Subgroups — #60682

Order 256.

939 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3455	C₂	1	2	yes	maximal, complemented
2	#3177	C₂	1	2	yes	maximal, complemented
3	#3176	C₂	1	2	yes	maximal, complemented
4	#3175	C₂	1	4	yes	maximal, complemented
5	#3174	C₂	1	4	yes	maximal, complemented
6	#3173	C₂	1	4	yes	maximal, complemented
7	#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	#3135	C₂	1	2	yes	maximal, complemented
10	#3054	C₂	1	2	yes	maximal, complemented
11	#3050	C₂	1	2	yes	maximal, complemented
12	#2943	C₂	1	2	yes	maximal, complemented
13	C₄×#135	C₂	1	2	yes	maximal, complemented
14	C₂×#405	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#135	C₂²	1	2	yes	—
16	C₂×#135	C₂²	1	2	yes	—
17	C₂²×M₁₆	C₂²	1	1	yes	characteristic
18	C₂³×C₈	C₂²	1	1	yes	characteristic
19	#542	C₂²	1	8	yes	—
20	#541	C₂²	1	4	yes	—
21	#540	C₂²	1	4	yes	—
22	#539	—	2	4	no	—
23	#538	—	2	4	no	—
24	#533	C₂²	1	2	yes	—
25	#532	C₂²	1	2	yes	—
26	#529	—	2	8	no	—
27	#525	C₂²	1	4	yes	—
28	C₄×#41	C₂²	1	4	yes	—
29	C₂×#117	C₂²	1	1	yes	characteristic
30	#500	C₂²	1	2	yes	complemented
31	#499	C₂²	1	2	yes	complemented
32	#499	C₂²	1	2	yes	complemented
33	#498	C₂²	1	2	yes	complemented
34	#495	C₂²	1	2	yes	complemented
35	#495	C₂²	1	2	yes	complemented
36	#494	C₂²	1	1	yes	characteristic, complemented
37	#493	C₂²	1	1	yes	characteristic, complemented
38	#492	C₂²	1	2	yes	complemented
39	#491	C₂²	1	4	yes	complemented
40	#490	C₂²	1	4	yes	complemented
41	#489	C₂²	1	8	yes	complemented
42	#470	C₂²	1	4	yes	—
43	#469	C₂²	1	4	yes	—
44	#468	C₂²	1	8	yes	—
45	#467	C₂²	1	4	yes	—
46	#466	C₂²	1	4	yes	—
47	#465	C₂²	1	8	yes	—
48	#442	C₂²	1	4	yes	complemented
49	#441	C₂²	1	4	yes	complemented
50	#440	C₂²	1	8	yes	complemented
51	C₄×Q₁₆	C₂²	1	4	yes	complemented
52	C₄×SD₁₆	C₂²	1	8	yes	complemented
53	C₄×D₈	C₂²	1	4	yes	complemented
54	#427	C₂²	1	2	yes	—
55	#427	C₂²	1	2	yes	—
56	#416	C₂²	1	4	yes	—
57	#416	C₂²	1	4	yes	—
58	#405	C₂²	1	2	yes	—
59	#405	C₂²	1	2	yes	—
60	#405	C₂²	1	2	yes	—
61	#405	C₂²	1	2	yes	—
62	C₂×#97	C₂²	1	1	yes	characteristic
63	C₂×#97	C₂²	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
66	C₂×#41	C₂³	1	4	yes	—
67	C₂³×C₄	C₂³	1	1	yes	characteristic
68	#135	—	2	8	no	—
69	#135	—	2	8	no	—
70	C₂×Q₁₆	C₂³	1	2	yes	—
71	C₂×Q₁₆	C₂³	1	2	yes	—
72	C₂×SD₁₆	C₂³	1	4	yes	—
73	C₂×SD₁₆	C₂³	1	4	yes	—
74	C₂×D₈	C₂³	1	2	yes	—
75	C₂×D₈	C₂³	1	2	yes	—
76	C₂×M₁₆	—	2	4	no	—
77	C₂×M₁₆	C₂³	1	2	yes	—
78	C₂×M₁₆	C₂³	1	2	yes	—
79	C₂²×C₈	—	2	4	no	—
80	C₂²×C₈	C₂³	1	1	yes	characteristic
81	C₂²×C₈	C₂³	1	2	yes	—
82	C₂²×C₈	C₂³	1	2	yes	—
