Subgroups — C₂²×#63543 #11899369

Order 1152.

849 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	C₂²×C₃×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63543	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#812	—	3	1	no	maximal
6	C₂²×#798	—	3	1	no	maximal
7	C₂²×C₃×C₄×D₃	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₃⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₃×D₃×C₈	C₄	1	6	yes	—
11	C₂×C₃×D₃×C₈	C₂²	1	6	yes	—
12	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	6	yes	complemented
13	C₂×C₃×C₃⋊C₁₆	C₂²	1	6	yes	complemented
14	#63543	C₂²	1	16	yes	direct product factor
15	C₂³×C₃×C₈	—	3	1	no	—
16	C₂²×D₃×C₈	D₃	1	1	yes	characteristic
17	C₂²×C₃×C₁₆	—	3	1	no	—
18	C₂×#812	—	3	12	no	—
19	C₂²×C₃⋊C₁₆	—	3	1	no	—
20	C₂²×C₃⋊C₁₆	—	3	1	no	—
21	C₂²×C₃⋊C₁₆	—	6	1	no	—
22	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
23	C₂×#798	—	3	12	no	—
24	C₂³×C₃×D₃	C₈	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×Dic₃	C₈	1	1	yes	characteristic
27	C₂×C₃×C₄×D₃	C₈	1	6	yes	—
28	C₂×C₃×C₄×D₃	C₂×C₄	1	6	yes	—
29	C₂×C₃²×C₈	C₂×C₄	1	3	yes	—
30	C₂×C₃²×C₈	C₂³	1	3	yes	—
31	C₂×C₃×C₃⋊C₈	C₂×C₄	1	6	yes	—
32	C₃×D₃×C₈	C₂×C₄	1	12	yes	—
33	C₃×D₃×C₈	C₂³	1	4	yes	—
34	C₃⋊(C₃⋊C₁₆)	C₂³	1	4	yes	complemented
35	C₃×C₃⋊C₁₆	C₂³	1	4	yes	complemented
36	C₂²×M₃₂	—	9	1	no	Sylow
37	C₂³×C₃×C₄	—	3	1	no	—
38	C₂²×C₄×D₃	Dic₃	1	1	yes	characteristic
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	—	3	1	no	—
41	C₂²×C₃×C₈	—	3	6	no	—
42	C₂²×C₃×C₈	—	3	6	no	—
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃⋊C₈	Dic₃	1	1	yes	characteristic
46	C₂×D₃×C₈	Dic₃	1	6	yes	—
47	C₂×D₃×C₈	C₂×D₃	1	6	yes	—
48	C₂×C₃×C₁₆	—	3	6	no	—
49	#812	—	3	16	no	—
50	C₂×C₃⋊C₁₆	—	3	6	no	—
51	C₂×C₃⋊C₁₆	—	3	6	no	—
52	C₂×C₃⋊C₁₆	—	6	6	no	—
53	C₂×C₃⋊C₁₆	C₂×D₃	1	6	yes	complemented
54	#798	—	3	16	no	—
55	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₃×D₃	—	2	4	no	—
57	C₂²×C₃×D₃	C₂×C₈	1	6	yes	—
58	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
59	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
60	C₂×C₃×Dic₃	C₂×C₈	1	6	yes	—
61	C₃×C₄×D₃	C₂×C₈	1	12	yes	—
62	C₃×C₄×D₃	C₂²×C₄	1	4	yes	—
63	C₃²×C₈	C₂²×C₄	1	3	yes	—
64	C₃²×C₈	C₂⁴	1	1	yes	characteristic
65	C₃×C₃⋊C₈	C₂²×C₄	1	4	yes	—
66	C₂³×C₈	—	3	1	no	—
67	C₂×M₃₂	—	9	12	no	—
68	C₂²×C₁₆	—	3	1	no	—
69	C₂²×C₁₆	—	9	1	no	—
70	C₂⁴×C₃	—	3	1	no	—
71	C₂³×D₃	C₃⋊C₈	1	1	yes	characteristic
72	C₂²×C₃×C₄	—	2	1	no	—
73	C₂²×C₃×C₄	—	3	1	no	—
74	C₂²×C₃×C₄	—	3	6	no	—
75	C₂²×C₃×C₄	—	3	6	no	—
76	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
78	C₂²×Dic₃	C₃⋊C₈	1	1	yes	characteristic
