Subgroups — C₃²×#2383 #11769432

Order 1152.

1368 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#126	C₂	1	1	yes	maximal, characteristic
2	C₃²×#389	C₂	1	3	yes	maximal, complemented
3	C₃²×#388	C₂	1	3	yes	maximal
4	C₃²×#387	C₂	1	3	yes	maximal, complemented
5	C₃²×#385	C₂	1	3	yes	maximal
6	C₃²×#384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2383	C₃	1	4	yes	maximal, direct product factor
9	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₃²×#126	C₄	1	8	yes	complemented
11	C₂×C₃²×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₃²×C₄⋊C₄	C₂²	1	3	yes	—
13	C₂×C₃²×C₄⋊C₄	C₂²	1	3	yes	—
14	C₂×C₃²×C₄⋊C₄	C₂²	1	3	yes	—
15	C₂×C₃²×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂×C₃²×C₂²⋊C₄	C₂²	1	3	yes	—
17	C₂×C₃²×C₄²	C₂²	1	3	yes	—
18	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
19	C₃²×#95	C₂²	1	3	yes	—
20	C₃²×#95	C₂²	1	3	yes	—
21	C₃²×#95	C₂²	1	3	yes	—
22	C₃²×#95	C₂²	1	3	yes	—
23	C₂×C₃×#126	C₂×C₃	1	4	yes	—
24	C₃×#389	C₂×C₃	1	12	yes	complemented
25	C₃×#388	C₂×C₃	1	12	yes	—
26	C₃×#387	C₂×C₃	1	12	yes	complemented
27	C₃×#385	C₂×C₃	1	12	yes	—
28	C₃×#384	C₂×C₃	1	4	yes	complemented
29	C₃×#383	C₂×C₃	1	4	yes	complemented
30	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
31	C₂²×C₃²×C₄	—	4	3	no	—
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃²×C₄	C₂³	1	3	yes	—
35	C₂²×C₃²×C₄	C₂³	1	3	yes	—
36	C₂²×C₃²×C₄	C₂³	1	3	yes	—
37	C₂²×C₃²×C₄	C₂³	1	3	yes	—
38	C₃²×C₄⋊C₄	—	4	3	no	—
39	C₃²×C₄⋊C₄	—	4	3	no	—
40	C₃²×C₄⋊C₄	—	4	3	no	—
41	C₃²×C₄⋊C₄	C₂×C₄	1	12	yes	complemented
42	C₃²×C₂²⋊C₄	—	4	3	no	—
43	C₃²×C₂²⋊C₄	C₂×C₄	1	12	yes	—
44	C₃²×C₄²	—	4	3	no	—
45	C₃²×C₄²	C₂×C₄	1	4	yes	complemented
46	#2383	C₃²	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
48	C₃×#126	C₃×C₄	1	32	yes	complemented
49	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	12	yes	—
50	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	12	yes	—
51	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	12	yes	—
52	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	12	yes	—
53	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
54	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
55	C₂×C₃×C₄²	C₂²×C₃	1	12	yes	—
56	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
57	C₃×#95	C₂²×C₃	1	12	yes	—
58	C₃×#95	C₂²×C₃	1	12	yes	—
59	C₃×#95	C₂²×C₃	1	12	yes	—
60	C₃×#95	C₂²×C₃	1	12	yes	—
61	C₂³×C₃²	—	2	3	no	—
62	C₂³×C₃²	—	2	3	no	—
63	C₂³×C₃²	C₂²×C₄	1	2	yes	—
64	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
65	C₂×C₃²×C₄	—	2	3	no	—
66	C₂×C₃²×C₄	—	2	3	no	—
67	C₂×C₃²×C₄	—	2	3	no	—
68	C₂×C₃²×C₄	—	2	3	no	—
69	C₂×C₃²×C₄	—	2	3	no	—
70	C₂×C₃²×C₄	—	2	3	no	—
71	C₂×C₃²×C₄	—	2	3	no	—
72	C₂×C₃²×C₄	—	2	3	no	—
73	C₂×C₃²×C₄	—	2	3	no	—
74	C₂×C₃²×C₄	—	2	3	no	—
75	C₂×C₃²×C₄	—	2	6	no	—
