Subgroups — C₃×#74356 #11857081

Order 1152.

1389 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×C₃×#888	C₂	1	6	yes	maximal
2	C₃²×#384	C₂	1	1	yes	maximal, characteristic
3	C₃×#4975	C₂	1	6	yes	maximal
4	C₃×#4935	C₂	1	1	yes	maximal, characteristic
5	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₃×#2868	—	3	1	no	maximal
7	#74356	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×Dic₆	C₂²	1	4	yes	—
9	C₂×C₃²×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	6	yes	—
12	C₂×C₃×Dic₃⋊C₄	C₂²	1	12	yes	—
13	C₂×C₃×C₄×Dic₃	C₂²	1	3	yes	—
14	C₂×C₃×C₄×Dic₃	C₂²	1	3	yes	—
15	C₃×#888	—	2	24	no	—
16	C₂×C₃×C₄⋊Q₈	—	3	6	no	—
17	C₂×#888	C₂×C₃	1	6	yes	—
18	C₃×#384	—	2	1	no	—
19	C₃×#384	—	3	6	no	—
20	C₃×#384	—	3	1	no	—
21	C₃×#384	D₃	1	1	yes	characteristic
22	C₃×#384	C₂×C₃	1	1	yes	characteristic
23	C₃×C₄³	—	3	1	no	—
24	#4975	C₂×C₃	1	6	yes	—
25	#4935	C₂×C₃	1	1	yes	characteristic
26	C₄²×Dic₃	C₂×C₃	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂³	1	4	yes	—
28	C₂²×C₃²×C₄	C₂³	1	3	yes	—
29	C₂²×C₃×Dic₃	C₂³	1	4	yes	—
30	C₂²×C₃×Dic₃	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	C₂³	1	3	yes	—
32	C₂×C₃×Dic₆	—	2	24	no	—
33	C₃²×C₄⋊C₄	—	2	12	no	—
34	C₃²×C₄²	Q₈	1	4	yes	complemented
35	C₃×C₄⋊Dic₃	—	2	12	no	—
36	C₃×Dic₃⋊C₄	—	2	24	no	—
37	C₃×C₄×Dic₃	D₄	1	12	yes	—
38	C₃×C₄×Dic₃	Q₈	1	12	yes	complemented
39	#2868	—	3	1	no	Sylow
40	C₂²×C₃×Q₈	—	3	4	no	—
41	C₂²×Dic₆	C₂²×C₃	1	4	yes	—
42	C₃×C₄⋊Q₈	—	6	24	no	—
43	C₂×C₃×C₄⋊C₄	—	2	6	no	—
44	C₂×C₃×C₄⋊C₄	—	3	12	no	—
45	C₂×C₃×C₄⋊C₄	—	3	6	no	—
46	C₂×C₃×C₄⋊C₄	C₂×D₃	1	6	yes	—
47	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	6	yes	—
48	C₂×C₃×C₄²	—	2	1	no	—
49	C₂×C₃×C₄²	—	3	3	no	—
50	C₂×C₃×C₄²	—	3	3	no	—
51	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄⋊Dic₃	C₂²×C₃	1	6	yes	—
54	C₂×Dic₃⋊C₄	C₂²×C₃	1	12	yes	—
55	C₂×C₄×Dic₃	C₂²×C₃	1	3	yes	—
56	C₂×C₄×Dic₃	C₂²×C₃	1	3	yes	—
57	#888	—	2	24	no	—
58	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	12	no	—
60	C₂×C₃²×C₄	C₂×D₄	1	6	yes	—
61	C₂×C₃²×C₄	C₂×Q₈	1	6	yes	—
62	C₂×C₃²×C₄	C₂×Q₈	1	6	yes	—
63	C₂×C₃×Dic₃	—	2	12	no	—
64	C₂×C₃×Dic₃	C₂×D₄	1	6	yes	—
65	C₂×C₃×Dic₃	C₂×D₄	1	6	yes	—
66	C₂×C₃×Dic₃	C₂×Q₈	1	12	yes	—
67	C₃×Dic₆	—	4	16	no	—
68	C₂×C₄⋊Q₈	—	3	6	no	—
69	#384	—	3	6	no	—
70	#384	—	3	1	no	—
71	#384	C₃×D₃	1	1	yes	characteristic
72	C₄³	—	3	1	no	—
73	C₂×C₃×Q₈	—	6	24	no	—
74	C₂²×C₃×C₄	—	2	3	no	—
75	C₂²×C₃×C₄	—	2	4	no	—
76	C₂²×C₃×C₄	—	3	3	no	—
77	C₂²×C₃×C₄	—	3	1	no	—
78	C₂²×C₃×C₄	—	3	4	no	—
79	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
80	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
