Subgroups — C₃²×#1666 #11768715

Order 1152.

516 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#492	C₂	1	1	yes	maximal, characteristic
2	C₃²×#475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#460	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#331	C₂	1	1	yes	maximal, characteristic
8	C₃×#1666	C₃	1	4	yes	maximal, direct product factor
9	C₂×C₃²×SD₁₆	—	2	1	no	—
10	C₂×C₃²×SD₁₆	—	2	1	no	—
11	C₂×C₃²×SD₁₆	—	2	1	no	—
12	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃²×#127	C₂²	1	1	yes	characteristic
14	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
15	C₃²×#106	—	2	1	no	—
16	C₃²×C₄⋊C₈	—	2	1	no	—
17	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃²×#103	—	2	1	no	—
21	C₃²×#102	—	2	1	no	—
22	C₃²×C₄×C₈	—	2	1	no	—
23	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×#492	C₂×C₃	1	4	yes	—
25	C₃×#475	C₂×C₃	1	4	yes	complemented
26	C₃×#460	C₂×C₃	1	4	yes	—
27	C₃×C₈×Q₈	C₂×C₃	1	4	yes	complemented
28	C₃×#334	C₂×C₃	1	4	yes	complemented
29	C₃×#332	C₂×C₃	1	4	yes	complemented
30	C₃×#331	C₂×C₃	1	4	yes	—
31	C₂×C₃²×Q₈	—	2	1	no	—
32	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
33	C₂×C₃²×D₄	—	2	1	no	—
34	C₂×C₃²×D₄	—	2	1	no	—
35	C₂×C₃²×D₄	—	2	1	no	—
36	C₃²×SD₁₆	—	4	2	no	—
37	C₃²×SD₁₆	—	4	2	no	—
38	C₃²×SD₁₆	—	4	2	no	—
39	C₂×C₃²×C₈	—	2	1	no	—
40	C₂×C₃²×C₈	—	2	1	no	—
41	C₂×C₃²×C₈	—	2	1	no	—
42	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
44	C₃²×C₄⋊C₄	—	2	1	no	—
45	C₃²×C₄⋊C₄	—	2	1	no	—
46	C₃²×C₄⋊C₄	—	2	1	no	—
47	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₃²×C₄²	—	2	1	no	—
49	C₃²×C₄²	C₂³	1	1	yes	characteristic
50	#1666	C₃²	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×SD₁₆	—	2	4	no	—
52	C₂×C₃×SD₁₆	—	2	4	no	—
53	C₂×C₃×SD₁₆	—	2	4	no	—
54	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
55	C₃×#127	C₂²×C₃	1	4	yes	—
56	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
57	C₃×#106	—	2	4	no	—
58	C₃×C₄⋊C₈	—	2	4	no	—
59	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
60	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
61	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
62	C₃×#103	—	2	4	no	—
63	C₃×#102	—	2	4	no	—
64	C₃×C₄×C₈	—	2	4	no	—
65	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
66	C₂³×C₃²	—	4	1	no	—
67	C₃²×Q₈	—	2	1	no	—
68	C₃²×Q₈	—	4	1	no	—
69	C₃²×Q₈	—	4	1	no	—
70	C₃²×Q₈	SD₁₆	1	2	yes	complemented
71	C₃²×D₄	—	4	1	no	—
72	C₃²×D₄	—	4	1	no	—
73	C₃²×D₄	—	4	1	no	—
74	C₃²×D₄	—	4	1	no	—
75	C₃²×D₄	—	8	1	no	—
76	C₂×C₃²×C₄	—	2	1	no	—
77	C₂×C₃²×C₄	—	2	1	no	—
78	C₂×C₃²×C₄	—	4	1	no	—
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	#41	1	1	yes	characteristic
82	C₃²×C₈	—	2	1	no	—
83	C₃²×C₈	—	4	1	no	—
84	C₃²×C₈	—	4	1	no	—
85	C₃²×C₈	—	4	1	no	—
