Subgroups — #11785985

Order 1152.

501 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	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×#4932	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589155	C₂	1	4	yes	maximal
5	#70444	—	3	1	no	maximal
6	#69437	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
12	C₃×C₃⋊#95	C₄	1	2	yes	—
13	C₃×C₃⋊#95	C₄	1	2	yes	—
14	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
15	C₂×C₃×C₄⋊C₈	—	3	1	no	—
16	C₃×#376	—	3	1	no	—
17	#4932	D₃	1	1	yes	characteristic
18	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
19	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
21	C₂×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
22	#4550	—	3	4	no	—
23	#4524	—	3	4	no	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
27	C₂²×C₃×Dic₃	C₂×C₄	1	4	yes	—
28	C₃²×C₄²	D₄	1	2	yes	—
29	C₃²×C₄²	Q₈	1	2	yes	—
30	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
32	C₃×C₄×Dic₃	—	2	2	no	—
33	C₃×C₄×Dic₃	—	2	2	no	—
34	C₂×C₃×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	—	2	4	no	—
36	#1832	—	9	1	no	Sylow
37	C₂²×C₃×C₈	—	3	2	no	—
38	C₂×C₃×C₄²	—	2	1	no	—
39	C₂×C₃×C₄²	—	3	2	no	—
40	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄×Dic₃	C₂×D₃	1	2	yes	—
43	C₂²×C₃⋊C₈	—	3	2	no	—
44	C₂²×C₃⋊C₈	—	3	2	no	—
45	C₂²×C₃⋊C₈	—	6	2	no	—
46	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
47	C₃×C₄⋊C₈	—	6	2	no	—
48	C₃×#95	—	3	2	no	—
49	C₃×#95	—	3	2	no	—
50	C₃⋊#95	Dic₃	1	2	yes	—
51	C₃⋊#95	Dic₃	1	2	yes	—
52	C₄⋊(C₃⋊C₈)	—	2	2	no	—
53	C₄⋊(C₃⋊C₈)	—	6	4	no	—
54	C₄⋊(C₃⋊C₈)	—	6	2	no	—
55	C₄⋊(C₃⋊C₈)	—	6	2	no	—
56	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	—	2	2	no	—
58	C₂×C₃²×C₄	—	2	2	no	—
59	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
60	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	#41	1	2	yes	—
64	C₂×C₃²×C₄	#41	1	2	yes	—
65	C₂×C₃×Dic₃	—	2	4	no	—
66	C₂×C₃×Dic₃	—	2	4	no	—
67	C₂×C₃×Dic₃	—	2	4	no	—
68	C₃⋊(C₃⋊C₈)	—	4	2	no	—
69	C₃×C₃⋊C₈	—	4	2	no	—
70	C₂×C₄⋊C₈	—	3	1	no	—
71	C₂×C₄⋊C₈	—	9	1	no	—
72	#376	—	3	1	no	—
73	#336	—	9	4	no	—
74	C₂²×C₃×C₄	—	2	1	no	—
75	C₂²×C₃×C₄	—	2	1	no	—
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	—	3	4	no	—
78	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂²×Dic₃	C₂×Dic₃	1	4	yes	—
85	C₂×C₃×C₈	—	6	2	no	—
86	C₂×C₃×C₈	—	6	4	no	—
87	C₃×C₄²	—	2	2	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	—	6	2	no	—
90	C₃×C₄²	—	6	2	no	—
91	C₃×C₄²	Dic₆	1	2	yes	—
92	C₃×C₄²	Dic₆	1	2	yes	—
93	C₃×C₄²	D₁₂	1	2	yes	—
94	C₃×C₄²	#67	1	2	yes	—
95	C₄×Dic₃	—	2	2	no	—
96	C₄×Dic₃	—	2	2	no	—
97	C₂×C₃⋊C₈	—	2	4	no	—
98	C₂×C₃⋊C₈	—	2	2	no	—
99	C₂×C₃⋊C₈	—	6	4	no	—
100	C₂×C₃⋊C₈	—	6	2	no	—
101	C₂×C₃⋊C₈	—	6	4	no	—
102	C₂×C₃⋊C₈	—	6	4	no	—
103	C₂×C₃⋊C₈	—	6	2	no	—
104	C₂×C₃⋊C₈	—	12	4	no	—
105	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₃²	#117	1	2	yes	—
107	C₂²×C₃²	#131	1	2	yes	—
108	C₂²×C₃²	#131	1	2	yes	—
109	C₃²×C₄	—	2	2	no	—
110	C₃²×C₄	—	2	2	no	—
111	C₃²×C₄	#124	1	1	yes	characteristic
112	C₃²×C₄	#124	1	1	yes	characteristic
113	C₃²×C₄	#125	1	2	yes	—
114	C₃×Dic₃	—	4	4	no	—
115	C₂²×C₈	—	3	2	no	—
116	C₂²×C₈	—	9	2	no	—
117	C₂×C₄²	—	3	2	no	—
118	C₂×C₄²	D₃²	1	1	yes	characteristic
119	C₄⋊C₈	—	6	2	no	—
120	C₄⋊C₈	—	18	2	no	—
121	#95	—	3	2	no	—
122	#95	—	3	2	no	—
123	C₂³×C₃	—	2	1	no	—
124	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
125	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	6	4	no	—
139	C₂×C₃×C₄	—	6	4	no	—
140	C₂×C₃×C₄	—	6	4	no	—
141	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
