Subgroups — #61370

Order 256.

909 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3501	C₂	1	2	yes	maximal, complemented
2	#3389	C₂	1	4	yes	maximal, complemented
3	#3384	C₂	1	4	yes	maximal, complemented
4	#3380	C₂	1	2	yes	maximal, complemented
5	#3380	C₂	1	4	yes	maximal, complemented
6	Q₈×D₈	C₂	1	2	yes	maximal, complemented
7	#3164	C₂	1	2	yes	maximal, complemented
8	#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	#3081	C₂	1	4	yes	maximal, complemented
10	#3077	C₂	1	2	yes	maximal, complemented
11	#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	#2948	C₂	1	2	yes	maximal
13	C₂×C₄×D₈	C₂	1	1	yes	maximal, characteristic
14	C₂²×D₈	C₂²	1	1	yes	characteristic
15	#554	—	2	2	no	—
16	#554	—	2	4	no	—
17	#554	C₂²	1	4	yes	complemented
18	#554	C₂²	1	8	yes	complemented
19	D₄×Q₈	—	2	2	no	—
20	D₄×Q₈	C₂²	1	4	yes	complemented
21	#544	C₂²	1	4	yes	—
22	#543	—	2	4	no	—
23	#533	C₂²	1	2	yes	—
24	#533	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈	—	2	4	no	—
26	#518	C₂²	1	4	yes	—
27	C₂×C₄×D₄	C₂²	1	2	yes	—
28	#501	C₂²	1	8	yes	complemented
29	#500	C₂²	1	1	yes	characteristic, complemented
30	#500	C₂²	1	4	yes	complemented
31	#500	C₂²	1	1	yes	characteristic, complemented
32	#499	C₂²	1	2	yes	complemented
33	#499	C₂²	1	4	yes	complemented
34	#499	C₂²	1	2	yes	complemented
35	#498	C₂²	1	2	yes	complemented
36	#478	C₂²	1	8	yes	complemented
37	#478	C₂²	1	8	yes	complemented
38	#478	C₂²	1	8	yes	complemented
39	#476	C₂²	1	4	yes	complemented
40	#476	C₂²	1	8	yes	complemented
41	#474	C₂²	1	4	yes	complemented
42	#474	C₂²	1	8	yes	complemented
43	#446	C₂²	1	2	yes	complemented
44	#446	C₂²	1	2	yes	complemented
45	C₈×Q₈	C₂²	1	2	yes	complemented
46	C₈×Q₈	C₂²	1	2	yes	complemented
47	#442	C₂²	1	8	yes	—
48	C₄×D₈	C₂²	1	2	yes	—
49	C₄×D₈	C₂²	1	4	yes	—
50	C₄×D₈	C₂²	1	2	yes	—
51	C₄×D₈	C₂²	1	4	yes	—
52	C₄×D₈	C₂²	1	2	yes	—
53	C₄×D₈	C₂²	1	2	yes	—
54	#428	C₂²	1	4	yes	—
55	#427	C₂²	1	2	yes	—
56	#427	C₂²	1	2	yes	—
57	C₂×#107	C₂²	1	1	yes	characteristic
58	#424	C₂²	1	2	yes	—
59	#423	C₂²	1	1	yes	characteristic
60	#418	C₂²	1	4	yes	—
61	C₂×#102	C₂²	1	2	yes	—
62	#405	C₂²	1	2	yes	—
63	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
64	C₂²×Q₈	—	2	4	no	—
65	C₂²×D₄	C₂³	1	2	yes	—
66	C₂³×C₄	—	2	2	no	—
67	C₂×D₈	Q₈	1	8	yes	complemented
68	C₂×D₈	C₂³	1	2	yes	—
69	C₂×D₈	C₂³	1	2	yes	—
70	C₂×M₁₆	C₂³	1	2	yes	—
71	C₂²×C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈	—	2	4	no	—
74	C₄⋊Q₈	D₄	1	1	yes	characteristic
75	C₄⋊Q₈	D₄	1	1	yes	characteristic
76	C₄⋊Q₈	C₂³	1	2	yes	—
77	C₄⋊Q₈	C₂³	1	2	yes	—
78	C₄⋊Q₈	C₂³	1	8	yes	—
79	#125	—	2	4	no	—
80	#125	D₄	1	2	yes	—
