Subgroups — C₂³×C₃×#208 #11918438

Order 1152.

4121 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#208	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₃×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂³×C₃×M₁₆	—	3	1	no	maximal
5	C₂³×#208	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂⁵×C₃²	C₄	1	1	yes	characteristic
7	C₂³×C₃²×C₄	C₄	1	14	yes	—
8	C₂³×C₃²×C₄	C₄	1	1	yes	characteristic
9	C₂³×C₃²×C₄	C₂²	1	14	yes	—
10	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	112	yes	direct product factor
12	C₂²×C₃×C₃⋊C₈	C₂²	1	28	yes	complemented
13	C₂⁴×C₃×C₄	—	2	1	no	—
14	C₂⁴×C₃×C₄	D₃	1	1	yes	characteristic
15	C₂⁴×C₃×C₄	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₃×M₁₆	—	3	28	no	—
17	C₂³×C₃×C₈	—	3	2	no	—
18	C₂²×#208	C₂×C₃	1	28	yes	direct product factor
19	C₂³×C₃⋊C₈	C₂×C₃	1	2	yes	complemented
20	C₂⁴×C₃²	—	2	8	no	—
21	C₂⁴×C₃²	C₂×C₄	1	14	yes	—
22	C₂⁴×C₃²	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	84	yes	—
24	C₂²×C₃²×C₄	C₂×C₄	1	14	yes	—
25	C₂²×C₃²×C₄	C₂×C₄	1	7	yes	—
26	C₂²×C₃²×C₄	C₂³	1	7	yes	—
27	C₂²×C₃²×C₄	C₂³	1	28	yes	—
28	C₃×#208	C₂³	1	64	yes	direct product factor
29	C₂×C₃×C₃⋊C₈	C₂³	1	56	yes	complemented
30	C₂³×M₁₆	—	3	1	no	Sylow
31	C₂⁵×C₃	—	2	1	no	—
32	C₂⁵×C₃	Dic₃	1	1	yes	characteristic
33	C₂⁵×C₃	C₃×C₄	1	1	yes	characteristic
34	C₂³×C₃×C₄	—	2	14	no	—
35	C₂³×C₃×C₄	—	2	1	no	—
36	C₂³×C₃×C₄	—	2	1	no	—
37	C₂³×C₃×C₄	—	2	14	no	—
38	C₂³×C₃×C₄	Dic₃	1	1	yes	characteristic
39	C₂³×C₃×C₄	Dic₃	1	14	yes	—
40	C₂³×C₃×C₄	C₃×C₄	1	14	yes	—
41	C₂³×C₃×C₄	C₃×C₄	1	1	yes	characteristic
42	C₂³×C₃×C₄	C₂×D₃	1	14	yes	—
43	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂³×C₃×C₄	C₂²×C₃	1	14	yes	—
46	C₂×C₃×M₁₆	—	3	112	no	—
47	C₂²×C₃×C₈	—	3	28	no	—
48	C₂×#208	C₂²×C₃	1	112	yes	direct product factor
49	C₂²×C₃⋊C₈	C₂²×C₃	1	28	yes	complemented
50	C₂³×C₃²	—	2	56	no	—
51	C₂³×C₃²	M₁₆	1	8	yes	complemented
52	C₂³×C₃²	C₂²×C₄	1	28	yes	—
53	C₂³×C₃²	C₂²×C₄	1	7	yes	—
54	C₂×C₃²×C₄	C₂²×C₄	1	56	yes	—
55	C₂×C₃²×C₄	C₂²×C₄	1	21	yes	—
56	C₂×C₃²×C₄	C₂²×C₄	1	28	yes	—
