Subgroups — C₄×D₃×M₁₆ #78513

Order 384.

656 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₄²×D₃	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃×M₁₆	C₂	1	2	yes	maximal, complemented
5	C₄×#208	C₂	1	1	yes	maximal, characteristic, complemented
6	#4707	C₂	1	2	yes	maximal, complemented
7	D₃×#97	C₂	1	2	yes	maximal, complemented
8	C₄×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal, complemented
9	C₄×D₃×C₈	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×M₁₆	—	3	1	no	maximal, Sylow
11	C₂²×C₄×D₃	C₄	1	2	yes	—
12	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×M₁₆	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₃	C₄	1	2	yes	—
16	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×#208	C₂²	1	2	yes	—
18	D₃×M₁₆	C₄	1	16	yes	direct product factor
19	C₂×ΓL(1,𝔽₂₅)	C₂²	1	4	yes	—
20	C₂×D₃×C₈	C₂²	1	4	yes	—
21	C₄²×D₃	C₄	1	2	yes	—
22	C₄²×D₃	C₄	1	2	yes	—
23	C₄²×D₃	C₂²	1	1	yes	characteristic
24	C₄²×D₃	C₂²	1	1	yes	characteristic
25	C₄²×D₃	C₂²	1	1	yes	characteristic
26	C₄²×D₃	C₂²	1	1	yes	characteristic
27	C₃×#97	C₂²	1	2	yes	complemented
28	C₃×C₄×C₈	C₂²	1	2	yes	complemented
29	#815	C₂²	1	4	yes	complemented
30	C₈×Dic₃	C₂²	1	4	yes	complemented
31	#803	C₂²	1	2	yes	complemented
32	C₄×C₃⋊C₈	C₂²	1	2	yes	complemented
33	C₂²×M₁₆	—	3	2	no	—
34	C₂²×C₄²	—	3	1	no	—
35	C₄×M₁₆	—	3	2	no	—
36	C₄×M₁₆	—	3	1	no	—
37	C₄×M₁₆	—	3	2	no	—
38	C₄×M₁₆	—	3	2	no	—
39	C₄×M₁₆	D₃	1	1	yes	characteristic, direct product factor
40	C₂×#97	—	3	2	no	—
41	C₂×C₄×C₈	—	3	2	no	—
42	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂²×Dic₃	C₂×C₄	1	2	yes	—
46	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₃	—	2	8	no	—
48	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₃	C₂×C₄	1	8	yes	—
54	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
55	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
57	C₃×M₁₆	C₂×C₄	1	8	yes	complemented
58	C₂×C₃×C₈	C₂³	1	4	yes	—
59	C₃×C₄²	C₂×C₄	1	2	yes	—
60	C₃×C₄²	C₂³	1	1	yes	characteristic
61	C₃×C₄²	C₂³	1	1	yes	characteristic
62	C₄×Dic₃	C₂×C₄	1	2	yes	—
63	C₄×Dic₃	C₂×C₄	1	2	yes	—
64	C₄×Dic₃	C₂×C₄	1	2	yes	—
65	C₄×Dic₃	C₂×C₄	1	2	yes	—
66	C₄×Dic₃	C₂×C₄	1	2	yes	—
67	C₄×Dic₃	C₂³	1	1	yes	characteristic
68	C₄×Dic₃	C₂³	1	1	yes	characteristic
69	#208	C₂×C₄	1	8	yes	complemented
70	C₂×C₃⋊C₈	C₂³	1	4	yes	—
71	ΓL(1,𝔽₂₅)	C₂×C₄	1	16	yes	complemented
72	D₃×C₈	C₂×C₄	1	16	yes	complemented
73	C₂³×C₄	—	3	1	no	—
74	C₂³×C₄	—	3	2	no	—
75	C₂×M₁₆	—	3	2	no	—
76	C₂×M₁₆	—	3	16	no	—
77	C₂×M₁₆	—	3	4	no	—
78	C₂×M₁₆	C₂×D₃	1	2	yes	—
79	C₂²×C₈	—	3	4	no	—
80	C₂×C₄²	—	3	2	no	—
81	C₂×C₄²	—	3	1	no	—
