Subgroups — #72225903

Order 1536.

1016 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10713448	C₂	1	4	yes	maximal, complemented
2	#10704180	C₂	1	1	yes	maximal, characteristic, complemented
3	#10688988	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#1712	C₂	1	1	yes	maximal, characteristic
5	#141754	—	3	1	no	maximal, Sylow
6	C₂×C₃×#493	—	2	2	no	—
7	C₂×C₃×#493	C₂²	1	2	yes	—
8	C₃×#1712	—	2	2	no	—
9	C₃×#1712	C₄	1	2	yes	—
10	C₃×#1712	C₄	1	2	yes	—
11	C₂×C₃×C₈²	C₂²	1	1	yes	characteristic
12	C₂×#4563	—	2	8	no	—
13	#69792	C₂²	1	2	yes	—
14	C₄×#812	C₂²	1	2	yes	—
15	#69730	—	2	2	no	—
16	#67438	—	2	2	no	—
17	C₂×#1712	D₃	1	1	yes	characteristic
18	#13213	—	3	4	no	—
19	#11402	—	3	1	no	—
20	#8822	—	3	1	no	—
21	C₂²×C₃×D₈	—	2	4	no	—
22	C₂²×C₃×D₈	—	4	4	no	—
23	C₂×C₃×#127	—	2	2	no	—
24	C₃×#493	—	2	4	no	—
25	C₃×#493	—	2	4	no	—
26	C₃×#493	—	2	2	no	—
27	C₃×#493	—	2	4	no	—
28	C₃×#493	—	4	4	no	—
29	C₃×#493	C₂×C₄	1	4	yes	—
30	C₂×C₃×C₄×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
32	C₂×#819	—	2	4	no	—
33	C₂×#812	D₄	1	4	yes	—
34	#4924	—	2	2	no	—
35	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈²	D₄	1	2	yes	—
38	#4563	—	4	16	no	—
39	#4461	—	2	2	no	—
40	C₄×C₃⋊C₁₆	—	2	2	no	—
41	#4457	—	4	4	no	—
42	C₂×#493	—	2	2	no	—
43	C₂×#493	C₂×D₃	1	2	yes	—
44	C₂×#361	—	6	8	no	—
45	C₄×M₃₂	—	3	2	no	—
46	#1848	—	3	2	no	—
47	#1712	—	2	2	no	—
48	#1712	Dic₃	1	2	yes	—
49	#1712	Dic₃	1	2	yes	—
50	#1566	—	6	2	no	—
51	C₂×C₈²	C₂×D₃	1	1	yes	characteristic
52	#1312	—	6	2	no	—
53	C₂²×C₃×D₄	—	4	2	no	—
54	C₂²×C₃×D₄	—	4	4	no	—
55	C₂×C₃×D₈	—	2	8	no	—
56	C₂×C₃×D₈	—	4	8	no	—
57	C₂×C₃×D₈	—	4	8	no	—
58	C₂×C₃×D₈	—	4	4	no	—
59	C₂×C₃×D₈	—	4	8	no	—
60	C₂×C₃×D₈	—	4	8	no	—
61	C₂×C₃×D₈	—	8	16	no	—
62	C₂²×C₃×C₈	SD₁₆	1	4	yes	—
63	C₂²×C₃×C₈	C₂×D₄	1	2	yes	—
64	C₃×#127	—	2	4	no	—
65	C₃×#127	—	4	2	no	—
66	C₃×#127	—	4	4	no	—
67	C₂×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
68	C₂×#208	—	4	2	no	—
69	C₃×C₄×C₈	C₂²⋊C₄	1	2	yes	—
70	C₃×C₄×C₈	C₂²⋊C₄	1	2	yes	—
71	C₃×C₄×C₈	D₈	1	4	yes	—
72	C₃×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
74	C₃×C₄×C₈	#41	1	2	yes	—
75	#819	—	4	8	no	—
76	#812	—	2	8	no	—
77	C₂×C₃⋊C₁₆	—	2	4	no	—
78	C₄⋊(C₃⋊C₈)	—	4	4	no	—
79	C₂²×D₈	—	2	4	no	—
80	C₂²×D₈	—	4	4	no	—
81	C₂×#127	—	2	2	no	—
82	C₂×M₃₂	—	3	4	no	—
83	#493	—	2	4	no	—
84	#493	—	2	4	no	—
85	#493	—	2	4	no	—
86	#493	—	2	2	no	—
87	#493	—	4	4	no	—
88	#493	C₂×Dic₃	1	4	yes	—
89	C₂×#108	—	6	4	no	—
90	#423	—	6	2	no	—
91	C₂×C₄×C₈	#67	1	2	yes	—
