Subgroups — #512436

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 261 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#504	C₂	1	1	yes	maximal, characteristic
3	#33222	C₂	1	2	yes	maximal
4	#13290	C₂	1	2	yes	maximal, complemented
5	#13171	C₂	1	4	yes	maximal, complemented
6	#12909	C₂	1	1	yes	maximal, characteristic, complemented
7	#12842	C₂	1	4	yes	maximal
8	C₂×#504	C₂²	1	2	yes	—
9	C₂×#500	C₂²	1	2	yes	—
10	#3079	—	2	2	no	—
11	C₄×#131	C₂²	1	1	yes	characteristic
12	#2252	—	2	1	no	—
13	#2252	—	2	1	no	—
14	#2252	—	2	2	no	—
15	C₂×#368	C₂²	1	4	yes	—
16	C₂×#366	C₂²	1	4	yes	—
17	#2108	—	2	2	no	—
18	#2108	C₂²	1	2	yes	—
19	C₄×M₃₂	—	2	1	no	—
20	C₄×M₃₂	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	—	2	1	no	—
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1997	—	2	2	no	—
24	#1952	—	2	2	no	—
25	#1952	C₂²	1	4	yes	—
26	#1848	C₂²	1	2	yes	—
27	#1816	—	2	2	no	—
28	#1816	C₂²	1	4	yes	—
29	#1387	C₂²	1	8	yes	—
30	C₂×#131	D₄	1	1	yes	characteristic
31	C₂×#131	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Q₈	—	2	2	no	—
33	C₄×#41	D₄	1	1	yes	characteristic
34	#504	—	2	4	no	—
35	#504	Q₈	1	8	yes	complemented
36	C₂×M₃₂	—	2	2	no	—
37	C₂×M₃₂	C₂³	1	2	yes	—
38	C₂²×C₁₆	—	2	2	no	—
39	C₂²×C₁₆	C₂³	1	2	yes	—
40	#500	—	2	2	no	—
41	#500	—	2	2	no	—
42	#500	—	2	2	no	—
43	#500	—	2	2	no	—
44	#477	—	4	2	no	—
45	#476	—	4	2	no	—
46	#475	—	4	2	no	—
47	#474	—	4	2	no	—
48	C₂×#108	—	2	2	no	—
49	C₂×#108	C₂³	1	4	yes	—
50	#428	—	2	4	no	—
51	C₂×#107	—	2	2	no	—
52	C₂×#107	C₂³	1	4	yes	—
53	#423	—	2	2	no	—
54	#417	—	2	4	no	—
55	#405	—	2	1	no	—
56	#405	—	2	1	no	—
57	#405	D₄	1	1	yes	characteristic
58	#405	C₂³	1	1	yes	characteristic
59	C₄×M₁₆	—	2	1	no	—
60	C₄×M₁₆	D₄	1	1	yes	characteristic
61	C₂×C₄×C₈	—	2	1	no	—
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	#368	—	2	8	no	—
64	#366	—	2	4	no	—
65	#366	—	2	4	no	—
66	#346	—	2	2	no	—
67	#346	—	2	1	no	—
68	C₄×C₁₆	—	2	2	no	—
69	C₄×C₁₆	—	2	1	no	—
70	C₄×C₁₆	D₄	1	1	yes	characteristic
71	C₄×C₁₆	D₄	1	1	yes	characteristic
72	C₄×C₁₆	D₄	1	2	yes	—
73	#343	—	2	4	no	—
74	#343	—	2	4	no	—
75	C₂×#41	C₂×D₄	1	1	yes	characteristic
76	C₂²×Q₈	—	4	2	no	—
77	#131	—	2	2	no	—
78	#131	—	2	2	no	—
79	#131	Q₁₆	1	4	yes	—
80	#131	C₂×Q₈	1	4	yes	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂×M₁₆	—	2	1	no	—
83	C₂×M₁₆	—	2	4	no	—
84	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₈	C₂⁴	1	1	yes	characteristic
90	C₄⋊Q₈	—	4	2	no	—
91	C₄⋊Q₈	—	4	2	no	—
92	C₄⋊Q₈	—	4	2	no	—
93	C₄⋊Q₈	—	4	2	no	—
94	C₄×Q₈	D₈	1	1	yes	characteristic
95	C₄×Q₈	D₈	1	1	yes	characteristic
96	C₄×D₄	—	2	1	no	—
97	C₄×D₄	—	2	1	no	—
98	C₄×D₄	D₈	1	1	yes	characteristic
99	C₄×D₄	D₈	1	1	yes	characteristic
100	#117	—	2	1	no	—
101	#117	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	2	4	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄²	—	2	1	no	—
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	8	no	—
108	M₃₂	C₂×Q₈	1	4	yes	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	8	no	—
111	C₂×C₁₆	C₂×D₄	1	2	yes	—
112	C₂×C₁₆	C₂×D₄	1	2	yes	—
113	C₂×C₁₆	C₂×Q₈	1	4	yes	—
114	C₂×C₁₆	#41	1	4	yes	—
115	#108	—	2	8	no	—
116	#108	—	4	4	no	—
117	#107	—	2	4	no	—
118	#107	—	2	4	no	—
119	#107	—	4	2	no	—
120	#107	—	4	2	no	—
121	#107	—	4	4	no	—
122	#106	—	4	4	no	—