57	C₂×C₃²×C₄	C₂⁴	1	7	yes	—
58	C₂×C₃²×C₄	C₂⁴	1	8	yes	—
59	C₃×C₃⋊C₈	C₂⁴	1	16	yes	complemented
60	C₂⁴×C₄	C₃×D₃	1	1	yes	characteristic
61	C₂²×M₁₆	—	3	28	no	—
62	C₂³×C₈	—	3	2	no	—
63	C₂⁴×C₃	—	2	16	no	—
64	C₂⁴×C₃	—	2	8	no	—
65	C₂⁴×C₃	—	2	8	no	—
66	C₂⁴×C₃	—	2	14	no	—
67	C₂⁴×C₃	—	2	1	no	—
68	C₂⁴×C₃	C₂×Dic₃	1	14	yes	—
69	C₂⁴×C₃	C₂×Dic₃	1	1	yes	characteristic
70	C₂⁴×C₃	C₂×C₃×C₄	1	14	yes	—
71	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	—	2	84	no	—
73	C₂²×C₃×C₄	—	2	7	no	—
74	C₂²×C₃×C₄	—	2	14	no	—
75	C₂²×C₃×C₄	—	2	7	no	—
76	C₂²×C₃×C₄	—	2	28	no	—
77	C₂²×C₃×C₄	C₂×Dic₃	1	84	yes	—
78	C₂²×C₃×C₄	C₂×Dic₃	1	14	yes	—
79	C₂²×C₃×C₄	C₂×Dic₃	1	7	yes	—
80	C₂²×C₃×C₄	C₂×C₃×C₄	1	84	yes	—
81	C₂²×C₃×C₄	C₂×C₃×C₄	1	7	yes	—
82	C₂²×C₃×C₄	C₂×C₃×C₄	1	14	yes	—
83	C₂²×C₃×C₄	C₂²×D₃	1	7	yes	—
84	C₂²×C₃×C₄	C₂²×D₃	1	28	yes	—
85	C₂²×C₃×C₄	C₂³×C₃	1	7	yes	—
86	C₂²×C₃×C₄	C₂³×C₃	1	28	yes	—
87	C₃×M₁₆	—	3	64	no	—
88	C₂×C₃×C₈	—	3	56	no	—
89	#208	C₂³×C₃	1	64	yes	direct product factor
90	C₂×C₃⋊C₈	C₂³×C₃	1	56	yes	complemented
91	C₂²×C₃²	—	2	56	no	—
92	C₂²×C₃²	C₂×M₁₆	1	28	yes	complemented
93	C₂²×C₃²	C₂³×C₄	1	8	yes	—
94	C₂²×C₃²	C₂³×C₄	1	7	yes	—
95	C₃²×C₄	C₂³×C₄	1	7	yes	—
96	C₃²×C₄	C₂³×C₄	1	8	yes	—
97	C₃²×C₄	C₂⁵	1	1	yes	characteristic
98	C₂⁵	C₃×Dic₃	1	1	yes	characteristic
99	C₂³×C₄	C₃×Dic₃	1	14	yes	—
100	C₂³×C₄	C₃×Dic₃	1	1	yes	characteristic
101	C₂³×C₄	C₂×C₃×D₃	1	1	yes	characteristic
102	C₂³×C₄	C₂×C₃×D₃	1	14	yes	—
103	C₂×M₁₆	—	3	112	no	—
104	C₂²×C₈	—	3	28	no	—
105	C₂³×C₃	—	2	112	no	—
106	C₂³×C₃	—	2	56	no	—
107	C₂³×C₃	—	2	56	no	—
108	C₂³×C₃	—	2	8	no	—
109	C₂³×C₃	—	2	28	no	—
110	C₂³×C₃	—	2	7	no	—
111	C₂³×C₃	#208	1	8	yes	direct product factor
112	C₂³×C₃	C₃×M₁₆	1	8	yes	complemented
113	C₂³×C₃	C₂²×Dic₃	1	28	yes	—
114	C₂³×C₃	C₂²×Dic₃	1	7	yes	—
115	C₂³×C₃	C₂²×C₃×C₄	1	28	yes	—
116	C₂³×C₃	C₂²×C₃×C₄	1	7	yes	—
117	C₂×C₃×C₄	—	2	7	no	—
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	56	no	—
120	C₂×C₃×C₄	—	2	21	no	—
121	C₂×C₃×C₄	—	2	28	no	—
122	C₂×C₃×C₄	C₂²×Dic₃	1	56	yes	—
123	C₂×C₃×C₄	C₂²×Dic₃	1	28	yes	—
124	C₂×C₃×C₄	C₂²×Dic₃	1	21	yes	—
