Subgroups — #22341

Order 256.

363 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#126	C₂	1	1	yes	maximal, characteristic
2	#1939	C₂	1	1	yes	maximal, characteristic, complemented
3	#1939	C₂	1	1	yes	maximal, characteristic, complemented
4	#1923	C₂	1	1	yes	maximal, characteristic
5	#1923	C₂	1	1	yes	maximal, characteristic
6	#1923	C₂	1	1	yes	maximal, characteristic
7	#1916	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	#1850	C₂	1	1	yes	maximal, characteristic
11	#1850	C₂	1	1	yes	maximal, characteristic
12	#1831	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈	C₂²	1	1	yes	characteristic
17	C₂×#126	C₄	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈	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×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	#388	C₄	1	1	yes	characteristic
34	#388	C₄	1	1	yes	characteristic
35	#388	C₄	1	1	yes	characteristic
36	#387	C₄	1	1	yes	characteristic
37	#387	C₄	1	1	yes	characteristic
38	#387	C₄	1	1	yes	characteristic
39	#383	C₄	1	1	yes	characteristic
40	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
46	C₄³	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	C₂³×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	—	4	1	no	—
62	C₂²×C₈	—	4	1	no	—
63	C₂²×C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₈	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	#126	C₈	1	8	yes	complemented
72	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²	C₂³	1	1	yes	characteristic
89	C₂×C₄²	C₂³	1	1	yes	characteristic
90	C₂×C₄²	C₂³	1	1	yes	characteristic
91	C₂×C₄²	C₂³	1	1	yes	characteristic
92	C₂×C₄²	C₂³	1	1	yes	characteristic
93	C₂×C₄²	C₂³	1	1	yes	characteristic
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	4	1	no	—
98	C₄⋊C₈	—	4	1	no	—
99	C₄⋊C₈	—	4	1	no	—
100	C₄⋊C₈	—	4	1	no	—
101	C₂²⋊C₈	—	2	2	no	—
102	C₂²⋊C₈	—	4	1	no	—
103	C₂²⋊C₈	—	4	1	no	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	—	4	1	no	—
107	C₄×C₈	—	4	1	no	—
108	C₄×C₈	—	4	1	no	—
109	#95	C₂×C₄	1	1	yes	characteristic
110	#95	C₂×C₄	1	1	yes	characteristic
111	#95	C₂×C₄	1	1	yes	characteristic
112	#95	C₂×C₄	1	1	yes	characteristic
113	#95	C₂×C₄	1	1	yes	characteristic
114	#95	C₂×C₄	1	1	yes	characteristic
115	#95	C₂×C₄	1	1	yes	characteristic
116	#95	C₂×C₄	1	1	yes	characteristic
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	C₂⁴	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄	C₂⁴	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	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	—	4	4	no	—
165	C₂×C₈	#41	1	2	yes	—
166	C₂×C₈	#41	1	2	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄⋊C₄	C₂×C₈	1	4	yes	complemented
171	C₄⋊C₄	C₂×C₈	1	4	yes	complemented
172	C₄⋊C₄	C₂×C₈	1	4	yes	complemented
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	4	1	no	—
175	C₂²⋊C₄	—	4	1	no	—
176	C₂²⋊C₄	C₂×C₈	1	4	yes	—
177	C₂²⋊C₄	C₂×C₈	1	4	yes	—
178	C₂²⋊C₄	C₂×C₈	1	4	yes	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	C₂×C₈	1	4	yes	complemented
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₂²×C₈	1	2	yes	—
202	C₂³	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
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	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	C₂²×C₈	1	2	yes	—
240	C₂×C₄	C₂²×C₈	1	2	yes	—
241	C₂×C₄	C₂²×C₈	1	2	yes	—
242	C₂×C₄	C₂²×C₈	1	2	yes	—
243	C₂×C₄	C₂²×C₈	1	2	yes	—
244	C₂×C₄	C₂²×C₈	1	2	yes	—
245	C₂×C₄	#131	1	2	yes	—
246	C₂×C₄	C₂×#41	1	1	yes	characteristic
247	C₂×C₄	C₂×#41	1	1	yes	characteristic
248	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
249	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
251	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	—	4	1	no	—
255	C₈	—	4	1	no	—
256	C₈	—	4	1	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₄×#41	1	1	yes	characteristic
268	C₂²	#520	1	1	yes	characteristic
269	C₂²	#520	1	1	yes	characteristic
270	C₂²	C₂³×C₈	1	1	yes	characteristic
271	C₂²	C₂×#131	1	1	yes	characteristic
272	C₂²	#568	1	1	yes	characteristic
273	C₂²	#568	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	#538	1	1	yes	characteristic
288	C₄	#539	1	1	yes	characteristic
289	C₄	#541	1	1	yes	characteristic
290	C₄	#542	1	1	yes	characteristic
291	C₂	—	4	2	no	—
292	C₂	#2334	1	1	yes	characteristic
293	C₂	C₈×#41	1	1	yes	characteristic
294	C₂	#2967	1	1	yes	characteristic
295	C₂	#2988	1	1	yes	characteristic
296	C₂	#2988	1	1	yes	characteristic
297	C₂	#2992	1	1	yes	characteristic
298	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	C₂⁴	C₂²×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
2. #1939	C₂	semidirect
3. #1939	C₂	semidirect
7. #1916	C₂	semidirect
8. #1916	C₂	semidirect
9. #1916	C₂	semidirect
12. #1831	C₂	semidirect
13. C₈×C₄⋊C₄	C₂	semidirect
14. C₄×C₄⋊C₈	C₂	semidirect
71. #126	C₈	semidirect
170. C₄⋊C₄	C₂×C₈	semidirect
171. C₄⋊C₄	C₂×C₈	semidirect
172. C₄⋊C₄	C₂×C₈	semidirect
194. C₄²	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×#126	2
#1939	2
#1923	2
#1916	2
#1850	2
#1831	2
C₈×C₄⋊C₄	2
C₄×C₄⋊C₈	2
C₈×C₂²⋊C₄	2