Subgroups — C₄×Dic₃×M₁₆ #10792517

Order 768.

1734 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₃×M₁₆	C₂	1	4	yes	maximal
3	C₂×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#208	C₂	1	1	yes	maximal, characteristic
5	Dic₃×#97	C₂	1	4	yes	maximal, complemented
6	C₄×#815	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
8	C₄²×M₁₆	—	3	1	no	maximal, Sylow
9	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₃	C₄	1	4	yes	—
12	C₂²×C₄×Dic₃	C₂²	1	2	yes	—
13	C₂²×#208	C₂²	1	2	yes	—
14	C₃×C₄×M₁₆	C₄	1	8	yes	complemented
15	C₂×C₃×#97	C₂²	1	2	yes	—
16	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
17	Dic₃×M₁₆	C₄	1	32	yes	direct product factor
18	C₂×#815	C₂²	1	8	yes	—
19	C₂×C₈×Dic₃	C₂²	1	8	yes	—
20	C₄²×Dic₃	C₄	1	4	yes	—
21	C₄²×Dic₃	C₂²	1	1	yes	characteristic
22	C₄²×Dic₃	C₂²	1	1	yes	characteristic
23	C₄²×Dic₃	C₂²	1	2	yes	—
24	C₄×#208	C₄	1	8	yes	complemented
25	C₂×#803	C₂²	1	2	yes	—
26	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄×M₁₆	—	3	1	no	—
28	C₂×C₄×M₁₆	—	3	4	no	—
29	C₂×C₄×M₁₆	D₃	1	1	yes	characteristic
30	C₂×C₄³	—	3	1	no	—
31	C₄×#97	—	3	4	no	—
32	C₄×#97	—	3	2	no	—
33	C₄²×C₈	—	3	2	no	—
34	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
35	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂³×Dic₃	C₂×C₄	1	4	yes	—
37	C₂×C₃×M₁₆	C₂×C₄	1	16	yes	—
38	C₂×C₃×M₁₆	C₂×C₄	1	8	yes	—
39	C₂²×C₃×C₈	C₂³	1	4	yes	—
40	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
42	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
43	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₄×Dic₃	—	2	16	no	—
46	C₂×C₄×Dic₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₃	C₂×C₄	1	8	yes	—
48	C₂×C₄×Dic₃	C₂×C₄	1	4	yes	—
49	C₂×C₄×Dic₃	C₂×C₄	1	16	yes	—
50	C₂×C₄×Dic₃	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₃	C₂³	1	2	yes	—
52	C₂×C₄×Dic₃	C₂³	1	2	yes	—
53	C₂×#208	C₂×C₄	1	8	yes	—
54	C₂×#208	C₂×C₄	1	16	yes	—
55	C₂²×C₃⋊C₈	C₂³	1	4	yes	—
56	C₃×#97	C₂×C₄	1	8	yes	complemented
57	C₃×C₄×C₈	C₂×C₄	1	8	yes	complemented
58	#815	C₂×C₄	1	32	yes	complemented
59	C₈×Dic₃	C₂×C₄	1	32	yes	complemented
60	#803	C₂×C₄	1	8	yes	complemented
61	C₄×C₃⋊C₈	C₂×C₄	1	8	yes	complemented
62	C₂²×M₁₆	—	3	2	no	—
63	C₂²×M₁₆	C₂×D₃	1	2	yes	—
64	C₂²×C₄²	—	3	2	no	—
65	C₂²×C₄²	—	3	4	no	—
66	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
67	C₄×M₁₆	—	3	32	no	—
68	C₄×M₁₆	—	3	8	no	—
69	C₄×M₁₆	Dic₃	1	8	yes	direct product factor
70	C₂×#97	—	3	8	no	—
71	C₂×#97	—	3	2	no	—
72	C₂×#97	C₂×D₃	1	2	yes	—
73	C₂×C₄×C₈	—	3	8	no	—
74	C₂×C₄×C₈	—	3	2	no	—
75	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
76	C₄³	—	3	4	no	—
77	C₄³	—	3	2	no	—
78	C₄³	—	3	1	no	—
79	C₄³	—	3	1	no	—
80	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	—	2	8	no	—
82	C₂²×C₃×C₄	C₄²	1	8	yes	—
83	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
84	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
86	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
87	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
88	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₃	—	2	16	no	—