125	C₂×C₃×C₄	C₂²×C₃×C₄	1	56	yes	—
126	C₂×C₃×C₄	C₂²×C₃×C₄	1	21	yes	—
127	C₂×C₃×C₄	C₂²×C₃×C₄	1	28	yes	—
128	C₂×C₃×C₄	C₂³×D₃	1	7	yes	—
129	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
130	C₂×C₃×C₄	C₂⁴×C₃	1	7	yes	—
131	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
132	C₃×C₈	—	3	16	no	—
133	C₃⋊C₈	C₂⁴×C₃	1	16	yes	complemented
134	C₂×C₃²	—	2	8	no	—
135	C₂×C₃²	C₂²×M₁₆	1	14	yes	complemented
136	C₂×C₃²	C₂⁴×C₄	1	1	yes	characteristic
137	C₂⁴	—	2	8	no	—
138	C₂⁴	C₂×C₃×Dic₃	1	14	yes	—
139	C₂⁴	C₂×C₃×Dic₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₃×Dic₃	1	84	yes	—
141	C₂²×C₄	C₂×C₃×Dic₃	1	7	yes	—
142	C₂²×C₄	C₂×C₃×Dic₃	1	14	yes	—
143	C₂²×C₄	C₂²×C₃×D₃	1	7	yes	—
144	C₂²×C₄	C₂²×C₃×D₃	1	28	yes	—
145	M₁₆	—	3	64	no	—
146	C₂×C₈	—	3	56	no	—
147	C₂²×C₃	—	2	112	no	—
148	C₂²×C₃	—	2	56	no	—
149	C₂²×C₃	—	2	56	no	—
150	C₂²×C₃	—	2	28	no	—
151	C₂²×C₃	—	2	8	no	—
152	C₂²×C₃	—	2	7	no	—
153	C₂²×C₃	C₂×#208	1	28	yes	direct product factor
154	C₂²×C₃	C₂×C₃×M₁₆	1	28	yes	complemented
155	C₂²×C₃	C₂³×Dic₃	1	8	yes	—
156	C₂²×C₃	C₂³×Dic₃	1	7	yes	—
157	C₂²×C₃	C₂³×C₃×C₄	1	8	yes	—
158	C₂²×C₃	C₂³×C₃×C₄	1	7	yes	—
159	C₃×C₄	—	2	7	no	—
160	C₃×C₄	—	2	8	no	—
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	C₂³×Dic₃	1	7	yes	—
163	C₃×C₄	C₂³×Dic₃	1	8	yes	—
164	C₃×C₄	C₂³×C₃×C₄	1	7	yes	—
165	C₃×C₄	C₂³×C₃×C₄	1	8	yes	—
166	C₃×C₄	C₂⁴×D₃	1	1	yes	characteristic
167	C₃×C₄	C₂⁵×C₃	1	1	yes	characteristic
168	C₃²	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
169	C₂³	—	2	56	no	—
170	C₂³	C₃×#208	1	8	yes	direct product factor
171	C₂³	C₂²×C₃×Dic₃	1	28	yes	—
172	C₂³	C₂²×C₃×Dic₃	1	7	yes	—
173	C₂×C₄	C₂²×C₃×Dic₃	1	56	yes	—
174	C₂×C₄	C₂²×C₃×Dic₃	1	21	yes	—
175	C₂×C₄	C₂²×C₃×Dic₃	1	28	yes	—
176	C₂×C₄	C₂³×C₃×D₃	1	7	yes	—
177	C₂×C₄	C₂³×C₃×D₃	1	8	yes	—
178	C₈	—	3	16	no	—
179	C₂×C₃	—	2	16	no	—
180	C₂×C₃	—	2	8	no	—
181	C₂×C₃	—	2	8	no	—
182	C₂×C₃	—	2	14	no	—
183	C₂×C₃	—	2	1	no	—
184	C₂×C₃	C₂²×#208	1	14	yes	direct product factor
185	C₂×C₃	C₂²×C₃×M₁₆	1	14	yes	complemented
186	C₂×C₃	C₂⁴×Dic₃	1	1	yes	characteristic