81	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
82	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
83	C₂²×Dic₃	C₂³×C₃	1	4	yes	—
84	C₂²×Dic₃	C₂³×C₃	1	1	yes	characteristic
85	C₂²×Dic₃	C₂³×C₃	1	3	yes	—
86	C₂×Dic₆	—	2	24	no	—
87	C₃×C₄⋊C₄	—	2	12	no	—
88	C₃×C₄⋊C₄	—	2	12	no	—
89	C₃×C₄⋊C₄	—	4	12	no	—
90	C₃×C₄⋊C₄	—	6	24	no	—
91	C₃×C₄⋊C₄	—	6	12	no	—
92	C₃×C₄²	—	2	4	no	—
93	C₃×C₄²	—	3	12	no	—
94	C₃×C₄²	—	3	12	no	—
95	C₃×C₄²	Dic₆	1	4	yes	complemented
96	C₃×C₄²	C₃×Q₈	1	4	yes	complemented
97	C₄⋊Dic₃	—	2	12	no	—
98	Dic₃⋊C₄	—	2	24	no	—
99	C₄×Dic₃	C₃×D₄	1	12	yes	—
100	C₄×Dic₃	C₃×Q₈	1	12	yes	complemented
101	C₂²×C₃²	C₂²×D₄	1	3	yes	—
102	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×C₃²	C₂²×Q₈	1	3	yes	—
104	C₃²×C₄	—	4	4	no	—
105	C₃²×C₄	C₄⋊Q₈	1	12	yes	complemented
106	C₃×Dic₃	—	4	4	no	—
107	C₃×Dic₃	#127	1	4	yes	—
108	C₃×Dic₃	C₄⋊Q₈	1	12	yes	complemented
109	C₂²×Q₈	—	3	4	no	—
110	C₄⋊Q₈	—	6	24	no	—
111	C₂×C₄⋊C₄	—	3	12	no	—
112	C₂×C₄⋊C₄	—	3	6	no	—
113	C₂×C₄⋊C₄	C₂×C₃×D₃	1	6	yes	—
114	C₂×C₄²	—	3	3	no	—
115	C₂×C₄²	—	3	3	no	—
116	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
117	C₂³×C₃	—	2	1	no	—
118	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
119	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
120	C₃×Q₈	—	12	16	no	—
121	C₂×C₃×C₄	—	2	12	no	—
122	C₂×C₃×C₄	—	2	12	no	—
123	C₂×C₃×C₄	—	2	6	no	—
124	C₂×C₃×C₄	—	2	6	no	—
125	C₂×C₃×C₄	—	2	6	no	—
126	C₂×C₃×C₄	—	3	6	no	—
127	C₂×C₃×C₄	—	3	6	no	—
128	C₂×C₃×C₄	—	3	12	no	—
129	C₂×C₃×C₄	—	4	12	no	—
130	C₂×C₃×C₄	—	6	12	no	—
131	C₂×C₃×C₄	C₂×Dic₆	1	6	yes	—
132	C₂×C₃×C₄	D₃×D₄	1	6	yes	—
133	C₂×C₃×C₄	D₃×Q₈	1	6	yes	—
134	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
135	C₂×C₃×C₄	C₂×C₃×Q₈	1	6	yes	—
136	C₂×C₃×C₄	C₂×C₃×Q₈	1	6	yes	—
137	C₂×Dic₃	—	2	12	no	—
138	C₂×Dic₃	C₂×C₃×D₄	1	6	yes	—
139	C₂×Dic₃	C₂×C₃×D₄	1	6	yes	—
140	C₂×Dic₃	C₂×C₃×Q₈	1	12	yes	—
141	Dic₆	—	4	16	no	—
142	C₂×C₃²	C₂×#127	1	1	yes	characteristic
143	C₂×C₃²	C₂×C₄⋊Q₈	1	3	yes	—
144	C₂×C₃²	C₂×C₄⋊Q₈	1	3	yes	—
145	C₂×Q₈	—	6	24	no	—
146	C₂²×C₄	—	3	3	no	—
147	C₂²×C₄	—	3	1	no	—
148	C₂²×C₄	—	3	4	no	—
149	C₂²×C₄	C₂²×C₃×D₃	1	4	yes	—
150	C₂²×C₄	C₂²×C₃×D₃	1	3	yes	—
151	C₄⋊C₄	—	2	12	no	—
152	C₄⋊C₄	—	6	24	no	—
153	C₄⋊C₄	—	6	12	no	—
154	C₄²	—	3	12	no	—
155	C₄²	—	3	12	no	—
156	C₄²	C₃×Dic₆	1	4	yes	complemented
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	2	3	no	—
159	C₂²×C₃	—	2	3	no	—
160	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
161	C₂²×C₃	C₂×D₃×D₄	1	3	yes	—
162	C₂²×C₃	C₂×D₃×Q₈	1	3	yes	—
163	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
164	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