91	C₂²×Dic₃	C₄²	1	16	yes	—
92	C₂²×Dic₃	C₂²×C₄	1	4	yes	—
93	C₂²×Dic₃	C₂²×C₄	1	4	yes	—
94	C₃×M₁₆	C₄²	1	32	yes	complemented
95	C₂×C₃×C₈	C₂²×C₄	1	16	yes	—
96	C₂×C₃×C₈	C₂²×C₄	1	8	yes	—
97	C₃×C₄²	C₄²	1	8	yes	—
98	C₃×C₄²	C₂²×C₄	1	2	yes	—
99	C₃×C₄²	C₂²×C₄	1	2	yes	—
100	C₃×C₄²	C₂²×C₄	1	2	yes	—
101	C₃×C₄²	C₂²×C₄	1	2	yes	—
102	C₄×Dic₃	—	2	8	no	—
103	C₄×Dic₃	—	2	16	no	—
104	C₄×Dic₃	C₄²	1	16	yes	—
105	C₄×Dic₃	M₁₆	1	16	yes	direct product factor
106	C₄×Dic₃	C₂²×C₄	1	8	yes	—
107	C₄×Dic₃	C₂²×C₄	1	8	yes	—
108	#208	C₄²	1	32	yes	complemented
109	C₂×C₃⋊C₈	C₂²×C₄	1	8	yes	—
110	C₂×C₃⋊C₈	C₂²×C₄	1	16	yes	—
111	C₂³×C₄	—	3	4	no	—
112	C₂³×C₄	C₄×D₃	1	2	yes	—
113	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₂×M₁₆	—	3	8	no	—
115	C₂×M₁₆	—	3	16	no	—
116	C₂×M₁₆	C₄×D₃	1	16	yes	—
117	C₂×M₁₆	C₂×Dic₃	1	8	yes	—
118	C₂²×C₈	—	3	4	no	—
119	C₂²×C₈	C₂²×D₃	1	4	yes	—
120	C₂×C₄²	—	3	4	no	—
121	C₂×C₄²	—	3	4	no	—
122	C₂×C₄²	—	3	16	no	—
123	C₂×C₄²	—	3	2	no	—
124	C₂×C₄²	—	3	2	no	—
125	C₂×C₄²	—	3	8	no	—
126	C₂×C₄²	—	3	4	no	—
127	C₂×C₄²	—	6	16	no	—
128	C₂×C₄²	C₄×D₃	1	2	yes	—
129	C₂×C₄²	C₄×D₃	1	4	yes	—
130	C₂×C₄²	C₂×Dic₃	1	4	yes	—
131	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
133	#97	—	3	32	no	—
134	#97	—	3	8	no	—
135	#97	C₂×Dic₃	1	8	yes	complemented
136	C₄×C₈	—	3	32	no	—
137	C₄×C₈	—	3	8	no	—
138	C₄×C₈	C₂×Dic₃	1	8	yes	complemented
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	C₂×C₄²	1	4	yes	—
141	C₂³×C₃	C₂×C₄²	1	2	yes	—
142	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	16	no	—
145	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
146	C₂×C₃×C₄	C₂×C₄²	1	8	yes	—
147	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
148	C₂×C₃×C₄	C₂×C₄²	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₄²	1	8	yes	—
150	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
151	C₂×C₃×C₄	C₂×M₁₆	1	8	yes	—
152	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
154	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
155	C₂×Dic₃	—	2	8	no	—
156	C₂×Dic₃	—	2	32	no	—
157	C₂×Dic₃	C₂×C₄²	1	8	yes	—
158	C₂×Dic₃	C₂×C₄²	1	8	yes	—
159	C₂×Dic₃	C₂×M₁₆	1	16	yes	—
160	C₃×C₈	C₂×C₄²	1	16	yes	complemented
161	C₃⋊C₈	C₂×C₄²	1	16	yes	complemented
162	C₂⁴	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	3	4	no	—
165	C₂²×C₄	—	3	16	no	—
166	C₂²×C₄	—	3	4	no	—
167	C₂²×C₄	—	6	16	no	—
168	C₂²×C₄	C₄×Dic₃	1	8	yes	—
169	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
170	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
172	C₂²×C₄	C₂×C₄×D₃	1	8	yes	—
173	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
175	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
176	M₁₆	—	3	32	no	—
177	M₁₆	C₄×Dic₃	1	32	yes	direct product factor
178	C₂×C₈	—	3	8	no	—
179	C₂×C₈	—	3	16	no	—
180	C₂×C₈	C₂×C₄×D₃	1	16	yes	—
181	C₂×C₈	C₂²×Dic₃	1	8	yes	—
182	C₄²	—	3	16	no	—
183	C₄²	—	3	16	no	—
184	C₄²	—	3	8	no	—
185	C₄²	—	3	8	no	—
186	C₄²	—	6	16	no	—
187	C₄²	—	6	8	no	—
188	C₄²	C₄×Dic₃	1	8	yes	—
189	C₄²	C₂×C₄×D₃	1	2	yes	—
190	C₄²	C₂×C₄×D₃	1	2	yes	—
191	C₄²	C₂²×Dic₃	1	2	yes	—
192	C₄²	C₂²×Dic₃	1	2	yes	—
