Subgroups — #72225900

Order 1536.

872 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10713436	C₂	1	4	yes	maximal, complemented
2	#10704180	C₂	1	1	yes	maximal, characteristic, complemented
3	#10688954	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#19603	C₂	1	1	yes	maximal, characteristic
5	#141751	—	3	1	no	maximal, Sylow
6	C₃×#3169	—	2	2	no	—
7	C₃×#3147	C₂²	1	2	yes	—
8	C₃×#1713	C₄	1	1	yes	characteristic, complemented
9	C₃×#1712	C₄	1	1	yes	characteristic, complemented
10	C₃×#1709	C₄	1	2	yes	complemented
11	C₃×#1706	—	2	2	no	—
12	C₂×C₃×C₈²	C₂²	1	1	yes	characteristic
13	C₂×#4566	—	2	8	no	—
14	#69764	—	2	2	no	—
15	#69764	C₂²	1	2	yes	—
16	C₄×#812	C₂²	1	2	yes	—
17	#67438	—	2	2	no	—
18	#19603	D₃	1	1	yes	characteristic
19	#13210	—	3	4	no	—
20	#11402	—	3	1	no	—
21	#8813	—	3	1	no	—
22	C₂×C₃×#135	—	2	4	no	—
23	C₃×#532	—	2	2	no	—
24	C₃×#500	—	2	2	no	—
25	C₃×#495	—	2	4	no	—
26	C₃×#494	C₂×C₄	1	2	yes	—
27	C₃×#493	—	2	2	no	—
28	C₃×#493	C₂×C₄	1	2	yes	—
29	C₃×#492	—	2	2	no	—
30	C₃×#486	—	2	4	no	—
31	C₃×#466	—	4	4	no	—
32	C₃×C₄×D₈	—	4	4	no	—
33	C₂×C₃×C₄×C₈	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₂×#817	—	2	4	no	—
36	C₂×#817	—	4	4	no	—
37	C₂×#812	D₄	1	4	yes	—
38	#4935	—	2	2	no	—
39	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
40	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
41	C₃×C₈²	D₄	1	2	yes	—
42	#4566	—	4	16	no	—
43	#4461	—	2	2	no	—
44	C₄×C₃⋊C₁₆	—	2	2	no	—
45	#3169	—	2	2	no	—
46	#3147	C₂×D₃	1	2	yes	—
47	C₂×#360	—	6	8	no	—
48	C₄×M₃₂	—	3	2	no	—
49	#1844	—	3	2	no	—
50	#1844	—	6	2	no	—
51	#1713	Dic₃	1	1	yes	characteristic, complemented
52	#1712	Dic₃	1	1	yes	characteristic, complemented
53	#1709	Dic₃	1	2	yes	complemented
54	#1706	—	2	2	no	—
55	C₂×C₈²	C₂×D₃	1	1	yes	characteristic
56	#1312	—	6	2	no	—
57	C₂×C₃×#41	—	4	2	no	—
58	C₃×#135	—	4	8	no	—
59	C₃×#135	—	4	8	no	—
60	C₂×C₃×Q₁₆	—	2	4	no	—
61	C₂×C₃×SD₁₆	—	4	4	no	—
62	C₂×C₃×D₈	—	2	4	no	—
63	C₂×C₃×D₈	—	4	4	no	—
64	C₂²×C₃×C₈	D₈	1	4	yes	—
65	C₂²×C₃×C₈	C₂×D₄	1	2	yes	—
66	C₃×C₄⋊Q₈	—	2	2	no	—
67	C₃×#127	—	2	2	no	—
68	C₃×#124	—	4	2	no	—
69	C₃×#121	—	4	4	no	—
70	C₃×C₄×D₄	—	4	4	no	—
71	C₂×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄⋊Dic₃	—	4	2	no	—
73	C₂×C₄⋊Dic₃	—	4	4	no	—
74	C₃×#107	—	4	4	no	—
75	C₃×#102	—	4	8	no	—
76	C₃×C₄×C₈	C₂²⋊C₄	1	2	yes	—
77	C₃×C₄×C₈	C₂²⋊C₄	1	2	yes	—
78	C₃×C₄×C₈	SD₁₆	1	4	yes	—
79	C₃×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄×C₈	#41	1	2	yes	—
82	#817	—	4	8	no	—
83	#817	—	8	8	no	—
84	#812	—	2	8	no	—
85	C₂×C₃⋊C₁₆	—	2	4	no	—
86	C₂×#135	—	2	4	no	—
87	#532	—	2	2	no	—
88	C₂×M₃₂	—	3	4	no	—
89	#500	—	2	2	no	—
90	#495	—	2	4	no	—
91	#494	C₂×Dic₃	1	2	yes	—
92	#493	—	2	2	no	—
93	#493	C₂×Dic₃	1	2	yes	—
94	#492	—	2	2	no	—