76	C₂×C₃²×C₄	—	4	4	no	—
77	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
78	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
79	C₂×#126	C₂×C₃²	1	1	yes	characteristic
80	#389	C₂×C₃²	1	3	yes	complemented
81	#388	C₂×C₃²	1	3	yes	—
82	#387	C₂×C₃²	1	3	yes	complemented
83	#385	C₂×C₃²	1	3	yes	—
84	#384	C₂×C₃²	1	1	yes	characteristic, complemented
85	#383	C₂×C₃²	1	1	yes	characteristic, complemented
86	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
87	C₂²×C₃×C₄	—	4	12	no	—
88	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
89	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
90	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
91	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
92	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
93	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
94	C₃×C₄⋊C₄	—	4	12	no	—
95	C₃×C₄⋊C₄	—	4	12	no	—
96	C₃×C₄⋊C₄	—	4	12	no	—
97	C₃×C₄⋊C₄	C₂×C₃×C₄	1	48	yes	complemented
98	C₃×C₂²⋊C₄	—	4	12	no	—
99	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	48	yes	—
100	C₃×C₄²	—	4	12	no	—
101	C₃×C₄²	C₂×C₃×C₄	1	16	yes	complemented
102	C₂²×C₃²	—	2	6	no	—
103	C₂²×C₃²	—	4	1	no	—
104	C₂²×C₃²	—	4	3	no	—
105	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
106	C₂²×C₃²	H(5,𝔽₂)	1	3	yes	—
107	C₂²×C₃²	2₋¹⁺⁴	1	3	yes	—
108	C₃²×C₄	—	2	6	no	—
109	C₃²×C₄	—	2	6	no	—
110	C₃²×C₄	—	4	3	no	—
111	C₃²×C₄	—	4	1	no	—
112	C₃²×C₄	—	4	1	no	—
113	C₃²×C₄	—	4	3	no	—
114	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
115	#126	C₃²×C₄	1	8	yes	complemented
116	C₂×C₄⋊C₄	C₂²×C₃²	1	3	yes	—
117	C₂×C₄⋊C₄	C₂²×C₃²	1	3	yes	—
118	C₂×C₄⋊C₄	C₂²×C₃²	1	3	yes	—
119	C₂×C₄⋊C₄	C₂²×C₃²	1	3	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₃²	1	3	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₃²	1	3	yes	—
122	C₂×C₄²	C₂²×C₃²	1	3	yes	—
123	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
124	#95	C₂²×C₃²	1	3	yes	—
125	#95	C₂²×C₃²	1	3	yes	—
126	#95	C₂²×C₃²	1	3	yes	—
127	#95	C₂²×C₃²	1	3	yes	—
128	C₂³×C₃	—	2	12	no	—
129	C₂³×C₃	—	2	12	no	—
130	C₂³×C₃	C₂²×C₃×C₄	1	8	yes	—
131	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
132	C₂×C₃×C₄	—	2	12	no	—
133	C₂×C₃×C₄	—	2	12	no	—
134	C₂×C₃×C₄	—	2	12	no	—
135	C₂×C₃×C₄	—	2	12	no	—
136	C₂×C₃×C₄	—	2	12	no	—
137	C₂×C₃×C₄	—	2	12	no	—
138	C₂×C₃×C₄	—	2	12	no	—
139	C₂×C₃×C₄	—	2	12	no	—
140	C₂×C₃×C₄	—	2	12	no	—
141	C₂×C₃×C₄	—	2	12	no	—
142	C₂×C₃×C₄	—	2	24	no	—
143	C₂×C₃×C₄	—	4	16	no	—
144	C₂×C₃×C₄	C₂²×C₃×C₄	1	24	yes	—
145	C₂×C₃×C₄	C₂²×C₃×C₄	1	24	yes	—
146	C₂×C₃²	—	4	2	no	—
147	C₂×C₃²	#520	1	3	yes	—
148	C₂×C₃²	#560	1	1	yes	characteristic
149	C₂×C₃²	#563	1	3	yes	—
150	C₂⁴	C₂³×C₃²	1	1	yes	characteristic
151	C₂²×C₄	—	4	3	no	—
152	C₂²×C₄	C₂³×C₃²	1	3	yes	—
153	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₃²	1	3	yes	—