81	#125	C₂³	1	4	yes	—
82	#125	C₂³	1	8	yes	—
83	C₂²⋊Q₈	—	2	4	no	—
84	C₂²⋊Q₈	—	2	8	no	—
85	C₂²⋊Q₈	—	2	2	no	—
86	C₂²⋊Q₈	—	2	2	no	—
87	C₂²⋊Q₈	—	2	4	no	—
88	C₂²⋊Q₈	—	2	8	no	—
89	C₂²⋊Q₈	—	2	8	no	—
90	C₂²⋊Q₈	—	2	8	no	—
91	C₂²⋊Q₈	—	2	8	no	—
92	C₂²⋊Q₈	—	2	8	no	—
93	C₂²⋊Q₈	D₄	1	2	yes	—
94	C₂²⋊Q₈	D₄	1	2	yes	—
95	C₄×Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	C₄×Q₈	C₂³	1	2	yes	—
98	C₄×Q₈	C₂³	1	2	yes	—
99	C₄×D₄	—	2	4	no	—
100	C₄×D₄	—	2	4	no	—
101	C₄×D₄	—	2	2	no	—
102	C₄×D₄	—	2	2	no	—
103	C₄×D₄	C₂³	1	4	yes	—
104	C₄×D₄	C₂³	1	4	yes	—
105	C₄×D₄	C₂³	1	8	yes	—
106	C₄×D₄	C₂³	1	8	yes	—
107	#117	C₂³	1	4	yes	—
108	#117	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄	—	2	8	no	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₂²⋊C₄	—	2	8	no	—
114	C₂×C₄²	C₂³	1	1	yes	characteristic
115	#107	C₂³	1	2	yes	—
116	#107	C₂³	1	2	yes	—
117	#107	C₂³	1	2	yes	—
118	#107	C₂³	1	2	yes	—
119	#107	C₂³	1	2	yes	—
120	#107	C₂³	1	2	yes	—
121	#107	C₂³	1	8	yes	—
122	#106	C₂³	1	4	yes	—
123	#106	C₂³	1	4	yes	—
124	#106	C₂³	1	8	yes	—
125	C₄⋊C₈	C₂³	1	2	yes	—
126	C₄⋊C₈	C₂³	1	2	yes	—
127	C₄⋊C₈	C₂³	1	4	yes	—
128	C₄⋊C₈	C₂³	1	4	yes	—
129	#102	C₂³	1	4	yes	—
130	#102	C₂³	1	4	yes	—
131	#102	C₂³	1	8	yes	—
132	#102	C₂³	1	8	yes	—
133	#97	C₂³	1	4	yes	—
134	C₄×C₈	C₂³	1	1	yes	characteristic
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	4	yes	—
139	C₂⁴	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	8	no	—
143	C₂×Q₈	—	8	8	no	—
144	C₂×Q₈	C₂×D₄	1	2	yes	—
145	C₂×Q₈	C₂×D₄	1	2	yes	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	C₂×Q₈	1	8	yes	complemented
149	C₂×D₄	C₂⁴	1	4	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	8	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	8	4	no	—
156	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄	C₂⁴	1	1	yes	characteristic
159	D₈	C₂×Q₈	1	8	yes	complemented
160	D₈	C₂×Q₈	1	8	yes	complemented
161	M₁₆	—	2	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂×Q₈	1	4	yes	complemented
164	C₂×C₈	C₂⁴	1	1	yes	characteristic
165	C₂×C₈	C₂⁴	1	1	yes	characteristic
166	C₂×C₈	C₂⁴	1	4	yes	—
167	C₂×C₈	C₂⁴	1	1	yes	characteristic
168	C₂×C₈	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	2	8	no	—
173	C₄⋊C₄	—	2	8	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	C₂×D₄	1	4	yes	—
178	C₄⋊C₄	C₂×D₄	1	2	yes	—
179	C₄⋊C₄	C₂×D₄	1	2	yes	—
180	C₄⋊C₄	C₂×D₄	1	2	yes	—
181	C₄⋊C₄	C₂×D₄	1	4	yes	—
182	C₄⋊C₄	C₂×D₄	1	2	yes	—
183	C₄⋊C₄	C₂⁴	1	4	yes	—
184	C₄⋊C₄	C₂⁴	1	4	yes	—
185	C₄⋊C₄	C₂⁴	1	4	yes	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	8	no	—
