Subgroups — #18028

Order 256.

308 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3101	C₂	1	1	yes	maximal, characteristic
2	#2986	C₂	1	1	yes	maximal, characteristic, complemented
3	#2974	C₂	1	1	yes	maximal, characteristic
4	#1592	C₂	1	1	yes	maximal, characteristic, complemented
5	#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	#1583	C₂	1	1	yes	maximal, characteristic, complemented
9	#1582	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	#1572	C₂	1	1	yes	maximal, characteristic, complemented
13	#1497	C₂	1	1	yes	maximal, characteristic, complemented
14	#1478	C₂	1	1	yes	maximal, characteristic
15	#1454	C₂	1	1	yes	maximal, characteristic, complemented
16	#533	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
18	#483	C₄	1	1	yes	characteristic
19	#483	C₄	1	1	yes	characteristic
20	#476	C₄	1	1	yes	characteristic
21	#476	C₄	1	1	yes	characteristic
22	#461	C₄	1	1	yes	characteristic
23	#461	C₄	1	1	yes	characteristic
24	#450	C₄	1	1	yes	characteristic
25	#450	C₄	1	1	yes	characteristic
26	#446	—	2	1	no	—
27	#446	C₂²	1	1	yes	characteristic, complemented
28	#446	C₂²	1	1	yes	characteristic, complemented
29	C₈×Q₈	—	2	1	no	—
30	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
31	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
32	C₄×SD₁₆	C₂²	1	1	yes	characteristic
33	C₄×SD₁₆	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆	C₂²	1	1	yes	characteristic
35	C₄×SD₁₆	C₂²	1	1	yes	characteristic
36	#436	C₂²	1	1	yes	characteristic
37	#436	C₂²	1	1	yes	characteristic
38	#435	—	2	1	no	—
39	#435	—	2	1	no	—
40	C₈×D₄	C₂²	1	1	yes	characteristic
41	C₈×D₄	C₂²	1	1	yes	characteristic
42	#433	—	2	1	no	—
43	#433	—	2	1	no	—
44	#432	C₂²	1	1	yes	characteristic
45	#431	C₂²	1	1	yes	characteristic
46	#431	C₂²	1	1	yes	characteristic
47	#431	C₂²	1	1	yes	characteristic
48	#431	C₂²	1	1	yes	characteristic
49	#407	—	2	1	no	—
50	C₂×C₂²⋊C₈	—	2	1	no	—
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	Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
58	Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
59	D₄⋊C₈	C₂²	1	1	yes	characteristic
60	D₄⋊C₈	C₂²	1	1	yes	characteristic
61	D₄⋊C₈	C₂²	1	1	yes	characteristic
62	D₄⋊C₈	C₂²	1	1	yes	characteristic
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	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄	—	2	1	no	—
69	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
71	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
72	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆	—	2	1	no	—
74	C₂×M₁₆	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
78	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
79	#125	—	2	1	no	—
80	C₂²⋊Q₈	—	2	1	no	—
81	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	C₂³	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	C₂×C₄	1	1	yes	characteristic
89	C₄×D₄	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄	C₂³	1	1	yes	characteristic
91	C₄×D₄	C₂³	1	1	yes	characteristic
92	#117	—	2	1	no	—
93	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄	—	2	1	no	—
95	C₂×C₄²	C₂³	1	1	yes	characteristic
96	#106	C₂×C₄	1	1	yes	characteristic
97	#106	C₂×C₄	1	1	yes	characteristic
98	#106	C₂×C₄	1	1	yes	characteristic
99	#106	C₂×C₄	1	1	yes	characteristic
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	2	1	no	—
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	—	2	1	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
106	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
107	C₄⋊C₈	D₄	1	1	yes	characteristic
108	C₄⋊C₈	D₄	1	1	yes	characteristic
109	C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₄⋊C₈	C₂³	1	1	yes	characteristic
113	#103	C₂×C₄	1	1	yes	characteristic
114	#103	C₂×C₄	1	1	yes	characteristic
115	#103	C₂×C₄	1	1	yes	characteristic
116	#103	C₂×C₄	1	1	yes	characteristic
117	#102	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	C₂²⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	—	2	1	no	—
123	C₂²⋊C₈	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
129	C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
130	C₂²⋊C₈	D₄	1	1	yes	characteristic
131	C₂²⋊C₈	D₄	1	1	yes	characteristic
132	#97	—	2	1	no	—
133	#97	—	2	1	no	—
134	#97	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	C₂³	1	1	yes	characteristic
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	C₂⁴	—	2	1	no	—
145	C₂×Q₈	—	2	1	no	—
146	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
147	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	1	no	—
150	C₂×D₄	—	4	1	no	—
151	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	8	1	no	—
159	C₂²×C₄	—	8	1	no	—
160	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄	#41	1	1	yes	characteristic
163	SD₁₆	—	2	2	no	—
164	SD₁₆	—	2	2	no	—
165	SD₁₆	—	4	1	no	—
166	SD₁₆	—	4	1	no	—
167	M₁₆	—	8	1	no	—
168	M₁₆	—	8	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	8	1	no	—
176	C₂×C₈	—	8	1	no	—
177	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
179	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
180	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈	C₂×D₄	1	1	yes	characteristic
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
194	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
195	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	C₂²×C₄	1	1	yes	characteristic
206	C₄²	C₂×D₄	1	1	yes	characteristic
207	C₄²	#41	1	1	yes	characteristic
208	C₄²	C₂⁴	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	8	1	no	—
214	C₂³	C₄×D₄	1	1	yes	characteristic
215	Q₈	—	2	1	no	—
216	Q₈	—	2	1	no	—
217	Q₈	—	2	1	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	2	1	no	—
221	D₄	—	2	1	no	—
222	D₄	—	2	1	no	—
223	D₄	—	4	2	no	—
224	D₄	#131	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	8	1	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	C₄×D₄	1	1	yes	characteristic
244	C₂×C₄	C₄×D₄	1	1	yes	characteristic
245	C₂×C₄	C₄×D₄	1	1	yes	characteristic
246	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄	C₂×#41	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	8	4	no	—
273	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	C₂×#131	1	1	yes	characteristic
285	C₄	#568	1	1	yes	characteristic
286	C₄	#577	1	1	yes	characteristic
287	C₄	#577	1	1	yes	characteristic
288	C₂	—	4	1	no	—
289	C₂	—	4	2	no	—
290	C₂	—	8	1	no	—
291	C₂	#2933	1	1	yes	characteristic
292	C₂	#2949	1	1	yes	characteristic
293	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	C₂²×D₄	C₂×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
2. #2986	C₂	semidirect
4. #1592	C₂	semidirect
5. #1590	C₂	semidirect
6. #1588	C₂	semidirect
7. #1585	C₂	semidirect
8. #1583	C₂	semidirect
9. #1582	C₂	semidirect
10. SD₁₆⋊C₈	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
12. #1572	C₂	semidirect
13. #1497	C₂	semidirect
15. #1454	C₂	semidirect
27. #446	C₂²	semidirect
28. #446	C₂²	semidirect
30. C₈×Q₈	C₂²	semidirect
31. C₈×Q₈	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
57. Q₈⋊C₈	C₂²	semidirect
58. Q₈⋊C₈	C₂²	semidirect
63. #322	C₂²	semidirect
64. #322	C₂²	semidirect
65. #322	C₂²	semidirect
66. C₈²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3101	2
#2986	2
#2974	2
#1592	2
#1590	2
#1588	2
#1585	2
#1583	2
#1582	2
SD₁₆⋊C₈	2
C₈×SD₁₆	2
#1572	2
#1497	2
#1478	2
#1454	2