156	C₂²×C₄	C₂³×C₃²	1	3	yes	—
157	C₂²×C₄	C₂³×C₃²	1	3	yes	—
158	C₄⋊C₄	—	4	3	no	—
159	C₄⋊C₄	—	4	3	no	—
160	C₄⋊C₄	—	4	3	no	—
161	C₄⋊C₄	C₂×C₃²×C₄	1	12	yes	complemented
162	C₂²⋊C₄	—	4	3	no	—
163	C₂²⋊C₄	C₂×C₃²×C₄	1	12	yes	—
164	C₄²	—	4	3	no	—
165	C₄²	C₂×C₃²×C₄	1	4	yes	complemented
166	C₂²×C₃	—	2	24	no	—
167	C₂²×C₃	—	4	4	no	—
168	C₂²×C₃	—	4	12	no	—
169	C₂²×C₃	C₂³×C₃×C₄	1	4	yes	—
170	C₂²×C₃	C₃×H(5,𝔽₂)	1	12	yes	—
171	C₂²×C₃	C₃×2₋¹⁺⁴	1	12	yes	—
172	C₃×C₄	—	2	24	no	—
173	C₃×C₄	—	2	24	no	—
174	C₃×C₄	—	4	4	no	—
175	C₃×C₄	—	4	4	no	—
176	C₃×C₄	—	4	12	no	—
177	C₃×C₄	—	4	12	no	—
178	C₃²	#2383	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³	—	2	3	no	—
180	C₂³	—	2	3	no	—
181	C₂³	C₂²×C₃²×C₄	1	2	yes	—
182	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
183	C₂×C₄	—	2	3	no	—
184	C₂×C₄	—	2	3	no	—
185	C₂×C₄	—	2	3	no	—
186	C₂×C₄	—	2	3	no	—
187	C₂×C₄	—	2	3	no	—
188	C₂×C₄	—	2	3	no	—
189	C₂×C₄	—	2	3	no	—
190	C₂×C₄	—	2	3	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	C₂²×C₃²×C₄	1	6	yes	—
196	C₂×C₄	C₂²×C₃²×C₄	1	6	yes	—
197	C₂×C₃	—	4	8	no	—
198	C₂×C₃	C₃×#520	1	12	yes	—
199	C₂×C₃	C₃×#560	1	4	yes	—
200	C₂×C₃	C₃×#563	1	12	yes	—
201	C₂²	—	2	6	no	—
202	C₂²	—	4	1	no	—
203	C₂²	—	4	3	no	—
204	C₂²	C₂³×C₃²×C₄	1	1	yes	characteristic
205	C₂²	C₃²×H(5,𝔽₂)	1	3	yes	—
206	C₂²	C₃²×2₋¹⁺⁴	1	3	yes	—
207	C₄	—	2	6	no	—
208	C₄	—	2	6	no	—
209	C₄	—	4	1	no	—
210	C₄	—	4	1	no	—
211	C₄	—	4	3	no	—
212	C₄	—	4	3	no	—
213	C₃	C₃×#2383	1	4	yes	direct product factor
214	C₂	—	4	2	no	—
215	C₂	C₃²×#520	1	3	yes	—
216	C₂	C₃²×#560	1	1	yes	characteristic
217	C₂	C₃²×#563	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃²	C₂⁴
commutator	C₂²	C₂³×C₃²×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃²
prime radical	C₂⁴×C₃²	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×#389	C₂	semidirect
4. C₃²×#387	C₂	semidirect
6. C₃²×#384	C₂	semidirect
7. C₃²×#383	C₂	semidirect
8. C₃×#2383	C₃	direct (N × H)
10. C₃²×#126	C₄	semidirect
24. C₃×#389	C₂×C₃	semidirect
26. C₃×#387	C₂×C₃	semidirect
28. C₃×#384	C₂×C₃	semidirect
29. C₃×#383	C₂×C₃	semidirect
41. C₃²×C₄⋊C₄	C₂×C₄	semidirect
45. C₃²×C₄²	C₂×C₄	semidirect
46. #2383	C₃²	direct (N × H)
48. C₃×#126	C₃×C₄	semidirect
80. #389	C₂×C₃²	semidirect
82. #387	C₂×C₃²	semidirect
84. #384	C₂×C₃²	semidirect
85. #383	C₂×C₃²	semidirect
97. C₃×C₄⋊C₄	C₂×C₃×C₄	semidirect
101. C₃×C₄²	C₂×C₃×C₄	semidirect
115. #126	C₃²×C₄	semidirect
161. C₄⋊C₄	C₂×C₃²×C₄	semidirect
165. C₄²	C₂×C₃²×C₄	semidirect
178. C₃²	#2383	direct (N × H)
213. C₃	C₃×#2383	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×#126	2
C₃²×#389	2
C₃²×#388	2
C₃²×#387	2
C₃²×#385	2
C₃²×#384	2
C₃²×#383	2
C₃×#2383	3

Subgroups — C32×#2383 #11769432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2383 #11769432