Subgroups — C₃×#69277 #11797433

Order 1152.

373 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃×C₁₆	C₂	1	2	yes	maximal, complemented
2	C₃×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×(C₂×D₃)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₄×C₁₆	—	3	1	no	maximal
7	#69277	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₃×C₈	C₂²	1	2	yes	—
9	C₃×C₄×D₁₂	C₄	1	1	yes	characteristic
10	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×(C₂×D₃)⋊C₈	C₄	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×D₃×C₁₆	—	2	4	no	—
17	C₂²×C₃×C₁₆	—	3	2	no	—
18	C₃×C₈×D₄	—	3	1	no	—
19	C₂×D₃×C₁₆	C₂×C₃	1	2	yes	complemented
20	C₈×D₁₂	C₂×C₃	1	1	yes	characteristic
21	C₃×C₄⋊C₁₆	—	2	1	no	—
22	C₃×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
23	C₃×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₃×C₂²⋊C₁₆	—	3	2	no	—
25	C₃×C₄×C₁₆	—	3	1	no	—
26	(C₂×D₃)⋊C₁₆	C₂×C₃	1	2	yes	complemented
27	C₄×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃×D₁₂	C₈	1	1	yes	characteristic
29	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
32	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₃×(C₂×D₃)⋊C₄	C₈	1	2	yes	—
34	C₃×C₄⋊Dic₃	C₈	1	1	yes	characteristic
35	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₃×D₃×C₈	—	2	2	no	—
37	C₃×D₃×C₈	—	2	2	no	—
38	C₃²×C₁₆	—	2	2	no	—
39	C₃×C₃⋊C₁₆	—	2	1	no	—
40	C₃×C₃⋊C₁₆	D₄	1	2	yes	complemented
41	D₄×C₁₆	—	3	1	no	Sylow
42	C₂²×C₃×C₈	—	3	2	no	—
43	C₃×C₄×D₄	—	3	1	no	—
44	C₂×D₃×C₈	C₂²×C₃	1	2	yes	—
45	C₄×D₁₂	C₃×C₄	1	1	yes	characteristic
46	C₂×C₃×C₁₆	—	2	2	no	—
47	C₂×C₃×C₁₆	—	3	1	no	—
48	C₂×C₃×C₁₆	—	3	1	no	—
49	C₂×C₃×C₁₆	—	6	4	no	—
50	C₂×C₃×C₁₆	C₂×D₃	1	2	yes	—
51	C₂×C₃×C₁₆	C₂²×C₃	1	2	yes	—
52	C₃×C₄⋊C₈	—	3	1	no	—
53	C₃×C₂²⋊C₈	—	3	2	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
57	(C₂×D₃)⋊C₈	C₃×C₄	1	2	yes	—
58	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
60	C₄⋊(C₃⋊C₈)	C₃×C₄	1	1	yes	characteristic
61	D₃×C₁₆	—	2	4	no	—
62	C₂²×C₃×D₃	C₂×C₈	1	2	yes	—
63	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
67	C₃×D₁₂	C₁₆	1	4	yes	complemented
68	C₃×C₄×D₃	—	2	2	no	—
69	C₃×C₄×D₃	—	2	2	no	—
70	C₃²×C₈	—	2	1	no	—
71	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₃²×C₈	#41	1	1	yes	characteristic
73	C₃×C₃⋊C₈	—	2	2	no	—
74	C₂²×C₁₆	—	3	2	no	—
75	C₈×D₄	—	3	1	no	—
76	C₄⋊C₁₆	C₃×D₃	1	1	yes	characteristic, complemented
77	C₂²⋊C₁₆	—	3	2	no	—
78	C₄×C₁₆	—	3	1	no	—
79	C₂×C₃×D₄	—	3	1	no	—
80	C₂²×C₃×C₄	—	3	2	no	—
81	C₂×D₁₂	C₃×C₈	1	1	yes	characteristic
82	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
83	C₂×C₃×C₈	—	2	1	no	—
84	C₂×C₃×C₈	—	2	1	no	—
85	C₂×C₃×C₈	—	3	2	no	—
86	C₂×C₃×C₈	—	6	2	no	—
87	C₂×C₃×C₈	—	6	2	no	—
88	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
92	C₃×C₄⋊C₄	—	3	1	no	—
93	C₃×C₂²⋊C₄	—	3	2	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