95	#486	—	2	4	no	—
96	#466	—	4	4	no	—
97	C₄×D₈	—	4	4	no	—
98	C₂×#106	—	6	4	no	—
99	C₂×#106	—	12	4	no	—
100	C₂×C₄×C₈	#67	1	2	yes	—
101	C₂×C₄×C₈	C₂²×D₃	1	1	yes	characteristic
102	#384	—	6	2	no	—
103	#360	—	12	16	no	—
104	#346	—	6	2	no	—
105	C₄×C₁₆	—	6	2	no	—
106	C₈²	C₂×Dic₃	1	1	yes	characteristic
107	C₈²	C₂×Dic₃	1	1	yes	characteristic
108	C₈²	#67	1	2	yes	—
109	C₃×#41	—	8	4	no	—
110	C₃×#41	—	8	4	no	—
111	C₂×C₃×Q₈	—	4	2	no	—
112	C₂×C₃×D₄	—	4	2	no	—
113	C₂×C₃×D₄	—	4	4	no	—
114	C₂×C₃×D₄	—	8	2	no	—
115	C₂²×C₃×C₄	—	8	2	no	—
116	C₂²×C₃×C₄	#127	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂×D₈	1	2	yes	—
118	C₂²×Dic₃	—	8	2	no	—
119	C₃×Q₁₆	—	4	8	no	—
120	C₃×SD₁₆	—	8	8	no	—
121	C₃×D₈	—	4	8	no	—
122	C₃×D₈	—	8	8	no	—
123	C₂×C₃×C₈	—	2	8	no	—
124	C₂×C₃×C₈	#102	1	4	yes	—
125	C₂×C₃×C₈	#102	1	4	yes	—
126	C₂×C₃×C₈	C₄×D₄	1	2	yes	—
127	C₂×C₃×C₈	#121	1	2	yes	—
128	C₂×C₃×C₈	AGL(1,ℤ/8)	1	8	yes	—
129	C₃×C₄⋊C₄	—	4	4	no	—
130	C₃×C₂²⋊C₄	—	8	4	no	—
131	C₃×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₃×C₄²	#124	1	1	yes	characteristic
133	C₃×C₄²	C₂×SD₁₆	1	2	yes	—
134	C₄⋊Dic₃	—	8	8	no	—
135	C₄⋊Dic₃	—	8	4	no	—
136	C₃⋊C₁₆	—	4	4	no	—
137	C₂×#41	—	4	2	no	—
138	#135	—	4	8	no	—
139	#135	—	4	8	no	—
140	C₂×Q₁₆	—	2	4	no	—
141	C₂×SD₁₆	—	4	4	no	—
142	C₂×D₈	—	2	4	no	—
143	C₂×D₈	—	4	4	no	—
144	C₂²×C₈	#213	1	4	yes	—
145	C₂²×C₈	D₃×D₄	1	2	yes	—
146	C₄⋊Q₈	—	2	2	no	—
147	#127	—	2	2	no	—
148	#124	—	4	2	no	—
149	#121	—	4	4	no	—
150	C₄×D₄	—	4	4	no	—
151	C₂×C₄⋊C₄	—	12	4	no	—
152	C₂×C₄⋊C₄	—	12	2	no	—
153	C₂×C₄²	C₂×#67	1	1	yes	characteristic
154	M₃₂	—	6	8	no	—
155	C₂×C₁₆	—	6	4	no	—
156	#107	—	4	4	no	—
157	#106	—	12	8	no	—
158	#106	—	24	8	no	—
159	#102	—	4	8	no	—
160	C₄×C₈	#214	1	4	yes	—
161	C₄×C₈	#217	1	2	yes	—
162	C₄×C₈	#217	1	2	yes	—
163	C₄×C₈	#237	1	2	yes	—
164	C₄×C₈	C₂²×Dic₃	1	1	yes	characteristic
165	C₄×C₈	C₂×#67	1	1	yes	characteristic
166	C₂³×C₃	—	8	2	no	—
167	C₂³×C₃	#493	1	1	yes	characteristic
168	C₃×Q₈	—	8	4	no	—
169	C₃×D₄	—	8	8	no	—
170	C₃×D₄	—	8	4	no	—
171	C₃×D₄	—	16	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	8	2	no	—
174	C₂×C₃×C₄	—	16	2	no	—
175	C₂×C₃×C₄	—	16	2	no	—
176	C₂×C₃×C₄	#390	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×#102	1	2	yes	—
178	C₂×C₃×C₄	#486	1	2	yes	—
179	C₂×C₃×C₄	#492	1	1	yes	characteristic
180	C₂×C₃×C₄	#496	1	4	yes	—
181	C₂×Dic₃	—	16	4	no	—
182	C₂×Dic₃	—	16	2	no	—
183	C₃×C₈	—	2	4	no	—
184	C₃×C₈	#360	1	8	yes	complemented
185	C₃×C₈	#442	1	4	yes	—
186	C₃×C₈	#469	1	4	yes	—
187	#41	—	8	4	no	—
188	#41	—	8	4	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂×D₄	—	8	2	no	—
193	C₂²×C₄	—	8	2	no	—
194	C₂²×C₄	—	24	2	no	—
195	C₂²×C₄	C₂×#213	1	2	yes	—
196	C₂²×C₄	#940	1	1	yes	characteristic
197	Q₁₆	—	4	8	no	—
198	SD₁₆	—	8	8	no	—