165	C₂²×C₃	C₂²×C₃×Q₈	1	3	yes	—
166	C₃×C₄	—	2	12	no	—
167	C₃×C₄	—	3	4	no	—
168	C₃×C₄	—	3	12	no	—
169	C₃×C₄	—	4	4	no	—
170	C₃×C₄	—	4	4	no	—
171	C₃×C₄	—	8	4	no	—
172	C₃×C₄	—	12	4	no	—
173	C₃×C₄	#888	1	12	yes	complemented
174	C₃×C₄	C₃×C₄⋊Q₈	1	12	yes	complemented
175	Dic₃	—	4	4	no	—
176	Dic₃	C₃×#127	1	4	yes	—
177	Dic₃	C₃×C₄⋊Q₈	1	12	yes	complemented
178	C₃²	#2868	1	1	yes	characteristic, complemented, Sylow
179	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
180	Q₈	—	12	16	no	—
181	C₂×C₄	—	2	12	no	—
182	C₂×C₄	—	3	6	no	—
183	C₂×C₄	—	3	6	no	—
184	C₂×C₄	—	3	12	no	—
185	C₂×C₄	—	6	12	no	—
186	C₂×C₄	C₂×C₃×Dic₆	1	6	yes	—
187	C₂×C₄	C₃×D₃×D₄	1	6	yes	—
188	C₂×C₄	C₃×D₃×Q₈	1	6	yes	—
189	C₂×C₃	—	2	3	no	—
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	—	2	3	no	—
192	C₂×C₃	C₂×#888	1	3	yes	—
193	C₂×C₃	D₃×#127	1	1	yes	characteristic
194	C₂×C₃	D₃×C₄⋊Q₈	1	3	yes	—
195	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
196	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	3	yes	—
197	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	3	yes	—
198	C₂²	C₂²×C₃×Dic₆	1	1	yes	characteristic
199	C₂²	C₂×C₃×D₃×D₄	1	3	yes	—
200	C₂²	C₂×C₃×D₃×Q₈	1	3	yes	—
201	C₄	—	3	4	no	—
202	C₄	—	3	12	no	—
203	C₄	—	4	4	no	—
204	C₄	—	12	4	no	—
205	C₄	C₃×#888	1	12	yes	complemented
206	C₃	—	2	1	no	—
207	C₃	#74356	1	1	yes	characteristic, direct product factor
208	C₃	C₃×#2868	1	1	yes	characteristic, complemented
209	C₂	C₂×C₃×#888	1	3	yes	—
210	C₂	C₃×D₃×#127	1	1	yes	characteristic
211	C₂	C₃×D₃×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₃
commutator, cosocle	C₂³×C₃	C₂⁴×C₃
Frattini	C₂³	C₂³×C₃×D₃
Fitting	C₃²×#384	C₂
socle, prime radical	C₂³×C₃²	C₂⁴
hypercenter	C₃×#384	D₃
hypocenter	C₃	C₃×#2868
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#384

Complements

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

N	H ≅ G/N	type
7. #74356	C₃	direct (N × H)
34. C₃²×C₄²	Q₈	semidirect
38. C₃×C₄×Dic₃	Q₈	semidirect
95. C₃×C₄²	Dic₆	semidirect
96. C₃×C₄²	C₃×Q₈	semidirect
100. C₄×Dic₃	C₃×Q₈	semidirect
105. C₃²×C₄	C₄⋊Q₈	semidirect
108. C₃×Dic₃	C₄⋊Q₈	semidirect
156. C₄²	C₃×Dic₆	semidirect
173. C₃×C₄	#888	semidirect
174. C₃×C₄	C₃×C₄⋊Q₈	semidirect
177. Dic₃	C₃×C₄⋊Q₈	semidirect
178. C₃²	#2868	semidirect
205. C₄	C₃×#888	semidirect
207. C₃	#74356	direct (N × H)
208. C₃	C₃×#2868	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#888	2
C₃²×#384	2
C₃×#4975	2
C₃×#4935	2
C₃×C₄²×Dic₃	2
C₃×#2868	3
#74356	3

Subgroups — C3×#74356 #11857081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#74356 #11857081