Subgroups — C₃²×#1813 #11768862

Order 1152.

756 conjugacy classes of subgroups.

The subgroup lattice has 186 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	C₂×C₃²×#108	C₂	1	2	yes	maximal, complemented
3	C₃²×#413	C₂	1	2	yes	maximal
4	C₃²×#338	C₂	1	2	yes	maximal, complemented
5	C₃×#1813	C₃	1	4	yes	maximal, direct product factor
6	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
7	C₃²×#131	—	2	4	no	—
8	C₂×C₃²×M₁₆	C₄	1	2	yes	—
9	C₂×C₃²×M₁₆	C₂²	1	4	yes	—
10	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₄	1	2	yes	—
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃²×#108	—	2	4	no	—
14	C₃²×#101	—	2	2	no	—
15	C₃²×#100	—	2	2	no	—
16	C₂×C₃×#131	C₂×C₃	1	4	yes	—
17	C₂×C₃×#108	C₂×C₃	1	8	yes	complemented
18	C₃×#413	C₂×C₃	1	8	yes	—
19	C₃×#338	C₂×C₃	1	8	yes	complemented
20	C₃²×#41	—	2	2	no	—
21	C₃²×#41	—	2	2	no	—
22	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
23	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
24	C₂×C₃²×D₄	Q₈	1	2	yes	—
25	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
27	C₃²×M₁₆	—	2	2	no	—
28	C₃²×M₁₆	—	2	4	no	—
29	C₃²×M₁₆	—	2	4	no	—
30	C₃²×M₁₆	—	2	4	no	—
31	C₂×C₃²×C₈	—	2	4	no	—
32	C₂×C₃²×C₈	—	2	4	no	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	D₄	1	4	yes	—
37	#1813	C₃²	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
39	C₃×#131	—	2	16	no	—
40	C₂×C₃×M₁₆	C₃×C₄	1	8	yes	—
41	C₂×C₃×M₁₆	C₂²×C₃	1	16	yes	—
42	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
43	C₂²×C₃×C₈	C₃×C₄	1	8	yes	—
44	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
45	C₃×#108	—	2	16	no	—
46	C₃×#101	—	2	8	no	—
47	C₃×#100	—	2	8	no	—
48	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
49	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
50	C₃²×Q₈	—	2	2	no	—
51	C₃²×D₄	—	2	4	no	—
52	C₃²×D₄	—	2	2	no	—
53	C₂×C₃²×C₄	—	2	2	no	—
54	C₂×C₃²×C₄	—	2	2	no	—
55	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
56	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	#41	1	2	yes	—
60	C₃²×C₈	—	2	2	no	—
61	C₃²×C₈	—	4	4	no	—
62	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
63	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×#131	C₂×C₃²	1	1	yes	characteristic
65	C₂×#108	C₂×C₃²	1	2	yes	complemented
66	#413	C₂×C₃²	1	2	yes	—
67	#338	C₂×C₃²	1	2	yes	complemented
68	C₃×#41	—	2	8	no	—
69	C₃×#41	—	2	8	no	—
70	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
71	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
72	C₂×C₃×D₄	C₃×Q₈	1	8	yes	—
73	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
74	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
75	C₃×M₁₆	—	2	8	no	—
76	C₃×M₁₆	—	2	16	no	—
77	C₃×M₁₆	—	2	16	no	—
78	C₃×M₁₆	—	2	16	no	—
79	C₂×C₃×C₈	—	2	16	no	—
80	C₂×C₃×C₈	—	2	16	no	—
81	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
82	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
83	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
84	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
85	C₂²×C₃²	—	2	2	no	—
86	C₂²×C₃²	—	4	1	no	—
87	C₂²×C₃²	—	4	2	no	—
88	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
90	C₃²×C₄	—	2	2	no	—
91	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₃²×C₄	#117	1	1	yes	characteristic
93	C₃²×C₄	#121	1	2	yes	—
94	C₂×#41	C₂²×C₃²	1	1	yes	characteristic
95	#131	—	2	4	no	—
96	C₂×M₁₆	C₃²×C₄	1	2	yes	—
