Subgroups — C₃²×#1671 #11768720

Order 1152.

498 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#476	C₂	1	1	yes	maximal, characteristic
3	C₃²×#461	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic
5	C₃²×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1671	C₃	1	4	yes	maximal, direct product factor
9	C₂×C₃²×SD₁₆	—	2	1	no	—
10	C₂²×C₃²×C₈	—	2	1	no	—
11	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
14	C₃²×#106	—	2	1	no	—
15	C₃²×#106	—	2	1	no	—
16	C₃²×#106	—	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×C₂²⋊C₈	—	2	1	no	—
23	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×#498	C₂×C₃	1	4	yes	complemented
25	C₃×#476	C₂×C₃	1	4	yes	—
26	C₃×#461	C₂×C₃	1	4	yes	—
27	C₃×C₈×D₄	C₂×C₃	1	4	yes	—
28	C₃×#334	C₂×C₃	1	4	yes	complemented
29	C₃×#333	C₂×C₃	1	4	yes	complemented
30	C₃×#332	C₂×C₃	1	4	yes	complemented
31	C₂×C₃²×Q₈	—	2	1	no	—
32	C₂×C₃²×Q₈	—	2	1	no	—
33	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
34	C₂²×C₃²×C₄	—	2	1	no	—
35	C₃²×SD₁₆	—	4	2	no	—
36	C₂×C₃²×C₈	—	2	1	no	—
37	C₂×C₃²×C₈	—	2	1	no	—
38	C₂×C₃²×C₈	—	2	1	no	—
39	C₂×C₃²×C₈	—	4	2	no	—
40	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
42	C₃²×C₄⋊C₄	—	2	1	no	—
43	C₃²×C₄⋊C₄	—	2	1	no	—
44	C₃²×C₄⋊C₄	—	2	1	no	—
45	C₃²×C₄⋊C₄	—	2	1	no	—
46	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₃²×C₂²⋊C₄	—	2	1	no	—
48	C₃²×C₄²	C₂³	1	1	yes	characteristic
49	#1671	C₃²	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₃×SD₁₆	—	2	4	no	—
51	C₂²×C₃×C₈	—	2	4	no	—
52	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
53	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
54	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
55	C₃×#106	—	2	4	no	—
56	C₃×#106	—	2	4	no	—
57	C₃×#106	—	2	4	no	—
58	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
59	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
60	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
61	C₃×#103	—	2	4	no	—
62	C₃×#102	—	2	4	no	—
63	C₃×C₂²⋊C₈	—	2	4	no	—
64	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
65	C₂³×C₃²	—	2	1	no	—
66	C₃²×Q₈	—	4	1	no	—
67	C₃²×Q₈	—	4	1	no	—
68	C₃²×Q₈	—	8	1	no	—
69	C₃²×D₄	—	2	1	no	—
70	C₃²×D₄	SD₁₆	1	2	yes	—
71	C₂×C₃²×C₄	—	2	1	no	—
72	C₂×C₃²×C₄	—	4	1	no	—
73	C₂×C₃²×C₄	—	4	1	no	—
74	C₂×C₃²×C₄	—	4	1	no	—
75	C₂×C₃²×C₄	—	4	1	no	—
76	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	#41	1	1	yes	characteristic
79	C₃²×C₈	—	2	1	no	—
80	C₃²×C₈	—	4	1	no	—
81	C₃²×C₈	—	4	1	no	—
82	C₃²×C₈	—	4	1	no	—
83	C₃²×C₈	SD₁₆	1	2	yes	complemented
84	#498	C₂×C₃²	1	1	yes	characteristic, complemented
85	#476	C₂×C₃²	1	1	yes	characteristic
86	#461	C₂×C₃²	1	1	yes	characteristic
87	C₈×D₄	C₂×C₃²	1	1	yes	characteristic
88	#334	C₂×C₃²	1	1	yes	characteristic, complemented
89	#333	C₂×C₃²	1	1	yes	characteristic, complemented
90	#332	C₂×C₃²	1	1	yes	characteristic, complemented
91	C₂×C₃×Q₈	—	2	4	no	—
92	C₂×C₃×Q₈	—	2	4	no	—
93	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₃×SD₁₆	—	4	8	no	—
96	C₂×C₃×C₈	—	2	4	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	2	4	no	—
99	C₂×C₃×C₈	—	4	8	no	—
100	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
101	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
102	C₃×C₄⋊C₄	—	2	4	no	—
103	C₃×C₄⋊C₄	—	2	4	no	—
104	C₃×C₄⋊C₄	—	2	4	no	—
105	C₃×C₄⋊C₄	—	2	4	no	—
106	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
107	C₃×C₂²⋊C₄	—	2	4	no	—
108	C₃×C₄²	C₂³×C₃	1	4	yes	—
109	C₂²×C₃²	—	2	2	no	—
110	C₂²×C₃²	—	4	1	no	—
111	C₂²×C₃²	—	4	1	no	—
112	C₂²×C₃²	#121	1	1	yes	characteristic
113	C₃²×C₄	—	2	1	no	—
114	C₃²×C₄	—	4	1	no	—
115	C₃²×C₄	—	8	1	no	—
116	C₃²×C₄	—	8	1	no	—
117	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
118	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
119	C₃²×C₄	#135	1	1	yes	characteristic
120	C₃²×C₄	#137	1	1	yes	characteristic
