Subgroups — #20478

Order 256.

463 conjugacy classes of subgroups.

The subgroup lattice has 256 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
2	C₂×#405	C₂	1	1	yes	maximal, characteristic
3	#2298	C₂	1	1	yes	maximal, characteristic
4	#1771	C₂	1	1	yes	maximal, characteristic, complemented
5	#1771	C₂	1	1	yes	maximal, characteristic, complemented
6	#1770	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#1753	C₂	1	1	yes	maximal, characteristic
9	#1753	C₂	1	1	yes	maximal, characteristic
10	#1752	C₂	1	1	yes	maximal, characteristic
11	C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	#1746	C₂	1	1	yes	maximal, characteristic, complemented
13	#1744	C₂	1	1	yes	maximal, characteristic, complemented
14	#1726	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#97	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×#117	C₂²	1	1	yes	characteristic
19	#433	C₄	1	8	yes	complemented
20	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	#405	C₄	1	8	yes	complemented
25	C₂×#97	C₂²	1	1	yes	characteristic
26	C₂×#97	C₂²	1	1	yes	characteristic
27	C₂×#97	C₂²	1	1	yes	characteristic
28	C₂×#97	C₂²	1	1	yes	characteristic
29	C₂×#97	C₂²	1	1	yes	characteristic
30	C₂×#97	C₂²	1	1	yes	characteristic
31	C₂×#97	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	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#388	C₄	1	2	yes	—
40	#388	C₄	1	2	yes	—
41	#387	C₄	1	2	yes	—
42	#387	C₄	1	2	yes	—
43	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
47	#376	C₂²	1	1	yes	characteristic
48	C₄³	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	C₂³×C₄	C₂³	1	1	yes	characteristic
58	C₂×M₁₆	—	2	2	no	—
59	C₂×M₁₆	—	2	2	no	—
60	C₂×M₁₆	C₂×C₄	1	4	yes	—
61	C₂²×C₈	—	2	2	no	—
62	C₂²×C₈	—	2	2	no	—
63	C₂²×C₈	C₂×C₄	1	4	yes	—
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	C₂²×C₈	C₂³	1	1	yes	characteristic
72	#117	C₂×C₄	1	4	yes	—
73	#117	C₂×C₄	1	4	yes	—
74	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
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	2	yes	—
79	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×C₄	1	2	yes	—
83	C₂×C₄²	C₂×C₄	1	2	yes	—
84	C₂×C₄²	C₂³	1	1	yes	characteristic
85	C₂×C₄²	C₂³	1	1	yes	characteristic
86	C₂×C₄²	C₂³	1	1	yes	characteristic
87	C₂×C₄²	C₂³	1	1	yes	characteristic
88	C₂×C₄²	C₂³	1	1	yes	characteristic
89	C₂×C₄²	C₂³	1	1	yes	characteristic
90	C₄⋊C₈	C₂×C₄	1	4	yes	—
91	C₄⋊C₈	C₂×C₄	1	4	yes	—
92	C₂²⋊C₈	C₂×C₄	1	4	yes	—
93	C₂²⋊C₈	C₂×C₄	1	4	yes	—
94	#97	—	2	2	no	—
95	#97	—	2	2	no	—
96	#97	—	2	2	no	—
97	#97	—	2	2	no	—
98	#97	C₂×C₄	1	4	yes	—
99	#97	C₂×C₄	1	4	yes	—
100	#97	C₂×C₄	1	4	yes	—
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	C₂×C₄	1	4	yes	—
106	C₄×C₈	C₂×C₄	1	4	yes	—
107	C₄×C₈	C₂×C₄	1	4	yes	—
108	#95	C₂×C₄	1	2	yes	—
109	#95	C₂×C₄	1	2	yes	—
110	#95	C₂×C₄	1	2	yes	—
