Subgroups — #17340

Order 256.

352 conjugacy classes of subgroups.

The subgroup lattice has 251 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2965	C₂	1	1	yes	maximal, characteristic
2	#2965	C₂	1	1	yes	maximal, characteristic
3	#2922	C₂	1	1	yes	maximal, characteristic
4	#1455	C₂	1	1	yes	maximal, characteristic, complemented
5	#1454	C₂	1	1	yes	maximal, characteristic, complemented
6	#1454	C₂	1	1	yes	maximal, characteristic, complemented
7	#1453	C₂	1	1	yes	maximal, characteristic, complemented
8	#1453	C₂	1	1	yes	maximal, characteristic, complemented
9	#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	#1452	C₂	1	1	yes	maximal, characteristic, complemented
11	#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	#1451	C₂	1	1	yes	maximal, characteristic, complemented
13	#1450	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#41	C₂²	1	1	yes	characteristic
17	#446	C₄	1	2	yes	—
18	#446	C₄	1	2	yes	—
19	#436	C₄	1	2	yes	—
20	#436	C₄	1	2	yes	—
21	#435	C₄	1	2	yes	—
22	#435	C₄	1	2	yes	—
23	#435	C₄	1	2	yes	—
24	#435	C₄	1	2	yes	—
25	#433	C₄	1	2	yes	—
26	#433	C₄	1	2	yes	—
27	#431	C₂²	1	1	yes	characteristic
28	#431	C₂²	1	1	yes	characteristic
29	#431	C₂²	1	1	yes	characteristic
30	#431	C₂²	1	1	yes	characteristic
31	#431	C₂²	1	1	yes	characteristic
32	#431	C₂²	1	1	yes	characteristic
33	#431	C₂²	1	1	yes	characteristic
34	#431	C₂²	1	1	yes	characteristic
35	#431	C₂²	1	1	yes	characteristic
36	#431	C₂²	1	1	yes	characteristic
37	#424	C₄	1	2	yes	—
38	#408	C₄	1	2	yes	—
39	C₄×M₁₆	C₂²	1	1	yes	characteristic
40	C₄×M₁₆	C₂²	1	1	yes	characteristic
41	C₄×M₁₆	C₂²	1	1	yes	characteristic
42	C₄×M₁₆	C₂²	1	1	yes	characteristic
43	C₄×M₁₆	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
47	#322	C₂²	1	1	yes	characteristic
48	#322	C₂²	1	1	yes	characteristic
49	#322	C₂²	1	1	yes	characteristic
50	#322	C₂²	1	1	yes	characteristic
51	#322	C₂²	1	1	yes	characteristic
52	#322	C₂²	1	1	yes	characteristic
53	#322	C₂²	1	1	yes	characteristic
54	#322	C₂²	1	1	yes	characteristic
55	#322	C₂²	1	1	yes	characteristic
56	#322	C₂²	1	1	yes	characteristic
57	#322	C₂²	1	1	yes	characteristic
58	#322	C₂²	1	1	yes	characteristic
59	C₈²	C₂²	1	1	yes	characteristic
60	C₈²	C₂²	1	1	yes	characteristic
61	C₈²	C₂²	1	1	yes	characteristic
62	C₈²	C₂²	1	1	yes	characteristic
63	C₂×#41	C₂×C₄	1	1	yes	characteristic
64	C₂×M₁₆	C₂×C₄	1	2	yes	—
65	C₂×M₁₆	C₂×C₄	1	2	yes	—
66	C₂×M₁₆	C₂×C₄	1	2	yes	—
67	C₂×M₁₆	C₂×C₄	1	2	yes	—
68	C₂×M₁₆	C₂×C₄	1	2	yes	—
69	C₂²×C₈	C₂×C₄	1	2	yes	—
70	C₂²×C₈	C₂×C₄	1	2	yes	—
71	C₂²×C₈	C₂×C₄	1	2	yes	—
72	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
73	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
74	C₄×D₄	C₂×C₄	1	1	yes	characteristic
75	C₄×D₄	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄	C₂×C₄	1	1	yes	characteristic
77	C₄×D₄	C₂×C₄	1	1	yes	characteristic
78	C₄×D₄	C₂×C₄	1	1	yes	characteristic
79	C₄×D₄	C₂×C₄	1	1	yes	characteristic
80	#117	C₂×C₄	1	1	yes	characteristic
81	#117	C₂×C₄	1	1	yes	characteristic
82	#117	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²	C₂³	1	1	yes	characteristic
84	C₂×C₄²	C₂³	1	1	yes	characteristic
85	C₂×C₄²	C₂³	1	1	yes	characteristic
86	C₄⋊C₈	C₂×C₄	1	2	yes	—
87	C₄⋊C₈	C₂×C₄	1	2	yes	—
88	C₄⋊C₈	C₂×C₄	1	2	yes	—
89	C₄⋊C₈	C₂×C₄	1	2	yes	—
90	C₄⋊C₈	C₂×C₄	1	2	yes	—
91	C₄⋊C₈	C₂×C₄	1	2	yes	—
92	C₄⋊C₈	C₂×C₄	1	2	yes	—
93	C₄⋊C₈	C₂×C₄	1	2	yes	—
94	C₄⋊C₈	C₂×C₄	1	2	yes	—
95	C₄⋊C₈	C₂×C₄	1	2	yes	—
96	C₂²⋊C₈	C₂×C₄	1	2	yes	—
97	C₂²⋊C₈	C₂×C₄	1	2	yes	—
98	C₂²⋊C₈	C₂×C₄	1	2	yes	—