121	C₂×SD₁₆	—	2	1	no	—
122	C₂²×C₈	—	2	1	no	—
123	C₄⋊Q₈	C₂²×C₃²	1	1	yes	characteristic
124	C₄⋊Q₈	C₂²×C₃²	1	1	yes	characteristic
125	C₄×D₄	C₂²×C₃²	1	1	yes	characteristic
126	#106	—	2	1	no	—
127	#106	—	2	1	no	—
128	#106	—	2	1	no	—
129	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
130	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
131	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
132	#103	—	2	1	no	—
133	#102	—	2	1	no	—
134	C₂²⋊C₈	—	2	1	no	—
135	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
136	C₂³×C₃	—	2	4	no	—
137	C₃×Q₈	—	4	4	no	—
138	C₃×Q₈	—	4	4	no	—
139	C₃×Q₈	—	8	4	no	—
140	C₃×D₄	—	2	4	no	—
141	C₃×D₄	C₃×SD₁₆	1	8	yes	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	4	4	no	—
144	C₂×C₃×C₄	—	4	4	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	—	4	4	no	—
147	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
148	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
149	C₂×C₃×C₄	C₃×#41	1	4	yes	—
150	C₃×C₈	—	2	4	no	—
151	C₃×C₈	—	4	4	no	—
152	C₃×C₈	—	4	4	no	—
153	C₃×C₈	—	4	4	no	—
154	C₃×C₈	C₃×SD₁₆	1	8	yes	complemented
155	C₂×C₃²	—	4	2	no	—
156	C₂×C₃²	#461	1	1	yes	characteristic
157	C₂×C₃²	#465	1	1	yes	characteristic
158	C₂×C₃²	#471	1	1	yes	characteristic
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	SD₁₆	—	4	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
169	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	C₃²×D₄	1	1	yes	characteristic
175	C₂²⋊C₄	—	2	1	no	—
176	C₄²	C₂³×C₃²	1	1	yes	characteristic
177	C₂²×C₃	—	2	8	no	—
178	C₂²×C₃	—	4	4	no	—
179	C₂²×C₃	—	4	4	no	—
180	C₂²×C₃	C₃×#121	1	4	yes	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	4	4	no	—
183	C₃×C₄	—	8	4	no	—
184	C₃×C₄	—	8	4	no	—
185	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
186	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
187	C₃×C₄	C₃×#135	1	4	yes	—
188	C₃×C₄	C₃×#137	1	4	yes	—
189	C₃²	#1671	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	—	2	1	no	—
191	Q₈	—	4	1	no	—
192	Q₈	—	4	1	no	—
193	Q₈	—	8	1	no	—
194	D₄	—	2	1	no	—
195	D₄	C₃²×SD₁₆	1	2	yes	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
203	C₂×C₄	C₃²×#41	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	4	1	no	—
206	C₈	—	4	1	no	—
207	C₈	—	4	1	no	—
208	C₈	C₃²×SD₁₆	1	2	yes	complemented
209	C₂×C₃	—	4	8	no	—
210	C₂×C₃	C₃×#461	1	4	yes	—
211	C₂×C₃	C₃×#465	1	4	yes	—
212	C₂×C₃	C₃×#471	1	4	yes	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	C₃²×#121	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	4	1	no	—
219	C₄	—	8	1	no	—
220	C₄	—	8	1	no	—
221	C₄	C₂×C₃²×SD₁₆	1	1	yes	characteristic
222	C₄	C₂×C₃²×SD₁₆	1	1	yes	characteristic
223	C₄	C₃²×#135	1	1	yes	characteristic
224	C₄	C₃²×#137	1	1	yes	characteristic
225	C₃	C₃×#1671	1	4	yes	direct product factor
226	C₂	—	4	2	no	—
227	C₂	C₃²×#461	1	1	yes	characteristic
228	C₂	C₃²×#465	1	1	yes	characteristic
229	C₂	C₃²×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
prime radical	C₂×C₃²×D₄	D₄
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
1. C₃²×#498	C₂	semidirect
5. C₃²×#334	C₂	semidirect
6. C₃²×#333	C₂	semidirect
7. C₃²×#332	C₂	semidirect
8. C₃×#1671	C₃	direct (N × H)
24. C₃×#498	C₂×C₃	semidirect
28. C₃×#334	C₂×C₃	semidirect
29. C₃×#333	C₂×C₃	semidirect
30. C₃×#332	C₂×C₃	semidirect
49. #1671	C₃²	direct (N × H)
83. C₃²×C₈	SD₁₆	semidirect
84. #498	C₂×C₃²	semidirect
88. #334	C₂×C₃²	semidirect
89. #333	C₂×C₃²	semidirect
90. #332	C₂×C₃²	semidirect
154. C₃×C₈	C₃×SD₁₆	semidirect
189. C₃²	#1671	direct (N × H)
208. C₈	C₃²×SD₁₆	semidirect
225. C₃	C₃×#1671	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×#498	2
C₃²×#476	2
C₃²×#461	2
C₃²×C₈×D₄	2
C₃²×#334	2
C₃²×#333	2
C₃²×#332	2
C₃×#1671	3

Subgroups — C32×#1671 #11768720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1671 #11768720