Subgroups — C₃×#83789 #11858670

Order 1152.

1013 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	C₃×#5819	C₂	1	2	yes	maximal
3	C₃×#5125	C₂	1	6	yes	maximal, complemented
4	C₂×C₃×#819	C₂	1	6	yes	maximal, complemented
5	C₃×#2916	—	3	1	no	maximal
6	#83789	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
8	C₃²×#131	C₄	1	8	yes	—
9	C₂×C₃²×M₁₆	C₂²	1	3	yes	—
10	C₂²×C₃²×C₈	C₂²	1	3	yes	—
11	C₃×#948	—	2	8	no	—
12	C₂×C₃×#208	C₂²	1	12	yes	—
13	C₃×#819	C₂²	1	4	yes	—
14	C₃×#819	C₂²	1	12	yes	—
15	C₃×#568	—	3	2	no	—
16	C₂×C₃×#131	—	2	1	no	—
17	C₂×C₃×#131	D₃	1	1	yes	characteristic
18	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
19	#5819	C₂×C₃	1	2	yes	—
20	C₃×#430	—	3	6	no	—
21	C₂×C₃×#108	—	3	6	no	—
22	#5125	C₂×C₃	1	6	yes	complemented
23	C₂×#819	C₂×C₃	1	6	yes	complemented
24	C₃²×#41	C₂×C₄	1	4	yes	—
25	C₃²×#41	D₄	1	4	yes	—
26	C₂×C₃²×Q₈	Q₈	1	1	yes	characteristic
27	C₂×C₃²×D₄	Q₈	1	3	yes	—
28	C₂²×C₃²×C₄	C₂³	1	3	yes	—
29	C₃²×M₁₆	C₂×C₄	1	12	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	12	yes	—
31	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂³	1	3	yes	—
33	C₃×#208	—	2	12	no	—
34	C₃×#208	C₂³	1	8	yes	—
35	C₂×C₃×C₃⋊C₈	—	2	12	no	—
36	#2916	—	3	1	no	Sylow
37	C₂×C₃×#41	—	2	1	no	—
38	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
40	C₃×#131	—	2	8	no	—
41	C₃×#131	—	6	8	no	—
42	C₃×#131	Dic₃	1	8	yes	—
43	C₃×#131	C₃×C₄	1	8	yes	—
44	C₂×C₃×M₁₆	—	2	3	no	—
45	C₂×C₃×M₁₆	—	3	12	no	—
46	C₂×C₃×M₁₆	C₂×D₃	1	3	yes	—
47	C₂×C₃×M₁₆	C₂²×C₃	1	3	yes	—
48	C₂²×C₃×C₈	—	2	3	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	3	yes	—
50	C₂²×C₃×C₈	C₂²×C₃	1	3	yes	—
51	#948	—	2	8	no	—
52	C₂×#208	C₂²×C₃	1	12	yes	—
53	C₃×#108	—	3	4	no	—
54	C₃×#108	—	3	12	no	—
55	#819	C₂²×C₃	1	12	yes	—
56	#819	C₂²×C₃	1	4	yes	—
57	C₂³×C₃²	C₂×Q₈	1	3	yes	—
58	C₃²×Q₈	C₄⋊C₄	1	4	yes	—
59	C₃²×D₄	C₄⋊C₄	1	12	yes	—
60	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
61	C₂×C₃²×C₄	C₂×D₄	1	6	yes	—
62	C₂×C₃²×C₄	C₂×Q₈	1	3	yes	—
63	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
64	C₃²×C₈	C₂²×C₄	1	2	yes	—
65	C₃²×C₈	C₂²×C₄	1	6	yes	—
66	C₃×C₃⋊C₈	—	2	8	no	—
67	#568	—	3	2	no	—
68	C₂×#131	C₃×D₃	1	1	yes	characteristic
69	#430	—	3	6	no	—
70	C₂×#108	—	3	6	no	—
71	C₃×#41	—	2	4	no	—
72	C₃×#41	—	2	4	no	—
73	C₃×#41	D₁₂	1	4	yes	—
74	C₃×#41	C₂×Dic₃	1	4	yes	—
75	C₃×#41	C₂×C₃×C₄	1	4	yes	—
76	C₃×#41	C₃×D₄	1	4	yes	—
77	C₂×C₃×Q₈	—	2	1	no	—
78	C₂×C₃×Q₈	Dic₆	1	1	yes	characteristic
79	C₂×C₃×Q₈	C₃×Q₈	1	1	yes	characteristic
80	C₂×C₃×D₄	—	2	3	no	—
81	C₂×C₃×D₄	Dic₆	1	3	yes	—
82	C₂×C₃×D₄	C₃×Q₈	1	3	yes	—
83	C₂²×C₃×C₄	—	2	3	no	—