86	C₃²×C₈	SD₁₆	1	2	yes	complemented
87	#492	C₂×C₃²	1	1	yes	characteristic
88	#475	C₂×C₃²	1	1	yes	characteristic, complemented
89	#460	C₂×C₃²	1	1	yes	characteristic
90	C₈×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
91	#334	C₂×C₃²	1	1	yes	characteristic, complemented
92	#332	C₂×C₃²	1	1	yes	characteristic, complemented
93	#331	C₂×C₃²	1	1	yes	characteristic
94	C₂×C₃×Q₈	—	2	4	no	—
95	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
96	C₂×C₃×D₄	—	2	4	no	—
97	C₂×C₃×D₄	—	2	4	no	—
98	C₂×C₃×D₄	—	2	4	no	—
99	C₃×SD₁₆	—	4	8	no	—
100	C₃×SD₁₆	—	4	8	no	—
101	C₃×SD₁₆	—	4	8	no	—
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	—	2	4	no	—
104	C₂×C₃×C₈	—	2	4	no	—
105	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
106	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
107	C₃×C₄⋊C₄	—	2	4	no	—
108	C₃×C₄⋊C₄	—	2	4	no	—
109	C₃×C₄⋊C₄	—	2	4	no	—
110	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
111	C₃×C₄²	—	2	4	no	—
112	C₃×C₄²	C₂³×C₃	1	4	yes	—
113	C₂²×C₃²	—	8	1	no	—
114	C₂²×C₃²	—	8	1	no	—
115	C₂²×C₃²	—	8	1	no	—
116	C₂²×C₃²	#121	1	1	yes	characteristic
117	C₃²×C₄	—	2	2	no	—
118	C₃²×C₄	—	2	1	no	—
119	C₃²×C₄	—	4	1	no	—
120	C₃²×C₄	—	8	1	no	—
121	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
122	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
123	C₃²×C₄	#135	1	1	yes	characteristic
124	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×SD₁₆	—	2	1	no	—
126	C₂×SD₁₆	—	2	1	no	—
127	C₂×SD₁₆	—	2	1	no	—
128	C₄⋊Q₈	C₂²×C₃²	1	1	yes	characteristic
129	#127	C₂²×C₃²	1	1	yes	characteristic
130	C₄×Q₈	C₂²×C₃²	1	1	yes	characteristic
131	#106	—	2	1	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
134	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
135	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
136	#103	—	2	1	no	—
137	#102	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
140	C₂³×C₃	—	4	4	no	—
141	C₃×Q₈	—	2	4	no	—
142	C₃×Q₈	—	4	4	no	—
143	C₃×Q₈	—	4	4	no	—
144	C₃×Q₈	C₃×SD₁₆	1	8	yes	complemented
145	C₃×D₄	—	4	4	no	—
146	C₃×D₄	—	4	4	no	—
147	C₃×D₄	—	4	4	no	—
148	C₃×D₄	—	4	4	no	—
149	C₃×D₄	—	8	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	4	4	no	—
153	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
154	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
155	C₂×C₃×C₄	C₃×#41	1	4	yes	—
156	C₃×C₈	—	2	4	no	—
157	C₃×C₈	—	4	4	no	—
158	C₃×C₈	—	4	4	no	—
159	C₃×C₈	—	4	4	no	—
160	C₃×C₈	C₃×SD₁₆	1	8	yes	complemented
161	C₂×C₃²	—	16	1	no	—
162	C₂×C₃²	#460	1	1	yes	characteristic
163	C₂×C₃²	#465	1	1	yes	characteristic
164	C₂×C₃²	#471	1	1	yes	characteristic
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	C₃²×D₄	1	1	yes	characteristic
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	—	2	1	no	—
