Subgroups — C₂²×C₃×#812 #11899360

Order 1152.

969 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×M₃₂	—	3	1	no	maximal
5	C₂²×#812	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂³×C₃²×C₄	C₄	1	1	yes	characteristic
7	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₄	1	6	yes	—
9	C₂²×C₃²×C₈	C₂²	1	6	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃×#812	C₂²	1	16	yes	direct product factor
12	C₂×C₃×C₃⋊C₁₆	C₂²	1	12	yes	complemented
13	C₂³×C₃×C₈	—	2	1	no	—
14	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
15	C₂³×C₃×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×M₃₂	—	3	12	no	—
17	C₂²×C₃×C₁₆	—	3	2	no	—
18	C₂×#812	C₂×C₃	1	12	yes	direct product factor
19	C₂²×C₃⋊C₁₆	C₂×C₃	1	2	yes	complemented
20	C₂⁴×C₃²	C₈	1	1	yes	characteristic
21	C₂²×C₃²×C₄	C₈	1	6	yes	—
22	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	6	yes	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	12	yes	—
26	C₂×C₃²×C₈	C₂×C₄	1	6	yes	—
27	C₂×C₃²×C₈	C₂×C₄	1	3	yes	—
28	C₂×C₃²×C₈	C₂³	1	3	yes	—
29	C₂×C₃²×C₈	C₂³	1	4	yes	—
30	C₃×C₃⋊C₁₆	C₂³	1	8	yes	complemented
31	C₂²×M₃₂	—	3	1	no	Sylow
32	C₂³×C₃×C₄	—	2	1	no	—
33	C₂³×C₃×C₄	Dic₃	1	1	yes	characteristic
34	C₂³×C₃×C₄	C₃×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₈	—	2	1	no	—
36	C₂²×C₃×C₈	—	2	6	no	—
37	C₂²×C₃×C₈	—	2	6	no	—
38	C₂²×C₃×C₈	—	2	1	no	—
39	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
40	C₂²×C₃×C₈	Dic₃	1	6	yes	—
41	C₂²×C₃×C₈	C₃×C₄	1	6	yes	—
42	C₂²×C₃×C₈	C₃×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂×D₃	1	6	yes	—
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₃×C₈	C₂²×C₃	1	6	yes	—
47	C₃×M₃₂	—	3	16	no	—
48	C₂×C₃×C₁₆	—	3	12	no	—
49	#812	C₂²×C₃	1	16	yes	direct product factor
50	C₂×C₃⋊C₁₆	C₂²×C₃	1	12	yes	complemented
51	C₂³×C₃²	—	2	4	no	—
52	C₂³×C₃²	C₂×C₈	1	6	yes	—
53	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×C₈	1	12	yes	—
55	C₂×C₃²×C₄	C₂×C₈	1	6	yes	—
56	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
57	C₂×C₃²×C₄	C₂²×C₄	1	4	yes	—
58	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
59	C₃²×C₈	C₂²×C₄	1	3	yes	—
60	C₃²×C₈	C₂²×C₄	1	4	yes	—
61	C₃²×C₈	C₂⁴	1	1	yes	characteristic
62	C₂³×C₈	C₃×D₃	1	1	yes	characteristic
63	C₂×M₃₂	—	3	12	no	—
64	C₂²×C₁₆	—	3	2	no	—
65	C₂⁴×C₃	—	2	1	no	—
66	C₂⁴×C₃	C₃⋊C₈	1	1	yes	characteristic
67	C₂⁴×C₃	C₃×C₈	1	1	yes	characteristic
68	C₂²×C₃×C₄	—	2	6	no	—
69	C₂²×C₃×C₄	—	2	1	no	—
70	C₂²×C₃×C₄	—	2	6	no	—
71	C₂²×C₃×C₄	—	2	1	no	—
72	C₂²×C₃×C₄	C₃⋊C₈	1	6	yes	—
73	C₂²×C₃×C₄	C₃⋊C₈	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₃×C₈	1	6	yes	—
75	C₂²×C₃×C₄	C₃×C₈	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂×Dic₃	1	6	yes	—
77	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×C₃×C₄	1	6	yes	—
79	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₃×C₈	—	2	12	no	—
81	C₂×C₃×C₈	—	2	6	no	—
82	C₂×C₃×C₈	—	2	3	no	—
83	C₂×C₃×C₈	—	2	3	no	—
84	C₂×C₃×C₈	—	2	4	no	—