92	C₂×C₄×C₈	C₂²×D₃	1	1	yes	characteristic
93	#361	—	12	16	no	—
94	#346	—	6	2	no	—
95	C₄×C₁₆	—	6	2	no	—
96	#334	—	12	4	no	—
97	C₈²	C₂×Dic₃	1	1	yes	characteristic
98	C₈²	C₂×Dic₃	1	1	yes	characteristic
99	C₈²	#67	1	2	yes	—
100	C₂⁴×C₃	—	8	2	no	—
101	C₂×C₃×D₄	—	4	4	no	—
102	C₂×C₃×D₄	—	4	8	no	—
103	C₂×C₃×D₄	—	8	16	no	—
104	C₂×C₃×D₄	—	8	4	no	—
105	C₂×C₃×D₄	—	8	2	no	—
106	C₂×C₃×D₄	—	8	4	no	—
107	C₂×C₃×D₄	—	8	4	no	—
108	C₂²×C₃×C₄	#127	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×SD₁₆	1	2	yes	—
110	C₃×D₈	—	4	16	no	—
111	C₃×D₈	—	8	16	no	—
112	C₃×D₈	—	8	16	no	—
113	C₃×D₈	—	8	8	no	—
114	C₂×C₃×C₈	—	2	8	no	—
115	C₂×C₃×C₈	#102	1	4	yes	—
116	C₂×C₃×C₈	#102	1	4	yes	—
117	C₂×C₃×C₈	C₄×D₄	1	2	yes	—
118	C₂×C₃×C₈	#121	1	2	yes	—
119	C₂×C₃×C₈	AGL(1,ℤ/8)	1	8	yes	—
120	C₃×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₃×C₄²	#124	1	1	yes	characteristic
122	C₃×C₄²	C₂×D₈	1	2	yes	—
123	#208	—	8	4	no	—
124	C₂×C₃⋊C₈	—	8	2	no	—
125	C₃⋊C₁₆	—	4	4	no	—
126	C₂²×D₄	—	4	4	no	—
127	C₂²×D₄	—	4	2	no	—
128	C₂×D₈	—	2	8	no	—
129	C₂×D₈	—	4	8	no	—
130	C₂×D₈	—	4	8	no	—
131	C₂×D₈	—	4	8	no	—
132	C₂×D₈	—	4	8	no	—
133	C₂×D₈	—	4	4	no	—
134	C₂×D₈	—	8	16	no	—
135	C₂×M₁₆	—	12	2	no	—
136	C₂²×C₈	#214	1	4	yes	—
137	C₂²×C₈	D₃×D₄	1	2	yes	—
138	#127	—	2	4	no	—
139	#127	—	4	4	no	—
140	#127	—	4	2	no	—
141	C₂×C₄²	C₂×#67	1	1	yes	characteristic
142	M₃₂	—	6	8	no	—
143	C₂×C₁₆	—	6	4	no	—
144	#108	—	12	8	no	—
145	C₄⋊C₈	—	12	4	no	—
146	C₄×C₈	#213	1	4	yes	—
147	C₄×C₈	#217	1	2	yes	—
148	C₄×C₈	#217	1	2	yes	—
149	C₄×C₈	#237	1	2	yes	—
150	C₄×C₈	C₂²×Dic₃	1	1	yes	characteristic
151	C₄×C₈	C₂×#67	1	1	yes	characteristic
152	C₂³×C₃	—	8	4	no	—
153	C₂³×C₃	—	16	8	no	—
154	C₂³×C₃	—	16	2	no	—
155	C₂³×C₃	—	16	2	no	—
156	C₂³×C₃	#492	1	1	yes	characteristic
157	C₃×D₄	—	8	16	no	—
158	C₃×D₄	—	8	8	no	—
159	C₃×D₄	—	16	8	no	—
160	C₃×D₄	—	16	4	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	#390	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×#102	1	2	yes	—
164	C₂×C₃×C₄	#486	1	2	yes	—
165	C₂×C₃×C₄	#493	1	1	yes	characteristic
166	C₂×C₃×C₄	#496	1	4	yes	—
167	C₃×C₈	—	2	4	no	—
168	C₃×C₈	#361	1	8	yes	complemented
169	C₃×C₈	#442	1	4	yes	—
170	C₃×C₈	#469	1	4	yes	—
171	C₃⋊C₈	—	16	2	no	—
172	C₂⁴	—	8	2	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	8	16	no	—
176	C₂×D₄	—	8	4	no	—
177	C₂×D₄	—	8	4	no	—
178	C₂×D₄	—	8	4	no	—
179	C₂×D₄	—	8	2	no	—
180	C₂²×C₄	C₂×#214	1	2	yes	—
181	C₂²×C₄	#940	1	1	yes	characteristic
182	D₈	—	4	16	no	—
183	D₈	—	8	16	no	—