123	C₄⋊C₈	—	4	2	no	—
124	C₄⋊C₈	—	4	2	no	—
125	#103	—	4	4	no	—
126	#102	—	4	4	no	—
127	#97	—	2	1	no	—
128	#97	—	2	1	no	—
129	#97	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
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	#41	—	2	1	no	—
138	#41	—	2	1	no	—
139	#41	C₄⋊Q₈	1	2	yes	—
140	#41	C₂×Q₁₆	1	2	yes	—
141	C₂×Q₈	—	8	2	no	—
142	C₂×Q₈	—	8	2	no	—
143	C₂×Q₈	—	16	4	no	—
144	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	C₂×D₈	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	#135	1	1	yes	characteristic
152	C₂²×C₄	#135	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
154	M₁₆	—	2	1	no	—
155	M₁₆	—	2	1	no	—
156	M₁₆	—	2	4	no	—
157	M₁₆	—	2	4	no	—
158	M₁₆	—	4	4	no	—
159	M₁₆	—	4	4	no	—
160	M₁₆	C₄⋊Q₈	1	2	yes	—
161	M₁₆	C₂×Q₁₆	1	2	yes	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	4	4	no	—
167	C₂×C₈	#121	1	1	yes	characteristic
168	C₂×C₈	#121	1	1	yes	characteristic
169	C₂×C₈	C₂×Q₁₆	1	2	yes	—
170	C₂×C₈	#135	1	2	yes	—
171	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₈	C₂²×Q₈	1	2	yes	—
173	C₂×C₈	C₂×#41	1	2	yes	—
174	C₂×C₈	C₂×#41	1	1	yes	characteristic
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	8	2	no	—
180	C₄⋊C₄	—	8	2	no	—
181	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
182	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	#121	1	1	yes	characteristic
190	C₄²	#121	1	1	yes	characteristic
191	C₄²	C₂×D₈	1	1	yes	characteristic
192	C₄²	C₂×D₈	1	1	yes	characteristic
193	C₁₆	—	2	4	no	—
194	C₁₆	C₂²⋊Q₈	1	4	yes	—
195	C₁₆	C₂²⋊Q₈	1	4	yes	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂×#135	1	1	yes	characteristic
198	Q₈	—	2	1	no	—
199	Q₈	—	16	4	no	—
200	Q₈	—	16	4	no	—
201	Q₈	#500	1	2	yes	—
202	D₄	—	2	2	no	—
203	D₄	—	2	2	no	—
204	D₄	—	2	1	no	—
205	D₄	#500	1	2	yes	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	—	16	4	no	—
217	C₂×C₄	#466	1	1	yes	characteristic
218	C₂×C₄	#466	1	1	yes	characteristic
219	C₂×C₄	#466	1	1	yes	characteristic
220	C₂×C₄	#466	1	1	yes	characteristic
221	C₂×C₄	C₂×#121	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
223	C₂×C₄	#532	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
225	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
226	C₂×C₄	C₂×#135	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	8	4	no	—
230	C₈	#467	1	2	yes	—
231	C₈	#467	1	2	yes	—
232	C₈	C₂×C₂²⋊Q₈	1	2	yes	—
233	C₈	#533	1	2	yes	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	C₂×#466	1	1	yes	characteristic
240	C₂²	#3160	1	1	yes	characteristic
241	C₂²	#3170	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	—	16	4	no	—
248	C₄	#2221	1	2	yes	—
249	C₄	#2222	1	2	yes	—
250	C₄	#2857	1	1	yes	characteristic
251	C₄	C₂×#467	1	1	yes	characteristic
252	C₄	C₂×#500	1	1	yes	characteristic
253	C₄	#3169	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	#33209	1	1	yes	characteristic
258	C₂	C₂×#2221	1	1	yes	characteristic
259	C₂	C₂×#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ 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. C₂×#2252	C₂	semidirect
4. #13290	C₂	semidirect
5. #13171	C₂	semidirect
6. #12909	C₂	semidirect
35. #504	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2252	2
C₄×#504	2
#33222	2
#13290	2
#13171	2
#12909	2
#12842	2