85	C₂×C₃×C₈	C₂×Dic₃	1	6	yes	—
86	C₂×C₃×C₈	C₂×Dic₃	1	12	yes	—
87	C₂×C₃×C₈	C₂×Dic₃	1	3	yes	—
88	C₂×C₃×C₈	C₂×C₃×C₄	1	3	yes	—
89	C₂×C₃×C₈	C₂×C₃×C₄	1	12	yes	—
90	C₂×C₃×C₈	C₂×C₃×C₄	1	6	yes	—
91	C₂×C₃×C₈	C₂²×D₃	1	3	yes	—
92	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
93	C₂×C₃×C₈	C₂³×C₃	1	3	yes	—
94	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
95	C₃×C₁₆	—	3	8	no	—
96	C₃⋊C₁₆	C₂³×C₃	1	8	yes	complemented
97	C₂²×C₃²	—	2	12	no	—
98	C₂²×C₃²	M₃₂	1	4	yes	complemented
99	C₂²×C₃²	C₂²×C₈	1	4	yes	—
100	C₂²×C₃²	C₂²×C₈	1	3	yes	—
101	C₃²×C₄	C₂²×C₈	1	3	yes	—
102	C₃²×C₄	C₂²×C₈	1	4	yes	—
103	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
104	C₂³×C₄	C₃×Dic₃	1	1	yes	characteristic
105	C₂²×C₈	C₃×Dic₃	1	1	yes	characteristic
106	C₂²×C₈	C₃×Dic₃	1	6	yes	—
107	C₂²×C₈	C₂×C₃×D₃	1	6	yes	—
108	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
109	M₃₂	—	3	16	no	—
110	C₂×C₁₆	—	3	12	no	—
111	C₂³×C₃	—	2	8	no	—
112	C₂³×C₃	—	2	4	no	—
113	C₂³×C₃	—	2	4	no	—
114	C₂³×C₃	—	2	6	no	—
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	C₂×C₃⋊C₈	1	6	yes	—
117	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
118	C₂³×C₃	C₂×C₃×C₈	1	6	yes	—
119	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
120	C₂×C₃×C₄	—	2	12	no	—
121	C₂×C₃×C₄	—	2	3	no	—
122	C₂×C₃×C₄	—	2	6	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	3	no	—
125	C₂×C₃×C₄	C₂×C₃⋊C₈	1	12	yes	—
126	C₂×C₃×C₄	C₂×C₃⋊C₈	1	6	yes	—
127	C₂×C₃×C₄	C₂×C₃⋊C₈	1	3	yes	—
128	C₂×C₃×C₄	C₂×C₃×C₈	1	12	yes	—
129	C₂×C₃×C₄	C₂×C₃×C₈	1	3	yes	—
130	C₂×C₃×C₄	C₂×C₃×C₈	1	6	yes	—
131	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
132	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
133	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
134	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	—	2	3	no	—
137	C₃×C₈	—	2	4	no	—
138	C₃×C₈	C₂²×Dic₃	1	4	yes	—
139	C₃×C₈	C₂²×Dic₃	1	3	yes	—
140	C₃×C₈	C₂²×C₃×C₄	1	3	yes	—
141	C₃×C₈	C₂²×C₃×C₄	1	4	yes	—
142	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
143	C₃×C₈	C₂⁴×C₃	1	1	yes	characteristic
144	C₂×C₃²	—	2	4	no	—
145	C₂×C₃²	C₂×M₃₂	1	6	yes	complemented
146	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
147	C₂⁴	C₃×C₃⋊C₈	1	1	yes	characteristic
148	C₂²×C₄	C₃×C₃⋊C₈	1	6	yes	—
149	C₂²×C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₃×Dic₃	1	6	yes	—
151	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
152	C₂×C₈	C₂×C₃×Dic₃	1	12	yes	—
153	C₂×C₈	C₂×C₃×Dic₃	1	6	yes	—
154	C₂×C₈	C₂×C₃×Dic₃	1	3	yes	—
155	C₂×C₈	C₂²×C₃×D₃	1	3	yes	—
156	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
157	C₁₆	—	3	8	no	—
158	C₂²×C₃	—	2	24	no	—
159	C₂²×C₃	—	2	12	no	—
160	C₂²×C₃	—	2	12	no	—
161	C₂²×C₃	—	2	4	no	—
162	C₂²×C₃	—	2	3	no	—
163	C₂²×C₃	—	2	4	no	—
164	C₂²×C₃	#812	1	4	yes	direct product factor
165	C₂²×C₃	C₃×M₃₂	1	4	yes	complemented
166	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
167	C₂²×C₃	C₂²×C₃⋊C₈	1	3	yes	—