199	D₈	—	4	8	no	—
200	D₈	—	8	8	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	#832	1	4	yes	—
203	C₂×C₈	#832	1	4	yes	—
204	C₂×C₈	#911	1	8	yes	—
205	C₂×C₈	D₄×Dic₃	1	2	yes	—
206	C₂×C₈	#938	1	2	yes	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	24	8	no	—
209	C₄⋊C₄	—	24	4	no	—
210	C₂²⋊C₄	—	8	4	no	—
211	C₄²	C₂×#214	1	2	yes	—
212	C₄²	#936	1	1	yes	characteristic
213	C₄²	C₂×#217	1	1	yes	characteristic
214	C₁₆	—	12	4	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	16	2	no	—
217	C₂²×C₃	—	16	4	no	—
218	C₂²×C₃	#1721	1	2	yes	—
219	C₂²×C₃	#1957	1	1	yes	characteristic
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	16	2	no	—
222	C₃×C₄	#1715	1	2	yes	—
223	C₃×C₄	#1972	1	2	yes	—
224	C₃×C₄	C₂×#360	1	4	yes	—
225	Dic₃	—	32	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	#5073	1	1	yes	characteristic
228	Q₈	—	8	4	no	—
229	D₄	—	8	4	no	—
230	D₄	—	8	8	no	—
231	D₄	—	16	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	16	2	no	—
235	C₂×C₄	—	16	2	no	—
236	C₂×C₄	—	48	4	no	—
237	C₂×C₄	—	48	2	no	—
238	C₂×C₄	#5070	1	2	yes	—
239	C₂×C₄	#5076	1	1	yes	characteristic
240	C₂×C₄	#5154	1	4	yes	—
241	C₂×C₄	C₂×#832	1	2	yes	—
242	C₂×C₄	#5221	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	#4566	1	8	yes	complemented
245	C₈	#5152	1	4	yes	—
246	C₈	#5159	1	4	yes	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	32	2	no	—
249	C₂×C₃	#9837	1	1	yes	characteristic
250	C₂×C₃	#13210	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	16	4	no	—
253	C₂²	—	16	2	no	—
254	C₂²	#71280	1	2	yes	—
255	C₂²	#71764	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	16	2	no	—
258	C₄	—	96	2	no	—
259	C₄	#71277	1	2	yes	—
260	C₄	#72039	1	2	yes	—
261	C₄	C₂×#4566	1	4	yes	—
262	C₃	#141751	1	1	yes	characteristic, complemented, Sylow
263	C₂	—	2	2	no	—
264	C₂	—	32	2	no	—
265	C₂	#10695026	1	1	yes	characteristic
266	C₂	#10713436	1	2	yes	—

	subgroup	G/H
center	C₂²	#71764
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting, prime radical	C₃×#19603	C₂
socle	C₂²×C₃	#1957
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	#19603	D₃
hypocenter	C₃	#141751
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #19603

Complements

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

N	H ≅ G/N	type
1. #10713436	C₂	semidirect
2. #10704180	C₂	semidirect
3. #10688954	C₂	semidirect
8. C₃×#1713	C₄	semidirect
9. C₃×#1712	C₄	semidirect
10. C₃×#1709	C₄	semidirect
51. #1713	Dic₃	semidirect
52. #1712	Dic₃	semidirect
53. #1709	Dic₃	semidirect
184. C₃×C₈	#360	semidirect
244. C₈	#4566	semidirect
262. C₃	#141751	semidirect

Maximal subgroups

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

subgroup	index
#10713436	2
#10704180	2
#10688954	2
C₃×#19603	2
#141751	3