Subgroups — #18048

Order 256.

359 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2971	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	D₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
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	#1490	C₂	1	1	yes	maximal, characteristic, complemented
14	#1489	C₂	1	1	yes	maximal, characteristic, complemented
15	#1455	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic
17	C₂²×M₁₆	—	2	1	no	—
18	C₄×#41	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
20	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
22	#442	C₄	1	1	yes	characteristic
23	#442	C₄	1	1	yes	characteristic
24	#440	C₄	1	1	yes	characteristic
25	#440	C₄	1	1	yes	characteristic
26	C₄×SD₁₆	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆	C₂²	1	1	yes	characteristic
28	C₄×D₈	C₂²	1	1	yes	characteristic
29	C₄×D₈	C₂²	1	1	yes	characteristic
30	#436	—	2	1	no	—
31	#436	—	2	1	no	—
32	#436	C₂²	1	1	yes	characteristic
33	#436	C₂²	1	1	yes	characteristic
34	#436	C₂²	1	1	yes	characteristic, complemented
35	#436	C₂²	1	1	yes	characteristic, complemented
36	#435	—	2	1	no	—
37	#435	—	2	1	no	—
38	C₈×D₄	C₂²	1	1	yes	characteristic
39	C₈×D₄	C₂²	1	1	yes	characteristic
40	#432	—	2	1	no	—
41	#432	C₂²	1	1	yes	characteristic
42	#428	C₄	1	1	yes	characteristic
43	#423	C₂²	1	1	yes	characteristic
44	#418	C₄	1	1	yes	characteristic
45	#417	C₄	1	1	yes	characteristic
46	#407	—	2	1	no	—
47	C₄×M₁₆	—	2	1	no	—
48	C₄×M₁₆	C₂²	1	1	yes	characteristic
49	C₄×M₁₆	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
81	C₂×M₁₆	—	4	4	no	—
82	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
83	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
84	C₂×M₁₆	D₄	1	1	yes	characteristic
85	C₂×M₁₆	D₄	1	1	yes	characteristic
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
89	C₄×Q₈	C₂³	1	1	yes	characteristic
90	C₄×D₄	—	2	1	no	—
91	C₄×D₄	—	2	1	no	—
92	C₄×D₄	—	2	1	no	—
93	C₄×D₄	—	2	1	no	—
94	C₄×D₄	C₂×C₄	1	1	yes	characteristic
95	C₄×D₄	C₂×C₄	1	1	yes	characteristic
96	C₄×D₄	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄	C₂³	1	1	yes	characteristic
98	C₄×D₄	C₂³	1	1	yes	characteristic
99	C₄×D₄	C₂³	1	1	yes	characteristic
100	#117	—	2	1	no	—
101	#117	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	—	2	1	no	—
104	C₂×C₄²	—	2	1	no	—
105	C₂×C₄²	C₂³	1	1	yes	characteristic
106	#107	C₂×C₄	1	1	yes	characteristic
107	#107	C₂×C₄	1	1	yes	characteristic
108	#106	C₂×C₄	1	1	yes	characteristic
109	#106	C₂×C₄	1	1	yes	characteristic
110	C₄⋊C₈	—	2	1	no	—
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	2	1	no	—
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	C₄⋊C₈	C₂³	1	1	yes	characteristic
117	#103	C₂×C₄	1	1	yes	characteristic
118	#103	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	#102	C₂×C₄	1	1	yes	characteristic
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	#97	—	2	1	no	—
133	#97	—	2	1	no	—
134	#97	C₂×C₄	1	1	yes	characteristic
135	#97	C₂×C₄	1	1	yes	characteristic
136	#97	D₄	1	1	yes	characteristic
137	#97	D₄	1	1	yes	characteristic
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	—	2	1	no	—
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	C₄×C₈	C₂³	1	1	yes	characteristic
146	C₂⁴	—	2	1	no	—
147	#41	—	2	2	no	—
148	#41	—	4	1	no	—
149	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
156	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
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	—	2	1	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄	#41	1	1	yes	characteristic
170	SD₁₆	—	2	2	no	—
171	SD₁₆	M₁₆	1	4	yes	complemented
172	D₈	—	2	2	no	—
173	D₈	—	2	2	no	—
174	M₁₆	—	2	2	no	—
175	M₁₆	—	2	2	no	—
176	M₁₆	—	4	1	no	—
177	M₁₆	—	4	1	no	—
178	M₁₆	—	4	2	no	—
179	M₁₆	—	4	2	no	—
180	M₁₆	—	4	2	no	—
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	M₁₆	—	4	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
194	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
195	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
196	C₂×C₈	C₂×D₄	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄	1	1	yes	characteristic
198	C₂×C₈	C₂×D₄	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
204	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
205	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
206	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	C₂²×C₄	1	1	yes	characteristic
218	C₄²	C₂×D₄	1	1	yes	characteristic
219	C₄²	#41	1	1	yes	characteristic
220	C₄²	C₂⁴	1	1	yes	characteristic
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	1	no	—
227	C₂³	C₄×D₄	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	C₂×M₁₆	1	2	yes	complemented
230	D₄	—	2	1	no	—
231	D₄	—	2	1	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	1	no	—
234	D₄	—	2	1	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	C₂×M₁₆	1	2	yes	—
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	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	8	1	no	—
259	C₂×C₄	C₄×D₄	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄	1	1	yes	characteristic
261	C₂×C₄	C₄×D₄	1	1	yes	characteristic
262	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄	C₂×#41	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	C₂×M₁₆	1	2	yes	complemented
278	C₈	AGL(1,ℤ/8)	1	2	yes	complemented
279	C₂²	—	2	1	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	4	no	—
292	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	C₂²×M₁₆	1	1	yes	characteristic
303	C₄	#568	1	1	yes	characteristic
304	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₄	#575	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	4	2	no	—
308	C₂	—	8	1	no	—
309	C₂	—	8	1	no	—
310	C₂	D₄×M₁₆	1	1	yes	characteristic
311	C₂	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₂	#2956	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 ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2971	C₂	semidirect
3. D₄×M₁₆	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. #1490	C₂	semidirect
14. #1489	C₂	semidirect
15. #1455	C₂	semidirect
20. C₈×Q₈	C₂²	semidirect
21. C₈×Q₈	C₂²	semidirect
34. #436	C₂²	semidirect
35. #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
171. SD₁₆	M₁₆	semidirect
229. Q₈	C₂×M₁₆	semidirect
277. C₈	C₂×M₁₆	semidirect
278. C₈	AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
#2971	2
C₄×AGL(1,ℤ/8)	2
D₄×M₁₆	2
#1594	2
#1589	2
#1588	2
#1586	2
#1583	2
#1581	2
SD₁₆⋊C₈	2
C₈×SD₁₆	2
#1569	2
#1490	2
#1489	2
#1455	2