79	C₂×C₄×D₃	C₃⋊C₈	1	6	yes	—
80	C₂×C₄×D₃	C₂×Dic₃	1	6	yes	—
81	C₂×C₃×C₈	—	2	3	no	—
82	C₂×C₃×C₈	—	2	3	no	—
83	C₂×C₃×C₈	—	3	6	no	—
84	C₂×C₃×C₈	—	3	12	no	—
85	C₂×C₃×C₈	—	3	4	no	—
86	C₂×C₃×C₈	C₄×D₃	1	3	yes	—
87	C₂×C₃×C₈	C₂×Dic₃	1	3	yes	—
88	C₂×C₃×C₈	C₂²×D₃	1	3	yes	—
89	C₂×C₃×C₈	C₂²×D₃	1	3	yes	—
90	C₂×C₃⋊C₈	C₂×Dic₃	1	6	yes	—
91	D₃×C₈	C₂×Dic₃	1	12	yes	—
92	D₃×C₈	C₂²×D₃	1	4	yes	—
93	C₃×C₁₆	—	3	4	no	—
94	C₃⋊C₁₆	—	3	4	no	—
95	C₃⋊C₁₆	—	3	4	no	—
96	C₃⋊C₁₆	—	6	4	no	—
97	C₃⋊C₁₆	C₂²×D₃	1	4	yes	complemented
98	C₂²×C₃²	M₃₂	1	4	yes	complemented
99	C₂²×C₃²	C₂²×C₈	1	3	yes	—
100	C₂×C₃×D₃	—	2	12	no	—
101	C₂×C₃×D₃	C₂²×C₈	1	4	yes	—
102	C₃²×C₄	C₂²×C₈	1	3	yes	—
103	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
104	C₃×Dic₃	C₂²×C₈	1	4	yes	—
105	C₂³×C₄	—	3	1	no	—
106	C₂²×C₈	—	3	1	no	—
107	C₂²×C₈	—	3	6	no	—
108	C₂²×C₈	—	3	6	no	—
109	C₂²×C₈	D₃²	1	1	yes	characteristic
110	M₃₂	—	9	16	no	—
111	C₂×C₁₆	—	3	6	no	—
112	C₂×C₁₆	—	9	6	no	—
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	—	3	6	no	—
115	C₂³×C₃	—	6	4	no	—
116	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
117	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
118	C₂²×D₃	—	2	4	no	—
119	C₂²×D₃	C₂×C₃⋊C₈	1	6	yes	—
120	C₂×C₃×C₄	—	2	3	no	—
121	C₂×C₃×C₄	—	2	3	no	—
122	C₂×C₃×C₄	—	3	6	no	—
123	C₂×C₃×C₄	—	3	12	no	—
124	C₂×C₃×C₄	—	3	4	no	—
125	C₂×C₃×C₄	D₃×C₈	1	3	yes	—
126	C₂×C₃×C₄	C₂×C₃⋊C₈	1	3	yes	—
127	C₂×C₃×C₄	C₂×C₄×D₃	1	3	yes	—
128	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
129	C₂×Dic₃	C₂×C₃⋊C₈	1	6	yes	—
130	C₄×D₃	C₂×C₃⋊C₈	1	12	yes	—
131	C₄×D₃	C₂²×Dic₃	1	4	yes	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	—	2	3	no	—
134	C₃×C₈	—	3	4	no	—
135	C₃×C₈	C₂×C₄×D₃	1	3	yes	—
136	C₃×C₈	C₂²×Dic₃	1	3	yes	—
137	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
138	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
139	C₃⋊C₈	C₂²×Dic₃	1	4	yes	—
140	C₂×C₃²	C₂×M₃₂	1	6	yes	complemented
141	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
142	C₃×D₃	—	2	4	no	—
143	C₂⁴	—	3	1	no	—
144	C₂²×C₄	—	3	1	no	—
145	C₂²×C₄	—	3	6	no	—
146	C₂²×C₄	—	3	6	no	—
147	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
148	C₂×C₈	—	3	6	no	—
149	C₂×C₈	—	3	4	no	—
150	C₂×C₈	—	3	12	no	—
151	C₂×C₈	D₃×Dic₃	1	3	yes	—
152	C₂×C₈	C₂×D₃²	1	3	yes	—
153	C₁₆	—	3	4	no	—
154	C₁₆	—	9	4	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	—	2	3	no	—
157	C₂²×C₃	—	3	4	no	—
158	C₂²×C₃	—	6	12	no	—
159	C₂²×C₃	#798	1	4	yes	complemented
160	C₂²×C₃	#812	1	4	yes	complemented
161	C₂²×C₃	C₂×D₃×C₈	1	3	yes	—
162	C₂²×C₃	C₂²×C₃⋊C₈	1	3	yes	—