82	C₂×C₄²	—	3	2	no	—
83	C₂×C₄²	—	3	2	no	—
84	C₂×C₄²	—	3	1	no	—
85	C₂×C₄²	—	3	1	no	—
86	C₂×C₄²	—	3	1	no	—
87	C₂×C₄²	—	3	1	no	—
88	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
89	#97	—	3	4	no	—
90	#97	—	3	2	no	—
91	#97	C₂×D₃	1	2	yes	complemented
92	C₄×C₈	—	3	4	no	—
93	C₄×C₈	—	3	2	no	—
94	C₄×C₈	C₂×D₃	1	2	yes	complemented
95	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×D₃	—	2	4	no	—
97	C₂²×D₃	C₄²	1	4	yes	—
98	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
102	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
104	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
105	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
106	C₂×Dic₃	—	2	4	no	—
107	C₂×Dic₃	C₄²	1	4	yes	—
108	C₂×Dic₃	C₂²×C₄	1	2	yes	—
109	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₃	C₂²×C₄	1	2	yes	—
112	C₄×D₃	—	2	4	no	—
113	C₄×D₃	—	2	4	no	—
114	C₄×D₃	—	2	4	no	—
115	C₄×D₃	C₄²	1	4	yes	—
116	C₄×D₃	C₄²	1	4	yes	—
117	C₄×D₃	M₁₆	1	8	yes	direct product factor
118	C₄×D₃	C₂²×C₄	1	4	yes	—
119	C₄×D₃	C₂²×C₄	1	4	yes	—
120	C₃×C₈	C₂²×C₄	1	8	yes	complemented
121	C₃⋊C₈	C₂²×C₄	1	8	yes	complemented
122	C₂⁴	—	3	1	no	—
123	C₂²×C₄	—	3	1	no	—
124	C₂²×C₄	—	3	1	no	—
125	C₂²×C₄	—	3	1	no	—
126	C₂²×C₄	—	3	2	no	—
127	C₂²×C₄	—	3	8	no	—
128	C₂²×C₄	—	3	1	no	—
129	C₂²×C₄	—	3	1	no	—
130	C₂²×C₄	—	3	2	no	—
131	C₂²×C₄	—	3	2	no	—
132	C₂²×C₄	—	3	2	no	—
133	C₂²×C₄	—	3	2	no	—
134	C₂²×C₄	—	6	8	no	—
135	C₂²×C₄	C₄×D₃	1	2	yes	—
136	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
137	M₁₆	—	3	8	no	—
138	M₁₆	—	3	16	no	—
139	M₁₆	C₄×D₃	1	8	yes	direct product factor
140	C₂×C₈	—	3	4	no	—
141	C₂×C₈	—	3	16	no	—
142	C₂×C₈	C₂²×D₃	1	4	yes	—
143	C₄²	—	3	2	no	—
144	C₄²	—	3	2	no	—
145	C₄²	—	3	2	no	—
146	C₄²	—	3	2	no	—
147	C₄²	—	3	2	no	—
148	C₄²	—	3	1	no	—
149	C₄²	—	3	1	no	—
150	C₄²	C₄×D₃	1	2	yes	—
151	C₄²	C₂²×D₃	1	1	yes	characteristic
152	C₄²	C₂²×D₃	1	1	yes	characteristic
153	C₂²×C₃	—	2	2	no	—
154	C₂²×C₃	C₂×C₄²	1	2	yes	—
155	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×D₃	—	2	2	no	—
157	C₂×D₃	—	2	8	no	—
158	C₂×D₃	C₂×C₄²	1	2	yes	—
159	C₂×D₃	C₂×C₄²	1	2	yes	—
160	C₂×D₃	C₂×M₁₆	1	4	yes	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	C₂×C₄²	1	2	yes	—
163	C₃×C₄	C₂×M₁₆	1	4	yes	complemented
164	C₃×C₄	C₂³×C₄	1	2	yes	—
165	Dic₃	—	2	2	no	—
166	Dic₃	C₂×C₄²	1	2	yes	—
167	Dic₃	C₂×C₄²	1	2	yes	—
168	Dic₃	C₂×M₁₆	1	4	yes	complemented
169	C₂³	—	3	1	no	—
170	C₂³	—	3	4	no	—
171	C₂³	—	3	1	no	—
172	C₂³	—	6	4	no	—
173	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	3	4	no	—
176	C₂×C₄	—	3	1	no	—
177	C₂×C₄	—	3	1	no	—
