Subgroups — #60713

Order 256.

939 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3510	C₂	1	2	yes	maximal, complemented
2	#3183	C₂	1	4	yes	maximal, complemented
3	#3175	C₂	1	4	yes	maximal, complemented
4	#3173	C₂	1	4	yes	maximal, complemented
5	#3168	C₂	1	4	yes	maximal, complemented
6	#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#495	C₂	1	1	yes	maximal, characteristic, complemented
8	#3140	C₂	1	2	yes	maximal, complemented
9	#3059	C₂	1	1	yes	maximal, characteristic, complemented
10	#3058	C₂	1	1	yes	maximal, characteristic
11	#3057	C₂	1	2	yes	maximal, complemented
12	#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	#2949	C₂	1	2	yes	maximal, complemented
14	#2948	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#131	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆	C₂²	1	2	yes	—
18	C₂²×D₈	C₂²	1	1	yes	characteristic
19	C₂×#131	C₂²	1	1	yes	characteristic
20	C₂×#131	C₂²	1	1	yes	characteristic
21	#559	—	2	4	no	—
22	#556	—	2	4	no	—
23	#551	—	2	8	no	—
24	#542	C₂²	1	8	yes	complemented
25	#536	C₂²	1	2	yes	—
26	#535	C₂²	1	2	yes	—
27	#533	C₂²	1	2	yes	—
28	#529	C₂²	1	8	yes	complemented
29	C₂×#124	C₂²	1	2	yes	—
30	#519	C₂²	1	2	yes	—
31	#518	C₂²	1	2	yes	—
32	C₄×#41	C₂²	1	1	yes	characteristic
33	#500	C₂²	1	2	yes	complemented
34	#499	C₂²	1	2	yes	complemented
35	#499	C₂²	1	2	yes	complemented
36	#498	C₂²	1	2	yes	complemented
37	#495	C₂²	1	2	yes	—
38	#495	C₂²	1	2	yes	—
39	#495	C₂²	1	2	yes	—
40	#495	C₂²	1	2	yes	—
41	#489	C₂²	1	8	yes	complemented
42	#488	C₂²	1	8	yes	complemented
43	#470	C₂²	1	4	yes	complemented
44	#469	C₂²	1	4	yes	—
45	#468	C₂²	1	8	yes	complemented
46	#467	C₂²	1	4	yes	complemented
47	#466	C₂²	1	4	yes	—
48	#465	C₂²	1	8	yes	complemented
49	#442	C₂²	1	4	yes	—
50	#441	C₂²	1	4	yes	complemented
51	#440	C₂²	1	8	yes	complemented
52	C₄×Q₁₆	C₂²	1	4	yes	complemented
53	C₄×SD₁₆	C₂²	1	8	yes	complemented
54	C₄×D₈	C₂²	1	4	yes	—
55	#427	C₂²	1	1	yes	characteristic
56	#427	C₂²	1	2	yes	—
57	#427	C₂²	1	1	yes	characteristic
58	#419	C₂²	1	2	yes	—
59	#419	C₂²	1	2	yes	—
60	#418	C₂²	1	2	yes	—
61	#418	C₂²	1	2	yes	—
62	#405	C₂²	1	1	yes	characteristic
63	#405	C₂²	1	1	yes	characteristic
64	#405	C₂²	1	2	yes	complemented
65	#405	C₂²	1	2	yes	complemented
66	C₄×M₁₆	C₂²	1	1	yes	characteristic
67	C₄×M₁₆	C₂²	1	2	yes	complemented
68	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂²	1	2	yes	complemented
70	C₂×#41	C₂³	1	1	yes	characteristic
71	C₂²×Q₈	C₂³	1	2	yes	—
72	C₂²×D₄	C₂³	1	2	yes	—
73	C₂×Q₁₆	—	2	4	no	—
74	C₂×Q₁₆	C₂³	1	2	yes	—
75	C₂×Q₁₆	C₂³	1	2	yes	—
76	C₂×SD₁₆	—	2	8	no	—
77	C₂×SD₁₆	C₂³	1	4	yes	—
78	C₂×SD₁₆	C₂³	1	4	yes	—
79	C₂×D₈	—	2	4	no	—
80	C₂×D₈	C₂³	1	2	yes	—
81	C₂×D₈	C₂³	1	2	yes	—
82	#131	—	2	4	no	—
83	#131	—	2	4	no	—
84	C₂×M₁₆	C₂³	1	1	yes	characteristic
