Subgroups — #18083

Order 256.

286 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3172	C₂	1	1	yes	maximal, characteristic
2	#2994	C₂	1	1	yes	maximal, characteristic
3	#2986	C₂	1	1	yes	maximal, characteristic, complemented
4	#1593	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	#1587	C₂	1	1	yes	maximal, characteristic, complemented
8	#1584	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×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#1565	C₂	1	1	yes	maximal, characteristic, complemented
13	#1482	C₂	1	1	yes	maximal, characteristic
14	#1481	C₂	1	1	yes	maximal, characteristic, complemented
15	#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	#533	C₂²	1	1	yes	characteristic
17	#529	C₂²	1	1	yes	characteristic
18	#501	C₄	1	1	yes	characteristic
19	#497	C₄	1	1	yes	characteristic
20	#491	C₄	1	1	yes	characteristic
21	#489	C₄	1	1	yes	characteristic
22	#470	C₄	1	1	yes	characteristic
23	#470	C₄	1	1	yes	characteristic
24	#468	C₄	1	1	yes	characteristic
25	#468	C₄	1	1	yes	characteristic
26	#446	—	2	1	no	—
27	#446	C₂²	1	1	yes	characteristic
28	#446	C₂²	1	1	yes	characteristic
29	#446	C₂²	1	1	yes	characteristic
30	C₈×Q₈	—	2	1	no	—
31	C₈×Q₈	C₂²	1	1	yes	characteristic
32	C₈×Q₈	C₂²	1	1	yes	characteristic
33	C₈×Q₈	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆	C₂²	1	1	yes	characteristic
37	C₄×SD₁₆	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	#433	—	2	1	no	—
44	#433	—	2	1	no	—
45	#431	C₂²	1	1	yes	characteristic
46	#431	C₂²	1	1	yes	characteristic
47	#431	C₂²	1	1	yes	characteristic
48	#424	—	2	1	no	—
49	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₄×M₁₆	C₂²	1	1	yes	characteristic
51	#335	C₂²	1	1	yes	characteristic
52	#335	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	Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	Q₈⋊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×Q₁₆	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×M₁₆	—	2	1	no	—
72	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
77	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
78	#126	—	2	1	no	—
79	#125	—	2	1	no	—
80	#125	C₂×C₄	1	1	yes	characteristic
81	#124	C₂×C₄	1	1	yes	characteristic
82	#123	—	2	1	no	—
83	C₂²⋊Q₈	—	2	1	no	—
84	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
87	#121	C₂×C₄	1	1	yes	characteristic
88	C₄×Q₈	—	2	1	no	—
89	C₄×Q₈	C₂³	1	1	yes	characteristic
90	C₄×Q₈	C₂³	1	1	yes	characteristic
91	C₄×Q₈	C₂³	1	1	yes	characteristic
92	C₄×D₄	—	2	1	no	—
93	C₄×D₄	C₂³	1	1	yes	characteristic
94	#117	—	2	1	no	—
95	#117	—	2	1	no	—
96	C₂×C₄²	C₂³	1	1	yes	characteristic
97	#107	C₂×C₄	1	1	yes	characteristic
98	#107	C₂×C₄	1	1	yes	characteristic
99	#106	C₂×C₄	1	1	yes	characteristic
100	#106	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	D₄	1	1	yes	characteristic
110	C₄⋊C₈	D₄	1	1	yes	characteristic
111	C₄⋊C₈	C₂³	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	#103	C₂×C₄	1	1	yes	characteristic
116	#103	C₂×C₄	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	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	D₄	1	1	yes	characteristic
128	C₂²⋊C₈	D₄	1	1	yes	characteristic
129	#97	—	2	1	no	—
130	#97	—	2	1	no	—
131	#97	—	2	1	no	—
132	#97	C₂×C₄	1	1	yes	characteristic
133	#97	C₂×C₄	1	1	yes	characteristic
134	C₄×C₈	—	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	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄	#41	1	1	yes	characteristic
154	Q₁₆	—	2	2	no	—
155	Q₁₆	—	4	1	no	—
156	SD₁₆	—	2	2	no	—
157	SD₁₆	—	4	1	no	—
158	M₁₆	—	2	2	no	—
159	M₁₆	—	4	1	no	—
160	M₁₆	—	8	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	8	1	no	—
168	C₂×C₈	—	8	1	no	—
169	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄	1	1	yes	characteristic
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
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	C₂²⋊C₄	—	2	1	no	—
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	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	C₂²×C₄	1	1	yes	characteristic
203	C₄²	C₂×D₄	1	1	yes	characteristic
204	C₄²	#41	1	1	yes	characteristic
205	C₄²	C₂⁴	1	1	yes	characteristic
206	C₂³	—	2	1	no	—
207	C₂³	C₄×D₄	1	1	yes	characteristic
208	Q₈	—	2	1	no	—
209	Q₈	—	2	1	no	—
210	Q₈	—	2	1	no	—
211	Q₈	—	2	1	no	—
212	Q₈	—	2	1	no	—
213	Q₈	—	2	1	no	—
214	Q₈	—	4	2	no	—
215	D₄	—	2	1	no	—
216	D₄	—	2	1	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	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	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	C₄×D₄	1	1	yes	characteristic
233	C₂×C₄	C₄×D₄	1	1	yes	characteristic
234	C₂×C₄	C₄×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂×#41	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	1	no	—
246	C₈	—	4	1	no	—
247	C₈	—	4	1	no	—
248	C₈	—	4	1	no	—
249	C₈	—	4	1	no	—
250	C₈	—	4	1	no	—
251	C₈	#131	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	C₂×#131	1	1	yes	characteristic
271	C₄	#568	1	1	yes	characteristic
272	C₄	#577	1	1	yes	characteristic
273	C₄	#578	1	1	yes	characteristic
274	C₂	—	4	1	no	—
275	C₂	—	8	1	no	—
276	C₂	#2935	1	1	yes	characteristic
277	C₂	#2947	1	1	yes	characteristic
278	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	#121	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
3. #2986	C₂	semidirect
4. #1593	C₂	semidirect
5. #1590	C₂	semidirect
6. #1588	C₂	semidirect
7. #1587	C₂	semidirect
8. #1584	C₂	semidirect
9. #1582	C₂	semidirect
10. SD₁₆⋊C₈	C₂	semidirect
11. C₈×Q₁₆	C₂	semidirect
12. #1565	C₂	semidirect
14. #1481	C₂	semidirect
15. #1453	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3172	2
#2994	2
#2986	2
#1593	2
#1590	2
#1588	2
#1587	2
#1584	2
#1582	2
SD₁₆⋊C₈	2
C₈×Q₁₆	2
#1565	2
#1482	2
#1481	2
#1453	2