193	C₂²×C₃	—	2	8	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	C₄³	1	4	yes	—
196	C₂²×C₃	C₂²×C₄²	1	2	yes	—
197	C₂²×C₃	C₂²×C₄²	1	1	yes	characteristic
198	C₂²×C₃	C₂²×M₁₆	1	4	yes	—
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	C₄³	1	4	yes	—
201	C₃×C₄	C₄×M₁₆	1	8	yes	complemented
202	C₃×C₄	C₂²×C₄²	1	2	yes	—
203	C₃×C₄	C₂²×C₄²	1	2	yes	—
204	Dic₃	—	2	8	no	—
205	Dic₃	C₄×M₁₆	1	16	yes	direct product factor
206	C₂³	—	2	4	no	—
207	C₂³	C₄²×D₃	1	4	yes	—
208	C₂³	C₂×C₄×Dic₃	1	2	yes	—
209	C₂³	C₂²×C₄×D₃	1	1	yes	characteristic
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	3	8	no	—
213	C₂×C₄	—	3	16	no	—
214	C₂×C₄	—	3	8	no	—
215	C₂×C₄	—	6	32	no	—
216	C₂×C₄	—	6	8	no	—
217	C₂×C₄	C₄²×D₃	1	4	yes	—
218	C₂×C₄	C₄²×D₃	1	8	yes	—
219	C₂×C₄	D₃×M₁₆	1	8	yes	—
220	C₂×C₄	C₂×C₄×Dic₃	1	4	yes	—
221	C₂×C₄	C₂×C₄×Dic₃	1	2	yes	—
222	C₂×C₄	C₂×C₄×Dic₃	1	8	yes	—
223	C₂×C₄	C₂×C₄×Dic₃	1	4	yes	—
224	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₄×D₃	1	4	yes	—
226	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
227	C₈	—	3	16	no	—
228	C₈	C₂×C₄×Dic₃	1	16	yes	complemented
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	C₂×C₄³	1	1	yes	characteristic
231	C₂×C₃	C₂×C₄×M₁₆	1	2	yes	—
232	C₂×C₃	C₂×C₄×M₁₆	1	4	yes	—
233	C₂²	—	2	4	no	—
234	C₂²	—	2	8	no	—
235	C₂²	C₄²×Dic₃	1	4	yes	—
236	C₂²	C₂×C₄²×D₃	1	2	yes	—
237	C₂²	C₂×D₃×M₁₆	1	4	yes	—
238	C₂²	C₂²×C₄×Dic₃	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	3	16	no	—
241	C₄	—	6	8	no	—
242	C₄	C₄²×Dic₃	1	4	yes	—
243	C₄	Dic₃×M₁₆	1	8	yes	direct product factor
244	C₄	C₂×C₄²×D₃	1	2	yes	—
245	C₄	C₂²×C₄×Dic₃	1	2	yes	—
246	C₃	C₄²×M₁₆	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	2	4	no	—
248	C₂	C₄×D₃×M₁₆	1	4	yes	—
249	C₂	C₂×C₄²×Dic₃	1	1	yes	characteristic
250	C₂	C₂×Dic₃×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₃
commutator	C₂×C₃	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×M₁₆	C₂
socle	C₂³×C₃	C₂³×C₄
prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₃
hypocenter	C₃	C₄²×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
5. Dic₃×#97	C₂	semidirect
6. C₄×#815	C₂	semidirect
7. C₄×C₈×Dic₃	C₂	semidirect
14. C₃×C₄×M₁₆	C₄	semidirect
17. Dic₃×M₁₆	C₄	direct (N × H)
24. C₄×#208	C₄	semidirect
56. C₃×#97	C₂×C₄	semidirect
57. C₃×C₄×C₈	C₂×C₄	semidirect
58. #815	C₂×C₄	semidirect
59. C₈×Dic₃	C₂×C₄	semidirect
60. #803	C₂×C₄	semidirect
61. C₄×C₃⋊C₈	C₂×C₄	semidirect
69. C₄×M₁₆	Dic₃	direct (N × H)
94. C₃×M₁₆	C₄²	semidirect
105. C₄×Dic₃	M₁₆	direct (N × H)
108. #208	C₄²	semidirect
135. #97	C₂×Dic₃	semidirect
138. C₄×C₈	C₂×Dic₃	semidirect
160. C₃×C₈	C₂×C₄²	semidirect
161. C₃⋊C₈	C₂×C₄²	semidirect
177. M₁₆	C₄×Dic₃	direct (N × H)
201. C₃×C₄	C₄×M₁₆	semidirect
205. Dic₃	C₄×M₁₆	direct (N × H)
228. C₈	C₂×C₄×Dic₃	semidirect
243. C₄	Dic₃×M₁₆	direct (N × H)
246. C₃	C₄²×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×M₁₆	2
C₂×Dic₃×M₁₆	2
C₂×C₄²×Dic₃	2
C₂×C₄×#208	2
Dic₃×#97	2
C₄×#815	2
C₄×C₈×Dic₃	2
C₄²×M₁₆	3

Subgroups — C4×Dic3×M16 #10792517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₃×M₁₆ #10792517