Subgroups — #18052

Order 256.

337 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2991	C₂	1	1	yes	maximal, characteristic, complemented
2	#2974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
4	#1594	C₂	1	1	yes	maximal, characteristic, complemented
5	#1589	C₂	1	1	yes	maximal, characteristic, complemented
6	#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	#1586	C₂	1	1	yes	maximal, characteristic, complemented
8	#1583	C₂	1	1	yes	maximal, characteristic, complemented
9	#1581	C₂	1	1	yes	maximal, characteristic, complemented
10	SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#1569	C₂	1	1	yes	maximal, characteristic, complemented
13	#1488	C₂	1	1	yes	maximal, characteristic, complemented
14	#1486	C₂	1	1	yes	maximal, characteristic, complemented
15	#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic
17	C₄×#41	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
19	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
21	#442	C₄	1	1	yes	characteristic
22	#442	C₄	1	1	yes	characteristic
23	#440	C₄	1	1	yes	characteristic
24	#440	C₄	1	1	yes	characteristic
25	C₄×SD₁₆	C₂²	1	1	yes	characteristic
26	C₄×SD₁₆	C₂²	1	1	yes	characteristic
27	C₄×D₈	C₂²	1	1	yes	characteristic
28	C₄×D₈	C₂²	1	1	yes	characteristic
29	#436	—	2	1	no	—
30	#436	—	2	1	no	—
31	#436	C₂²	1	1	yes	characteristic
32	#436	C₂²	1	1	yes	characteristic
33	#436	C₂²	1	1	yes	characteristic, complemented
34	#436	C₂²	1	1	yes	characteristic, complemented
35	#435	—	2	1	no	—
36	#435	—	2	1	no	—
37	C₈×D₄	C₂²	1	1	yes	characteristic
38	C₈×D₄	C₂²	1	1	yes	characteristic
39	#432	C₂²	1	1	yes	characteristic
40	#431	—	2	1	no	—
41	#431	C₂²	1	1	yes	characteristic
42	#431	C₂²	1	1	yes	characteristic
43	#428	C₄	1	1	yes	characteristic
44	#423	—	2	1	no	—
45	#423	C₂²	1	1	yes	characteristic
46	#418	C₄	1	1	yes	characteristic
47	#417	C₄	1	1	yes	characteristic
48	#407	—	2	1	no	—
49	C₂×C₂²⋊C₈	—	2	1	no	—
50	C₄×M₁₆	C₂²	1	1	yes	characteristic
51	#335	C₂²	1	1	yes	characteristic, complemented
52	#335	C₂²	1	1	yes	characteristic, complemented
53	#334	C₂²	1	1	yes	characteristic, complemented
54	#334	C₂²	1	1	yes	characteristic, complemented
55	Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	D₄⋊C₈	C₂²	1	1	yes	characteristic
58	D₄⋊C₈	C₂²	1	1	yes	characteristic
59	D₄⋊C₈	C₂²	1	1	yes	characteristic
60	D₄⋊C₈	C₂²	1	1	yes	characteristic
61	D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
62	D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	#322	C₂²	1	1	yes	characteristic, complemented
64	#322	C₂²	1	1	yes	characteristic, complemented
65	#322	C₂²	1	1	yes	characteristic, complemented
66	C₈²	C₂²	1	1	yes	characteristic, complemented
67	C₂×#41	C₂×C₄	1	1	yes	characteristic
68	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
69	C₂³×C₄	—	2	1	no	—
70	AGL(1,ℤ/8)	—	2	4	no	—
71	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
72	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
73	C₂×D₈	C₂×C₄	1	1	yes	characteristic
74	C₂×D₈	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆	—	2	1	no	—
76	C₂×M₁₆	—	2	1	no	—
77	C₂×M₁₆	—	2	1	no	—
78	C₂×M₁₆	—	2	1	no	—
79	C₂×M₁₆	—	2	1	no	—
80	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈	C₂³	1	1	yes	characteristic
86	C₄×D₄	—	2	1	no	—
87	C₄×D₄	—	2	1	no	—
88	C₄×D₄	—	2	1	no	—
89	C₄×D₄	—	2	1	no	—
90	C₄×D₄	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄	C₂×C₄	1	1	yes	characteristic
92	C₄×D₄	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄	C₂³	1	1	yes	characteristic
94	C₄×D₄	C₂³	1	1	yes	characteristic
95	C₄×D₄	C₂³	1	1	yes	characteristic
96	#117	—	2	1	no	—
97	#117	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	—	2	1	no	—
100	C₂×C₄²	—	2	1	no	—
101	C₂×C₄²	C₂³	1	1	yes	characteristic
102	#107	C₂×C₄	1	1	yes	characteristic
103	#107	C₂×C₄	1	1	yes	characteristic
104	#106	C₂×C₄	1	1	yes	characteristic
105	#106	C₂×C₄	1	1	yes	characteristic
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	D₄	1	1	yes	characteristic
111	C₄⋊C₈	D₄	1	1	yes	characteristic
112	C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₄⋊C₈	C₂³	1	1	yes	characteristic
116	#103	C₂×C₄	1	1	yes	characteristic
117	#103	C₂×C₄	1	1	yes	characteristic
118	#102	C₂×C₄	1	1	yes	characteristic
