Subgroups — #18031

Order 256.

299 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3102	C₂	1	1	yes	maximal, characteristic
2	#2994	C₂	1	1	yes	maximal, characteristic, complemented
3	#2991	C₂	1	1	yes	maximal, characteristic, complemented
4	#1594	C₂	1	1	yes	maximal, characteristic, complemented
5	#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	#1587	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	#1583	C₂	1	1	yes	maximal, characteristic, complemented
10	#1579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#1570	C₂	1	1	yes	maximal, characteristic, complemented
13	#1495	C₂	1	1	yes	maximal, characteristic, complemented
14	#1477	C₂	1	1	yes	maximal, characteristic, complemented
15	#1454	C₂	1	1	yes	maximal, characteristic, complemented
16	#529	C₂²	1	1	yes	characteristic
17	C₄×#41	C₂²	1	1	yes	characteristic
18	#485	C₄	1	1	yes	characteristic
19	#482	C₄	1	1	yes	characteristic
20	#479	C₄	1	1	yes	characteristic
21	#478	C₄	1	1	yes	characteristic
22	#464	C₄	1	1	yes	characteristic
23	#463	C₄	1	1	yes	characteristic
24	#452	C₄	1	1	yes	characteristic
25	#449	C₄	1	1	yes	characteristic
26	#446	C₂²	1	1	yes	characteristic
27	C₈×Q₈	C₂²	1	1	yes	characteristic
28	C₈×Q₈	C₂²	1	1	yes	characteristic
29	C₈×Q₈	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆	C₂²	1	1	yes	characteristic
32	C₄×SD₁₆	C₂²	1	1	yes	characteristic
33	C₄×D₈	C₂²	1	1	yes	characteristic
34	#436	—	2	1	no	—
35	#436	—	2	1	no	—
36	#436	C₂²	1	1	yes	characteristic
37	#436	C₂²	1	1	yes	characteristic
38	#436	C₂²	1	1	yes	characteristic
39	#435	—	2	1	no	—
40	C₈×D₄	—	2	1	no	—
41	C₈×D₄	C₂²	1	1	yes	characteristic
42	#433	—	2	1	no	—
43	#431	—	2	1	no	—
44	#431	C₂²	1	1	yes	characteristic
45	#431	C₂²	1	1	yes	characteristic
46	#431	C₂²	1	1	yes	characteristic
47	#424	—	2	1	no	—
48	#423	—	2	1	no	—
49	#423	C₂²	1	1	yes	characteristic
50	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	#334	C₂²	1	1	yes	characteristic
52	#334	C₂²	1	1	yes	characteristic
53	#334	C₂²	1	1	yes	characteristic
54	#334	C₂²	1	1	yes	characteristic
55	Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	Q₈⋊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
62	D₄⋊C₈	C₂²	1	1	yes	characteristic
63	#322	C₂²	1	1	yes	characteristic
64	#322	C₂²	1	1	yes	characteristic
65	#322	C₂²	1	1	yes	characteristic
66	C₈²	C₂²	1	1	yes	characteristic
67	C₂×#41	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
69	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
71	C₂×D₈	C₂×C₄	1	1	yes	characteristic
72	C₂×M₁₆	—	2	1	no	—
73	C₂×M₁₆	—	2	1	no	—
74	C₂×M₁₆	—	2	1	no	—
75	C₂×M₁₆	—	2	1	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	#126	—	2	1	no	—
79	#125	C₂×C₄	1	1	yes	characteristic
80	#124	C₂×C₄	1	1	yes	characteristic
81	#123	—	2	1	no	—
82	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
83	#121	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈	C₂³	1	1	yes	characteristic
86	C₄×Q₈	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
92	C₄×D₄	C₂³	1	1	yes	characteristic
93	#117	—	2	1	no	—
94	#117	—	2	1	no	—
95	#117	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂³	1	1	yes	characteristic
98	#107	C₂×C₄	1	1	yes	characteristic
99	#107	C₂×C₄	1	1	yes	characteristic
100	#106	C₂×C₄	1	1	yes	characteristic
101	#106	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	—	2	1	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
110	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
111	C₄⋊C₈	D₄	1	1	yes	characteristic
112	C₄⋊C₈	D₄	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	C₄⋊C₈	C₂³	1	1	yes	characteristic
117	#103	C₂×C₄	1	1	yes	characteristic
118	#103	C₂×C₄	1	1	yes	characteristic
119	#103	C₂×C₄	1	1	yes	characteristic
120	#103	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	C₂×C₄	1	1	yes	characteristic
131	C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
132	C₂²⋊C₈	D₄	1	1	yes	characteristic
133	C₂²⋊C₈	D₄	1	1	yes	characteristic
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	#41	—	2	2	no	—
145	#41	—	4	1	no	—
146	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
147	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	—	2	1	no	—
150	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
151	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄	#41	1	1	yes	characteristic
159	Q₁₆	—	2	2	no	—
160	SD₁₆	—	2	2	no	—
161	SD₁₆	—	4	1	no	—
162	D₈	—	4	1	no	—
163	M₁₆	—	8	1	no	—
164	M₁₆	—	8	1	no	—
165	M₁₆	—	8	1	no	—
166	M₁₆	—	8	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	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	—	8	1	no	—
174	C₂×C₈	—	8	1	no	—
175	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄	1	1	yes	characteristic
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
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	C₄²	—	2	1	no	—
202	C₄²	—	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	C₄×D₄	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	2	1	no	—
214	Q₈	—	2	1	no	—
215	Q₈	#131	1	2	yes	—
216	D₄	—	2	1	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	1	no	—
219	D₄	—	2	1	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
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	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	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	C₄×D₄	1	1	yes	characteristic
240	C₂×C₄	C₄×D₄	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×#41	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	1	no	—
255	C₈	—	4	1	no	—
256	C₈	—	4	1	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	C₂×#131	1	1	yes	characteristic
280	C₄	#568	1	1	yes	characteristic
281	C₄	#577	1	1	yes	characteristic
282	C₄	#577	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	8	1	no	—
285	C₂	—	8	1	no	—
286	C₂	#2933	1	1	yes	characteristic
287	C₂	#2952	1	1	yes	characteristic
288	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	#449	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. #2994	C₂	semidirect
3. #2991	C₂	semidirect
4. #1594	C₂	semidirect
5. #1590	C₂	semidirect
6. #1587	C₂	semidirect
7. #1586	C₂	semidirect
8. #1583	C₂	semidirect
9. #1583	C₂	semidirect
10. #1579	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
12. #1570	C₂	semidirect
13. #1495	C₂	semidirect
14. #1477	C₂	semidirect
15. #1454	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3102	2
#2994	2
#2991	2
#1594	2
#1590	2
#1587	2
#1586	2
#1583	2
#1579	2
C₈×SD₁₆	2
#1570	2
#1495	2
#1477	2
#1454	2