Subgroups — C₃×#69281 #11798011

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	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#4502	C₂	1	2	yes	maximal
4	C₃×Dic₃⋊C₁₆	C₂	1	2	yes	maximal
5	C₃×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₃×#2183	—	3	1	no	maximal
7	#69281	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×C₄×Dic₆	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
10	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
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	C₂×C₃	1	1	yes	characteristic
18	C₃×C₄⋊C₁₆	—	2	1	no	—
19	C₃×C₄⋊C₁₆	—	3	2	no	—
20	C₃×C₄⋊C₁₆	D₃	1	1	yes	characteristic
21	C₃×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
22	C₃×#346	—	3	2	no	—
23	C₃×C₄×C₁₆	—	3	1	no	—
24	#4502	C₂×C₃	1	2	yes	—
25	Dic₃⋊C₁₆	C₂×C₃	1	2	yes	—
26	C₄×C₃⋊C₁₆	C₂×C₃	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	C₃²×C₁₆	—	2	2	no	—
36	C₃×C₃⋊C₁₆	—	2	1	no	—
37	C₃×C₃⋊C₁₆	Q₈	1	2	yes	complemented
38	#2183	—	3	1	no	Sylow
39	C₃×C₄×Q₈	—	3	1	no	—
40	C₄×Dic₆	C₃×C₄	1	1	yes	characteristic
41	C₂×C₃×C₁₆	—	2	2	no	—
42	C₂×C₃×C₁₆	—	3	1	no	—
43	C₂×C₃×C₁₆	—	3	1	no	—
44	C₂×C₃×C₁₆	C₂×D₃	1	2	yes	—
45	C₂×C₃×C₁₆	C₂²×C₃	1	2	yes	—
46	C₃×C₄⋊C₈	—	3	2	no	—
47	C₃×C₄⋊C₈	—	3	1	no	—
48	C₃×C₄×C₈	—	2	1	no	—
49	C₃×C₄×C₈	—	3	2	no	—
50	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
52	Dic₃⋊C₈	C₃×C₄	1	2	yes	—
53	C₈×Dic₃	C₂²×C₃	1	2	yes	—
54	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
56	C₄⋊(C₃⋊C₈)	C₃×C₄	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	—	3	2	no	—
69	C₄⋊C₁₆	C₃×D₃	1	1	yes	characteristic
70	#346	—	3	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×C₃×C₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂³×C₃	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×C₃×C₄	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×C₃×C₄	1	2	yes	—
90	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
91	C₃×C₁₆	—	2	2	no	—
92	C₃×C₁₆	—	2	2	no	—
93	C₃×C₁₆	—	3	2	no	—
94	C₃×C₁₆	—	4	2	no	—
95	C₃×C₁₆	—	6	1	no	—
96	C₃⋊C₁₆	—	2	1	no	—
97	C₃⋊C₁₆	C₃×Q₈	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	C₂×C₃×D₃	1	2	yes	—
109	C₄⋊C₈	—	3	2	no	—
110	C₄⋊C₈	—	3	1	no	—
111	C₄×C₈	—	3	2	no	—
112	C₄×C₈	C₂×C₃×D₃	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×C₃×C₄	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	#237	1	1	yes	characteristic
135	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
136	C₃×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
137	C₃×C₈	C₃×#41	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	C₃×C₄×D₃	1	1	yes	characteristic
145	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
146	C₄⋊C₄	—	3	2	no	—
147	C₄⋊C₄	—	3	1	no	—
148	C₄²	—	3	2	no	—
149	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
150	C₁₆	—	2	2	no	—
151	C₁₆	—	3	2	no	—
152	C₁₆	—	6	1	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	1	no	—
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	—	2	2	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	C₃×M₃₂	1	2	yes	—
166	C₃×C₄	#887	1	1	yes	characteristic
167	C₃×C₄	#907	1	1	yes	characteristic
168	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
169	C₃×C₄	C₃×#131	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	C₃×D₃×C₈	1	1	yes	characteristic
177	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
178	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
179	C₈	—	2	1	no	—
180	C₈	—	6	2	no	—
181	C₈	C₃×#237	1	1	yes	characteristic
182	C₈	C₃×D₃×Q₈	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	#4830	1	1	yes	characteristic
187	C₂×C₃	C₂×#798	1	1	yes	characteristic
188	C₂×C₃	#4907	1	1	yes	characteristic
189	C₂×C₃	C₃×C₈×Q₈	1	1	yes	characteristic
190	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
191	C₂×C₃	C₃×#504	1	1	yes	characteristic
192	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	6	2	no	—
195	C₄	—	6	2	no	—
196	C₄	C₃×#798	1	2	yes	—
197	C₄	C₃×#887	1	1	yes	characteristic
198	C₄	C₃×#907	1	1	yes	characteristic
199	C₃	—	2	1	no	—
200	C₃	#69281	1	1	yes	characteristic, direct product factor
201	C₃	C₃×#2183	1	1	yes	characteristic, complemented
202	C₂	C₃×#4830	1	1	yes	characteristic
203	C₂	C₂×C₃×#798	1	1	yes	characteristic
204	C₂	C₃×#4907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃²	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₄⋊C₁₆	D₃
hypocenter	C₃	C₃×#2183
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
7. #69281	C₃	direct (N × H)
37. C₃×C₃⋊C₁₆	Q₈	semidirect
97. C₃⋊C₁₆	C₃×Q₈	semidirect
172. C₃²	#2183	semidirect
200. C₃	#69281	direct (N × H)
201. C₃	C₃×#2183	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
C₃²×C₄⋊C₁₆	2
C₃×#4502	2
C₃×Dic₃⋊C₁₆	2
C₃×C₄×C₃⋊C₁₆	2
C₃×#2183	3
#69281	3

Subgroups — C3×#69281 #11798011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69281 #11798011