99	C₂²⋊C₈	C₂×C₄	1	2	yes	—
100	C₂²⋊C₈	C₂×C₄	1	2	yes	—
101	C₂²⋊C₈	C₂×C₄	1	2	yes	—
102	C₂²⋊C₈	C₂×C₄	1	2	yes	—
103	C₂²⋊C₈	C₂×C₄	1	2	yes	—
104	C₂²⋊C₈	C₂×C₄	1	2	yes	—
105	C₂²⋊C₈	C₂×C₄	1	2	yes	—
106	#97	C₂×C₄	1	2	yes	—
107	#97	C₂×C₄	1	2	yes	—
108	#97	C₂×C₄	1	2	yes	—
109	#97	C₂×C₄	1	2	yes	—
110	#97	C₂×C₄	1	2	yes	—
111	C₄×C₈	C₂×C₄	1	2	yes	—
112	C₄×C₈	C₂×C₄	1	2	yes	—
113	C₄×C₈	C₂×C₄	1	2	yes	—
114	C₄×C₈	C₂³	1	1	yes	characteristic
115	C₄×C₈	C₂³	1	1	yes	characteristic
116	C₄×C₈	C₂³	1	1	yes	characteristic
117	C₄×C₈	C₂³	1	1	yes	characteristic
118	C₄×C₈	C₂³	1	1	yes	characteristic
119	C₄×C₈	C₂³	1	1	yes	characteristic
120	C₄×C₈	C₂³	1	1	yes	characteristic
121	C₄×C₈	C₂³	1	1	yes	characteristic
122	C₄×C₈	C₂³	1	1	yes	characteristic
123	C₄×C₈	C₂³	1	1	yes	characteristic
124	C₄×C₈	C₂³	1	1	yes	characteristic
125	C₄×C₈	C₂³	1	1	yes	characteristic
126	#41	—	4	2	no	—
127	C₂×Q₈	C₄²	1	1	yes	characteristic
128	C₂×D₄	C₄²	1	1	yes	characteristic
129	C₂×D₄	C₄²	1	1	yes	characteristic
130	C₂×D₄	C₄²	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	M₁₆	—	2	4	no	—
141	M₁₆	—	2	4	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	4	2	no	—
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	C₂²×C₄	1	2	yes	—
154	C₂×C₈	C₂²×C₄	1	2	yes	—
155	C₂×C₈	C₂²×C₄	1	2	yes	—
156	C₂×C₈	C₂²×C₄	1	2	yes	—
157	C₂×C₈	C₂²×C₄	1	2	yes	—
158	C₂×C₈	C₂²×C₄	1	2	yes	—
159	C₂×C₈	C₂²×C₄	1	2	yes	—
160	C₄⋊C₄	C₄²	1	1	yes	characteristic
161	C₄⋊C₄	C₄²	1	1	yes	characteristic
162	C₄⋊C₄	C₄²	1	1	yes	characteristic
163	C₄⋊C₄	C₄²	1	1	yes	characteristic
164	C₄⋊C₄	C₄²	1	1	yes	characteristic
165	C₄⋊C₄	C₄²	1	1	yes	characteristic
166	C₂²⋊C₄	C₄²	1	1	yes	characteristic
167	C₂²⋊C₄	C₄²	1	1	yes	characteristic
168	C₂²⋊C₄	C₄²	1	1	yes	characteristic
169	C₂²⋊C₄	C₄²	1	1	yes	characteristic
170	C₂²⋊C₄	C₄²	1	1	yes	characteristic
171	C₂²⋊C₄	C₄²	1	1	yes	characteristic
172	C₄²	C₂²×C₄	1	1	yes	characteristic
173	C₄²	C₂²×C₄	1	1	yes	characteristic
174	C₄²	C₂²×C₄	1	1	yes	characteristic
175	C₄²	C₂²×C₄	1	1	yes	characteristic
176	C₄²	C₂²×C₄	1	1	yes	characteristic
177	C₄²	C₂²×C₄	1	1	yes	characteristic
178	C₄²	C₂²×C₄	1	1	yes	characteristic
179	C₄²	C₂²×C₄	1	1	yes	characteristic
180	C₄²	C₂²×C₄	1	1	yes	characteristic
181	C₄²	C₂⁴	1	1	yes	characteristic
182	C₂³	C₂×C₄²	1	1	yes	characteristic
183	C₂³	C₂×C₄²	1	1	yes	characteristic
184	C₂³	C₂×C₄²	1	1	yes	characteristic
185	Q₈	—	4	1	no	—
186	D₄	—	4	1	no	—
187	D₄	—	4	1	no	—
188	D₄	—	4	1	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	C₂×C₄²	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
207	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	#131	1	4	yes	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂²×C₄²	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	C₂×#131	1	2	yes	—
242	C₄	#568	1	2	yes	—
243	C₄	#568	1	2	yes	—
244	C₂	—	4	1	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	#2873	1	1	yes	characteristic
248	C₂	C₄×#131	1	1	yes	characteristic
249	C₂	#2875	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁴
commutator, socle	C₂²	C₂²×C₄²
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1455	C₂	semidirect
5. #1454	C₂	semidirect
6. #1454	C₂	semidirect
7. #1453	C₂	semidirect
8. #1453	C₂	semidirect
9. #1453	C₂	semidirect
10. #1452	C₂	semidirect
11. #1451	C₂	semidirect
12. #1451	C₂	semidirect
13. #1450	C₂	semidirect
14. C₈×M₁₆	C₂	semidirect
15. C₈×M₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2965	2
#2922	2
#1455	2
#1454	2
#1453	2
#1452	2
#1451	2
#1450	2
C₈×M₁₆	2