Subgroups — #72226198

Order 3072.

846 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72226180	C₂	1	2	yes	maximal
2	#72225854	C₂	1	2	yes	maximal
3	#72225829	C₂	1	1	yes	maximal, characteristic
4	#72225823	C₂	1	1	yes	maximal, characteristic
5	C₃×#135758	C₂	1	1	yes	maximal, characteristic
6	#72214006	—	3	1	no	maximal, Sylow
7	#10711612	—	2	4	no	—
8	#10706823	—	2	2	no	—
9	#10706822	—	2	4	no	—
10	C₈×#812	C₂²	1	2	yes	—
11	#10688988	C₂²	1	2	yes	—
12	#10681802	C₄	1	2	yes	complemented
13	#10681801	C₄	1	1	yes	characteristic, complemented
14	C₁₆×C₃⋊C₁₆	C₄	1	1	yes	characteristic, complemented
15	C₃×#12903	—	2	2	no	—
16	C₃×C₈×M₃₂	C₂²	1	1	yes	characteristic
17	C₃×#8822	C₂²	1	1	yes	characteristic
18	C₃×#8813	C₂²	1	1	yes	characteristic
19	#135792	—	3	1	no	—
20	#135792	—	3	2	no	—
21	#135758	—	3	2	no	—
22	#135758	D₃	1	1	yes	characteristic
23	#122211	—	3	1	no	—
24	C₂×C₃×#365	—	4	4	no	—
25	C₃×C₄×M₃₂	D₄	1	2	yes	—
26	C₃×#1848	—	2	1	no	—
27	C₃×#1844	—	2	2	no	—
28	C₃×#1844	—	2	1	no	—
29	C₃×#1566	—	2	2	no	—
30	C₂×C₃×C₈²	C₂³	1	1	yes	characteristic
31	C₂×#4487	—	4	8	no	—
32	#69792	—	2	2	no	—
33	C₄×#812	D₄	1	4	yes	—
34	#69730	—	2	2	no	—
35	#69730	—	2	2	no	—
36	C₃×#1311	Q₈	1	2	yes	—
37	C₃×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₃×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
39	#67513	—	4	4	no	—
40	#67464	—	4	8	no	—
41	#67437	Q₈	1	4	yes	—
42	C₈×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
43	C₈×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
44	#12903	—	2	2	no	—
45	#12903	—	6	4	no	—
46	#12020	—	6	4	no	—
47	#12020	—	6	2	no	—
48	C₈×M₃₂	—	3	2	no	—
49	C₈×M₃₂	C₂×D₃	1	1	yes	characteristic
50	#8822	—	3	2	no	—
51	#8822	C₂×D₃	1	1	yes	characteristic
52	#8813	C₂×D₃	1	1	yes	characteristic
53	#7052	—	3	1	no	—
54	#7052	—	3	2	no	—
55	C₁₆²	—	3	1	no	—
56	C₂×C₃×M₃₂	SD₁₆	1	4	yes	—
57	C₂×C₃×#108	—	4	2	no	—
58	C₂×C₃×#106	—	4	4	no	—
59	C₂×C₃×#106	—	4	2	no	—
60	C₃×#423	—	4	1	no	—
61	C₂×C₃×C₄×C₈	C₂×D₄	1	2	yes	—
62	C₂×C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
63	C₃×#384	—	4	1	no	—
64	C₂×#819	—	4	4	no	—
65	C₂×#812	SD₁₆	1	8	yes	—