168	C₂²×C₃	C₂²×C₃×C₈	1	4	yes	—
169	C₂²×C₃	C₂²×C₃×C₈	1	3	yes	—
170	C₃×C₄	—	2	3	no	—
171	C₃×C₄	—	2	4	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	C₂²×C₃⋊C₈	1	3	yes	—
174	C₃×C₄	C₂²×C₃⋊C₈	1	4	yes	—
175	C₃×C₄	C₂²×C₃×C₈	1	3	yes	—
176	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
177	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
178	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
179	C₃²	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
180	C₂³	—	2	4	no	—
181	C₂³	C₂×C₃×C₃⋊C₈	1	6	yes	—
182	C₂³	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
183	C₂×C₄	C₂×C₃×C₃⋊C₈	1	12	yes	—
184	C₂×C₄	C₂×C₃×C₃⋊C₈	1	3	yes	—
185	C₂×C₄	C₂×C₃×C₃⋊C₈	1	6	yes	—
186	C₂×C₄	C₂²×C₃×Dic₃	1	4	yes	—
187	C₂×C₄	C₂²×C₃×Dic₃	1	3	yes	—
188	C₈	C₂²×C₃×Dic₃	1	3	yes	—
189	C₈	C₂²×C₃×Dic₃	1	4	yes	—
190	C₈	C₂³×C₃×D₃	1	1	yes	characteristic
191	C₂×C₃	—	2	8	no	—
192	C₂×C₃	—	2	4	no	—
193	C₂×C₃	—	2	4	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	6	no	—
196	C₂×C₃	C₂×#812	1	6	yes	direct product factor
197	C₂×C₃	C₂×C₃×M₃₂	1	6	yes	complemented
198	C₂×C₃	C₂³×C₃⋊C₈	1	1	yes	characteristic
199	C₂×C₃	C₂³×C₃×C₈	1	1	yes	characteristic
200	C₂²	—	2	12	no	—
201	C₂²	C₃×#812	1	4	yes	direct product factor
202	C₂²	C₂²×C₃×C₃⋊C₈	1	4	yes	—
203	C₂²	C₂²×C₃×C₃⋊C₈	1	3	yes	—
204	C₄	C₂²×C₃×C₃⋊C₈	1	3	yes	—
205	C₄	C₂²×C₃×C₃⋊C₈	1	4	yes	—
206	C₄	C₂³×C₃×Dic₃	1	1	yes	characteristic
207	C₃	—	2	1	no	—
208	C₃	C₂²×#812	1	1	yes	characteristic, direct product factor
209	C₃	C₂²×C₃×M₃₂	1	1	yes	characteristic, complemented
210	C₂	—	2	4	no	—
211	C₂	C₂×C₃×#812	1	6	yes	direct product factor
212	C₂	C₂³×C₃×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₈	C₂×D₃
commutator	C₂×C₃	C₂³×C₃×C₈
Frattini	C₈	C₂³×C₃×D₃
Fitting	C₂³×C₃²×C₈	C₂
socle	C₂³×C₃²	C₂×C₈
prime radical	C₂⁴×C₃²	C₈
cosocle	C₃×C₈	C₂⁴×C₃
hypercenter	C₂³×C₃×C₈	D₃
hypocenter	C₃	C₂²×C₃×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₈ ◁ C₂³×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#812	C₂	direct (N × H)
3. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
5. C₂²×#812	C₃	direct (N × H)
11. C₃×#812	C₂²	direct (N × H)
12. C₂×C₃×C₃⋊C₁₆	C₂²	semidirect
18. C₂×#812	C₂×C₃	direct (N × H)
19. C₂²×C₃⋊C₁₆	C₂×C₃	semidirect
30. C₃×C₃⋊C₁₆	C₂³	semidirect
49. #812	C₂²×C₃	direct (N × H)
50. C₂×C₃⋊C₁₆	C₂²×C₃	semidirect
96. C₃⋊C₁₆	C₂³×C₃	semidirect
98. C₂²×C₃²	M₃₂	semidirect
145. C₂×C₃²	C₂×M₃₂	semidirect
164. C₂²×C₃	#812	direct (N × H)
165. C₂²×C₃	C₃×M₃₂	semidirect
179. C₃²	C₂²×M₃₂	semidirect
196. C₂×C₃	C₂×#812	direct (N × H)
197. C₂×C₃	C₂×C₃×M₃₂	semidirect
201. C₂²	C₃×#812	direct (N × H)
208. C₃	C₂²×#812	direct (N × H)
209. C₃	C₂²×C₃×M₃₂	semidirect
211. C₂	C₂×C₃×#812	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₈	2
C₂×C₃×#812	2
C₂²×C₃×C₃⋊C₁₆	2
C₂²×C₃×M₃₂	3
C₂²×#812	3

Subgroups — C22×C3×#812 #11899360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#812 #11899360