Subgroups — #11798030

Order 1152.

277 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal
5	#10589103	C₂	1	2	yes	maximal
6	#70949	—	3	1	no	maximal
7	#69048	—	3	1	no	maximal
8	C₃×C₄×Dic₆	C₄	1	1	yes	characteristic
9	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
11	C₃×Dic₃⋊C₈	C₄	1	2	yes	—
12	C₃×C₈×Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
16	C₃×C₈×Q₈	—	3	1	no	—
17	C₈×Dic₆	D₃	1	1	yes	characteristic
18	C₃×C₄×C₁₆	—	3	1	no	—
19	#4502	—	3	2	no	—
20	Dic₃⋊C₁₆	—	3	2	no	—
21	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
22	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
23	C₄⋊(C₃⋊C₁₆)	—	3	2	no	—
24	C₄⋊(C₃⋊C₁₆)	—	6	1	no	—
25	#4461	—	3	2	no	—
26	C₄×C₃⋊C₁₆	D₃	1	1	yes	characteristic
27	C₂×C₃×Dic₆	C₈	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄⋊Dic₃	C₈	1	1	yes	characteristic
32	C₃×Dic₃⋊C₄	C₈	1	2	yes	—
33	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
36	C₃×C₃⋊C₁₆	—	2	1	no	—
37	C₃×C₃⋊C₁₆	Q₈	1	2	yes	complemented
38	#2183	—	9	1	no	Sylow
39	C₃×C₄×Q₈	—	3	1	no	—
40	C₄×Dic₆	Dic₃	1	1	yes	characteristic
41	C₂×C₃×C₁₆	—	3	1	no	—
42	C₂×C₃×C₁₆	—	3	1	no	—
43	C₃×C₄⋊C₈	—	3	2	no	—
44	C₃×C₄⋊C₈	—	3	1	no	—
45	C₃×C₄×C₈	—	2	1	no	—
46	C₃×C₄×C₈	—	3	2	no	—
47	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
49	Dic₃⋊C₈	Dic₃	1	2	yes	—
50	C₈×Dic₃	C₂×D₃	1	2	yes	—
51	C₂×C₃⋊C₁₆	—	3	2	no	—
52	C₂×C₃⋊C₁₆	—	3	2	no	—
53	C₂×C₃⋊C₁₆	—	6	2	no	—
54	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
56	C₄⋊(C₃⋊C₈)	Dic₃	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
61	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
62	C₃×Dic₆	—	2	2	no	—
63	C₃²×C₈	—	2	1	no	—
64	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₃²×C₈	#41	1	1	yes	characteristic
66	C₃×C₃⋊C₈	—	2	2	no	—
67	C₈×Q₈	—	3	1	no	—
68	C₄⋊C₁₆	—	9	2	no	—
69	C₄⋊C₁₆	—	9	1	no	—
70	#346	—	9	2	no	—
71	C₄×C₁₆	—	3	1	no	—
72	C₂×C₃×Q₈	—	3	1	no	—
73	C₂×Dic₆	C₃⋊C₈	1	1	yes	characteristic
74	C₂×C₃×C₈	—	2	1	no	—
75	C₂×C₃×C₈	—	2	1	no	—
76	C₂×C₃×C₈	—	3	2	no	—
77	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
81	C₃×C₄⋊C₄	—	3	2	no	—
82	C₃×C₄⋊C₄	—	3	1	no	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	—	3	2	no	—
85	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
86	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
87	C₄⋊Dic₃	C₃⋊C₈	1	1	yes	characteristic
88	Dic₃⋊C₄	C₃⋊C₈	1	2	yes	—
89	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
90	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
91	C₃×C₁₆	—	3	2	no	—
92	C₃×C₁₆	—	6	1	no	—
93	C₃⋊C₁₆	—	2	1	no	—
94	C₃⋊C₁₆	—	6	2	no	—
95	C₃⋊C₁₆	—	6	2	no	—
96	C₃⋊C₁₆	—	12	2	no	—
97	C₃⋊C₁₆	Dic₆	1	2	yes	complemented
98	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
99	C₃²×C₄	—	2	1	no	—
100	C₃²×C₄	M₃₂	1	2	yes	—
101	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
102	C₃²×C₄	#131	1	1	yes	characteristic
103	C₃×Dic₃	—	2	2	no	—
104	C₃×Dic₃	—	2	2	no	—
105	C₄×Q₈	—	3	1	no	—
106	C₂×C₁₆	—	3	1	no	—
107	C₂×C₁₆	—	3	1	no	—
108	C₂×C₁₆	—	9	2	no	—
109	C₄⋊C₈	—	3	2	no	—
110	C₄⋊C₈	—	3	1	no	—
111	C₄×C₈	—	3	2	no	—
112	C₄×C₈	D₃²	1	1	yes	characteristic
113	C₃×Q₈	—	6	2	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	3	2	no	—
118	C₂×C₃×C₄	—	3	2	no	—
119	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
120	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