187	C₂×C₃	C₂⁴×C₃×C₄	1	1	yes	characteristic
188	C₂²	—	2	56	no	—
189	C₂²	C₂×C₃×#208	1	28	yes	direct product factor
190	C₂²	C₂³×C₃×Dic₃	1	8	yes	—
191	C₂²	C₂³×C₃×Dic₃	1	7	yes	—
192	C₄	C₂³×C₃×Dic₃	1	7	yes	—
193	C₄	C₂³×C₃×Dic₃	1	8	yes	—
194	C₄	C₂⁴×C₃×D₃	1	1	yes	characteristic
195	C₃	—	2	1	no	—
196	C₃	C₂³×#208	1	1	yes	characteristic, direct product factor
197	C₃	C₂³×C₃×M₁₆	1	1	yes	characteristic, complemented
198	C₂	—	2	8	no	—
199	C₂	C₂²×C₃×#208	1	14	yes	direct product factor
200	C₂	C₂⁴×C₃×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₄	C₂×D₃
commutator	C₂×C₃	C₂⁴×C₃×C₄
Frattini	C₄	C₂⁴×C₃×D₃
Fitting	C₂⁴×C₃²×C₄	C₂
socle	C₂⁴×C₃²	C₂×C₄
prime radical	C₂⁵×C₃²	C₄
cosocle	C₃×C₄	C₂⁵×C₃
hypercenter	C₂⁴×C₃×C₄	D₃
hypocenter	C₃	C₂³×C₃×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ C₂⁴×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×#208	C₂	direct (N × H)
3. C₂³×C₃×C₃⋊C₈	C₂	semidirect
5. C₂³×#208	C₃	direct (N × H)
11. C₂×C₃×#208	C₂²	direct (N × H)
12. C₂²×C₃×C₃⋊C₈	C₂²	semidirect
18. C₂²×#208	C₂×C₃	direct (N × H)
19. C₂³×C₃⋊C₈	C₂×C₃	semidirect
28. C₃×#208	C₂³	direct (N × H)
29. C₂×C₃×C₃⋊C₈	C₂³	semidirect
48. C₂×#208	C₂²×C₃	direct (N × H)
49. C₂²×C₃⋊C₈	C₂²×C₃	semidirect
51. C₂³×C₃²	M₁₆	semidirect
59. C₃×C₃⋊C₈	C₂⁴	semidirect
89. #208	C₂³×C₃	direct (N × H)
90. C₂×C₃⋊C₈	C₂³×C₃	semidirect
92. C₂²×C₃²	C₂×M₁₆	semidirect
111. C₂³×C₃	#208	direct (N × H)
112. C₂³×C₃	C₃×M₁₆	semidirect
133. C₃⋊C₈	C₂⁴×C₃	semidirect
135. C₂×C₃²	C₂²×M₁₆	semidirect
153. C₂²×C₃	C₂×#208	direct (N × H)
154. C₂²×C₃	C₂×C₃×M₁₆	semidirect
168. C₃²	C₂³×M₁₆	semidirect
170. C₂³	C₃×#208	direct (N × H)
184. C₂×C₃	C₂²×#208	direct (N × H)
185. C₂×C₃	C₂²×C₃×M₁₆	semidirect
189. C₂²	C₂×C₃×#208	direct (N × H)
196. C₃	C₂³×#208	direct (N × H)
197. C₃	C₂³×C₃×M₁₆	semidirect
199. C₂	C₂²×C₃×#208	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²×C₄	2
C₂²×C₃×#208	2
C₂³×C₃×C₃⋊C₈	2
C₂³×C₃×M₁₆	3
C₂³×#208	3

Subgroups — C23×C3×#208 #11918438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×#208 #11918438