111	#95	C₂×C₄	1	2	yes	—
112	#95	C₂×C₄	1	2	yes	—
113	#95	C₂×C₄	1	2	yes	—
114	C₂⁴	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	C₂²×C₄	1	2	yes	—
120	C₂²×C₄	C₂²×C₄	1	2	yes	—
121	C₂²×C₄	C₂²×C₄	1	2	yes	—
122	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄	C₂²×C₄	1	2	yes	—
128	C₂²×C₄	C₂²×C₄	1	2	yes	—
129	C₂²×C₄	C₂²×C₄	1	2	yes	—
130	C₂²×C₄	C₂⁴	1	1	yes	characteristic
131	M₁₆	—	2	8	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	C₂²×C₄	1	2	yes	—
146	C₂×C₈	C₂²×C₄	1	2	yes	—
147	C₂×C₈	C₂²×C₄	1	2	yes	—
148	C₂×C₈	C₂²×C₄	1	2	yes	—
149	C₂×C₈	C₂²×C₄	1	2	yes	—
150	C₂×C₈	C₂²×C₄	1	2	yes	—
151	C₂×C₈	C₂²×C₄	1	2	yes	—
152	C₂×C₈	C₂²×C₄	1	2	yes	—
153	C₂×C₈	#41	1	2	yes	—
154	C₂×C₈	#41	1	2	yes	—
155	C₂×C₈	#41	1	2	yes	—
156	C₂×C₈	#41	1	2	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	C₄²	1	4	yes	—
160	C₄⋊C₄	C₄²	1	4	yes	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	C₄²	1	4	yes	—
164	C₂²⋊C₄	C₄²	1	4	yes	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	4	2	no	—
175	C₄²	C₂²×C₄	1	2	yes	—
176	C₄²	C₂²×C₄	1	2	yes	—
177	C₄²	C₂²×C₄	1	2	yes	—
178	C₄²	C₂²×C₄	1	2	yes	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	C₂×C₄²	1	2	yes	—
184	C₂³	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂×C₄²	1	2	yes	—
207	C₂×C₄	C₂×C₄²	1	2	yes	—
208	C₂×C₄	C₂×C₄²	1	2	yes	—
209	C₂×C₄	C₂×C₄²	1	2	yes	—
210	C₂×C₄	C₂×C₄²	1	2	yes	—
211	C₂×C₄	#131	1	4	yes	—
212	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄	C₂×#41	1	1	yes	characteristic
215	C₂×C₄	C₂×#41	1	1	yes	characteristic
216	C₂×C₄	C₂×#41	1	1	yes	characteristic
217	C₂×C₄	C₂×#41	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	#117	1	4	yes	—
225	C₈	#117	1	4	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	C₂²×C₄²	1	1	yes	characteristic
232	C₂²	C₄×#41	1	1	yes	characteristic
233	C₂²	C₄×#41	1	1	yes	characteristic
234	C₂²	C₂×#131	1	2	yes	—
235	C₂²	#568	1	2	yes	—
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	C₂×#117	1	1	yes	characteristic
246	C₄	C₂×#117	1	1	yes	characteristic
247	C₄	C₄×#41	1	1	yes	characteristic
248	C₄	C₄×#41	1	1	yes	characteristic
249	C₂	—	4	2	no	—
250	C₂	C₄×#117	1	1	yes	characteristic
251	C₂	C₄×#131	1	1	yes	characteristic
252	C₂	#2875	1	1	yes	characteristic
253	C₂	#2968	1	2	yes	—
254	C₂	#2969	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator	C₂²	C₂²×C₄²
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
4. #1771	C₂	semidirect
5. #1771	C₂	semidirect
6. #1770	C₂	semidirect
7. C₈×C₄⋊C₄	C₂	semidirect
12. #1746	C₂	semidirect
13. #1744	C₂	semidirect
14. #1726	C₂	semidirect
15. C₄×#97	C₂	semidirect
19. #433	C₄	semidirect
24. #405	C₄	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×#405	2
#2298	2
#1771	2
#1770	2
C₈×C₄⋊C₄	2
#1753	2
#1752	2
C₈×C₂²⋊C₄	2
#1746	2
#1744	2
#1726	2
C₄×#97	2