83	C₂²×C₈	C₂³	1	1	yes	characteristic
84	C₂²×C₈	C₂³	1	1	yes	characteristic
85	C₂²×C₈	C₂³	1	1	yes	characteristic
86	C₄⋊Q₈	C₂³	1	2	yes	—
87	C₄⋊Q₈	C₂³	1	4	yes	—
88	#127	C₂³	1	2	yes	—
89	#126	—	2	8	no	—
90	#126	—	2	8	no	—
91	#125	—	2	8	no	—
92	#125	C₂³	1	4	yes	—
93	#124	—	2	8	no	—
94	#124	C₂³	1	4	yes	—
95	#123	—	2	4	no	—
96	#123	—	2	8	no	—
97	#123	—	2	8	no	—
98	C₂²⋊Q₈	—	2	4	no	—
99	C₂²⋊Q₈	—	2	8	no	—
100	C₂²⋊Q₈	C₂³	1	8	yes	—
101	#121	—	2	4	no	—
102	#121	—	2	8	no	—
103	#121	C₂³	1	8	yes	—
104	#120	—	2	4	no	—
105	C₄×Q₈	—	2	4	no	—
106	C₄×Q₈	C₂³	1	8	yes	—
107	C₄×D₄	—	2	8	no	—
108	C₄×D₄	—	2	8	no	—
109	C₄×D₄	—	2	4	no	—
110	C₄×D₄	C₂³	1	8	yes	—
111	#117	—	2	4	no	—
112	#117	D₄	1	4	yes	—
113	#117	C₂³	1	4	yes	—
114	#117	C₂³	1	2	yes	—
115	#117	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄	D₄	1	2	yes	—
117	C₂×C₂²⋊C₄	D₄	1	2	yes	—
118	C₂×C₄²	—	2	4	no	—
119	C₂×C₄²	C₂³	1	1	yes	characteristic
120	C₂×C₄²	C₂³	1	1	yes	characteristic
121	#107	C₂³	1	4	yes	—
122	#107	C₂³	1	4	yes	—
123	#106	C₂³	1	4	yes	—
124	#106	C₂³	1	4	yes	—
125	#103	C₂³	1	8	yes	—
126	#103	C₂³	1	8	yes	—
127	#102	C₂³	1	8	yes	—
128	#102	C₂³	1	8	yes	—
129	#97	C₂³	1	2	yes	—
130	#97	C₂³	1	2	yes	—
131	#97	C₂³	1	2	yes	—
132	#97	C₂³	1	2	yes	—
133	C₄×C₈	C₂³	1	2	yes	—
134	C₄×C₈	C₂³	1	2	yes	—
135	C₄×C₈	C₂³	1	1	yes	characteristic
136	C₄×C₈	C₂³	1	1	yes	characteristic
137	C₄×C₈	C₂³	1	1	yes	characteristic
138	C₄×C₈	C₂³	1	1	yes	characteristic
139	C₂⁴	C₂×D₄	1	1	yes	characteristic
140	#41	—	2	8	no	—
141	#41	—	2	8	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	C₂⁴	1	4	yes	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	8	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	C₂⁴	1	4	yes	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₄	1	2	yes	—
154	C₂²×C₄	C₂×D₄	1	2	yes	—
155	C₂²×C₄	C₂×D₄	1	2	yes	—
156	C₂²×C₄	C₂⁴	1	2	yes	—
157	C₂²×C₄	C₂⁴	1	1	yes	characteristic
158	Q₁₆	—	2	4	no	—
159	Q₁₆	—	2	4	no	—
160	SD₁₆	—	2	8	no	—
161	SD₁₆	—	2	8	no	—
162	D₈	—	2	4	no	—
163	D₈	—	2	4	no	—
164	M₁₆	—	2	4	no	—
165	M₁₆	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	#41	1	4	yes	—
171	C₂×C₈	#41	1	4	yes	—
172	C₂×C₈	C₂⁴	1	2	yes	—
173	C₂×C₈	C₂⁴	1	2	yes	—
174	C₂×C₈	C₂⁴	1	1	yes	characteristic
175	C₂×C₈	C₂⁴	1	1	yes	characteristic
176	C₂×C₈	C₂⁴	1	1	yes	characteristic
177	C₂×C₈	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄	—	2	8	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	8	no	—
181	C₄⋊C₄	—	2	8	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	C₂×D₄	1	4	yes	—
184	C₄⋊C₄	C₂×D₄	1	4	yes	—