163	C₂×D₃	—	2	12	no	—
164	C₂×D₃	C₂²×C₃⋊C₈	1	4	yes	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	3	no	—
167	C₃×C₄	—	3	4	no	—
168	C₃×C₄	C₂×D₃×C₈	1	3	yes	—
169	C₃×C₄	C₂²×C₃⋊C₈	1	3	yes	—
170	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
171	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
172	Dic₃	C₂²×C₃⋊C₈	1	4	yes	—
173	C₃²	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	3	6	no	—
175	C₂³	—	6	4	no	—
176	C₂³	D₃×C₃⋊C₈	1	1	yes	characteristic
177	C₂×C₄	—	3	6	no	—
178	C₂×C₄	—	3	4	no	—
179	C₂×C₄	—	3	12	no	—
180	C₂×C₄	D₃×C₃⋊C₈	1	3	yes	—
181	C₂×C₄	C₂×D₃×Dic₃	1	3	yes	—
182	C₈	—	3	4	no	—
183	C₈	C₂×D₃×Dic₃	1	3	yes	—
184	C₈	C₂²×D₃²	1	1	yes	characteristic
185	C₂×C₃	—	2	6	no	—
186	C₂×C₃	—	2	1	no	—
187	C₂×C₃	—	6	4	no	—
188	C₂×C₃	C₂×#798	1	6	yes	complemented
189	C₂×C₃	C₂×#812	1	6	yes	complemented
190	C₂×C₃	C₂²×D₃×C₈	1	1	yes	characteristic
191	C₂×C₃	C₂³×C₃⋊C₈	1	1	yes	characteristic
192	D₃	—	2	4	no	—
193	C₂²	—	3	4	no	—
194	C₂²	—	6	12	no	—
195	C₂²	#63543	1	4	yes	direct product factor
196	C₂²	C₂×D₃×C₃⋊C₈	1	3	yes	—
197	C₄	—	3	4	no	—
198	C₄	C₂×D₃×C₃⋊C₈	1	3	yes	—
199	C₄	C₂²×D₃×Dic₃	1	1	yes	characteristic
200	C₃	—	2	1	no	—
201	C₃	C₂²×#798	1	1	yes	characteristic, complemented
202	C₃	C₂²×#812	1	1	yes	characteristic, complemented
203	C₂	—	6	4	no	—
204	C₂	C₂×#63543	1	6	yes	direct product factor
205	C₂	C₂²×D₃×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₃²
commutator	C₂×C₃²	C₂³×C₈
Frattini	C₈	C₂²×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂³×C₃²	C₂×C₈
prime radical	C₂³×C₃×D₃	C₈
cosocle	C₃²×C₈	C₂⁴
hypocenter	C₃²	C₂²×M₃₂
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
3. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
4. C₂×#63543	C₂	direct (N × H)
12. C₂×C₃⋊(C₃⋊C₁₆)	C₂²	semidirect
13. C₂×C₃×C₃⋊C₁₆	C₂²	semidirect
14. #63543	C₂²	direct (N × H)
22. C₂²×C₃⋊C₁₆	D₃	semidirect
34. C₃⋊(C₃⋊C₁₆)	C₂³	semidirect
35. C₃×C₃⋊C₁₆	C₂³	semidirect
53. C₂×C₃⋊C₁₆	C₂×D₃	semidirect
97. C₃⋊C₁₆	C₂²×D₃	semidirect
98. C₂²×C₃²	M₃₂	semidirect
140. C₂×C₃²	C₂×M₃₂	semidirect
159. C₂²×C₃	#798	semidirect
160. C₂²×C₃	#812	semidirect
173. C₃²	C₂²×M₃₂	semidirect
188. C₂×C₃	C₂×#798	semidirect
189. C₂×C₃	C₂×#812	semidirect
195. C₂²	#63543	direct (N × H)
201. C₃	C₂²×#798	semidirect
202. C₃	C₂²×#812	semidirect
204. C₂	C₂×#63543	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₃×C₈	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₂²×C₃×C₃⋊C₁₆	2
C₂×#63543	2
C₂²×#812	3
C₂²×#798	3

Subgroups — C22×#63543 #11899369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63543 #11899369