66	#4924	—	4	2	no	—
67	C₃×#365	—	8	8	no	—
68	C₃×#346	SD₁₆	1	4	yes	—
69	C₃×C₄×C₁₆	C₄⋊C₄	1	2	yes	—
70	C₃×C₄×C₁₆	C₄⋊C₄	1	2	yes	—
71	C₃×#334	—	4	4	no	—
72	C₃×C₈²	C₂²×C₄	1	1	yes	characteristic
73	C₃×C₈²	C₂×Q₈	1	1	yes	characteristic
74	C₃×C₈²	C₂×Q₈	1	2	yes	—
75	#4487	—	8	16	no	—
76	#4461	SD₁₆	1	8	yes	—
77	C₄×C₃⋊C₁₆	C₄⋊C₄	1	4	yes	—
78	C₄×C₃⋊C₁₆	C₄⋊C₄	1	4	yes	—
79	#4457	—	4	4	no	—
80	#4457	—	4	4	no	—
81	C₂×#365	—	4	4	no	—
82	C₂×#365	—	12	8	no	—
83	C₄×M₃₂	—	3	4	no	—
84	C₄×M₃₂	#67	1	2	yes	—
85	#1848	—	2	1	no	—
86	#1848	—	6	2	no	—
87	#1844	—	2	2	no	—
88	#1844	—	2	1	no	—
89	#1566	—	2	2	no	—
90	#1566	—	6	2	no	—
91	#1566	—	6	2	no	—
92	C₂×C₈²	C₂²×D₃	1	1	yes	characteristic
93	#1368	—	12	8	no	—
94	#1368	—	12	4	no	—
95	#1311	—	3	4	no	—
96	#1311	Dic₆	1	2	yes	—
97	C₈×C₁₆	—	3	2	no	—
98	C₈×C₁₆	—	3	2	no	—
99	C₈×C₁₆	C₄×D₃	1	1	yes	characteristic
100	C₈×C₁₆	C₄×D₃	1	1	yes	characteristic
101	C₂×C₃×M₁₆	—	8	1	no	—
102	C₂²×C₃×C₈	C₂×SD₁₆	1	4	yes	—
103	C₂²×C₃×C₈	C₂×SD₁₆	1	2	yes	—
104	C₂×C₃×C₄⋊C₄	—	8	2	no	—
105	C₂×C₃×C₄⋊C₄	—	8	1	no	—
106	C₂×C₃×C₄²	#127	1	1	yes	characteristic
107	C₂×#208	—	8	2	no	—
108	C₃×M₃₂	—	2	8	no	—
109	C₂×C₃×C₁₆	#106	1	4	yes	—
110	C₂×C₃×C₁₆	#106	1	4	yes	—
111	C₃×#108	—	8	4	no	—
112	C₃×#106	—	8	4	no	—
113	C₃×#106	—	8	8	no	—
114	C₃×C₄⋊C₈	—	8	2	no	—
115	C₃×C₄×C₈	C₂×C₄⋊C₄	1	2	yes	—
116	C₃×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₃×C₄×C₈	C₄⋊Q₈	1	2	yes	—
118	C₃×C₄×C₈	C₄⋊Q₈	1	1	yes	characteristic
119	C₃×C₄×C₈	C₂×SD₁₆	1	4	yes	—
120	C₃×C₄×C₈	C₂×SD₁₆	1	2	yes	—
121	#819	—	8	8	no	—
122	#812	—	2	16	no	—
123	C₂×C₃⋊C₁₆	#106	1	8	yes	—
124	C₂×C₃⋊C₁₆	#106	1	8	yes	—
125	C₄⋊(C₃⋊C₈)	—	8	2	no	—
126	C₄⋊(C₃⋊C₈)	—	8	2	no	—
127	C₂×M₃₂	—	3	8	no	—
128	C₂×M₃₂	#214	1	4	yes	—
129	C₂×#108	—	4	2	no	—
130	C₂×#108	—	12	4	no	—
131	C₂×#106	—	4	4	no	—
132	C₂×#106	—	4	2	no	—
133	#423	—	4	1	no	—
134	#423	—	12	2	no	—
135	C₂×C₄×C₈	D₃×D₄	1	2	yes	—
136	C₂×C₄×C₈	C₂×#67	1	1	yes	characteristic
137	#384	—	4	1	no	—
138	#365	—	8	8	no	—
139	#365	—	24	16	no	—
140	#346	—	3	8	no	—
141	#346	#215	1	4	yes	—
142	C₄×C₁₆	—	3	4	no	—
143	C₄×C₁₆	—	3	4	no	—
144	C₄×C₁₆	Dic₃⋊C₄	1	2	yes	—
145	C₄×C₁₆	Dic₃⋊C₄	1	2	yes	—
