Subgroups — #18079

Order 256.

294 conjugacy classes of subgroups.

The subgroup lattice has 283 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	#2969	C₂	1	1	yes	maximal, characteristic
3	#2962	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	#1587	C₂	1	1	yes	maximal, characteristic, complemented
7	#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	#1583	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	#1485	C₂	1	1	yes	maximal, characteristic, complemented
14	#1483	C₂	1	1	yes	maximal, characteristic
15	#1455	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	—	2	1	no	—
28	#446	C₂²	1	1	yes	characteristic
29	#446	C₂²	1	1	yes	characteristic
30	#446	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
39	#436	C₂²	1	1	yes	characteristic
40	#435	—	2	1	no	—
41	C₈×D₄	C₂²	1	1	yes	characteristic
42	#433	—	2	1	no	—
43	#431	C₂²	1	1	yes	characteristic
44	#424	—	2	1	no	—
45	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	#405	—	2	1	no	—
47	#405	—	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
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	—	2	1	no	—
73	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
74	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆	D₄	1	1	yes	characteristic
76	C₂×M₁₆	D₄	1	1	yes	characteristic
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	—	2	1	no	—
79	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
81	#126	—	2	1	no	—
82	#125	—	2	1	no	—
83	#125	C₂×C₄	1	1	yes	characteristic
84	#124	C₂×C₄	1	1	yes	characteristic
85	#123	—	2	1	no	—
86	C₂²⋊Q₈	—	2	1	no	—
87	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
90	#121	C₂×C₄	1	1	yes	characteristic
91	C₄×Q₈	—	2	1	no	—
92	C₄×Q₈	C₂³	1	1	yes	characteristic
93	C₄×Q₈	C₂³	1	1	yes	characteristic
94	C₄×Q₈	C₂³	1	1	yes	characteristic
95	C₄×D₄	—	2	1	no	—
96	C₄×D₄	C₂³	1	1	yes	characteristic
97	#117	—	2	1	no	—
98	#117	—	2	1	no	—
99	C₂×C₄²	C₂³	1	1	yes	characteristic
100	#107	C₂×C₄	1	1	yes	characteristic
101	#107	C₂×C₄	1	1	yes	characteristic
102	#106	C₂×C₄	1	1	yes	characteristic
103	#106	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
110	C₄⋊C₈	—	2	1	no	—
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	#97	—	2	1	no	—
127	#97	—	2	1	no	—
128	#97	—	2	1	no	—
129	#97	—	2	1	no	—
130	#97	C₂×C₄	1	1	yes	characteristic
131	#97	C₂×C₄	1	1	yes	characteristic
132	#97	D₄	1	1	yes	characteristic
133	#97	D₄	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	—	2	2	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	4	1	no	—
162	M₁₆	—	4	2	no	—
163	M₁₆	—	4	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
193	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
194	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
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	C₂²⋊C₄	—	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	C₄×D₄	1	1	yes	characteristic
211	Q₈	—	2	1	no	—
212	Q₈	—	2	1	no	—
213	Q₈	—	2	1	no	—
214	Q₈	—	2	1	no	—
215	Q₈	—	2	1	no	—
216	Q₈	—	2	1	no	—
217	Q₈	—	8	1	no	—
218	D₄	—	2	1	no	—
219	D₄	—	2	1	no	—
220	D₄	—	8	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	—	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	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	C₄×D₄	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄	C₂×#41	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	4	1	no	—
249	C₈	—	4	1	no	—
250	C₈	—	4	1	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₈	#131	1	2	yes	—
254	C₈	#137	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	C₂×#131	1	1	yes	characteristic
274	C₄	#568	1	1	yes	characteristic
275	C₄	C₂×#137	1	1	yes	characteristic
276	C₄	#575	1	1	yes	characteristic
277	C₂	—	4	1	no	—
278	C₂	—	8	1	no	—
279	C₂	#2935	1	1	yes	characteristic
280	C₂	C₄×#137	1	1	yes	characteristic
281	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	#121	C₂×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
3. #2962	C₂	semidirect
4. #1593	C₂	semidirect
5. #1590	C₂	semidirect
6. #1587	C₂	semidirect
7. #1585	C₂	semidirect
8. #1584	C₂	semidirect
9. #1583	C₂	semidirect
10. SD₁₆⋊C₈	C₂	semidirect
11. C₈×Q₁₆	C₂	semidirect
12. #1565	C₂	semidirect
13. #1485	C₂	semidirect
15. #1455	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
#2969	2
#2962	2
#1593	2
#1590	2
#1587	2
#1585	2
#1584	2
#1583	2
SD₁₆⋊C₈	2
C₈×Q₁₆	2
#1565	2
#1485	2
#1483	2
#1455	2