Subgroups — #72254961

Order 3072.

342 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72254957	C₂	1	2	yes	maximal, complemented
2	#72245577	C₂	1	2	yes	maximal, complemented
3	C₃×#151086	C₂	1	1	yes	maximal, characteristic
4	#72245421	C₂	1	1	yes	maximal, characteristic, complemented
5	#72244890	C₂	1	1	yes	maximal, characteristic, complemented
6	#72224647	—	3	1	no	maximal, Sylow
7	#10715756	—	2	2	no	—
8	#10682855	C₂²	1	2	yes	—
9	#10682634	—	2	2	no	—
10	#10682520	C₂²	1	1	yes	characteristic
11	#10681803	C₂²	1	1	yes	characteristic
12	C₃×#13656	C₄	1	1	yes	characteristic
13	C₃×#12047	C₂²	1	2	yes	—
14	C₃×#7464	C₄	1	1	yes	characteristic
15	C₃×#7388	C₄	1	2	yes	—
16	C₃×#7330	C₂²	1	1	yes	characteristic
17	#151103	—	3	2	no	—
18	#151086	D₃	1	1	yes	characteristic
19	#93878	—	3	1	no	—
20	#93759	—	3	2	no	—
21	#93668	—	3	1	no	—
22	C₃×#2262	C₈	1	1	yes	characteristic
23	C₃×#2178	C₂×C₄	1	2	yes	—
24	C₃×#2170	—	2	2	no	—
25	#71451	—	4	2	no	—
26	C₃×#1425	C₈	1	1	yes	characteristic
27	C₃×#1417	C₈	1	2	yes	—
28	C₃×#1397	C₂×C₄	1	1	yes	characteristic
29	C₃×#1397	C₂×C₄	1	1	yes	characteristic
30	C₃×#1373	C₂×C₄	1	2	yes	—
31	C₃×#1335	—	2	2	no	—
32	C₃×#1311	C₂³	1	1	yes	characteristic
33	#67823	—	4	2	no	—
34	#67510	—	2	2	no	—
35	#67508	D₄	1	2	yes	—
36	#67438	—	2	1	no	—
37	#13656	Dic₃	1	1	yes	characteristic
38	#13648	—	6	2	no	—
39	#12047	C₂×D₃	1	2	yes	—
40	#7464	Dic₃	1	1	yes	characteristic
41	#7456	—	3	2	no	—
42	#7388	Dic₃	1	2	yes	—
43	#7382	—	6	2	no	—
44	#7332	—	3	1	no	—
45	#7330	C₂×D₃	1	1	yes	characteristic
46	#7053	—	3	1	no	—
47	C₃×#508	C₂×C₈	1	2	yes	—
48	C₃×#504	—	4	2	no	—
49	C₃×#443	—	2	2	no	—
50	C₃×Q₆₄	C₁₆	1	1	yes	characteristic, complemented
51	C₃×SD₆₄	C₁₆	1	2	yes	complemented
52	C₃×D₃₂	C₁₆	1	1	yes	characteristic, complemented
53	C₂×C₃×C₃₂	—	2	1	no	—
54	C₂×C₃×C₃₂	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃×C₃₂	C₂×C₈	1	1	yes	characteristic
56	C₃×#368	C₂×C₈	1	2	yes	—
57	C₃×#364	—	2	2	no	—
58	C₃×#359	—	2	2	no	—
59	C₃×#350	—	4	2	no	—
60	C₃×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
61	C₃×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
62	C₃×C₈²	#41	1	1	yes	characteristic
63	#4499	—	4	2	no	—
64	#4499	—	8	2	no	—
65	C₂×C₃⋊C₃₂	—	2	2	no	—
66	C₂×C₃⋊C₃₂	—	8	2	no	—
67	C₄×C₃⋊C₁₆	—	4	1	no	—
68	#2262	C₃⋊C₈	1	1	yes	characteristic
69	#2258	—	12	2	no	—
70	#2178	C₂×Dic₃	1	2	yes	—
71	#2170	—	2	2	no	—
72	#1425	C₃⋊C₈	1	1	yes	characteristic
73	#1422	—	6	2	no	—
74	#1417	C₃⋊C₈	1	2	yes	—
75	#1401	—	12	2	no	—
76	#1397	—	3	2	no	—
77	#1397	C₂×Dic₃	1	1	yes	characteristic
