Subgroups — C₃×#69067 #11797845

Order 1152.

321 conjugacy classes of subgroups.

The subgroup lattice has 223 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4904	C₂	1	1	yes	maximal, characteristic
2	C₃×#4700	C₂	1	2	yes	maximal, complemented
3	C₃²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4510	C₂	1	2	yes	maximal, complemented
5	C₃×#4506	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2178	—	3	1	no	maximal
7	#69067	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×#904	C₂²	1	2	yes	—
9	C₃×#901	—	2	2	no	—
10	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×#819	C₂²	1	2	yes	—
14	C₃×#805	—	2	2	no	—
15	C₃×Dic₂₄	C₄	1	1	yes	characteristic, complemented
16	C₃×#800	C₄	1	2	yes	complemented
17	C₃×D₄₈	C₄	1	1	yes	characteristic, complemented
18	C₃×#508	—	3	1	no	—
19	C₃×#443	—	3	2	no	—
20	#4904	C₂×C₃	1	1	yes	characteristic
21	#4700	C₂×C₃	1	2	yes	complemented
22	C₃×#368	—	3	1	no	—
23	C₃×#359	—	3	2	no	—
24	C₃×C₄×C₁₆	—	2	1	no	—
25	C₃×C₄×C₁₆	D₃	1	1	yes	characteristic, complemented
26	C₃×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
27	#4510	C₂×C₃	1	2	yes	complemented
28	#4506	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#235	—	2	2	no	—
30	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
32	C₃²×C₄²	D₄	1	1	yes	characteristic
33	C₃×#208	—	2	2	no	—
34	C₃×Dic₁₂	C₂×C₄	1	2	yes	—
35	C₃×D₂₄	C₂×C₄	1	2	yes	—
36	C₃×#204	—	2	2	no	—
37	C₃×ΓL(1,𝔽₂₅)	—	4	2	no	—
38	C₃×D₃×C₈	—	4	2	no	—
39	C₃²×C₁₆	—	2	1	no	—
40	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
42	#2178	—	3	1	no	Sylow
43	C₃×#135	—	3	2	no	—
44	C₃×#131	—	6	2	no	—
45	#904	C₂²×C₃	1	2	yes	—
46	#901	—	2	2	no	—
47	C₃×Q₃₂	—	3	1	no	—
48	C₃×SD₃₂	—	3	2	no	—
49	C₃×D₁₆	—	3	1	no	—
50	C₂×C₃×C₁₆	—	2	1	no	—
51	C₂×C₃×C₁₆	—	2	1	no	—
52	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
56	C₃×#108	—	3	2	no	—
57	C₃×#104	—	6	2	no	—
58	C₃×C₄×C₈	—	2	1	no	—
59	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	#819	C₂²×C₃	1	2	yes	—
62	#805	—	2	2	no	—
63	Dic₂₄	C₃×C₄	1	1	yes	characteristic, complemented
64	#800	C₃×C₄	1	2	yes	complemented
65	D₄₈	C₃×C₄	1	1	yes	characteristic, complemented
66	C₂×C₃²×C₄	—	2	1	no	—
67	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₃×#67	—	4	2	no	—
69	C₃×D₁₂	—	2	2	no	—
70	C₃×C₄×D₃	—	4	2	no	—
71	C₃×Dic₆	—	2	2	no	—
72	C₃²×C₈	D₈	1	1	yes	characteristic
73	C₃²×C₈	D₈	1	1	yes	characteristic
74	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₃²×C₈	#41	1	1	yes	characteristic
76	C₃×C₃⋊C₈	—	4	2	no	—
77	#508	—	3	1	no	—
78	#443	—	3	2	no	—
79	#368	—	3	1	no	—
80	#359	—	3	2	no	—
81	C₄×C₁₆	C₃×D₃	1	1	yes	characteristic, complemented
82	C₃×#41	—	6	2	no	—
83	#235	—	2	2	no	—
84	C₃×Q₁₆	—	3	2	no	—
85	C₃×SD₁₆	—	6	2	no	—
86	C₃×D₈	—	3	2	no	—
87	C₃×M₁₆	—	6	2	no	—
88	C₃×M₁₆	—	12	2	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	12	2	no	—
92	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
93	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	D₁₂	1	1	yes	characteristic
98	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
99	#208	—	2	2	no	—
100	Dic₁₂	C₂×C₃×C₄	1	2	yes	—
101	D₂₄	C₂×C₃×C₄	1	2	yes	—
102	#204	—	2	2	no	—
103	ΓL(1,𝔽₂₅)	—	4	2	no	—
104	D₃×C₈	—	4	2	no	—
105	C₃×C₁₆	—	2	2	no	—
106	C₃×C₁₆	—	2	1	no	—
107	C₃×C₁₆	—	2	1	no	—
108	C₃×C₁₆	—	2	1	no	—
109	C₃×C₁₆	—	2	1	no	—
110	C₃×C₁₆	C₄×D₃	1	1	yes	characteristic
111	C₃×C₁₆	C₄×D₃	1	1	yes	characteristic
112	C₃×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
113	C₃×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂²×C₃²	#135	1	1	yes	characteristic
115	C₂×C₃×D₃	—	4	2	no	—
116	C₃²×C₄	—	2	2	no	—
117	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
118	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
119	C₃×Dic₃	—	4	2	no	—
120	#135	—	3	2	no	—
121	#131	—	6	2	no	—
122	Q₃₂	—	3	1	no	—
123	SD₃₂	—	3	2	no	—
124	D₁₆	—	3	1	no	—
125	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
126	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