184	D₈	—	8	8	no	—
185	D₈	—	8	16	no	—
186	M₁₆	—	24	4	no	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	—	24	2	no	—
189	C₂×C₈	#832	1	4	yes	—
190	C₂×C₈	#832	1	4	yes	—
191	C₂×C₈	#911	1	8	yes	—
192	C₂×C₈	D₄×Dic₃	1	2	yes	—
193	C₂×C₈	#938	1	2	yes	—
194	C₄²	C₂×#213	1	2	yes	—
195	C₄²	#936	1	1	yes	characteristic
196	C₄²	C₂×#217	1	1	yes	characteristic
197	C₁₆	—	12	4	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	16	4	no	—
200	C₂²×C₃	—	16	8	no	—
201	C₂²×C₃	—	32	4	no	—
202	C₂²×C₃	—	32	4	no	—
203	C₂²×C₃	#1715	1	2	yes	—
204	C₂²×C₃	#1957	1	1	yes	characteristic
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	#1721	1	2	yes	—
207	C₃×C₄	#1972	1	2	yes	—
208	C₃×C₄	C₂×#361	1	4	yes	—
209	C₂³	—	8	4	no	—
210	C₂³	—	16	8	no	—
211	C₂³	—	16	2	no	—
212	C₂³	—	16	2	no	—
213	C₂³	#5076	1	1	yes	characteristic
214	D₄	—	8	16	no	—
215	D₄	—	8	8	no	—
216	D₄	—	16	8	no	—
217	D₄	—	16	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	#5070	1	2	yes	—
220	C₂×C₄	#5073	1	1	yes	characteristic
221	C₂×C₄	#5154	1	4	yes	—
222	C₂×C₄	C₂×#832	1	2	yes	—
223	C₂×C₄	#5221	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	—	48	2	no	—
226	C₈	#4563	1	8	yes	complemented
227	C₈	#5152	1	4	yes	—
228	C₈	#5159	1	4	yes	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	32	4	no	—
231	C₂×C₃	#9837	1	1	yes	characteristic
232	C₂×C₃	#13213	1	2	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	16	8	no	—
235	C₂²	—	16	4	no	—
236	C₂²	—	32	4	no	—
237	C₂²	—	32	4	no	—
238	C₂²	#71277	1	2	yes	—
239	C₂²	#71764	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	#71280	1	2	yes	—
242	C₄	C₂×#4563	1	4	yes	—
243	C₄	#72039	1	2	yes	—
244	C₃	#141754	1	1	yes	characteristic, complemented, Sylow
245	C₂	—	2	2	no	—
246	C₂	—	32	4	no	—
247	C₂	#10695026	1	1	yes	characteristic
248	C₂	#10713448	1	2	yes	—

	subgroup	G/H
center	C₂²	#71764
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×#1712	C₂
socle	C₂²×C₃	#1957
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	C₂×#1712	D₃
hypocenter	C₃	#141754
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1712

Complements

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

N	H ≅ G/N	type
1. #10713448	C₂	semidirect
2. #10704180	C₂	semidirect
3. #10688988	C₂	semidirect
168. C₃×C₈	#361	semidirect
226. C₈	#4563	semidirect
244. C₃	#141754	semidirect

Maximal subgroups

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

subgroup	index
#10713448	2
#10704180	2
#10688988	2
C₂×C₃×#1712	2
#141754	3