
71	C₂×#97	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	#376	D₄	1	1	yes	characteristic
76	#368	—	2	2	no	—
77	#368	—	2	2	no	—
78	#367	C₂×C₄	1	2	yes	—
79	#366	C₂×C₄	1	2	yes	—
80	#359	—	2	4	no	—
81	#359	—	2	4	no	—
82	#358	C₂×C₄	1	4	yes	—
83	#357	C₂×C₄	1	4	yes	—
84	#346	—	2	2	no	—
85	#346	—	2	2	no	—
86	#346	C₂×C₄	1	2	yes	—
87	#346	C₂×C₄	1	2	yes	—
88	C₄×C₁₆	—	2	2	no	—
89	C₄×C₁₆	—	2	2	no	—
90	C₄×C₁₆	—	2	2	no	—
91	#341	—	2	2	no	—
92	#339	—	2	2	no	—
93	#339	—	2	2	no	—
94	C₂×#41	—	2	2	no	—
95	#135	—	2	2	no	—
96	#135	—	2	2	no	—
97	#135	—	2	2	no	—
98	#135	—	2	2	no	—
99	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
100	C₂×SD₁₆	—	2	2	no	—
101	C₂×D₈	C₂²×C₄	1	2	yes	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	2	2	no	—
104	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₈	C₂⁴	1	1	yes	characteristic
108	C₄×Q₈	—	2	4	no	—
109	C₄×D₄	—	2	4	no	—
110	C₄×D₄	—	4	4	no	—
111	#117	—	2	2	no	—
112	#117	—	4	2	no	—
113	C₂×C₄²	—	4	2	no	—
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
117	Q₃₂	—	2	2	no	—
118	SD₃₂	—	2	4	no	—
119	D₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	C₂²×C₄	1	2	yes	—
127	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₁₆	#41	1	2	yes	—
130	C₂×C₁₆	#41	1	2	yes	—
131	C₂×C₁₆	#41	1	2	yes	—
132	C₂×C₁₆	#41	1	2	yes	—
133	#108	—	2	2	no	—
134	#108	—	2	2	no	—
135	#108	—	2	2	no	—
136	#108	—	2	2	no	—
137	#107	C₂²×C₄	1	4	yes	—
138	#106	—	2	2	no	—
139	C₄⋊C₈	—	4	4	no	—
140	C₄⋊C₈	—	4	4	no	—
141	#104	—	4	4	no	—
142	#104	—	4	4	no	—
143	#103	—	2	4	no	—
144	#102	—	2	4	no	—
145	C₂²⋊C₈	—	4	4	no	—
146	C₂²⋊C₈	—	4	4	no	—
147	#97	—	2	2	no	—
148	#97	—	2	2	no	—
149	#97	—	4	2	no	—
150	#97	D₈	1	4	yes	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	4	2	no	—
156	C₄×C₈	C₂²×C₄	1	2	yes	—
157	C₄×C₈	#41	1	2	yes	—
158	#95	—	2	2	no	—
159	#41	—	4	2	no	—
160	#41	—	4	2	no	—
161	#41	—	4	2	no	—
162	#41	—	4	2	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	#135	1	2	yes	—
171	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
172	Q₁₆	—	2	2	no	—
173	Q₁₆	—	2	2	no	—
174	SD₁₆	—	4	2	no	—
175	SD₁₆	—	4	2	no	—
176	D₈	—	2	2	no	—
177	D₈	—	2	2	no	—
178	M₁₆	—	4	2	no	—
179	M₁₆	—	4	2	no	—
180	M₁₆	—	4	2	no	—
181	M₁₆	—	4	2	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	C₄×D₄	1	2	yes	—
191	C₂×C₈	C₂×D₈	1	2	yes	—
192	C₂×C₈	C₂×D₈	1	2	yes	—
193	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₈	C₂×#41	1	1	yes	characteristic
195	C₂×C₈	C₂×#41	1	1	yes	characteristic
196	C₂×C₈	C₂×#41	1	1	yes	characteristic
197	C₂×C₈	C₂×#41	1	1	yes	characteristic
198	C₂×C₈	C₂×#41	1	1	yes	characteristic
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	4	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	4	2	no	—
206	C₄²	—	4	4	no	—
207	C₄²	C₄×D₄	1	2	yes	—
208	C₄²	C₂×D₈	1	2	yes	—
209	C₄²	#135	1	2	yes	—
210	C₄²	#135	1	2	yes	—
211	C₁₆	—	2	2	no	—
212	C₁₆	—	2	2	no	—
213	C₁₆	—	2	2	no	—
214	C₁₆	#117	1	2	yes	—
215	C₁₆	#117	1	2	yes	—
216	C₂³	—	4	2	no	—
217	C₂³	C₂×#135	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	16	4	no	—
234	C₂×C₄	C₄×D₈	1	4	yes	—
235	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂×C₄	#532	1	1	yes	characteristic
237	C₂×C₄	#532	1	1	yes	characteristic
238	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
239	C₂×C₄	C₂×#135	1	1	yes	characteristic
240	C₂×C₄	C₂×#135	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	8	2	no	—
245	C₈	—	8	2	no	—
246	C₈	C₂×#117	1	1	yes	characteristic
247	C₈	C₄×#41	1	1	yes	characteristic
248	C₈	#529	1	1	yes	characteristic
249	C₈	#529	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	4	no	—
253	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
254	C₂²	#3168	1	1	yes	characteristic
255	C₂²	#3168	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	4	no	—
261	C₄	#2293	1	1	yes	characteristic
262	C₄	C₄×#135	1	1	yes	characteristic
263	C₄	#3169	1	1	yes	characteristic
264	C₄	#3169	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	16	2	no	—
268	C₂	#21275	1	1	yes	characteristic
269	C₂	#33464	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #21294	C₂	semidirect
5. #13170	C₂	semidirect
6. #13169	C₂	semidirect
7. #13152	C₂	semidirect
8. #13151	C₂	semidirect
9. #12676	C₂	semidirect
10. C₄×#359	C₂	semidirect
11. #12539	C₂	semidirect
15. #2177	C₄	semidirect
16. #2176	C₄	semidirect
17. #2175	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33451	2
#21294	2
#13242	2
#13241	2
#13170	2
#13169	2
#13152	2
#13151	2
#12676	2
C₄×#359	2
#12539	2