Subgroups — C₃²×#1812 #11768861

Order 1152.

780 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	C₃²×#427	C₂	1	2	yes	maximal, complemented
3	C₃²×#410	C₂	1	2	yes	maximal
4	C₃²×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
5	C₃×#1812	C₃	1	4	yes	maximal, direct product factor
6	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
7	C₃²×#131	—	2	2	no	—
8	C₃²×#131	—	2	2	no	—
9	C₂×C₃²×M₁₆	C₄	1	2	yes	complemented
10	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₄	1	2	yes	complemented
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃²×#117	C₂²	1	4	yes	—
14	C₃²×#107	—	2	2	no	—
15	C₃²×#106	—	2	2	no	—
16	C₃²×#99	—	2	4	no	—
17	C₂×C₃×#131	C₂×C₃	1	4	yes	—
18	C₃×#427	C₂×C₃	1	8	yes	complemented
19	C₃×#410	C₂×C₃	1	8	yes	—
20	C₃×H(3,ℤ/4)	C₂×C₃	1	8	yes	complemented
21	C₃²×#41	—	2	2	no	—
22	C₃²×#41	—	2	2	no	—
23	C₂×C₃²×Q₈	Q₈	1	1	yes	characteristic
24	C₂×C₃²×D₄	D₄	1	2	yes	—
25	C₂×C₃²×D₄	Q₈	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₃²×M₁₆	—	2	2	no	—
29	C₃²×M₁₆	—	2	4	no	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	D₄	1	2	yes	—
36	C₂×C₃²×C₈	D₄	1	2	yes	—
37	C₃²×C₄⋊C₄	—	2	4	no	—
38	C₃²×C₂²⋊C₄	—	2	4	no	—
39	C₃²×C₄²	—	2	4	no	—
40	#1812	C₃²	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
42	C₃×#131	—	2	8	no	—
43	C₃×#131	—	2	8	no	—
44	C₂×C₃×M₁₆	C₃×C₄	1	8	yes	complemented
45	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
46	C₂²×C₃×C₈	C₃×C₄	1	8	yes	complemented
47	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
48	C₃×#117	C₂²×C₃	1	16	yes	—
49	C₃×#107	—	2	8	no	—
50	C₃×#106	—	2	8	no	—
51	C₃×#99	—	2	16	no	—
52	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
53	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
54	C₃²×Q₈	—	2	2	no	—
55	C₃²×D₄	—	2	4	no	—
56	C₃²×D₄	—	2	2	no	—
57	C₂×C₃²×C₄	—	2	2	no	—
58	C₂×C₃²×C₄	—	2	2	no	—
59	C₂×C₃²×C₄	—	4	4	no	—
60	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃²×C₄	#41	1	2	yes	—
65	C₃²×C₈	—	2	2	no	—
66	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×#131	C₂×C₃²	1	1	yes	characteristic
70	#427	C₂×C₃²	1	2	yes	complemented
71	#410	C₂×C₃²	1	2	yes	—
72	H(3,ℤ/4)	C₂×C₃²	1	2	yes	complemented
73	C₃×#41	—	2	8	no	—
74	C₃×#41	—	2	8	no	—
75	C₂×C₃×Q₈	C₃×Q₈	1	4	yes	—
76	C₂×C₃×D₄	C₃×D₄	1	8	yes	—
77	C₂×C₃×D₄	C₃×Q₈	1	4	yes	—
78	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
79	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
80	C₃×M₁₆	—	2	8	no	—
81	C₃×M₁₆	—	2	16	no	—
82	C₂×C₃×C₈	—	2	8	no	—
83	C₂×C₃×C₈	—	2	8	no	—
84	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
85	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
86	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
87	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
88	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
89	C₃×C₄⋊C₄	—	2	16	no	—
90	C₃×C₂²⋊C₄	—	2	16	no	—
91	C₃×C₄²	—	2	16	no	—
92	C₂²×C₃²	—	2	2	no	—
93	C₂²×C₃²	—	4	1	no	—
94	C₂²×C₃²	—	4	2	no	—
95	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₃²	#121	1	2	yes	—
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	—	8	4	no	—
99	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₃²×C₄	#117	1	1	yes	characteristic
101	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
102	C₂×#41	C₂²×C₃²	1	1	yes	characteristic
103	#131	—	2	2	no	—
104	#131	—	2	2	no	—
105	C₂×M₁₆	C₃²×C₄	1	2	yes	complemented
106	C₂×M₁₆	C₂²×C₃²	1	1	yes	characteristic
107	C₂²×C₈	C₃²×C₄	1	2	yes	complemented
108	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
109	#117	C₂²×C₃²	1	4	yes	—
110	#107	—	2	2	no	—
111	#106	—	2	2	no	—
112	#99	—	2	4	no	—
113	C₂³×C₃	C₃×C₄⋊C₄	1	8	yes	—