96	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
97	(C₂×D₃)⋊C₄	C₃×C₈	1	2	yes	—
98	C₄⋊Dic₃	C₃×C₈	1	1	yes	characteristic
99	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
100	D₃×C₈	—	2	2	no	—
101	D₃×C₈	—	2	2	no	—
102	C₃×C₁₆	—	2	2	no	—
103	C₃×C₁₆	—	2	2	no	—
104	C₃×C₁₆	—	3	2	no	—
105	C₃×C₁₆	—	4	2	no	—
106	C₃×C₁₆	—	6	1	no	—
107	C₃⋊C₁₆	—	2	1	no	—
108	C₃⋊C₁₆	C₃×D₄	1	2	yes	complemented
109	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₃×D₃	—	2	2	no	—
111	C₂×C₃×D₃	—	2	2	no	—
112	C₂×C₃×D₃	C₂×C₁₆	1	4	yes	complemented
113	C₃²×C₄	—	2	1	no	—
114	C₃²×C₄	C₂×C₁₆	1	2	yes	complemented
115	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
116	C₃²×C₄	#131	1	1	yes	characteristic
117	C₃×Dic₃	—	2	2	no	—
118	C₂²×C₈	—	3	2	no	—
119	C₄×D₄	—	3	1	no	—
120	C₂×C₁₆	—	3	1	no	—
121	C₂×C₁₆	—	3	1	no	—
122	C₂×C₁₆	—	6	4	no	—
123	C₂×C₁₆	C₂×C₃×D₃	1	2	yes	—
124	C₄⋊C₈	—	3	1	no	—
125	C₂²⋊C₈	—	3	2	no	—
126	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
127	C₂³×C₃	—	3	2	no	—
128	C₂²×D₃	C₂×C₃×C₈	1	2	yes	—
129	C₃×D₄	—	3	4	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	3	2	no	—
134	C₂×C₃×C₄	—	6	2	no	—
135	C₂×C₃×C₄	—	6	2	no	—
136	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
137	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂×Dic₃	C₂×C₃×C₈	1	2	yes	—
143	D₁₂	C₃×C₁₆	1	4	yes	complemented
144	C₄×D₃	—	2	2	no	—
145	C₄×D₃	—	2	2	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	2	1	no	—
148	C₃×C₈	—	2	1	no	—
149	C₃×C₈	—	2	1	no	—
150	C₃×C₈	—	4	1	no	—
151	C₃×C₈	—	6	2	no	—
152	C₃×C₈	D₃×D₄	1	1	yes	characteristic
153	C₃×C₈	#239	1	1	yes	characteristic
154	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
155	C₃×C₈	C₃×#41	1	1	yes	characteristic
156	C₃⋊C₈	—	2	2	no	—
157	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
158	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
159	C₂×C₃²	#504	1	1	yes	characteristic
160	C₃×D₃	—	2	4	no	—
161	C₂×D₄	—	3	1	no	—
162	C₂²×C₄	—	3	2	no	—
163	C₂×C₈	—	3	2	no	—
164	C₂×C₈	—	6	2	no	—
165	C₂×C₈	—	6	2	no	—
166	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
167	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
168	C₄⋊C₄	—	3	1	no	—
169	C₂²⋊C₄	—	3	2	no	—
170	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₁₆	—	3	2	no	—
173	C₁₆	—	6	1	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	3	4	no	—
176	C₂²×C₃	—	6	2	no	—
177	C₂²×C₃	—	6	2	no	—
178	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
179	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
180	C₂×D₃	—	2	2	no	—
181	C₂×D₃	—	2	2	no	—
182	C₂×D₃	C₂×C₃×C₁₆	1	4	yes	complemented
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	4	1	no	—
189	C₃×C₄	—	6	2	no	—
190	C₃×C₄	D₃×C₁₆	1	2	yes	complemented
191	C₃×C₄	C₂×C₃×C₁₆	1	2	yes	complemented