84	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
85	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
86	C₃×M₁₆	—	2	12	no	—
87	C₃×M₁₆	—	3	8	no	—
88	C₃×M₁₆	—	6	12	no	—
89	C₃×M₁₆	C₂×Dic₃	1	12	yes	—
90	C₃×M₁₆	C₂×C₃×C₄	1	12	yes	—
91	C₂×C₃×C₈	—	2	12	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	2	3	no	—
94	C₂×C₃×C₈	—	6	12	no	—
95	C₂×C₃×C₈	C₂×Dic₃	1	12	yes	—
96	C₂×C₃×C₈	C₂×C₃×C₄	1	12	yes	—
97	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂²×D₃	1	3	yes	—
99	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂³×C₃	1	3	yes	—
101	#208	—	2	12	no	—
102	#208	C₂³×C₃	1	8	yes	—
103	C₂×C₃⋊C₈	—	2	12	no	—
104	C₂²×C₃²	—	4	3	no	—
105	C₂²×C₃²	C₂×C₄⋊C₄	1	6	yes	—
106	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
107	C₃²×C₄	C₂×C₄⋊C₄	1	6	yes	—
108	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
109	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
110	C₂×#41	C₂×C₃×D₃	1	1	yes	characteristic
111	#131	—	6	8	no	—
112	#131	C₃×Dic₃	1	8	yes	—
113	C₂×M₁₆	—	3	12	no	—
114	C₂×M₁₆	C₂×C₃×D₃	1	3	yes	—
115	C₂²×C₈	C₂×C₃×D₃	1	3	yes	—
116	#108	—	3	4	no	—
117	#108	—	3	12	no	—
118	C₂³×C₃	—	2	3	no	—
119	C₂³×C₃	C₂×Dic₆	1	3	yes	—
120	C₂³×C₃	C₂×C₃×Q₈	1	3	yes	—
121	C₃×Q₈	—	2	4	no	—
122	C₃×Q₈	C₄⋊Dic₃	1	4	yes	—
123	C₃×Q₈	C₃×C₄⋊C₄	1	4	yes	—
124	C₃×D₄	—	2	12	no	—
125	C₃×D₄	C₄⋊Dic₃	1	12	yes	—
126	C₃×D₄	C₃×C₄⋊C₄	1	12	yes	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	3	no	—
129	C₂×C₃×C₄	—	2	6	no	—
130	C₂×C₃×C₄	—	2	6	no	—
131	C₂×C₃×C₄	C₂×Dic₆	1	3	yes	—
132	C₂×C₃×C₄	C₂×D₁₂	1	6	yes	—
133	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
134	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
135	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
136	C₂×C₃×C₄	C₂×C₃×Q₈	1	3	yes	—
137	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	—	2	6	no	—
141	C₃×C₈	—	6	8	no	—
142	C₃×C₈	C₂²×Dic₃	1	2	yes	—
143	C₃×C₈	C₂²×Dic₃	1	6	yes	—
144	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
145	C₃×C₈	C₂²×C₃×C₄	1	6	yes	—
146	C₃⋊C₈	—	2	8	no	—
147	C₂×C₃²	—	2	1	no	—
148	C₂×C₃²	—	2	6	no	—
149	C₂×C₃²	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	#41	C₃×D₁₂	1	4	yes	—
151	#41	C₂×C₃×Dic₃	1	4	yes	—
152	C₂×Q₈	C₃×Dic₆	1	1	yes	characteristic
153	C₂×D₄	C₃×Dic₆	1	3	yes	—
154	C₂²×C₄	C₂²×C₃×D₃	1	3	yes	—
155	M₁₆	—	3	8	no	—
156	M₁₆	—	6	12	no	—
157	M₁₆	C₂×C₃×Dic₃	1	12	yes	—
158	C₂×C₈	—	6	12	no	—
159	C₂×C₈	C₂×C₃×Dic₃	1	12	yes	—
160	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
161	C₂×C₈	C₂²×C₃×D₃	1	3	yes	—
162	C₂²×C₃	—	2	6	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	—	4	3	no	—