125	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
126	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
127	Dic₆	—	2	2	no	—
128	C₃×C₈	—	2	1	no	—
129	C₃×C₈	—	2	1	no	—
130	C₃×C₈	—	2	1	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	4	1	no	—
133	C₃×C₈	—	6	2	no	—
134	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
135	C₃×C₈	#235	1	1	yes	characteristic
136	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
137	C₃×C₈	#239	1	1	yes	characteristic
138	C₃⋊C₈	—	2	2	no	—
139	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
140	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
141	C₂×C₃²	#504	1	1	yes	characteristic
142	C₂×Q₈	—	3	1	no	—
143	C₂×C₈	—	3	2	no	—
144	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
145	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
146	C₄⋊C₄	—	3	2	no	—
147	C₄⋊C₄	—	3	1	no	—
148	C₄²	—	3	2	no	—
149	C₄²	D₃×Dic₃	1	1	yes	characteristic
150	C₁₆	—	3	2	no	—
151	C₁₆	—	6	1	no	—
152	C₁₆	—	18	2	no	—
153	C₂²×C₃	—	2	1	no	—
154	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
155	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
156	C₃×C₄	—	2	1	no	—
157	C₃×C₄	—	2	1	no	—
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	4	1	no	—
162	C₃×C₄	—	6	2	no	—
163	C₃×C₄	—	6	2	no	—
164	C₃×C₄	#798	1	2	yes	—
165	C₃×C₄	#812	1	2	yes	—
166	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
167	C₃×C₄	#901	1	1	yes	characteristic
168	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
169	C₃×C₄	#948	1	1	yes	characteristic
170	Dic₃	—	2	2	no	—
171	Dic₃	—	2	2	no	—
172	C₃²	#2183	1	1	yes	characteristic, complemented, Sylow
173	Q₈	—	6	2	no	—
174	C₂×C₄	—	3	2	no	—
175	C₂×C₄	—	3	2	no	—
176	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
177	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
178	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
179	C₈	—	2	1	no	—
180	C₈	—	6	2	no	—
181	C₈	D₃×Dic₆	1	1	yes	characteristic
182	C₈	#3799	1	1	yes	characteristic
183	C₂×C₃	—	2	1	no	—
184	C₂×C₃	—	2	1	no	—
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	C₈×Dic₆	1	1	yes	characteristic
187	C₂×C₃	C₂×#798	1	1	yes	characteristic
188	C₂×C₃	#4901	1	1	yes	characteristic
189	C₂×C₃	Q₈×C₃⋊C₈	1	1	yes	characteristic
190	C₂×C₃	C₂×#812	1	1	yes	characteristic
191	C₂×C₃	#5140	1	1	yes	characteristic
192	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	6	2	no	—
195	C₄	—	6	2	no	—
196	C₄	#63543	1	2	yes	—
197	C₄	#63814	1	1	yes	characteristic
198	C₄	Dic₃×Dic₆	1	1	yes	characteristic
199	C₃	—	2	1	no	—
200	C₃	#69048	1	1	yes	characteristic, complemented
201	C₃	#70949	1	1	yes	characteristic, complemented
202	C₂	C₃⋊C₈×Dic₆	1	1	yes	characteristic
203	C₂	#10590621	1	1	yes	characteristic
204	C₂	C₂×#63543	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator, socle, prime radical	C₂²×C₃²	C₂²×C₈
Fitting	C₃²×C₄×C₈	C₂²
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#2183
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
37. C₃×C₃⋊C₁₆	Q₈	semidirect
97. C₃⋊C₁₆	Dic₆	semidirect
172. C₃²	#2183	semidirect
200. C₃	#69048	semidirect
201. C₃	#70949	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×Dic₆	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
C₃×C₄×C₃⋊C₁₆	2
Dic₃⋊(C₃⋊C₁₆)	2
#10589103	2
#70949	3
#69048	3