119	#102	C₂×C₄	1	1	yes	characteristic
120	#102	C₂×C₄	1	1	yes	characteristic
121	#102	C₂×C₄	1	1	yes	characteristic
122	#102	C₂×C₄	1	1	yes	characteristic
123	#102	C₂×C₄	1	1	yes	characteristic
124	C₂²⋊C₈	—	2	1	no	—
125	C₂²⋊C₈	—	2	1	no	—
126	C₂²⋊C₈	—	2	1	no	—
127	C₂²⋊C₈	—	2	1	no	—
128	C₂²⋊C₈	—	2	1	no	—
129	C₂²⋊C₈	—	2	1	no	—
130	C₂²⋊C₈	—	2	1	no	—
131	C₂²⋊C₈	—	2	1	no	—
132	C₂²⋊C₈	D₄	1	1	yes	characteristic
133	C₂²⋊C₈	D₄	1	1	yes	characteristic
134	#97	—	2	1	no	—
135	#97	C₂×C₄	1	1	yes	characteristic
136	#97	C₂×C₄	1	1	yes	characteristic
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	C₂³	1	1	yes	characteristic
140	C₄×C₈	C₂³	1	1	yes	characteristic
141	C₄×C₈	C₂³	1	1	yes	characteristic
142	C₄×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₈	C₂³	1	1	yes	characteristic
145	C₂⁴	—	2	1	no	—
146	#41	—	2	2	no	—
147	#41	—	4	1	no	—
148	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	4	1	no	—
153	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
154	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	8	1	no	—
165	C₂²×C₄	—	8	1	no	—
166	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄	#41	1	1	yes	characteristic
169	SD₁₆	—	2	2	no	—
170	SD₁₆	M₁₆	1	4	yes	complemented
171	D₈	—	2	2	no	—
172	D₈	—	2	2	no	—
173	M₁₆	—	2	2	no	—
174	M₁₆	—	4	1	no	—
175	M₁₆	—	8	1	no	—
176	M₁₆	—	8	1	no	—
177	M₁₆	—	8	1	no	—
178	M₁₆	—	8	1	no	—
179	M₁₆	—	8	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	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	—	8	1	no	—
187	C₂×C₈	—	8	1	no	—
188	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
189	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
190	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
191	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄	1	1	yes	characteristic
193	C₂×C₈	C₂×D₄	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄	1	1	yes	characteristic
195	C₂×C₈	C₂×D₄	1	1	yes	characteristic
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	C₂²×C₄	1	1	yes	characteristic
214	C₄²	C₂×D₄	1	1	yes	characteristic
215	C₄²	#41	1	1	yes	characteristic
216	C₄²	C₂⁴	1	1	yes	characteristic
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	8	1	no	—
223	C₂³	C₄×D₄	1	1	yes	characteristic
224	Q₈	—	2	1	no	—
225	Q₈	C₂×M₁₆	1	2	yes	—
226	D₄	—	2	1	no	—
227	D₄	—	2	1	no	—
228	D₄	—	2	1	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	1	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	C₂×M₁₆	1	2	yes	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	8	1	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	C₄×D₄	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄	C₂×#41	1	1	yes	characteristic
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	C₂×M₁₆	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	8	4	no	—
288	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	C₂²×M₁₆	1	1	yes	characteristic
299	C₄	#568	1	1	yes	characteristic
300	C₄	#576	1	1	yes	characteristic
301	C₄	#577	1	1	yes	characteristic
302	C₂	—	4	1	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	1	no	—
305	C₂	—	8	1	no	—
306	C₂	D₄×M₁₆	1	1	yes	characteristic
307	C₂	#2946	1	1	yes	characteristic
308	C₂	#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×AGL(1,ℤ/8)	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2991	C₂	semidirect
2. #2974	C₂	semidirect
4. #1594	C₂	semidirect
5. #1589	C₂	semidirect
6. #1588	C₂	semidirect
7. #1586	C₂	semidirect
8. #1583	C₂	semidirect
9. #1581	C₂	semidirect
10. SD₁₆⋊C₈	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
12. #1569	C₂	semidirect
13. #1488	C₂	semidirect
14. #1486	C₂	semidirect
15. #1453	C₂	semidirect
19. C₈×Q₈	C₂²	semidirect
20. C₈×Q₈	C₂²	semidirect
33. #436	C₂²	semidirect
34. #436	C₂²	semidirect
51. #335	C₂²	semidirect
52. #335	C₂²	semidirect
53. #334	C₂²	semidirect
54. #334	C₂²	semidirect
55. Q₈⋊C₈	C₂²	semidirect
56. Q₈⋊C₈	C₂²	semidirect
61. D₄⋊C₈	C₂²	semidirect
62. D₄⋊C₈	C₂²	semidirect
63. #322	C₂²	semidirect
64. #322	C₂²	semidirect
65. #322	C₂²	semidirect
66. C₈²	C₂²	semidirect
170. SD₁₆	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2991	2
#2974	2
C₄×AGL(1,ℤ/8)	2
#1594	2
#1589	2
#1588	2
#1586	2
#1583	2
#1581	2
SD₁₆⋊C₈	2
C₈×SD₁₆	2
#1569	2
#1488	2
#1486	2
#1453	2