Subgroups — #131×C₃⋊Dic₃ #11857251

Order 1152.

1716 conjugacy classes of subgroups.

The subgroup lattice has 207 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	#41×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#64157	C₂	1	1	yes	maximal, characteristic
4	#10592821	C₂	1	3	yes	maximal, complemented
5	M₁₆×C₃⋊Dic₃	C₂	1	3	yes	maximal, complemented
6	#10592800	C₂	1	3	yes	maximal, complemented
7	C₂×C₈×C₃⋊Dic₃	C₂	1	3	yes	maximal, complemented
8	Dic₃×#131	—	3	4	no	maximal
9	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
10	C₃²×#131	C₄	1	8	yes	complemented
11	C₂×C₃²×M₁₆	C₂²	1	3	yes	—
12	C₂²×C₃²×C₈	C₂²	1	3	yes	—
13	#64157	C₄	1	8	yes	complemented
14	Q₈×C₃⋊Dic₃	C₄	1	2	yes	—
15	D₄×C₃⋊Dic₃	C₄	1	6	yes	—
16	#64135	C₂²	1	3	yes	—
17	C₂×C₄×C₃⋊Dic₃	C₂²	1	3	yes	—
18	C₂×#3748	C₂²	1	3	yes	—
19	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	3	yes	—
20	#63642	C₂²	1	6	yes	—
21	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₈×C₃⋊Dic₃	C₂²	1	3	yes	—
23	C₈×C₃⋊Dic₃	C₂²	1	3	yes	—
24	C₈×C₃⋊Dic₃	C₂²	1	3	yes	—
25	C₂×C₃×#131	D₃	1	4	yes	—
26	Dic₃×#41	—	3	4	no	—
27	C₂×#948	—	3	4	no	—
28	#5122	—	3	12	no	—
29	Dic₃×M₁₆	—	3	12	no	—
30	#5101	—	3	12	no	—
31	C₂×C₈×Dic₃	—	3	12	no	—
32	C₃²×#41	C₂×C₄	1	4	yes	—
33	C₃²×#41	C₂×C₄	1	4	yes	—
34	C₂×C₃²×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×D₄	C₂×C₄	1	3	yes	—
36	C₂²×C₃²×C₄	C₂³	1	3	yes	—
37	C₂²×C₃⋊Dic₃	C₂×C₄	1	6	yes	—
38	C₃²×M₁₆	C₂×C₄	1	12	yes	complemented
39	C₂×C₃²×C₈	C₂×C₄	1	12	yes	complemented
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂³	1	3	yes	—
42	#3757	C₂×C₄	1	6	yes	—
43	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	6	yes	—
44	C₄×C₃⋊Dic₃	C₂×C₄	1	6	yes	—
45	C₄×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
46	C₄×C₃⋊Dic₃	C₂³	1	3	yes	—
47	#3748	C₂×C₄	1	12	yes	complemented
48	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	12	yes	complemented
49	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
50	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	3	yes	—
51	C₄×#131	—	9	1	no	Sylow
52	C₂×C₃×#41	C₂×D₃	1	4	yes	—
53	C₃×#131	Dic₃	1	32	yes	complemented
54	C₂×C₃×M₁₆	C₂×D₃	1	12	yes	—
55	C₂²×C₃×C₈	C₂×D₃	1	12	yes	—
56	#948	—	3	32	no	—
57	Q₈×Dic₃	—	3	8	no	—
58	D₄×Dic₃	—	3	24	no	—
59	#926	—	3	12	no	—
60	C₂×C₄×Dic₃	—	3	12	no	—
61	C₂×#208	—	3	12	no	—
62	C₂²×C₃⋊C₈	—	3	12	no	—
63	#815	—	3	24	no	—
64	C₈×Dic₃	—	3	12	no	—
65	C₈×Dic₃	—	3	4	no	—
66	C₈×Dic₃	—	3	12	no	—
67	C₈×Dic₃	—	3	12	no	—
68	C₂³×C₃²	C₂²×C₄	1	3	yes	—
69	C₃²×Q₈	C₄²	1	4	yes	—
70	C₃²×D₄	C₄²	1	12	yes	—
71	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
72	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
