Subgroups — C₃×#68557 #11777222

Order 1152.

355 conjugacy classes of subgroups.

The subgroup lattice has 235 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4830	C₂	1	2	yes	maximal
2	C₃×C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃²×#335	C₂	1	1	yes	maximal, characteristic
4	C₃×#4484	C₂	1	2	yes	maximal
5	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃×C₈×Q₁₆	—	3	1	no	maximal
7	#68557	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Dic₁₂	C₄	1	1	yes	characteristic
9	C₃×C₄×Dic₆	C₂²	1	2	yes	—
10	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×#818	C₄	1	1	yes	characteristic
13	C₃×#816	C₄	1	2	yes	—
14	C₃×Dic₃⋊C₈	—	2	2	no	—
15	C₃×C₈×Dic₃	—	2	2	no	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₈×Q₈	—	3	2	no	—
19	C₃×C₄×Q₁₆	—	3	1	no	—
20	#4830	C₂×C₃	1	2	yes	—
21	C₄×Dic₁₂	C₂×C₃	1	1	yes	characteristic
22	C₃×#335	—	2	1	no	—
23	C₃×#335	D₃	1	1	yes	characteristic
24	C₃×#335	C₂×C₃	1	1	yes	characteristic
25	C₃×Q₈⋊C₈	—	3	2	no	—
26	C₃×C₈²	—	3	1	no	—
27	#4484	C₂×C₃	1	2	yes	—
28	C₈×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃²×C₄²	C₂³	1	1	yes	characteristic
34	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
35	C₃×Dic₃⋊C₄	—	2	2	no	—
36	C₃×C₄×Dic₃	—	2	2	no	—
37	C₂×C₃×C₃⋊C₈	—	2	1	no	—
38	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
40	C₃×Dic₁₂	C₈	1	4	yes	complemented
41	C₈×Q₁₆	—	3	1	no	Sylow
42	C₂×C₃×Q₁₆	—	3	1	no	—
43	C₃×C₄×Q₈	—	3	2	no	—
44	C₂×Dic₁₂	C₃×C₄	1	1	yes	characteristic
45	C₄×Dic₆	C₂²×C₃	1	2	yes	—
46	C₃×#107	—	3	1	no	—
47	C₃×C₄⋊C₈	—	2	2	no	—
48	C₃×C₄⋊C₈	—	6	2	no	—
49	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
50	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
51	C₃×#103	—	3	2	no	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	—	3	1	no	—
54	C₃×C₄×C₈	—	3	1	no	—
55	C₃×C₄×C₈	—	6	2	no	—
56	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
58	#818	C₃×C₄	1	1	yes	characteristic
59	#816	C₃×C₄	1	2	yes	—
60	Dic₃⋊C₈	—	2	2	no	—
61	C₈×Dic₃	—	2	2	no	—
62	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	#41	1	1	yes	characteristic
67	C₂×C₃×Dic₃	—	2	2	no	—
68	C₂×C₃×Dic₃	—	2	2	no	—
69	C₃×Dic₆	—	2	2	no	—
70	C₃×Dic₆	C₂×C₈	1	4	yes	—
71	C₃²×C₈	—	2	1	no	—
72	C₃²×C₈	—	4	2	no	—
73	C₃²×C₈	C₂×C₈	1	2	yes	—
74	C₃×C₃⋊C₈	—	2	2	no	—
75	C₃×C₃⋊C₈	—	2	1	no	—
76	C₃×C₃⋊C₈	Q₁₆	1	2	yes	complemented
77	C₈×Q₈	—	3	2	no	—
78	C₄×Q₁₆	—	3	1	no	—
79	#335	C₃×D₃	1	1	yes	characteristic
80	Q₈⋊C₈	—	3	2	no	—
81	C₈²	—	3	1	no	—
82	C₂×C₃×Q₈	—	3	2	no	—
83	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
84	C₃×Q₁₆	—	3	4	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	3	1	no	—
90	C₂×C₃×C₈	—	3	1	no	—
91	C₂×C₃×C₈	—	4	2	no	—
92	C₂×C₃×C₈	—	6	1	no	—
93	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
94	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
97	C₃×C₄⋊C₄	—	3	2	no	—
98	C₃×C₄⋊C₄	—	6	2	no	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	—	6	2	no	—
101	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
102	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
103	C₄⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
104	Dic₃⋊C₄	—	2	2	no	—
105	C₄×Dic₃	—	2	2	no	—
106	C₂×C₃⋊C₈	—	2	1	no	—
107	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
108	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
109	Dic₁₂	C₃×C₈	1	4	yes	complemented
110	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
111	C₃²×C₄	—	2	1	no	—
112	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
113	C₃²×C₄	#131	1	1	yes	characteristic
114	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
115	C₃²×C₄	#135	1	1	yes	characteristic
116	C₃×Dic₃	—	2	4	no	—
117	C₃×Dic₃	—	4	2	no	—
118	C₂×Q₁₆	—	3	1	no	—
119	C₄×Q₈	—	3	2	no	—
120	#107	—	3	1	no	—
121	C₄⋊C₈	—	6	2	no	—
122	C₄⋊C₈	C₂×C₃×D₃	1	2	yes	—
123	#103	—	3	2	no	—
124	C₄×C₈	—	3	1	no	—
125	C₄×C₈	—	3	1	no	—
126	C₄×C₈	—	6	2	no	—
