Subgroups — C₃×#68084 #11777311

Order 1152.

353 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×Dic₆	C₂	1	2	yes	maximal
3	C₃²×#322	C₂	1	1	yes	maximal, characteristic
4	C₃×#4457	C₂	1	1	yes	maximal, characteristic
5	C₃×#4456	C₂	1	2	yes	maximal
6	C₃×#1579	—	3	1	no	maximal
7	#68084	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	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×#818	C₄	1	1	yes	characteristic
14	C₃×#816	C₄	1	2	yes	—
15	C₃×Dic₃⋊C₈	—	2	2	no	—
16	C₃×C₈×Dic₃	—	2	2	no	—
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃×C₈×Q₈	—	3	2	no	—
19	C₃×C₄×Q₁₆	—	3	1	no	—
20	C₄×Dic₁₂	C₂×C₃	1	1	yes	characteristic
21	C₈×Dic₆	C₂×C₃	1	2	yes	—
22	C₃×#334	—	3	1	no	—
23	C₃×Q₈⋊C₈	—	3	2	no	—
24	C₃×#322	—	2	1	no	—
25	C₃×#322	D₃	1	1	yes	characteristic
26	C₃×#322	C₂×C₃	1	1	yes	characteristic
27	#4457	C₂×C₃	1	1	yes	characteristic
28	#4456	C₂×C₃	1	2	yes	—
29	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	—	2	1	no	—
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	D₄	1	2	yes	—
34	C₃²×C₄²	C₂³	1	1	yes	characteristic
35	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
36	C₃×Dic₃⋊C₄	—	2	2	no	—
37	C₃×C₄×Dic₃	—	2	2	no	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₃×Dic₁₂	C₈	1	4	yes	complemented
40	#1579	—	3	1	no	Sylow
41	C₂×C₃×Q₁₆	—	3	1	no	—
42	C₃×C₄×Q₈	—	3	2	no	—
43	C₂×Dic₁₂	C₃×C₄	1	1	yes	characteristic
44	C₄×Dic₆	C₂²×C₃	1	2	yes	—
45	C₃×#107	—	3	1	no	—
46	C₃×C₄⋊C₈	—	3	2	no	—
47	C₃×C₄⋊C₈	—	6	2	no	—
48	C₃×#103	—	3	2	no	—
49	C₃×C₄×C₈	—	2	1	no	—
50	C₃×C₄×C₈	—	2	1	no	—
51	C₃×C₄×C₈	—	2	1	no	—
52	C₃×C₄×C₈	—	6	2	no	—
53	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
59	#818	C₃×C₄	1	1	yes	characteristic
60	#816	C₃×C₄	1	2	yes	—
61	Dic₃⋊C₈	—	2	2	no	—
62	C₈×Dic₃	—	2	2	no	—
63	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
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	—	2	2	no	—
73	C₃²×C₈	—	4	1	no	—
74	C₃²×C₈	C₂×C₈	1	2	yes	—
75	C₃×C₃⋊C₈	—	4	2	no	—
76	C₈×Q₈	—	3	2	no	—
77	C₄×Q₁₆	—	3	1	no	—
78	#334	—	3	1	no	—
79	Q₈⋊C₈	—	3	2	no	—
80	#322	C₃×D₃	1	1	yes	characteristic
81	C₂×C₃×Q₈	—	3	2	no	—
82	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
83	C₃×Q₁₆	—	3	4	no	—
84	C₂×C₃×C₈	—	2	2	no	—
85	C₂×C₃×C₈	—	2	1	no	—
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	6	2	no	—
91	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
92	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃×C₈	D₁₂	1	2	yes	—
94	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
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	2	no	—
107	Dic₁₂	C₃×C₈	1	4	yes	complemented
108	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
109	C₃²×C₄	—	2	1	no	—
110	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
111	C₃²×C₄	#131	1	1	yes	characteristic
112	C₃²×C₄	#137	1	2	yes	—
113	C₃×Dic₃	—	2	4	no	—
114	C₃×Dic₃	—	4	2	no	—
115	C₂×Q₁₆	—	3	1	no	—
116	C₄×Q₈	—	3	2	no	—
117	#107	—	3	1	no	—
118	C₄⋊C₈	—	3	2	no	—
119	C₄⋊C₈	—	6	2	no	—
120	#103	—	3	2	no	—
121	C₄×C₈	—	6	2	no	—
122	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
123	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
124	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
125	C₃×Q₈	—	3	4	no	—
126	C₃×Q₈	—	6	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	6	2	no	—