73	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
74	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
75	C₂×C₃⋊Dic₃	—	2	12	no	—
76	C₂×C₃⋊Dic₃	C₂²×C₄	1	6	yes	—
77	C₂×C₃⋊Dic₃	C₂²×C₄	1	2	yes	—
78	C₃²×C₈	C₂²×C₄	1	2	yes	—
79	C₃²×C₈	C₂²×C₄	1	6	yes	—
80	C₃⋊(C₃⋊C₈)	C₂²×C₄	1	2	yes	—
81	C₃⋊(C₃⋊C₈)	C₂²×C₄	1	6	yes	—
82	C₂×#131	—	9	1	no	—
83	C₂×#131	C₃⋊D₃	1	1	yes	characteristic
84	C₄×#41	—	9	1	no	—
85	#405	—	9	3	no	—
86	#405	—	9	3	no	—
87	C₄×M₁₆	—	9	3	no	—
88	C₂×C₄×C₈	—	9	3	no	—
89	C₃×#41	C₄×D₃	1	16	yes	—
90	C₃×#41	C₂×Dic₃	1	16	yes	—
91	C₂×C₃×Q₈	C₄×D₃	1	4	yes	—
92	C₂×C₃×D₄	C₄×D₃	1	12	yes	—
93	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
94	C₂²×Dic₃	—	3	24	no	—
95	C₃×M₁₆	C₂×Dic₃	1	48	yes	complemented
96	C₂×C₃×C₈	C₂×Dic₃	1	48	yes	complemented
97	C₂×C₃×C₈	C₂²×D₃	1	12	yes	—
98	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
99	#217	—	3	24	no	—
100	C₄⋊Dic₃	—	3	24	no	—
101	C₄×Dic₃	—	3	24	no	—
102	C₄×Dic₃	—	3	4	no	—
103	C₄×Dic₃	—	3	12	no	—
104	#208	—	3	48	no	—
105	C₂×C₃⋊C₈	—	3	48	no	—
106	C₂×C₃⋊C₈	—	3	4	no	—
107	C₂×C₃⋊C₈	—	3	12	no	—
108	C₂²×C₃²	—	2	6	no	—
109	C₂²×C₃²	C₂×C₄²	1	6	yes	—
110	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
111	C₃²×C₄	C₂×C₄²	1	6	yes	—
112	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
113	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
114	C₃⋊Dic₃	—	2	6	no	—
115	C₃⋊Dic₃	#131	1	4	yes	direct product factor
116	C₂×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
117	#131	—	9	8	no	—
118	#131	C₃⋊Dic₃	1	8	yes	direct product factor
119	C₂×M₁₆	—	9	3	no	—
120	C₂×M₁₆	C₂×C₃⋊D₃	1	3	yes	—
121	C₂²×C₈	—	9	3	no	—
122	C₂²×C₈	C₂×C₃⋊D₃	1	3	yes	—
123	C₄×Q₈	—	9	2	no	—
124	C₄×D₄	—	9	6	no	—
125	#117	—	9	3	no	—
126	C₂×C₄²	—	9	3	no	—
127	#97	—	9	6	no	—
128	C₄×C₈	—	9	3	no	—
129	C₄×C₈	—	9	3	no	—
130	C₄×C₈	—	9	1	no	—
131	C₄×C₈	—	9	3	no	—
132	C₂³×C₃	C₂×C₄×D₃	1	12	yes	—
133	C₃×Q₈	C₄×Dic₃	1	16	yes	—
134	C₃×D₄	C₄×Dic₃	1	48	yes	—
135	C₂×C₃×C₄	C₂×C₄×D₃	1	12	yes	—
136	C₂×C₃×C₄	C₂×C₄×D₃	1	24	yes	—
137	C₂×C₃×C₄	C₂²×Dic₃	1	24	yes	—
138	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
139	C₂×Dic₃	—	3	8	no	—
140	C₂×Dic₃	—	3	24	no	—
141	C₂×Dic₃	—	6	48	no	—
142	C₃×C₈	C₂²×Dic₃	1	8	yes	—
143	C₃×C₈	C₂²×Dic₃	1	24	yes	—
144	C₃⋊C₈	—	3	8	no	—
145	C₃⋊C₈	—	3	24	no	—
146	C₂×C₃²	—	2	6	no	—
147	C₂×C₃²	C₂²×C₄²	1	1	yes	characteristic
148	C₂×C₃²	C₂×#131	1	2	yes	—
149	#41	C₄×C₃⋊D₃	1	4	yes	—
150	#41	C₂×C₃⋊Dic₃	1	4	yes	—
151	C₂×Q₈	C₄×C₃⋊D₃	1	1	yes	characteristic
152	C₂×D₄	C₄×C₃⋊D₃	1	3	yes	—
153	C₂²×C₄	—	9	6	no	—
154	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
155	M₁₆	—	9	12	no	—
156	M₁₆	C₂×C₃⋊Dic₃	1	12	yes	complemented