188	C₂²⋊C₄	—	2	8	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	C₂×D₄	1	4	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	C₂×D₄	1	1	yes	characteristic
194	C₄²	C₂×D₄	1	1	yes	characteristic
195	C₄²	C₂⁴	1	1	yes	characteristic
196	C₄²	C₂⁴	1	1	yes	characteristic
197	C₄²	C₂⁴	1	4	yes	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	8	no	—
200	C₂³	—	8	2	no	—
201	C₂³	—	8	2	no	—
202	C₂³	C₂²×D₄	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	4	4	no	—
206	Q₈	—	8	2	no	—
207	Q₈	—	8	8	no	—
208	D₄	—	2	8	no	—
209	D₄	—	2	4	no	—
210	D₄	C₂²×Q₈	1	8	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	8	8	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	8	8	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₄	1	4	yes	—
225	C₂×C₄	C₂²×D₄	1	4	yes	—
226	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄	C₂²×Q₈	1	2	yes	—
229	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
230	C₂×C₄	C₂⁵	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	4	no	—
234	C₈	C₂²×Q₈	1	2	yes	—
235	C₈	C₂²×Q₈	1	2	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	8	8	no	—
242	C₂²	D₄×Q₈	1	2	yes	—
243	C₂²	C₂³×D₄	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	4	no	—
249	C₄	—	4	4	no	—
250	C₄	—	4	4	no	—
251	C₄	—	4	4	no	—
252	C₄	D₄×Q₈	1	2	yes	—
253	C₄	#576	1	2	yes	—
254	C₄	C₂³×Q₈	1	1	yes	characteristic
255	C₄	C₂×2₋¹⁺⁴	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	8	no	—
258	C₂	C₂×D₄×Q₈	1	1	yes	characteristic
259	C₂	C₂×#576	1	1	yes	characteristic
260	C₂	#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂²×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3501	C₂	semidirect
2. #3389	C₂	semidirect
3. #3384	C₂	semidirect
4. #3380	C₂	semidirect
5. #3380	C₂	semidirect
6. Q₈×D₈	C₂	semidirect
7. #3164	C₂	semidirect
8. #3160	C₂	semidirect
9. #3081	C₂	semidirect
10. #3077	C₂	semidirect
11. #2960	C₂	semidirect
17. #554	C₂²	semidirect
18. #554	C₂²	semidirect
20. D₄×Q₈	C₂²	semidirect
28. #501	C₂²	semidirect
29. #500	C₂²	semidirect
30. #500	C₂²	semidirect
31. #500	C₂²	semidirect
32. #499	C₂²	semidirect
33. #499	C₂²	semidirect
34. #499	C₂²	semidirect
35. #498	C₂²	semidirect
36. #478	C₂²	semidirect
37. #478	C₂²	semidirect
38. #478	C₂²	semidirect
39. #476	C₂²	semidirect
40. #476	C₂²	semidirect
41. #474	C₂²	semidirect
42. #474	C₂²	semidirect
43. #446	C₂²	semidirect
44. #446	C₂²	semidirect
45. C₈×Q₈	C₂²	semidirect
46. C₈×Q₈	C₂²	semidirect
67. C₂×D₈	Q₈	semidirect
148. C₂×D₄	C₂×Q₈	semidirect
159. D₈	C₂×Q₈	semidirect
160. D₈	C₂×Q₈	semidirect
163. C₂×C₈	C₂×Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3501	2
#3389	2
#3384	2
#3380	2
Q₈×D₈	2
#3164	2
#3160	2
#3081	2
#3077	2
#2960	2
#2948	2
C₂×C₄×D₈	2