Subgroups — #21272

Order 256.

434 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#433	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	#2290	C₂	1	1	yes	maximal, characteristic
4	#1955	C₂	1	1	yes	maximal, characteristic
5	#1954	C₂	1	1	yes	maximal, characteristic
6	#1940	C₂	1	1	yes	maximal, characteristic, complemented
7	#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	#1924	C₂	1	1	yes	maximal, characteristic, complemented
9	#1924	C₂	1	1	yes	maximal, characteristic, complemented
10	#1923	C₂	1	1	yes	maximal, characteristic, complemented
11	#1923	C₂	1	1	yes	maximal, characteristic, complemented
12	#1768	C₂	1	1	yes	maximal, characteristic, complemented
13	#1750	C₂	1	1	yes	maximal, characteristic, complemented
14	#1749	C₂	1	1	yes	maximal, characteristic, complemented
15	#1726	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
17	C₂×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₄²	C₂²	1	1	yes	characteristic
19	#433	—	2	4	no	—
20	#431	—	2	4	no	—
21	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×#97	C₂²	1	1	yes	characteristic
36	C₂×#97	C₂²	1	1	yes	characteristic
37	C₂×#97	C₂²	1	1	yes	characteristic
38	C₂×#97	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
43	#388	C₄	1	1	yes	characteristic
44	#388	C₄	1	1	yes	characteristic
45	#387	C₄	1	1	yes	characteristic
46	#387	C₄	1	1	yes	characteristic
47	#386	C₄	1	1	yes	characteristic
48	#386	C₄	1	1	yes	characteristic
49	#385	C₄	1	1	yes	characteristic
50	#385	C₄	1	1	yes	characteristic
51	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
53	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
54	#376	C₂²	1	1	yes	characteristic
55	#336	C₂²	1	1	yes	characteristic
56	#336	C₂²	1	1	yes	characteristic
57	#336	C₂²	1	1	yes	characteristic
58	#336	C₂²	1	1	yes	characteristic
59	#336	C₂²	1	1	yes	characteristic
60	#336	C₂²	1	1	yes	characteristic
61	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄	C₂³	1	1	yes	characteristic
65	C₂³×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₈	C₂³	1	1	yes	characteristic
67	C₂²×C₈	C₂³	1	1	yes	characteristic
68	C₂²×C₈	C₂³	1	1	yes	characteristic
69	C₂²×C₈	C₂³	1	1	yes	characteristic
70	C₂²×C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₈	C₂³	1	1	yes	characteristic
74	C₄×D₄	—	2	2	no	—
75	C₄×D₄	—	2	2	no	—
76	#117	—	2	2	no	—
77	#117	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²	C₂³	1	1	yes	characteristic
96	C₂×C₄²	C₂³	1	1	yes	characteristic
97	C₂×C₄²	C₂³	1	1	yes	characteristic
98	C₂×C₄²	C₂³	1	1	yes	characteristic
99	C₂×C₄²	C₂³	1	1	yes	characteristic
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₂²⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₂²⋊C₈	—	2	2	no	—
111	C₂²⋊C₈	—	2	2	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	#97	—	2	2	no	—
115	#97	—	2	2	no	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	D₄	1	4	yes	complemented
122	#95	C₂×C₄	1	1	yes	characteristic
123	#95	C₂×C₄	1	1	yes	characteristic
124	#95	C₂×C₄	1	1	yes	characteristic
125	#95	C₂×C₄	1	1	yes	characteristic
126	#95	C₂×C₄	1	1	yes	characteristic
127	#95	C₂×C₄	1	1	yes	characteristic
128	#95	C₂×C₄	1	1	yes	characteristic
129	#95	C₂×C₄	1	1	yes	characteristic
130	#95	C₂×C₄	1	1	yes	characteristic
131	#95	C₂×C₄	1	1	yes	characteristic
132	C₂⁴	C₂²×C₄	1	1	yes	characteristic
133	C₂⁴	C₂²×C₄	1	1	yes	characteristic
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	—	4	1	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	C₂×D₄	1	2	yes	—
189	C₂×C₈	C₂×D₄	1	2	yes	—
190	C₂×C₈	#41	1	2	yes	—
191	C₂×C₈	#41	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	4	1	no	—
204	C₂²⋊C₄	—	4	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	C₂×D₄	1	2	yes	—
214	C₄²	#41	1	2	yes	—
215	C₄²	#41	1	2	yes	—
216	C₄²	#41	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	#131	1	2	yes	—
231	C₂³	C₂³×C₄	1	1	yes	characteristic
232	D₄	—	4	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	#117	1	2	yes	—
275	C₂×C₄	#117	1	2	yes	—
276	C₂×C₄	C₄×D₄	1	2	yes	—
277	C₂×C₄	C₄×D₄	1	2	yes	—
278	C₂×C₄	#131	1	2	yes	—
279	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄	C₂×#41	1	1	yes	characteristic
281	C₂×C₄	C₂×#41	1	1	yes	characteristic
282	C₂×C₄	C₂×#41	1	1	yes	characteristic
283	C₂×C₄	C₂×#41	1	1	yes	characteristic
284	C₂×C₄	C₂×#41	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	—	2	2	no	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	4	1	no	—
290	C₈	—	4	1	no	—
291	C₈	—	4	1	no	—
292	C₈	—	4	1	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₂×#117	1	1	yes	characteristic
308	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
309	C₂²	C₄×#41	1	1	yes	characteristic
310	C₂²	C₂×#131	1	1	yes	characteristic
311	C₂²	C₂×#131	1	1	yes	characteristic
312	C₂²	#568	1	1	yes	characteristic
313	C₂²	#568	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	#529	1	1	yes	characteristic
327	C₄	#529	1	1	yes	characteristic
328	C₄	#532	1	1	yes	characteristic
329	C₄	#532	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#2293	1	1	yes	characteristic
333	C₂	#2924	1	1	yes	characteristic
334	C₂	#2925	1	1	yes	characteristic
335	C₂	#2929	1	1	yes	characteristic
336	C₂	#2930	1	1	yes	characteristic
337	C₂	#2968	1	1	yes	characteristic
338	C₂	#2969	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator, socle	C₂³	C₂³×C₄
prime radical	C₂²×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#433	C₂	semidirect
2. C₂×#431	C₂	semidirect
6. #1940	C₂	semidirect
7. #1939	C₂	semidirect
8. #1924	C₂	semidirect
9. #1924	C₂	semidirect
10. #1923	C₂	semidirect
11. #1923	C₂	semidirect
12. #1768	C₂	semidirect
13. #1750	C₂	semidirect
14. #1749	C₂	semidirect
15. #1726	C₂	semidirect
121. C₄×C₈	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#433	2
C₂×#431	2
#2290	2
#1955	2
#1954	2
#1940	2
#1939	2
#1924	2
#1923	2
#1768	2
#1750	2
#1749	2
#1726	2