Subgroups — #22005

Order 256.

376 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2293	C₂	1	1	yes	maximal, characteristic
2	#1924	C₂	1	1	yes	maximal, characteristic, complemented
3	#1923	C₂	1	1	yes	maximal, characteristic, complemented
4	#1917	C₂	1	1	yes	maximal, characteristic, complemented
5	#1916	C₂	1	1	yes	maximal, characteristic, complemented
6	#1870	C₂	1	1	yes	maximal, characteristic
7	#1869	C₂	1	1	yes	maximal, characteristic
8	#1851	C₂	1	1	yes	maximal, characteristic, complemented
9	#1850	C₂	1	1	yes	maximal, characteristic, complemented
10	#1839	C₂	1	1	yes	maximal, characteristic, complemented
11	#1831	C₂	1	1	yes	maximal, characteristic, complemented
12	#1818	C₂	1	1	yes	maximal, characteristic, complemented
13	#1798	C₂	1	1	yes	maximal, characteristic, complemented
14	#1750	C₂	1	1	yes	maximal, characteristic, complemented
15	#1729	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆	C₂²	1	1	yes	characteristic
17	C₂³×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
19	C₂×#117	C₂²	1	1	yes	characteristic
20	C₂×#117	C₂²	1	1	yes	characteristic
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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊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×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	#390	C₄	1	1	yes	characteristic
34	#390	C₄	1	1	yes	characteristic
35	#389	C₄	1	1	yes	characteristic
36	#389	C₄	1	1	yes	characteristic
37	#388	C₄	1	1	yes	characteristic
38	#388	C₄	1	1	yes	characteristic
39	#388	C₄	1	1	yes	characteristic
40	#388	C₄	1	1	yes	characteristic
41	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
44	C₄³	C₂²	1	1	yes	characteristic
45	#336	C₂²	1	1	yes	characteristic
46	#336	C₂²	1	1	yes	characteristic
47	#336	C₂²	1	1	yes	characteristic
48	#336	C₂²	1	1	yes	characteristic
49	#336	C₂²	1	1	yes	characteristic
50	#336	C₂²	1	1	yes	characteristic
51	#336	C₂²	1	1	yes	characteristic
52	#336	C₂²	1	1	yes	characteristic
53	#336	C₂²	1	1	yes	characteristic
54	#336	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	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄	C₂³	1	1	yes	characteristic
61	C₂³×C₄	C₂³	1	1	yes	characteristic
62	C₂×M₁₆	—	2	2	no	—
63	C₂×M₁₆	—	4	1	no	—
64	C₂×M₁₆	—	4	1	no	—
65	C₂²×C₈	—	2	2	no	—
66	C₂²×C₈	—	4	1	no	—
67	C₂²×C₈	—	4	1	no	—
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	C₂²×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₈	C₂³	1	1	yes	characteristic
76	C₄×D₄	—	2	2	no	—
77	C₄×D₄	—	2	2	no	—
78	#117	—	4	1	no	—
79	#117	—	4	1	no	—
80	#117	—	4	1	no	—
81	#117	—	4	1	no	—
82	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄⋊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×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
98	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂³	1	1	yes	characteristic
104	C₂×C₄²	C₂³	1	1	yes	characteristic
105	C₂×C₄²	C₂³	1	1	yes	characteristic
106	C₂×C₄²	C₂³	1	1	yes	characteristic
107	C₂×C₄²	C₂³	1	1	yes	characteristic
108	C₄⋊C₈	—	4	1	no	—
109	C₄⋊C₈	—	4	1	no	—
110	C₄⋊C₈	—	4	1	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₂²⋊C₈	—	4	1	no	—
115	C₂²⋊C₈	—	4	1	no	—
116	C₂²⋊C₈	—	4	1	no	—
117	C₂²⋊C₈	—	4	1	no	—
118	#97	—	4	1	no	—
119	C₄×C₈	—	4	1	no	—
120	#95	C₂×C₄	1	1	yes	characteristic
121	#95	C₂×C₄	1	1	yes	characteristic
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	C₂⁴	C₂²×C₄	1	1	yes	characteristic
127	C₂⁴	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	—	4	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄	#41	1	2	yes	—
154	C₂²×C₄	#41	1	2	yes	—
155	C₂²×C₄	C₂⁴	1	1	yes	characteristic
156	M₁₆	—	4	4	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	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	—	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	—	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	—	4	4	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	4	1	no	—
194	C₂²⋊C₄	—	4	1	no	—
195	C₂²⋊C₄	—	4	1	no	—
196	C₂²⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	1	no	—
198	C₂²⋊C₄	—	4	1	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
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	#117	1	2	yes	—
227	C₂³	C₂³×C₄	1	1	yes	characteristic
228	D₄	—	4	4	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	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	—	2	1	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	#117	1	2	yes	—
271	C₂×C₄	#131	1	2	yes	—
272	C₂×C₄	#131	1	2	yes	—
273	C₂×C₄	C₂×#41	1	1	yes	characteristic
274	C₂×C₄	C₂×#41	1	1	yes	characteristic
275	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
277	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
278	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	4	1	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	4	1	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	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	C₂×#117	1	1	yes	characteristic
302	C₂²	#518	1	1	yes	characteristic
303	C₂²	#520	1	1	yes	characteristic
304	C₂²	C₂×#131	1	1	yes	characteristic
305	C₂²	C₂×#131	1	1	yes	characteristic
306	C₂²	#568	1	1	yes	characteristic
307	C₂²	#568	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	#538	1	1	yes	characteristic
321	C₄	#538	1	1	yes	characteristic
322	C₄	#539	1	1	yes	characteristic
323	C₄	#539	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	8	1	no	—
326	C₂	#2318	1	1	yes	characteristic
327	C₂	#2924	1	1	yes	characteristic
328	C₂	#2925	1	1	yes	characteristic
329	C₂	#2979	1	1	yes	characteristic
330	C₂	#2980	1	1	yes	characteristic
331	C₂	#2992	1	1	yes	characteristic
332	C₂	#2993	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
2. #1924	C₂	semidirect
3. #1923	C₂	semidirect
4. #1917	C₂	semidirect
5. #1916	C₂	semidirect
8. #1851	C₂	semidirect
9. #1850	C₂	semidirect
10. #1839	C₂	semidirect
11. #1831	C₂	semidirect
12. #1818	C₂	semidirect
13. #1798	C₂	semidirect
14. #1750	C₂	semidirect
15. #1729	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2293	2
#1924	2
#1923	2
#1917	2
#1916	2
#1870	2
#1869	2
#1851	2
#1850	2
#1839	2
#1831	2
#1818	2
#1798	2
#1750	2
#1729	2