178	C₂×C₄	—	3	2	no	—
179	C₂×C₄	—	3	4	no	—
180	C₂×C₄	—	3	4	no	—
181	C₂×C₄	—	3	8	no	—
182	C₂×C₄	—	3	4	no	—
183	C₂×C₄	—	3	4	no	—
184	C₂×C₄	—	3	2	no	—
185	C₂×C₄	—	6	4	no	—
186	C₂×C₄	—	6	4	no	—
187	C₂×C₄	—	6	4	no	—
188	C₂×C₄	—	6	4	no	—
189	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
190	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
192	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
193	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
194	C₈	—	3	8	no	—
195	C₈	C₂×C₄×D₃	1	8	yes	complemented
196	C₂×C₃	—	2	2	no	—
197	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
198	C₂×C₃	C₂²×M₁₆	1	2	yes	—
199	D₃	—	2	2	no	—
200	D₃	C₄×M₁₆	1	4	yes	direct product factor
201	C₂²	—	2	2	no	—
202	C₂²	—	3	2	no	—
203	C₂²	—	3	4	no	—
204	C₂²	—	3	2	no	—
205	C₂²	—	6	8	no	—
206	C₂²	—	6	2	no	—
207	C₂²	C₄²×D₃	1	2	yes	—
208	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	3	2	no	—
211	C₄	—	3	2	no	—
212	C₄	—	3	4	no	—
213	C₄	—	6	2	no	—
214	C₄	C₄²×D₃	1	2	yes	—
215	C₄	D₃×M₁₆	1	4	yes	direct product factor
216	C₄	C₂²×C₄×D₃	1	2	yes	—
217	C₃	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
218	C₂	—	2	2	no	—
219	C₂	—	3	4	no	—
220	C₂	—	6	2	no	—
221	C₂	C₂×C₄²×D₃	1	1	yes	characteristic
222	C₂	C₂×D₃×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₃
commutator	C₂×C₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×M₁₆	C₂
socle	C₂²×C₃	C₂³×C₄
prime radical	C₂³×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×M₁₆	C₂	semidirect
4. Dic₃×M₁₆	C₂	semidirect
5. C₄×#208	C₂	semidirect
6. #4707	C₂	semidirect
7. D₃×#97	C₂	semidirect
8. C₄×ΓL(1,𝔽₂₅)	C₂	semidirect
9. C₄×D₃×C₈	C₂	semidirect
18. D₃×M₁₆	C₄	direct (N × H)
27. C₃×#97	C₂²	semidirect
28. C₃×C₄×C₈	C₂²	semidirect
29. #815	C₂²	semidirect
30. C₈×Dic₃	C₂²	semidirect
31. #803	C₂²	semidirect
32. C₄×C₃⋊C₈	C₂²	semidirect
39. C₄×M₁₆	D₃	direct (N × H)
57. C₃×M₁₆	C₂×C₄	semidirect
69. #208	C₂×C₄	semidirect
71. ΓL(1,𝔽₂₅)	C₂×C₄	semidirect
72. D₃×C₈	C₂×C₄	semidirect
91. #97	C₂×D₃	semidirect
94. C₄×C₈	C₂×D₃	semidirect
117. C₄×D₃	M₁₆	direct (N × H)
120. C₃×C₈	C₂²×C₄	semidirect
121. C₃⋊C₈	C₂²×C₄	semidirect
139. M₁₆	C₄×D₃	direct (N × H)
163. C₃×C₄	C₂×M₁₆	semidirect
168. Dic₃	C₂×M₁₆	semidirect
195. C₈	C₂×C₄×D₃	semidirect
200. D₃	C₄×M₁₆	direct (N × H)
215. C₄	D₃×M₁₆	direct (N × H)
217. C₃	C₂×C₄×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×M₁₆	2
C₂×C₄²×D₃	2
C₃×C₄×M₁₆	2
Dic₃×M₁₆	2
C₄×#208	2
#4707	2
D₃×#97	2
C₄×ΓL(1,𝔽₂₅)	2
C₄×D₃×C₈	2
C₂×C₄×M₁₆	3

Subgroups — C4×D3×M16 #78513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₃×M₁₆ #78513