97	C₂×M₁₆	C₂²×C₃²	1	4	yes	—
98	C₂×M₁₆	C₂²×C₃²	1	1	yes	characteristic
99	C₂²×C₈	C₃²×C₄	1	2	yes	—
100	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
101	#108	—	2	4	no	—
102	#101	—	2	2	no	—
103	#100	—	2	2	no	—
104	C₂³×C₃	C₃×C₄⋊C₄	1	8	yes	—
105	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
106	C₃×Q₈	—	2	8	no	—
107	C₃×D₄	—	2	16	no	—
108	C₃×D₄	—	2	8	no	—
109	C₂×C₃×C₄	—	2	8	no	—
110	C₂×C₃×C₄	—	2	8	no	—
111	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
112	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
113	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
114	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
115	C₂×C₃×C₄	C₃×#41	1	8	yes	—
116	C₃×C₈	—	2	8	no	—
117	C₃×C₈	—	4	16	no	—
118	C₃×C₈	C₃×C₂²⋊C₄	1	8	yes	—
119	C₃×C₈	C₃×C₂²⋊C₄	1	8	yes	—
120	C₂×C₃²	—	2	1	no	—
121	C₂×C₃²	—	2	2	no	—
122	C₂×C₃²	—	4	2	no	—
123	C₂×C₃²	#380	1	1	yes	characteristic
124	#41	—	2	2	no	—
125	#41	—	2	2	no	—
126	C₂×Q₈	C₃²×D₄	1	1	yes	characteristic
127	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
128	C₂×D₄	C₃²×Q₈	1	2	yes	—
129	C₂²×C₄	C₂×C₃²×C₄	1	2	yes	—
130	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
131	M₁₆	—	2	2	no	—
132	M₁₆	—	2	4	no	—
133	M₁₆	—	2	4	no	—
134	M₁₆	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
138	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
139	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
140	C₂×C₈	C₃²×D₄	1	4	yes	—
141	C₂²×C₃	—	2	8	no	—
142	C₂²×C₃	—	4	4	no	—
143	C₂²×C₃	—	4	8	no	—
144	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
145	C₂²×C₃	C₃×C₂²⋊Q₈	1	8	yes	—
146	C₃×C₄	—	2	8	no	—
147	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
148	C₃×C₄	C₃×#117	1	4	yes	—
149	C₃×C₄	C₃×#121	1	8	yes	—
150	C₃²	#1813	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³	C₃²×C₄⋊C₄	1	2	yes	—
152	C₂³	C₂×C₃²×Q₈	1	1	yes	characteristic
153	Q₈	—	2	2	no	—
154	D₄	—	2	4	no	—
155	D₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
159	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
160	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
161	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
162	C₂×C₄	C₃²×#41	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	—	4	4	no	—
165	C₈	C₃²×C₂²⋊C₄	1	2	yes	—
166	C₈	C₃²×C₂²⋊C₄	1	2	yes	—
167	C₂×C₃	—	2	4	no	—
168	C₂×C₃	—	2	8	no	—
169	C₂×C₃	—	4	8	no	—
170	C₂×C₃	C₃×#380	1	4	yes	—
171	C₂²	—	2	2	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	4	2	no	—
174	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
175	C₂²	C₃²×C₂²⋊Q₈	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
178	C₄	C₃²×#117	1	1	yes	characteristic
179	C₄	C₃²×#121	1	2	yes	—
180	C₃	C₃×#1813	1	4	yes	direct product factor
181	C₂	—	2	1	no	—
182	C₂	—	2	2	no	—
183	C₂	—	4	2	no	—
184	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. C₂×C₃²×#108	C₂	semidirect
4. C₃²×#338	C₂	semidirect
5. C₃×#1813	C₃	direct (N × H)
17. C₂×C₃×#108	C₂×C₃	semidirect
19. C₃×#338	C₂×C₃	semidirect
37. #1813	C₃²	direct (N × H)
65. C₂×#108	C₂×C₃²	semidirect
67. #338	C₂×C₃²	semidirect
150. C₃²	#1813	direct (N × H)
180. C₃	C₃×#1813	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
C₂×C₃²×#108	2
C₃²×#413	2
C₃²×#338	2
C₃×#1813	3

Subgroups — C32×#1813 #11768862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1813 #11768862