114	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
115	C₃×Q₈	—	2	8	no	—
116	C₃×D₄	—	2	16	no	—
117	C₃×D₄	—	2	8	no	—
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	—	4	16	no	—
121	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
122	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
123	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
124	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
125	C₂×C₃×C₄	C₃×#41	1	8	yes	—
126	C₃×C₈	—	2	8	no	—
127	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
128	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
129	C₃×C₈	C₃×C₂²⋊C₄	1	8	yes	—
130	C₂×C₃²	—	2	2	no	—
131	C₂×C₃²	—	2	1	no	—
132	C₂×C₃²	—	4	2	no	—
133	C₂×C₃²	#380	1	1	yes	characteristic
134	#41	—	2	2	no	—
135	#41	—	2	2	no	—
136	C₂×Q₈	C₃²×Q₈	1	1	yes	characteristic
137	C₂×D₄	C₃²×D₄	1	2	yes	—
138	C₂×D₄	C₃²×Q₈	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₃²×C₄	1	2	yes	—
140	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
141	M₁₆	—	2	2	no	—
142	M₁₆	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
146	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
147	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
148	C₂×C₈	C₃²×D₄	1	2	yes	—
149	C₂×C₈	C₃²×D₄	1	2	yes	—
150	C₄⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₄²	—	2	4	no	—
153	C₂²×C₃	—	2	8	no	—
154	C₂²×C₃	—	4	4	no	—
155	C₂²×C₃	—	4	8	no	—
156	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
157	C₂²×C₃	C₃×#121	1	8	yes	—
158	C₃×C₄	—	2	8	no	—
159	C₃×C₄	—	8	16	no	—
160	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
161	C₃×C₄	C₃×#117	1	4	yes	—
162	C₃×C₄	C₃×C₂²⋊Q₈	1	8	yes	—
163	C₃²	#1812	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³	C₃²×C₄⋊C₄	1	2	yes	—
165	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
166	Q₈	—	2	2	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
173	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
174	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
175	C₂×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
176	C₂×C₄	C₃²×#41	1	2	yes	—
177	C₈	—	2	2	no	—
178	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
179	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
180	C₈	C₃²×C₂²⋊C₄	1	2	yes	—
181	C₂×C₃	—	2	8	no	—
182	C₂×C₃	—	2	4	no	—
183	C₂×C₃	—	4	8	no	—
184	C₂×C₃	C₃×#380	1	4	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
189	C₂²	C₃²×#121	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	8	4	no	—
192	C₄	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
193	C₄	C₃²×#117	1	1	yes	characteristic
194	C₄	C₃²×C₂²⋊Q₈	1	2	yes	—
195	C₃	C₃×#1812	1	4	yes	direct product factor
196	C₂	—	2	2	no	—
197	C₂	—	2	1	no	—
198	C₂	—	4	2	no	—
199	C₂	C₃²×#380	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	#117
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂×C₃²	#380
prime radical	C₂×C₃²×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×#427	C₂	semidirect
4. C₃²×H(3,ℤ/4)	C₂	semidirect
5. C₃×#1812	C₃	direct (N × H)
9. C₂×C₃²×M₁₆	C₄	semidirect
11. C₂²×C₃²×C₈	C₄	semidirect
18. C₃×#427	C₂×C₃	semidirect
20. C₃×H(3,ℤ/4)	C₂×C₃	semidirect
40. #1812	C₃²	direct (N × H)
44. C₂×C₃×M₁₆	C₃×C₄	semidirect
46. C₂²×C₃×C₈	C₃×C₄	semidirect
70. #427	C₂×C₃²	semidirect
72. H(3,ℤ/4)	C₂×C₃²	semidirect
105. C₂×M₁₆	C₃²×C₄	semidirect
107. C₂²×C₈	C₃²×C₄	semidirect
163. C₃²	#1812	direct (N × H)
195. C₃	C₃×#1812	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
C₃²×#427	2
C₃²×#410	2
C₃²×H(3,ℤ/4)	2
C₃×#1812	3

Subgroups — C32×#1812 #11768861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1812 #11768861