85	C₂×M₁₆	C₂³	1	1	yes	characteristic
86	C₂×M₁₆	C₂³	1	2	yes	—
87	C₂×M₁₆	C₂³	1	2	yes	—
88	C₂²×C₈	C₂³	1	2	yes	—
89	C₂²×C₈	C₂³	1	2	yes	—
90	C₂²×C₈	C₂³	1	1	yes	characteristic
91	C₂²×C₈	C₂³	1	1	yes	characteristic
92	C₄⋊Q₈	—	2	2	no	—
93	C₄⋊Q₈	C₂³	1	4	yes	—
94	#127	—	2	2	no	—
95	#126	—	2	8	no	—
96	#126	—	2	8	no	—
97	#125	C₂³	1	4	yes	—
98	#124	—	2	2	no	—
99	#124	—	2	2	no	—
100	#124	—	2	8	no	—
101	#124	—	2	4	no	—
102	#124	C₂³	1	4	yes	—
103	#124	C₂³	1	4	yes	—
104	#123	—	2	4	no	—
105	#123	—	2	8	no	—
106	#123	—	2	8	no	—
107	C₂²⋊Q₈	—	2	4	no	—
108	C₂²⋊Q₈	—	2	8	no	—
109	C₂²⋊Q₈	C₂³	1	8	yes	—
110	#121	C₂³	1	8	yes	—
111	#120	—	2	4	no	—
112	C₄×Q₈	—	2	4	no	—
113	C₄×Q₈	—	2	4	no	—
114	C₄×Q₈	D₄	1	2	yes	—
115	C₄×Q₈	C₂³	1	8	yes	—
116	C₄×D₄	—	2	8	no	—
117	C₄×D₄	—	2	4	no	—
118	C₄×D₄	D₄	1	4	yes	—
119	C₄×D₄	D₄	1	2	yes	—
120	C₄×D₄	C₂³	1	8	yes	—
121	#117	—	2	4	no	—
122	#117	—	2	8	no	—
123	#117	C₂³	1	2	yes	—
124	#117	C₂³	1	2	yes	—
125	#117	C₂³	1	1	yes	characteristic
126	#117	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄	—	2	4	no	—
128	C₂×C₂²⋊C₄	—	2	4	no	—
129	C₂×C₄²	C₂³	1	1	yes	characteristic
130	C₂×C₄²	C₂³	1	2	yes	—
131	#107	C₂³	1	2	yes	—
132	#107	C₂³	1	2	yes	—
133	#107	C₂³	1	4	yes	—
134	#106	C₂³	1	4	yes	—
135	#106	C₂³	1	2	yes	—
136	#106	C₂³	1	2	yes	—
137	#103	C₂³	1	8	yes	—
138	#103	C₂³	1	8	yes	—
139	#102	C₂³	1	8	yes	—
140	#102	C₂³	1	8	yes	—
141	#97	C₂³	1	2	yes	—
142	#97	C₂³	1	2	yes	—
143	#97	C₂³	1	2	yes	—
144	C₄×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₈	C₂³	1	1	yes	characteristic
146	C₄×C₈	C₂³	1	1	yes	characteristic
147	C₄×C₈	C₂³	1	1	yes	characteristic
148	C₄×C₈	C₂³	1	2	yes	—
149	C₄×C₈	C₂³	1	2	yes	—
150	C₄×C₈	C₂³	1	2	yes	—
151	C₂⁴	—	2	2	no	—
152	#41	—	2	2	no	—
153	#41	—	2	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
159	C₂×Q₈	C₂⁴	1	4	yes	—
160	C₂×D₄	—	2	8	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	4	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	C₂×D₄	1	2	yes	—
165	C₂×D₄	C₂×D₄	1	1	yes	characteristic
166	C₂×D₄	C₂⁴	1	4	yes	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	C₂×D₄	1	2	yes	—
170	C₂²×C₄	C₂×D₄	1	4	yes	—
171	C₂²×C₄	C₂⁴	1	2	yes	—
172	C₂²×C₄	C₂⁴	1	1	yes	characteristic
173	Q₁₆	—	2	4	no	—
174	Q₁₆	—	2	4	no	—
175	SD₁₆	—	2	8	no	—
176	SD₁₆	—	2	8	no	—
177	D₈	—	2	4	no	—
178	D₈	—	2	4	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	—	2	2	no	—
181	M₁₆	—	2	4	no	—
182	M₁₆	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	#41	1	4	yes	—
186	C₂×C₈	#41	1	4	yes	—
187	C₂×C₈	C₂⁴	1	2	yes	—
188	C₂×C₈	C₂⁴	1	2	yes	—
189	C₂×C₈	C₂⁴	1	1	yes	characteristic
190	C₂×C₈	C₂⁴	1	1	yes	characteristic
191	C₂×C₈	C₂⁴	1	1	yes	characteristic
192	C₂×C₈	C₂⁴	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	8	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	C₂×D₄	1	2	yes	—