157	C₂×C₈	—	9	1	no	—
158	C₂×C₈	—	9	3	no	—
159	C₂×C₈	—	9	12	no	—
160	C₂×C₈	C₂×C₃⋊Dic₃	1	12	yes	complemented
161	C₂×C₈	C₂²×C₃⋊D₃	1	3	yes	—
162	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
163	C₄⋊C₄	—	9	6	no	—
164	C₂²⋊C₄	—	9	6	no	—
165	C₄²	—	9	1	no	—
166	C₄²	—	9	3	no	—
167	C₄²	—	9	6	no	—
168	C₂²×C₃	—	2	24	no	—
169	C₂²×C₃	C₂×C₄×Dic₃	1	24	yes	—
170	C₂²×C₃	C₂²×C₄×D₃	1	4	yes	—
171	C₃×C₄	C₂×C₄×Dic₃	1	24	yes	—
172	C₃×C₄	C₂²×C₄×D₃	1	4	yes	—
173	C₃×C₄	C₂³×Dic₃	1	4	yes	—
174	Dic₃	—	3	16	no	—
175	Dic₃	—	6	24	no	—
176	C₃²	C₄×#131	1	1	yes	characteristic, complemented, Sylow
177	C₂³	C₂×C₄×C₃⋊D₃	1	3	yes	—
178	Q₈	C₄×C₃⋊Dic₃	1	4	yes	—
179	D₄	C₄×C₃⋊Dic₃	1	12	yes	—
180	C₂×C₄	—	9	2	no	—
181	C₂×C₄	—	9	6	no	—
182	C₂×C₄	—	18	12	no	—
183	C₂×C₄	C₂×C₄×C₃⋊D₃	1	6	yes	—
184	C₂×C₄	C₂×C₄×C₃⋊D₃	1	3	yes	—
185	C₂×C₄	C₂²×C₃⋊Dic₃	1	6	yes	—
186	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
187	C₈	—	9	2	no	—
188	C₈	—	9	6	no	—
189	C₈	C₂²×C₃⋊Dic₃	1	2	yes	—
190	C₈	C₂²×C₃⋊Dic₃	1	6	yes	—
191	C₂×C₃	—	2	24	no	—
192	C₂×C₃	D₃×#131	1	8	yes	—
193	C₂×C₃	C₂²×C₄×Dic₃	1	4	yes	—
194	C₂²	—	2	6	no	—
195	C₂²	C₂×C₄×C₃⋊Dic₃	1	6	yes	—
196	C₂²	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
197	C₄	—	9	4	no	—
198	C₄	—	18	6	no	—
199	C₄	C₂×C₄×C₃⋊Dic₃	1	6	yes	—
200	C₄	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
201	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
202	C₃	Dic₃×#131	1	4	yes	complemented
203	C₂	—	2	6	no	—
204	C₂	C₃⋊D₃×#131	1	2	yes	—
205	C₂	C₂²×C₄×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄²
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂²×C₃²	C₂³×C₄
prime radical	C₂×C₃²×#41	C₂²
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂×#131	C₃⋊D₃
hypocenter	C₃²	C₄×#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
4. #10592821	C₂	semidirect
5. M₁₆×C₃⋊Dic₃	C₂	semidirect
6. #10592800	C₂	semidirect
7. C₂×C₈×C₃⋊Dic₃	C₂	semidirect
10. C₃²×#131	C₄	semidirect
13. #64157	C₄	semidirect
38. C₃²×M₁₆	C₂×C₄	semidirect
39. C₂×C₃²×C₈	C₂×C₄	semidirect
47. #3748	C₂×C₄	semidirect
48. C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
53. C₃×#131	Dic₃	semidirect
95. C₃×M₁₆	C₂×Dic₃	semidirect
96. C₂×C₃×C₈	C₂×Dic₃	semidirect
115. C₃⋊Dic₃	#131	direct (N × H)
118. #131	C₃⋊Dic₃	direct (N × H)
156. M₁₆	C₂×C₃⋊Dic₃	semidirect
160. C₂×C₈	C₂×C₃⋊Dic₃	semidirect
176. C₃²	C₄×#131	semidirect
202. C₃	Dic₃×#131	semidirect

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
#41×C₃⋊Dic₃	2
C₂×#64157	2
#10592821	2
M₁₆×C₃⋊Dic₃	2
#10592800	2
C₂×C₈×C₃⋊Dic₃	2
Dic₃×#131	3

Subgroups — #131×C3⋊Dic3 #11857251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #131×C₃⋊Dic₃ #11857251