131	C₂×C₃×C₄	—	6	2	no	—
132	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
134	C₂×C₃×C₄	#235	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
138	C₂×Dic₃	—	2	2	no	—
139	C₂×Dic₃	—	2	2	no	—
140	Dic₆	—	2	2	no	—
141	Dic₆	C₂×C₃×C₈	1	4	yes	—
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	2	no	—
147	C₃×C₈	—	2	2	no	—
148	C₃×C₈	—	4	2	no	—
149	C₃×C₈	—	4	2	no	—
150	C₃×C₈	—	4	1	no	—
151	C₃×C₈	—	4	1	no	—
152	C₃×C₈	—	12	2	no	—
153	C₃×C₈	D₃×C₈	1	2	yes	—
154	C₃×C₈	C₂×C₃×C₈	1	2	yes	—
155	C₃⋊C₈	—	4	2	no	—
156	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
157	C₂×C₃²	#441	1	1	yes	characteristic
158	C₂×C₃²	#444	1	1	yes	characteristic
159	C₂×Q₈	—	3	2	no	—
160	Q₁₆	—	3	4	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	6	2	no	—
163	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
164	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
165	C₂×C₈	C₃×D₁₂	1	2	yes	—
166	C₄⋊C₄	—	3	2	no	—
167	C₄⋊C₄	—	6	2	no	—
168	C₄²	—	6	2	no	—
169	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
172	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	6	4	no	—
180	C₃×C₄	—	12	2	no	—
181	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
182	C₃×C₄	#901	1	1	yes	characteristic
183	C₃×C₄	#909	1	2	yes	—
184	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
185	C₃×C₄	C₃×#131	1	1	yes	characteristic
186	C₃×C₄	C₃×#137	1	2	yes	—
187	Dic₃	—	2	4	no	—
188	Dic₃	—	4	2	no	—
189	C₃²	#1579	1	1	yes	characteristic, complemented, Sylow
190	Q₈	—	3	4	no	—
191	Q₈	—	6	2	no	—
192	C₂×C₄	—	6	2	no	—
193	C₂×C₄	—	6	2	no	—
194	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
196	C₂×C₄	C₃×#235	1	1	yes	characteristic
197	C₈	—	2	1	no	—
198	C₈	—	2	2	no	—
199	C₈	—	4	1	no	—
200	C₈	—	12	2	no	—
201	C₈	C₃×D₃×C₈	1	2	yes	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
206	C₂×C₃	#4716	1	1	yes	characteristic
207	C₂×C₃	#4717	1	1	yes	characteristic
208	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
209	C₂×C₃	C₃×#441	1	1	yes	characteristic
210	C₂×C₃	C₃×#444	1	1	yes	characteristic
211	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	6	4	no	—
214	C₄	—	12	2	no	—
215	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
216	C₄	C₃×#901	1	1	yes	characteristic
217	C₄	C₃×#909	1	2	yes	—
218	C₃	—	2	1	no	—
219	C₃	#68084	1	1	yes	characteristic, direct product factor
220	C₃	C₃×#1579	1	1	yes	characteristic, complemented
221	C₂	C₃×C₈×D₁₂	1	1	yes	characteristic
222	C₂	C₃×#4716	1	1	yes	characteristic
223	C₂	C₃×#4717	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#322	C₂
socle, prime radical	C₂²×C₃²	C₄×D₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#322	D₃
hypocenter	C₃	C₃×#1579
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×#322

Complements

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

N	H ≅ G/N	type
7. #68084	C₃	direct (N × H)
39. C₃×Dic₁₂	C₈	semidirect
107. Dic₁₂	C₃×C₈	semidirect
189. C₃²	#1579	semidirect
219. C₃	#68084	direct (N × H)
220. C₃	C₃×#1579	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Dic₁₂	2
C₃×C₈×Dic₆	2
C₃²×#322	2
C₃×#4457	2
C₃×#4456	2
C₃×#1579	3
#68084	3

Subgroups — C3×#68084 #11777311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68084 #11777311