146	#334	—	4	4	no	—
147	#334	—	12	4	no	—
148	#334	—	12	4	no	—
149	C₈²	C₂×Dic₆	1	1	yes	characteristic
150	C₈²	C₂×C₄×D₃	1	1	yes	characteristic
151	C₈²	D₃×Q₈	1	2	yes	—
152	C₂²×C₃×C₄	—	16	1	no	—
153	C₂²×C₃×C₄	#492	1	2	yes	—
154	C₂²×C₃×C₄	#492	1	1	yes	characteristic
155	C₃×M₁₆	—	16	2	no	—
156	C₂×C₃×C₈	—	2	8	no	—
157	C₂×C₃×C₈	—	2	4	no	—
158	C₂×C₃×C₈	—	16	1	no	—
159	C₂×C₃×C₈	C₂×#106	1	2	yes	—
160	C₂×C₃×C₈	C₂×#106	1	4	yes	—
161	C₂×C₃×C₈	#498	1	2	yes	—
162	C₂×C₃×C₈	#498	1	4	yes	—
163	C₃×C₄⋊C₄	—	16	4	no	—
164	C₃×C₄⋊C₄	—	16	2	no	—
165	C₃×C₄²	#384	1	1	yes	characteristic
166	C₃×C₄²	#492	1	2	yes	—
167	C₃×C₄²	#492	1	1	yes	characteristic
168	#208	—	16	4	no	—
169	C₂×C₃⋊C₈	—	16	2	no	—
170	C₃×C₁₆	—	2	4	no	—
171	C₃×C₁₆	#365	1	8	yes	complemented
172	C₃⋊C₁₆	—	2	8	no	—
173	C₃⋊C₁₆	#365	1	16	yes	complemented
174	C₂×M₁₆	—	8	1	no	—
175	C₂×M₁₆	—	24	2	no	—
176	C₂²×C₈	D₃×SD₁₆	1	4	yes	—
177	C₂²×C₈	C₂×#214	1	2	yes	—
178	C₂×C₄⋊C₄	—	8	2	no	—
179	C₂×C₄⋊C₄	—	8	1	no	—
180	C₂×C₄²	#940	1	1	yes	characteristic
181	M₃₂	—	2	8	no	—
182	M₃₂	—	6	16	no	—
183	C₂×C₁₆	—	3	8	no	—
184	C₂×C₁₆	—	3	8	no	—
185	C₂×C₁₆	#808	1	4	yes	—
186	C₂×C₁₆	#808	1	4	yes	—
187	#108	—	8	4	no	—
188	#108	—	24	8	no	—
189	#106	—	8	4	no	—
190	#106	—	8	8	no	—
191	C₄⋊C₈	—	8	2	no	—
192	C₄⋊C₈	—	24	2	no	—
193	C₄⋊C₈	—	24	2	no	—
194	C₄×C₈	#888	1	2	yes	—
195	C₄×C₈	D₃×C₄⋊C₄	1	2	yes	—
196	C₄×C₈	D₃×SD₁₆	1	4	yes	—
197	C₄×C₈	C₂×Dic₃⋊C₄	1	1	yes	characteristic
198	C₄×C₈	C₂×#215	1	2	yes	—
199	C₄×C₈	#944	1	1	yes	characteristic
200	C₂³×C₃	#1705	1	1	yes	characteristic
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	32	1	no	—
204	C₂×C₃×C₄	—	32	2	no	—
205	C₂×C₃×C₄	#1705	1	1	yes	characteristic
206	C₂×C₃×C₄	#1705	1	2	yes	—
207	C₂×C₃×C₄	#1844	1	2	yes	—
208	C₂×C₃×C₄	#1844	1	1	yes	characteristic
209	C₃×C₈	—	2	4	no	—
210	C₃×C₈	—	2	2	no	—
211	C₃×C₈	—	32	1	no	—
212	C₃×C₈	C₂×#365	1	4	yes	—
213	C₃×C₈	C₂×#365	1	8	yes	—
214	C₃⋊C₈	—	32	2	no	—
215	C₂²×C₄	—	16	1	no	—
216	C₂²×C₄	#5076	1	1	yes	characteristic
217	C₂²×C₄	#5175	1	2	yes	—
218	M₁₆	—	16	2	no	—
219	M₁₆	—	48	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	16	1	no	—
223	C₂×C₈	—	48	2	no	—