165	C₂²×C₃	—	4	3	no	—
166	C₂²×C₃	—	4	6	no	—
167	C₂²×C₃	C₂×C₄⋊Dic₃	1	6	yes	—
168	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	6	yes	—
169	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
170	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
171	C₃×C₄	—	2	6	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	C₂×C₄⋊Dic₃	1	6	yes	—
175	C₃×C₄	C₂×C₃×C₄⋊C₄	1	6	yes	—
176	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
177	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
178	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
179	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
180	C₃²	#2916	1	1	yes	characteristic, complemented, Sylow
181	C₂³	C₂×C₃×Dic₆	1	3	yes	—
182	Q₈	C₃×C₄⋊Dic₃	1	4	yes	—
183	D₄	C₃×C₄⋊Dic₃	1	12	yes	—
184	C₂×C₄	C₂×C₃×Dic₆	1	3	yes	—
185	C₂×C₄	C₂×C₃×D₁₂	1	6	yes	—
186	C₂×C₄	C₂²×C₃×Dic₃	1	6	yes	—
187	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
188	C₈	—	6	8	no	—
189	C₈	C₂²×C₃×Dic₃	1	2	yes	—
190	C₈	C₂²×C₃×Dic₃	1	6	yes	—
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	6	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	6	no	—
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	—	4	6	no	—
198	C₂×C₃	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
199	C₂×C₃	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
200	C₂²	—	4	3	no	—
201	C₂²	C₂×C₃×C₄⋊Dic₃	1	6	yes	—
202	C₂²	C₂²×C₃×Dic₆	1	1	yes	characteristic
203	C₄	C₂×C₃×C₄⋊Dic₃	1	6	yes	—
204	C₄	C₂²×C₃×D₁₂	1	1	yes	characteristic
205	C₄	C₂³×C₃×Dic₃	1	1	yes	characteristic
206	C₃	—	2	1	no	—
207	C₃	#83789	1	1	yes	characteristic, direct product factor
208	C₃	C₃×#2916	1	1	yes	characteristic, complemented
209	C₂	—	2	1	no	—
210	C₂	—	2	6	no	—
211	C₂	C₂²×C₃×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂²×D₁₂
commutator	C₃×C₄	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂×C₃²	C₂²×C₄⋊C₄
prime radical	C₂×C₃²×#41	C₂²
cosocle	C₂×C₃×C₄	C₂⁴×C₃
hypercenter	C₂×C₃×#131	D₃
hypocenter	C₃	C₃×#2916
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×#41 ◁ C₂×C₃×#131

Complements

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

N	H ≅ G/N	type
3. C₃×#5125	C₂	semidirect
4. C₂×C₃×#819	C₂	semidirect
6. #83789	C₃	direct (N × H)
22. #5125	C₂×C₃	semidirect
23. C₂×#819	C₂×C₃	semidirect
180. C₃²	#2916	semidirect
207. C₃	#83789	direct (N × H)
208. C₃	C₃×#2916	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
C₃×#5819	2
C₃×#5125	2
C₂×C₃×#819	2
C₃×#2916	3
#83789	3

Subgroups — C3×#83789 #11858670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#83789 #11858670