127	#108	—	3	2	no	—
128	#104	—	6	2	no	—
129	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
130	C₃×Q₈	—	6	2	no	—
131	C₃×D₄	—	6	2	no	—
132	C₃×D₄	—	12	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	12	2	no	—
138	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
140	#67	—	4	2	no	—
141	D₁₂	—	2	2	no	—
142	C₄×D₃	—	4	2	no	—
143	Dic₆	—	2	2	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	2	1	no	—
148	C₃×C₈	—	12	2	no	—
149	C₃×C₈	D₂₄	1	1	yes	characteristic
150	C₃×C₈	D₂₄	1	1	yes	characteristic
151	C₃×C₈	C₃×D₈	1	1	yes	characteristic
152	C₃×C₈	C₃×D₈	1	1	yes	characteristic
153	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
154	C₃×C₈	#235	1	1	yes	characteristic
155	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₃×C₈	C₃×#41	1	1	yes	characteristic
157	C₃⋊C₈	—	4	2	no	—
158	C₂×C₃²	—	2	1	no	—
159	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
160	C₃×D₃	—	8	2	no	—
161	#41	—	6	2	no	—
162	Q₁₆	—	3	2	no	—
163	SD₁₆	—	6	2	no	—
164	D₈	—	3	2	no	—
165	M₁₆	—	6	2	no	—
166	M₁₆	—	12	2	no	—
167	C₂×C₈	—	12	2	no	—
168	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
169	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
170	C₄²	C₃×D₁₂	1	1	yes	characteristic
171	C₁₆	—	2	1	no	—
172	C₁₆	C₃×C₄×D₃	1	1	yes	characteristic
173	C₁₆	C₃×C₄×D₃	1	1	yes	characteristic
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	12	2	no	—
176	C₂²×C₃	#904	1	1	yes	characteristic
177	C₂²×C₃	C₃×#135	1	1	yes	characteristic
178	C₂×D₃	—	4	2	no	—
179	C₃×C₄	—	2	4	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	12	2	no	—
185	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
186	C₃×C₄	C₂×D₂₄	1	1	yes	characteristic
187	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
188	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
189	Dic₃	—	4	2	no	—
190	C₃²	#2178	1	1	yes	characteristic, complemented, Sylow
191	Q₈	—	6	2	no	—
192	D₄	—	6	2	no	—
193	D₄	—	12	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	12	2	no	—
196	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
197	C₈	—	12	2	no	—
198	C₈	C₃×D₂₄	1	1	yes	characteristic
199	C₈	C₃×D₂₄	1	1	yes	characteristic
200	C₈	C₂×C₃×C₄×D₃	1	1	yes	characteristic
201	C₈	C₃×#235	1	1	yes	characteristic
202	C₂×C₃	—	2	2	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	2	1	no	—
206	C₂×C₃	—	24	2	no	—
207	C₂×C₃	C₄×D₂₄	1	1	yes	characteristic
208	C₂×C₃	C₃×C₄×D₈	1	1	yes	characteristic
209	D₃	—	8	2	no	—
210	C₂²	—	12	2	no	—
211	C₂²	C₃×#904	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	12	2	no	—
214	C₄	C₃×C₄×D₁₂	1	1	yes	characteristic
215	C₄	C₂×C₃×D₂₄	1	1	yes	characteristic
216	C₃	—	2	1	no	—
217	C₃	#69067	1	1	yes	characteristic, direct product factor
218	C₃	C₃×#2178	1	1	yes	characteristic, complemented
219	C₂	—	2	1	no	—
220	C₂	—	24	2	no	—
221	C₂	C₃×C₄×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	D₂₄
commutator	C₃×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₄×C₁₆	C₂
socle	C₂×C₃²	C₄×D₈
prime radical	C₃×#4904	C₂
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₄×C₁₆	D₃
hypocenter	C₃	C₃×#2178
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈ ◁ C₃×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×#4700	C₂	semidirect
3. C₃²×C₄×C₁₆	C₂	semidirect
4. C₃×#4510	C₂	semidirect
5. C₃×#4506	C₂	semidirect
7. #69067	C₃	direct (N × H)
15. C₃×Dic₂₄	C₄	semidirect
16. C₃×#800	C₄	semidirect
17. C₃×D₄₈	C₄	semidirect
21. #4700	C₂×C₃	semidirect
25. C₃×C₄×C₁₆	D₃	semidirect
26. C₃×C₄×C₁₆	C₂×C₃	semidirect
27. #4510	C₂×C₃	semidirect
28. #4506	C₂×C₃	semidirect
63. Dic₂₄	C₃×C₄	semidirect
64. #800	C₃×C₄	semidirect
65. D₄₈	C₃×C₄	semidirect
81. C₄×C₁₆	C₃×D₃	semidirect
190. C₃²	#2178	semidirect
217. C₃	#69067	direct (N × H)
218. C₃	C₃×#2178	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4904	2
C₃×#4700	2
C₃²×C₄×C₁₆	2
C₃×#4510	2
C₃×#4506	2
C₃×#2178	3
#69067	3

Subgroups — C3×#69067 #11797845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69067 #11797845