185	C₄⋊C₄	C₂⁴	1	8	yes	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	8	no	—
188	C₂²⋊C₄	—	2	8	no	—
189	C₂²⋊C₄	—	2	8	no	—
190	C₂²⋊C₄	—	2	8	no	—
191	C₂²⋊C₄	C₂×D₄	1	4	yes	—
192	C₂²⋊C₄	C₂×D₄	1	4	yes	—
193	C₄²	—	2	8	no	—
194	C₄²	—	2	4	no	—
195	C₄²	C₂×D₄	1	2	yes	—
196	C₄²	C₂×D₄	1	2	yes	—
197	C₄²	C₂⁴	1	2	yes	—
198	C₄²	C₂⁴	1	1	yes	characteristic
199	C₄²	C₂⁴	1	1	yes	characteristic
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	—	4	2	no	—
203	C₂³	C₂²×D₄	1	2	yes	—
204	C₂³	C₂²×D₄	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	2	4	no	—
207	Q₈	—	8	4	no	—
208	D₄	—	2	4	no	—
209	D₄	—	2	4	no	—
210	D₄	—	4	8	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	8	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	8	8	no	—
223	C₂×C₄	C₂²×D₄	1	4	yes	—
224	C₂×C₄	C₂²×D₄	1	2	yes	—
225	C₂×C₄	C₂²×D₄	1	2	yes	—
226	C₂×C₄	C₂²×D₄	1	2	yes	—
227	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄	C₂×#41	1	2	yes	—
229	C₂×C₄	C₂×#41	1	2	yes	—
230	C₂×C₄	C₂⁵	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	2	no	—
233	C₈	C₂×#41	1	2	yes	—
234	C₈	C₂×#41	1	2	yes	—
235	C₈	C₂×#41	1	2	yes	—
236	C₈	C₂×#41	1	2	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	4	8	no	—
242	C₂²	—	4	4	no	—
243	C₂²	#532	1	2	yes	—
244	C₂²	C₂³×D₄	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	8	no	—
250	C₄	—	4	4	no	—
251	C₄	#532	1	2	yes	—
252	C₄	C₂²×#41	1	1	yes	characteristic
253	C₄	C₂²×#41	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	8	4	no	—
257	C₂	C₂×#532	1	1	yes	characteristic
258	C₂	#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂²	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3455	C₂	semidirect
2. #3177	C₂	semidirect
3. #3176	C₂	semidirect
4. #3175	C₂	semidirect
5. #3174	C₂	semidirect
6. #3173	C₂	semidirect
7. #3160	C₂	semidirect
8. #3147	C₂	semidirect
9. #3135	C₂	semidirect
10. #3054	C₂	semidirect
11. #3050	C₂	semidirect
12. #2943	C₂	semidirect
13. C₄×#135	C₂	semidirect
14. C₂×#405	C₂	semidirect
30. #500	C₂²	semidirect
31. #499	C₂²	semidirect
32. #499	C₂²	semidirect
33. #498	C₂²	semidirect
34. #495	C₂²	semidirect
35. #495	C₂²	semidirect
36. #494	C₂²	semidirect
37. #493	C₂²	semidirect
38. #492	C₂²	semidirect
39. #491	C₂²	semidirect
40. #490	C₂²	semidirect
41. #489	C₂²	semidirect
48. #442	C₂²	semidirect
49. #441	C₂²	semidirect
50. #440	C₂²	semidirect
51. C₄×Q₁₆	C₂²	semidirect
52. C₄×SD₁₆	C₂²	semidirect
53. C₄×D₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3455	2
#3177	2
#3176	2
#3175	2
#3174	2
#3173	2
#3160	2
#3147	2
#3135	2
#3054	2
#3050	2
#2943	2
C₄×#135	2
C₂×#405	2