192	C₃×C₄	#893	1	1	yes	characteristic
193	C₃×C₄	#907	1	1	yes	characteristic
194	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
195	C₃×C₄	C₃×#131	1	1	yes	characteristic
196	Dic₃	—	2	2	no	—
197	C₃²	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	3	2	no	—
199	D₄	—	3	4	no	—
200	C₂×C₄	—	3	2	no	—
201	C₂×C₄	—	6	2	no	—
202	C₂×C₄	—	6	2	no	—
203	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
204	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
205	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	6	2	no	—
208	C₈	C₃×D₃×D₄	1	1	yes	characteristic
209	C₈	C₃×#239	1	1	yes	characteristic
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	6	4	no	—
214	C₂×C₃	#4834	1	1	yes	characteristic
215	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
216	C₂×C₃	#4907	1	1	yes	characteristic
217	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
218	C₂×C₃	C₂²×C₃×C₁₆	1	1	yes	characteristic
219	C₂×C₃	C₃×#504	1	1	yes	characteristic
220	D₃	—	2	4	no	—
221	C₂²	—	3	4	no	—
222	C₂²	—	6	2	no	—
223	C₂²	—	6	2	no	—
224	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	6	2	no	—
227	C₄	C₃×D₃×C₁₆	1	2	yes	complemented
228	C₄	C₃×#893	1	1	yes	characteristic
229	C₄	C₃×#907	1	1	yes	characteristic
230	C₃	—	2	1	no	—
231	C₃	#69277	1	1	yes	characteristic, direct product factor
232	C₃	C₃×D₄×C₁₆	1	1	yes	characteristic, complemented
233	C₂	—	6	4	no	—
234	C₂	C₃×#4834	1	1	yes	characteristic
235	C₂	C₂×C₃×D₃×C₁₆	1	1	yes	characteristic
236	C₂	C₃×#4907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃×C₁₆
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₄⋊C₁₆	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃×D₁₂	C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₄⋊C₁₆	D₃
hypocenter	C₃	C₃×D₄×C₁₆
LCS	G ▷ C₂×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
1. C₂×C₃×D₃×C₁₆	C₂	semidirect
3. C₃²×C₄⋊C₁₆	C₂	semidirect
4. C₃×(C₂×D₃)⋊C₁₆	C₂	semidirect
5. C₃×C₄×C₃⋊C₁₆	C₂	semidirect
7. #69277	C₃	direct (N × H)
19. C₂×D₃×C₁₆	C₂×C₃	semidirect
22. C₃×C₄⋊C₁₆	D₃	semidirect
23. C₃×C₄⋊C₁₆	C₂×C₃	semidirect
26. (C₂×D₃)⋊C₁₆	C₂×C₃	semidirect
27. C₄×C₃⋊C₁₆	C₂×C₃	semidirect
40. C₃×C₃⋊C₁₆	D₄	semidirect
67. C₃×D₁₂	C₁₆	semidirect
76. C₄⋊C₁₆	C₃×D₃	semidirect
108. C₃⋊C₁₆	C₃×D₄	semidirect
112. C₂×C₃×D₃	C₂×C₁₆	semidirect
114. C₃²×C₄	C₂×C₁₆	semidirect
143. D₁₂	C₃×C₁₆	semidirect
182. C₂×D₃	C₂×C₃×C₁₆	semidirect
190. C₃×C₄	D₃×C₁₆	semidirect
191. C₃×C₄	C₂×C₃×C₁₆	semidirect
197. C₃²	D₄×C₁₆	semidirect
227. C₄	C₃×D₃×C₁₆	semidirect
231. C₃	#69277	direct (N × H)
232. C₃	C₃×D₄×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃×C₁₆	2
C₃×C₈×D₁₂	2
C₃²×C₄⋊C₁₆	2
C₃×(C₂×D₃)⋊C₁₆	2
C₃×C₄×C₃⋊C₁₆	2
C₃×D₄×C₁₆	3
#69277	3

Subgroups — C3×#69277 #11797433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69277 #11797433