170	SD₁₆	—	4	2	no	—
171	SD₁₆	—	4	2	no	—
172	SD₁₆	—	4	2	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
177	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	C₃²×D₄	1	1	yes	characteristic
182	C₄²	—	2	1	no	—
183	C₄²	C₂³×C₃²	1	1	yes	characteristic
184	C₂²×C₃	—	8	4	no	—
185	C₂²×C₃	—	8	4	no	—
186	C₂²×C₃	—	8	4	no	—
187	C₂²×C₃	C₃×#121	1	4	yes	—
188	C₃×C₄	—	2	8	no	—
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	—	4	4	no	—
191	C₃×C₄	—	8	4	no	—
192	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
193	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
194	C₃×C₄	C₃×#135	1	4	yes	—
195	C₃×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
196	C₃²	#1666	1	1	yes	characteristic, direct product factor, Sylow
197	C₂³	—	4	1	no	—
198	Q₈	—	2	1	no	—
199	Q₈	—	4	1	no	—
200	Q₈	—	4	1	no	—
201	Q₈	C₃²×SD₁₆	1	2	yes	complemented
202	D₄	—	4	1	no	—
203	D₄	—	4	1	no	—
204	D₄	—	4	1	no	—
205	D₄	—	4	1	no	—
206	D₄	—	8	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
212	C₂×C₄	C₃²×#41	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	4	1	no	—
215	C₈	—	4	1	no	—
216	C₈	—	4	1	no	—
217	C₈	C₃²×SD₁₆	1	2	yes	complemented
218	C₂×C₃	—	16	4	no	—
219	C₂×C₃	C₃×#460	1	4	yes	—
220	C₂×C₃	C₃×#465	1	4	yes	—
221	C₂×C₃	C₃×#471	1	4	yes	—
222	C₂²	—	8	1	no	—
223	C₂²	—	8	1	no	—
224	C₂²	—	8	1	no	—
225	C₂²	C₃²×#121	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	8	1	no	—
230	C₄	C₂×C₃²×SD₁₆	1	1	yes	characteristic
231	C₄	C₂×C₃²×SD₁₆	1	1	yes	characteristic
232	C₄	C₃²×#135	1	1	yes	characteristic
233	C₄	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₃	C₃×#1666	1	4	yes	direct product factor
235	C₂	—	16	1	no	—
236	C₂	C₃²×#460	1	1	yes	characteristic
237	C₂	C₃²×#465	1	1	yes	characteristic
238	C₂	C₃²×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
prime radical	C₃²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×#475	C₂	semidirect
4. C₃²×C₈×Q₈	C₂	semidirect
5. C₃²×#334	C₂	semidirect
6. C₃²×#332	C₂	semidirect
8. C₃×#1666	C₃	direct (N × H)
25. C₃×#475	C₂×C₃	semidirect
27. C₃×C₈×Q₈	C₂×C₃	semidirect
28. C₃×#334	C₂×C₃	semidirect
29. C₃×#332	C₂×C₃	semidirect
50. #1666	C₃²	direct (N × H)
70. C₃²×Q₈	SD₁₆	semidirect
86. C₃²×C₈	SD₁₆	semidirect
88. #475	C₂×C₃²	semidirect
90. C₈×Q₈	C₂×C₃²	semidirect
91. #334	C₂×C₃²	semidirect
92. #332	C₂×C₃²	semidirect
144. C₃×Q₈	C₃×SD₁₆	semidirect
160. C₃×C₈	C₃×SD₁₆	semidirect
196. C₃²	#1666	direct (N × H)
201. Q₈	C₃²×SD₁₆	semidirect
217. C₈	C₃²×SD₁₆	semidirect
234. C₃	C₃×#1666	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#492	2
C₃²×#475	2
C₃²×#460	2
C₃²×C₈×Q₈	2
C₃²×#334	2
C₃²×#332	2
C₃²×#331	2
C₃×#1666	3

Subgroups — C32×#1666 #11768715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1666 #11768715