78	#1397	C₂×Dic₃	1	1	yes	characteristic
79	#1373	C₂×Dic₃	1	2	yes	—
80	#1335	—	2	2	no	—
81	#1312	—	6	1	no	—
82	#1311	C₂²×D₃	1	1	yes	characteristic
83	C₃×#135	—	2	2	no	—
84	C₃×#131	—	4	2	no	—
85	C₃×Q₃₂	C₂×C₁₆	1	2	yes	—
86	C₃×SD₃₂	—	2	2	no	—
87	C₃×D₁₆	C₂×C₁₆	1	2	yes	—
88	C₃×M₃₂	—	4	2	no	—
89	C₃×M₃₂	—	8	2	no	—
90	C₂×C₃×C₁₆	—	8	2	no	—
91	C₂×C₃×C₁₆	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₃×C₁₆	#131	1	1	yes	characteristic
93	C₂×C₃×C₁₆	AGL(1,ℤ/8)	1	2	yes	—
94	C₃×#108	—	2	2	no	—
95	C₃×#104	—	4	2	no	—
96	C₃×C₄×C₈	—	2	1	no	—
97	C₃×C₄×C₈	C₄×D₄	1	1	yes	characteristic
98	C₂×C₃⋊C₁₆	—	8	1	no	—
99	C₃×C₃₂	—	2	1	no	—
100	C₃×C₃₂	—	4	1	no	—
101	C₃×C₃₂	C₂×C₁₆	1	1	yes	characteristic
102	C₃×C₃₂	C₂×C₁₆	1	1	yes	characteristic
103	C₃⋊C₃₂	—	4	2	no	—
104	C₃⋊C₃₂	—	8	2	no	—
105	#508	C₂×C₃⋊C₈	1	2	yes	—
106	#504	—	4	2	no	—
107	#443	—	2	2	no	—
108	Q₆₄	C₃⋊C₁₆	1	1	yes	characteristic, complemented
109	SD₆₄	C₃⋊C₁₆	1	2	yes	complemented
110	D₃₂	C₃⋊C₁₆	1	1	yes	characteristic, complemented
111	M₆₄	—	12	2	no	—
112	M₆₄	—	24	2	no	—
113	C₂×C₃₂	—	2	1	no	—
114	C₂×C₃₂	—	6	2	no	—
115	C₂×C₃₂	—	24	2	no	—
116	C₂×C₃₂	C₂×C₃⋊C₈	1	1	yes	characteristic
117	C₂×C₃₂	C₂×C₃⋊C₈	1	1	yes	characteristic
118	#368	C₂×C₃⋊C₈	1	2	yes	—
119	#364	—	2	2	no	—
120	#359	—	2	2	no	—
121	#350	—	4	2	no	—
122	C₄×C₁₆	—	12	1	no	—
123	C₄×C₁₆	D₃×D₄	1	1	yes	characteristic
124	C₄×C₁₆	C₂²×Dic₃	1	1	yes	characteristic
125	C₈²	#237	1	1	yes	characteristic
126	C₃×#41	—	4	2	no	—
127	C₃×Q₁₆	—	2	2	no	—
128	C₃×SD₁₆	—	4	2	no	—
129	C₃×D₈	—	2	2	no	—
130	C₃×M₁₆	—	4	2	no	—
131	C₃×M₁₆	—	8	2	no	—
132	C₂×C₃×C₈	—	4	1	no	—
133	C₂×C₃×C₈	—	8	2	no	—
134	C₂×C₃×C₈	C₈×D₄	1	1	yes	characteristic
135	C₂×C₃×C₈	#442	1	1	yes	characteristic
136	C₃×C₄²	#444	1	1	yes	characteristic
137	C₃×C₁₆	—	2	1	no	—
138	C₃×C₁₆	—	2	2	no	—
139	C₃×C₁₆	—	8	2	no	—
140	C₃×C₁₆	C₂²×C₁₆	1	1	yes	characteristic
141	C₃×C₁₆	#504	1	1	yes	characteristic
142	C₃⋊C₁₆	—	16	1	no	—
143	#135	—	2	2	no	—
144	#131	—	4	2	no	—
145	Q₃₂	C₂×C₃⋊C₁₆	1	2	yes	—
146	SD₃₂	—	2	2	no	—
147	D₁₆	C₂×C₃⋊C₁₆	1	2	yes	—
148	M₃₂	—	4	2	no	—
149	M₃₂	—	8	2	no	—
150	C₂×C₁₆	—	8	2	no	—
151	C₂×C₁₆	—	24	1	no	—
152	C₂×C₁₆	#911	1	2	yes	—
153	C₂×C₁₆	C₂²×C₃⋊C₈	1	1	yes	characteristic
154	C₂×C₁₆	#948	1	1	yes	characteristic
155	#108	—	2	2	no	—
156	#104	—	4	2	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	D₄×Dic₃	1	1	yes	characteristic
159	C₃₂	—	2	1	no	—
160	C₃₂	—	4	1	no	—
161	C₃₂	—	12	2	no	—
162	C₃₂	—	24	2	no	—
163	C₃₂	C₂×C₃⋊C₁₆	1	1	yes	characteristic
164	C₃₂	C₂×C₃⋊C₁₆	1	1	yes	characteristic