142	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
143	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
144	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
145	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
148	C₂×C₃×C₄	#235	1	2	yes	—
149	C₂×C₃×C₄	#235	1	2	yes	—
150	C₂×C₃×C₄	#237	1	2	yes	—
151	C₂×C₃×C₄	#239	1	2	yes	—
152	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
153	C₂×Dic₃	—	2	4	no	—
154	C₂×Dic₃	—	2	4	no	—
155	C₂×Dic₃	—	2	4	no	—
156	C₃×C₈	—	12	2	no	—
157	C₃⋊C₈	—	4	2	no	—
158	C₃⋊C₈	—	12	4	no	—
159	C₃⋊C₈	—	12	2	no	—
160	C₃⋊C₈	—	12	2	no	—
161	C₂×C₃²	#382	1	1	yes	characteristic
162	C₂×C₃²	#424	1	1	yes	characteristic
163	C₂×C₃²	#424	1	1	yes	characteristic
164	C₂×C₃²	#433	1	4	yes	—
165	C₂²×C₄	—	3	4	no	—
166	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
167	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
169	C₂×C₈	—	6	4	no	—
170	C₂×C₈	—	6	2	no	—
171	C₂×C₈	—	18	4	no	—
172	C₂×C₈	—	18	2	no	—
173	C₄²	—	6	2	no	—
174	C₄²	—	6	2	no	—
175	C₄²	#626	1	2	yes	—
176	C₄²	#627	1	2	yes	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	#892	1	2	yes	—
182	C₂²×C₃	#901	1	2	yes	—
183	C₂²×C₃	#907	1	2	yes	—
184	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
185	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
186	C₂²×C₃	#926	1	2	yes	—
187	C₂²×C₃	#948	1	2	yes	—
188	C₂²×C₃	#948	1	2	yes	—
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	—	12	4	no	—
199	C₃×C₄	#870	1	2	yes	—
200	C₃×C₄	#875	1	1	yes	characteristic
201	C₃×C₄	#875	1	1	yes	characteristic
202	C₃×C₄	#890	1	2	yes	—
203	C₃×C₄	#936	1	1	yes	characteristic
204	C₃×C₄	#947	1	1	yes	characteristic
205	Dic₃	—	4	4	no	—
206	C₃²	#1832	1	1	yes	characteristic, complemented, Sylow
207	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	6	4	no	—
211	C₂×C₄	—	6	4	no	—
212	C₂×C₄	—	6	4	no	—
213	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
214	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
215	C₂×C₄	#3795	1	2	yes	—
216	C₂×C₄	#3799	1	2	yes	—
217	C₂×C₄	C₂×#626	1	1	yes	characteristic
218	C₂×C₄	C₂×#627	1	1	yes	characteristic
219	C₈	—	12	2	no	—
220	C₈	—	36	2	no	—
221	C₂×C₃	—	2	4	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	#4840	1	4	yes	—
226	C₂×C₃	#4936	1	1	yes	characteristic
227	C₂×C₃	#4977	1	1	yes	characteristic
228	C₂×C₃	#4999	1	1	yes	characteristic
229	C₂×C₃	#4999	1	1	yes	characteristic
230	C₂×C₃	#5000	1	4	yes	—
231	C₂×C₃	#5102	1	1	yes	characteristic
232	C₂×C₃	#5119	1	1	yes	characteristic
233	C₂²	#63814	1	2	yes	—
234	C₂²	#63815	1	2	yes	—
235	C₂²	#63841	1	2	yes	—
236	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	12	4	no	—
240	C₄	#63860	1	1	yes	characteristic
241	C₄	#63864	1	1	yes	characteristic
242	C₄	#63869	1	2	yes	—
243	C₃	—	2	1	no	—
244	C₃	#69437	1	1	yes	characteristic, complemented
245	C₃	#70444	1	1	yes	characteristic, complemented
246	C₂	#10590253	1	4	yes	—
247	C₂	#10591116	1	1	yes	characteristic
248	C₂	#10591117	1	1	yes	characteristic
249	C₂	#10591224	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator, socle, prime radical	C₂³×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄²	C₂²
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1832
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
206. C₃²	#1832	semidirect
244. C₃	#69437	semidirect
245. C₃	#70444	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×#4932	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
#10589155	2
#70444	3
#69437	3