Subgroups — #22357

Order 256.

411 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#123	C₂	1	1	yes	maximal, characteristic
2	#1954	C₂	1	1	yes	maximal, characteristic
3	#1939	C₂	1	1	yes	maximal, characteristic, complemented
4	#1923	C₂	1	1	yes	maximal, characteristic, complemented
5	#1923	C₂	1	1	yes	maximal, characteristic, complemented
6	#1923	C₂	1	1	yes	maximal, characteristic, complemented
7	#1923	C₂	1	1	yes	maximal, characteristic, complemented
8	#1916	C₂	1	1	yes	maximal, characteristic, complemented
9	#1916	C₂	1	1	yes	maximal, characteristic, complemented
10	#1869	C₂	1	1	yes	maximal, characteristic
11	#1850	C₂	1	1	yes	maximal, characteristic, complemented
12	#1797	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₈	C₂²	1	1	yes	characteristic
17	C₂×#123	C₄	1	1	yes	characteristic
18	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
19	C₂²×C₄²	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈	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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#388	C₄	1	1	yes	characteristic
38	#388	C₄	1	1	yes	characteristic
39	#387	C₄	1	1	yes	characteristic
40	#387	C₄	1	1	yes	characteristic
41	#386	C₄	1	1	yes	characteristic
42	#385	C₄	1	1	yes	characteristic
43	#381	C₄	1	1	yes	characteristic
44	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
47	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₂²⋊C₄	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	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	—	4	1	no	—
66	C₂²×C₈	—	4	1	no	—
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	C₂²×C₈	C₂³	1	1	yes	characteristic
75	#123	C₈	1	8	yes	complemented
76	C₄×D₄	—	4	1	no	—
77	C₄×D₄	—	4	1	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	—	4	1	no	—
92	C₂×C₄²	—	4	1	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	—	4	1	no	—
102	C₄⋊C₈	—	4	1	no	—
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₂²⋊C₈	—	2	2	no	—
106	C₂²⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₂²⋊C₈	—	4	1	no	—
109	C₂²⋊C₈	—	4	1	no	—
110	C₂²⋊C₈	—	4	1	no	—
111	C₂²⋊C₈	—	4	1	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	—	4	1	no	—
115	C₄×C₈	—	4	1	no	—
116	C₄×C₈	—	4	1	no	—
117	#95	C₂×C₄	1	1	yes	characteristic
118	#95	C₂×C₄	1	1	yes	characteristic
119	#95	C₂×C₄	1	1	yes	characteristic
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	#95	C₂×C₄	1	1	yes	characteristic
127	C₂⁴	C₂²×C₄	1	1	yes	characteristic
128	C₂⁴	C₂²×C₄	1	1	yes	characteristic
129	C₂×D₄	—	4	1	no	—
130	C₂×D₄	—	4	1	no	—
131	C₂×D₄	C₂×C₈	1	4	yes	—
132	C₂²×C₄	—	2	2	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	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	C₂×C₈	1	4	yes	complemented
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	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	#41	1	2	yes	—
187	C₂×C₈	#41	1	2	yes	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	C₂×C₈	1	4	yes	complemented
192	C₄⋊C₄	C₂×C₈	1	4	yes	complemented
193	C₂²⋊C₄	—	2	2	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	C₂×C₈	1	4	yes	—
199	C₂²⋊C₄	C₂×C₈	1	4	yes	—
200	C₂²⋊C₄	C₂×C₈	1	4	yes	complemented
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	C₂²×C₈	1	2	yes	—
226	C₂³	C₂²×C₈	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	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	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	C₂²×C₈	1	2	yes	—
270	C₂×C₄	C₂²×C₈	1	2	yes	—
271	C₂×C₄	C₂²×C₈	1	2	yes	—
272	C₂×C₄	C₂²×C₈	1	2	yes	—
273	C₂×C₄	C₂²×C₈	1	2	yes	—
274	C₂×C₄	#131	1	2	yes	—
275	C₂×C₄	C₂×#41	1	1	yes	characteristic
276	C₂×C₄	C₂×#41	1	1	yes	characteristic
277	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
278	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
279	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	2	2	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	4	1	no	—
288	C₈	—	4	1	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	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	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	C₄×#41	1	1	yes	characteristic
304	C₂²	#518	1	1	yes	characteristic
305	C₂²	#520	1	1	yes	characteristic
306	C₂²	C₂³×C₈	1	1	yes	characteristic
307	C₂²	C₂×#131	1	1	yes	characteristic
308	C₂²	#568	1	1	yes	characteristic
309	C₂²	#568	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	#538	1	1	yes	characteristic
323	C₄	#539	1	1	yes	characteristic
324	C₄	#540	1	1	yes	characteristic
325	C₄	#542	1	1	yes	characteristic
326	C₂	—	4	2	no	—
327	C₂	—	4	2	no	—
328	C₂	#2335	1	1	yes	characteristic
329	C₂	C₈×#41	1	1	yes	characteristic
330	C₂	#2967	1	1	yes	characteristic
331	C₂	#2972	1	1	yes	characteristic
332	C₂	#2976	1	1	yes	characteristic
333	C₂	#2988	1	1	yes	characteristic
334	C₂	#2993	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator	C₂²	C₂³×C₈
socle	C₂³	C₂³×C₄
prime radical	C₂²×D₄	C₂×C₄
LCS	G ▷ C₂² ▷ 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
3. #1939	C₂	semidirect
4. #1923	C₂	semidirect
5. #1923	C₂	semidirect
6. #1923	C₂	semidirect
7. #1923	C₂	semidirect
8. #1916	C₂	semidirect
9. #1916	C₂	semidirect
11. #1850	C₂	semidirect
12. #1797	C₂	semidirect
13. C₈×C₄⋊C₄	C₂	semidirect
14. C₈×C₂²⋊C₄	C₂	semidirect
15. C₄×C₂²⋊C₈	C₂	semidirect
75. #123	C₈	semidirect
149. C₂²×C₄	C₂×C₈	semidirect
191. C₄⋊C₄	C₂×C₈	semidirect
192. C₄⋊C₄	C₂×C₈	semidirect
200. C₂²⋊C₄	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#123	2
#1954	2
#1939	2
#1923	2
#1916	2
#1869	2
#1850	2
#1797	2
C₈×C₄⋊C₄	2
C₈×C₂²⋊C₄	2
C₄×C₂²⋊C₈	2