165	C₃×Q₈	—	4	2	no	—
166	C₃×D₄	—	4	2	no	—
167	C₃×D₄	—	8	2	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	8	2	no	—
170	C₂×C₃×C₄	#1580	1	1	yes	characteristic
171	C₃×C₈	—	8	1	no	—
172	C₃×C₈	—	8	2	no	—
173	C₃×C₈	D₄×C₁₆	1	1	yes	characteristic
174	C₃×C₈	#2171	1	1	yes	characteristic
175	C₃×C₈	AGL(1,ℤ/16)	1	2	yes	—
176	#41	—	4	2	no	—
177	Q₁₆	—	2	2	no	—
178	SD₁₆	—	4	2	no	—
179	D₈	—	2	2	no	—
180	M₁₆	—	4	2	no	—
181	M₁₆	—	8	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	8	2	no	—
184	C₂×C₈	D₄×C₃⋊C₈	1	1	yes	characteristic
185	C₂×C₈	#5152	1	1	yes	characteristic
186	C₄²	#5197	1	1	yes	characteristic
187	C₁₆	—	2	1	no	—
188	C₁₆	—	2	2	no	—
189	C₁₆	—	8	2	no	—
190	C₁₆	—	48	1	no	—
191	C₁₆	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
192	C₁₆	#5140	1	1	yes	characteristic
193	C₂²×C₃	—	8	2	no	—
194	C₂²×C₃	#12068	1	1	yes	characteristic
195	C₃×C₄	—	4	1	no	—
196	C₃×C₄	—	8	2	no	—
197	C₃×C₄	#12062	1	1	yes	characteristic
198	C₃×C₄	#12067	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	8	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	#70878	1	1	yes	characteristic
205	C₈	—	8	1	no	—
206	C₈	—	8	2	no	—
207	C₈	D₄×C₃⋊C₁₆	1	1	yes	characteristic
208	C₈	#70924	1	1	yes	characteristic
209	C₈	#71515	1	2	yes	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	16	2	no	—
212	C₂×C₃	#122963	1	1	yes	characteristic
213	C₂²	—	8	2	no	—
214	C₂²	#10707077	1	1	yes	characteristic
215	C₄	—	4	1	no	—
216	C₄	—	8	2	no	—
217	C₄	#10707043	1	1	yes	characteristic
218	C₄	#10707072	1	1	yes	characteristic
219	C₃	#72224647	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	1	no	—
221	C₂	—	16	2	no	—
222	C₂	#72245110	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10707072
commutator	C₃×C₁₆	C₂²×C₁₆
Frattini	#1311	C₂²×D₃
Fitting	C₃×#151086	C₂
socle	C₂×C₃	#122963
prime radical	C₃×#2262	C₈
cosocle	C₃×#1311	C₂³
hypercenter	#151086	D₃
hypocenter	C₃	#72224647
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1311 ◁ #151086

Complements

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

N	H ≅ G/N	type
1. #72254957	C₂	semidirect
2. #72245577	C₂	semidirect
4. #72245421	C₂	semidirect
5. #72244890	C₂	semidirect
50. C₃×Q₆₄	C₁₆	semidirect
51. C₃×SD₆₄	C₁₆	semidirect
52. C₃×D₃₂	C₁₆	semidirect
108. Q₆₄	C₃⋊C₁₆	semidirect
109. SD₆₄	C₃⋊C₁₆	semidirect
110. D₃₂	C₃⋊C₁₆	semidirect
219. C₃	#72224647	semidirect

Maximal subgroups

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

subgroup	index
#72254957	2
#72245577	2
C₃×#151086	2
#72245421	2
#72244890	2
#72224647	3