Subgroups — #18009

Order 256.

297 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3107	C₂	1	1	yes	maximal, characteristic
2	#2970	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	#1591	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	#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	#1573	C₂	1	1	yes	maximal, characteristic, complemented
13	#1497	C₂	1	1	yes	maximal, characteristic, complemented
14	#1477	C₂	1	1	yes	maximal, characteristic
15	#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	#533	C₂²	1	1	yes	characteristic
17	C₄×#41	C₂²	1	1	yes	characteristic
18	#485	C₄	1	1	yes	characteristic
19	#485	C₄	1	1	yes	characteristic
20	#475	C₄	1	1	yes	characteristic
21	#474	C₄	1	1	yes	characteristic
22	#462	C₄	1	1	yes	characteristic
23	#461	C₄	1	1	yes	characteristic
24	#452	C₄	1	1	yes	characteristic
25	#452	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	#446	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	#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	#432	—	2	1	no	—
43	#432	C₂²	1	1	yes	characteristic
44	#431	C₂²	1	1	yes	characteristic
45	#431	C₂²	1	1	yes	characteristic
46	#424	—	2	1	no	—
47	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	#405	—	2	1	no	—
49	C₄×M₁₆	C₂²	1	1	yes	characteristic
50	C₂×#97	—	2	1	no	—
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×#41	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₁₆	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×M₁₆	D₄	1	1	yes	characteristic
73	C₂×M₁₆	D₄	1	1	yes	characteristic
74	C₂²×C₈	—	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	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈	C₂³	1	1	yes	characteristic
86	C₄×Q₈	C₂³	1	1	yes	characteristic
87	C₄×Q₈	C₂³	1	1	yes	characteristic
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	#117	—	2	1	no	—
93	#117	—	2	1	no	—
94	#117	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²	—	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	C₂×C₄	1	1	yes	characteristic
110	C₄⋊C₈	C₂×C₄	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	C₂×C₄	1	1	yes	characteristic
128	C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
129	#97	—	2	1	no	—
130	#97	—	2	1	no	—
131	#97	—	2	1	no	—
132	#97	—	2	1	no	—
133	#97	—	2	1	no	—
134	#97	D₄	1	1	yes	characteristic
135	#97	D₄	1	1	yes	characteristic
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	—	2	1	no	—
147	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄	#41	1	1	yes	characteristic
158	Q₁₆	—	4	1	no	—
159	Q₁₆	—	4	1	no	—
160	SD₁₆	—	2	2	no	—
161	SD₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	4	1	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	—	4	2	no	—
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	C₂²×C₄	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	C₂×C₈	C₂×D₄	1	1	yes	characteristic
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	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	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
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	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	C₂²×C₄	1	1	yes	characteristic
205	C₄²	C₂×D₄	1	1	yes	characteristic
206	C₄²	#41	1	1	yes	characteristic
207	C₄²	C₂⁴	1	1	yes	characteristic
208	C₂³	—	2	1	no	—
209	C₂³	C₄×D₄	1	1	yes	characteristic
210	Q₈	—	2	1	no	—
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	—	8	1	no	—
217	D₄	—	2	1	no	—
218	D₄	—	4	2	no	—
219	D₄	#131	1	2	yes	—
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	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	C₄×D₄	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄	1	1	yes	characteristic
239	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×#41	1	1	yes	characteristic
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	—	2	1	no	—
249	C₈	—	4	1	no	—
250	C₈	—	4	1	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	1	no	—
255	C₈	#137	1	2	yes	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	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	—	4	1	no	—
274	C₄	C₂×#131	1	1	yes	characteristic
275	C₄	#568	1	1	yes	characteristic
276	C₄	C₂×#137	1	1	yes	characteristic
277	C₄	#575	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	4	2	no	—
280	C₂	#2933	1	1	yes	characteristic
281	C₂	C₄×#137	1	1	yes	characteristic
282	C₂	#2957	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×#41	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. #1591	C₂	semidirect
6. #1587	C₂	semidirect
7. #1585	C₂	semidirect
8. #1584	C₂	semidirect
9. #1582	C₂	semidirect
10. SD₁₆⋊C₈	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
12. #1573	C₂	semidirect
13. #1497	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
#3107	2
#2970	2
#2962	2
#1593	2
#1591	2
#1587	2
#1585	2
#1584	2
#1582	2
SD₁₆⋊C₈	2
C₈×SD₁₆	2
#1573	2
#1497	2
#1477	2
#1453	2