199	C₄⋊C₄	C₂×D₄	1	4	yes	—
200	C₄⋊C₄	C₂⁴	1	4	yes	—
201	C₄⋊C₄	C₂⁴	1	4	yes	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	8	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₂²⋊C₄	—	2	8	no	—
207	C₂²⋊C₄	—	2	8	no	—
208	C₂²⋊C₄	—	2	8	no	—
209	C₂²⋊C₄	C₂×D₄	1	2	yes	—
210	C₂²⋊C₄	C₂×D₄	1	4	yes	—
211	C₄²	—	2	8	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	C₂×D₄	1	4	yes	—
215	C₄²	C₂×D₄	1	2	yes	—
216	C₄²	C₂⁴	1	1	yes	characteristic
217	C₄²	C₂⁴	1	2	yes	—
218	C₄²	C₂⁴	1	1	yes	characteristic
219	C₂³	—	2	4	no	—
220	C₂³	—	4	4	no	—
221	C₂³	—	8	4	no	—
222	C₂³	C₂²×D₄	1	2	yes	—
223	C₂³	C₂²×D₄	1	1	yes	characteristic
224	Q₈	—	2	2	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	2	4	no	—
227	Q₈	—	4	4	no	—
228	Q₈	—	8	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	4	no	—
233	D₄	—	4	4	no	—
234	D₄	—	8	8	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	8	no	—
244	C₂×C₄	C₂²×D₄	1	4	yes	—
245	C₂×C₄	C₂²×D₄	1	2	yes	—
246	C₂×C₄	C₂²×D₄	1	2	yes	—
247	C₂×C₄	C₂²×D₄	1	2	yes	—
248	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄	C₂×#41	1	2	yes	—
250	C₂×C₄	C₂×#41	1	2	yes	—
251	C₂×C₄	C₂⁵	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	C₂×#41	1	2	yes	—
255	C₈	C₂×#41	1	2	yes	—
256	C₈	C₂×#41	1	2	yes	—
257	C₈	C₂×#41	1	2	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	16	4	no	—
264	C₂²	#532	1	2	yes	—
265	C₂²	C₂³×D₄	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	4	no	—
271	C₄	—	4	4	no	—
272	C₄	—	4	4	no	—
273	C₄	#532	1	2	yes	—
274	C₄	C₂²×#41	1	1	yes	characteristic
275	C₄	C₂²×#41	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	4	2	no	—
278	C₂	—	8	4	no	—
279	C₂	C₂×#532	1	1	yes	characteristic
280	C₂	#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	#3058	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3510	C₂	semidirect
2. #3183	C₂	semidirect
3. #3175	C₂	semidirect
4. #3173	C₂	semidirect
5. #3168	C₂	semidirect
6. #3160	C₂	semidirect
7. C₂×#495	C₂	semidirect
8. #3140	C₂	semidirect
9. #3059	C₂	semidirect
11. #3057	C₂	semidirect
12. #2950	C₂	semidirect
13. #2949	C₂	semidirect
14. #2948	C₂	semidirect
15. C₄×#131	C₂	semidirect
24. #542	C₂²	semidirect
28. #529	C₂²	semidirect
33. #500	C₂²	semidirect
34. #499	C₂²	semidirect
35. #499	C₂²	semidirect
36. #498	C₂²	semidirect
41. #489	C₂²	semidirect
42. #488	C₂²	semidirect
43. #470	C₂²	semidirect
45. #468	C₂²	semidirect
46. #467	C₂²	semidirect
48. #465	C₂²	semidirect
50. #441	C₂²	semidirect
51. #440	C₂²	semidirect
52. C₄×Q₁₆	C₂²	semidirect
53. C₄×SD₁₆	C₂²	semidirect
64. #405	C₂²	semidirect
65. #405	C₂²	semidirect
67. C₄×M₁₆	C₂²	semidirect
69. C₂×C₄×C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3510	2
#3183	2
#3175	2
#3173	2
#3168	2
#3160	2
C₂×#495	2
#3140	2
#3059	2
#3058	2
#3057	2
#2950	2
#2949	2
#2948	2
C₄×#131	2