127	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
128	C₃×Q₈	—	3	4	no	—
129	C₃×Q₈	—	6	2	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	—	6	2	no	—
134	C₂×C₃×C₄	—	6	2	no	—
135	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	#239	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
141	C₂×Dic₃	—	2	2	no	—
142	C₂×Dic₃	—	2	2	no	—
143	Dic₆	—	2	2	no	—
144	Dic₆	C₂×C₃×C₈	1	4	yes	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	2	2	no	—
148	C₃×C₈	—	3	2	no	—
149	C₃×C₈	—	4	1	no	—
150	C₃×C₈	—	4	2	no	—
151	C₃×C₈	—	4	2	no	—
152	C₃×C₈	—	6	2	no	—
153	C₃×C₈	—	6	1	no	—
154	C₃×C₈	—	8	2	no	—
155	C₃×C₈	D₃×C₈	1	2	yes	—
156	C₃×C₈	C₂×C₃×C₈	1	2	yes	—
157	C₃⋊C₈	—	2	2	no	—
158	C₃⋊C₈	—	2	1	no	—
159	C₃⋊C₈	C₃×Q₁₆	1	2	yes	complemented
160	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
161	C₂×C₃²	C₄×Q₁₆	1	1	yes	characteristic
162	C₂×C₃²	#443	1	1	yes	characteristic
163	C₂×Q₈	—	3	2	no	—
164	Q₁₆	—	3	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	3	1	no	—
167	C₂×C₈	—	3	1	no	—
168	C₂×C₈	—	6	1	no	—
169	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
170	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
171	C₄⋊C₄	—	3	2	no	—
172	C₄⋊C₄	—	6	2	no	—
173	C₄²	—	6	2	no	—
174	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	#893	1	1	yes	characteristic
177	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	4	1	no	—
185	C₃×C₄	—	6	4	no	—
186	C₃×C₄	—	12	2	no	—
187	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
188	C₃×C₄	#907	1	1	yes	characteristic
189	C₃×C₄	#912	1	1	yes	characteristic
190	C₃×C₄	D₃×Q₁₆	1	1	yes	characteristic
191	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
192	C₃×C₄	C₃×#131	1	1	yes	characteristic
193	C₃×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
194	C₃×C₄	C₃×#135	1	1	yes	characteristic
195	Dic₃	—	2	4	no	—
196	Dic₃	—	4	2	no	—
197	C₃²	C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
198	Q₈	—	3	4	no	—
199	Q₈	—	6	2	no	—
200	C₂×C₄	—	6	2	no	—
201	C₂×C₄	—	6	2	no	—
202	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
203	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₃×#239	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	3	2	no	—
207	C₈	—	4	2	no	—
208	C₈	—	6	2	no	—
209	C₈	—	6	1	no	—
210	C₈	C₃×D₃×C₈	1	2	yes	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	#4834	1	1	yes	characteristic
215	C₂×C₃	#4873	1	1	yes	characteristic
216	C₂×C₃	#4895	1	1	yes	characteristic
217	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
218	C₂×C₃	C₃×C₄×Q₁₆	1	1	yes	characteristic
219	C₂×C₃	C₃×#443	1	1	yes	characteristic
220	C₂²	C₃×#893	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	6	4	no	—
223	C₄	—	12	2	no	—
224	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
225	C₄	C₃×#907	1	1	yes	characteristic
226	C₄	C₃×#912	1	1	yes	characteristic
227	C₄	C₃×D₃×Q₁₆	1	1	yes	characteristic
228	C₃	—	2	1	no	—
229	C₃	#68557	1	1	yes	characteristic, direct product factor
230	C₃	C₃×C₈×Q₁₆	1	1	yes	characteristic, complemented
231	C₂	C₃×#4834	1	1	yes	characteristic
232	C₂	C₃×#4873	1	1	yes	characteristic
233	C₂	C₃×#4895	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#335	C₂
socle, prime radical	C₂²×C₃²	C₄×D₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#335	D₃
hypocenter	C₃	C₃×C₈×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#335

Complements

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

N	H ≅ G/N	type
7. #68557	C₃	direct (N × H)
40. C₃×Dic₁₂	C₈	semidirect
76. C₃×C₃⋊C₈	Q₁₆	semidirect
109. Dic₁₂	C₃×C₈	semidirect
159. C₃⋊C₈	C₃×Q₁₆	semidirect
197. C₃²	C₈×Q₁₆	semidirect
229. C₃	#68557	direct (N × H)
230. C₃	C₃×C₈×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4830	2
C₃×C₄×Dic₁₂	2
C₃²×#335	2
C₃×#4484	2
C₃×C₈×C₃⋊C₈	2
C₃×C₈×Q₁₆	3
#68557	3

Subgroups — C3×#68557 #11777222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68557 #11777222