224	C₂×C₈	#4855	1	4	yes	—
225	C₂×C₈	D₃×#106	1	4	yes	—
226	C₂×C₈	C₂×#808	1	2	yes	—
227	C₂×C₈	#5080	1	2	yes	—
228	C₄⋊C₄	—	16	4	no	—
229	C₄⋊C₄	—	16	2	no	—
230	C₄²	#4975	1	1	yes	characteristic
231	C₄²	#5085	1	1	yes	characteristic
232	C₄²	#5175	1	2	yes	—
233	C₁₆	—	2	4	no	—
234	C₁₆	—	3	16	no	—
235	C₁₆	—	6	8	no	—
236	C₁₆	#4487	1	8	yes	complemented
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	#8813	1	1	yes	characteristic
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	64	1	no	—
243	C₃×C₄	#12903	1	4	yes	—
244	C₃×C₄	#12903	1	2	yes	—
245	C₂³	#71254	1	1	yes	characteristic
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	32	1	no	—
249	C₂×C₄	—	32	2	no	—
250	C₂×C₄	#69576	1	1	yes	characteristic
251	C₂×C₄	#70554	1	2	yes	—
252	C₂×C₄	#71254	1	1	yes	characteristic
253	C₂×C₄	#71332	1	2	yes	—
254	C₈	—	2	4	no	—
255	C₈	—	2	2	no	—
256	C₈	—	32	1	no	—
257	C₈	—	96	2	no	—
258	C₈	D₃×#365	1	8	yes	—
259	C₈	C₂×#4487	1	4	yes	—
260	C₂×C₃	—	4	1	no	—
261	C₂×C₃	#135758	1	1	yes	characteristic
262	C₂×C₃	#135758	1	2	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	#10688953	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	64	1	no	—
269	C₄	#10711611	1	4	yes	—
270	C₄	#10711612	1	2	yes	—
271	C₃	#72214006	1	1	yes	characteristic, complemented, Sylow
272	C₂	—	4	1	no	—
273	C₂	#72225854	1	1	yes	characteristic
274	C₂	#72225855	1	2	yes	—

	subgroup	G/H
center	C₂²	#10688953
commutator	C₃×C₈²	C₂²×C₄
Frattini	C₂×C₈²	C₂²×D₃
Fitting	C₃×#135758	C₂
socle	C₂²×C₃	#8813
prime radical	C₂³×C₃	#1705
cosocle	C₂×C₃×C₈²	C₂³
hypercenter	#135758	D₃
hypocenter	C₃	#72214006
LCS	G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #135758

Complements

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

N	H ≅ G/N	type
12. #10681802	C₄	semidirect
13. #10681801	C₄	semidirect
14. C₁₆×C₃⋊C₁₆	C₄	semidirect
171. C₃×C₁₆	#365	semidirect
173. C₃⋊C₁₆	#365	semidirect
236. C₁₆	#4487	semidirect
271. C₃	#72214006	semidirect

Maximal subgroups

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

subgroup	index
#72226180	2
#72225854	2
#72225829	2
#72225823	2
C₃×#135758	2
#72214006	3