Subgroups — C₄×D₃×Q₁₆ #79716

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃×Q₈	C₂	1	2	yes	maximal
3	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#216	C₂	1	2	yes	maximal, complemented
6	D₃×#107	C₂	1	1	yes	maximal, characteristic
7	#4873	C₂	1	1	yes	maximal, characteristic, complemented
8	D₃×#103	C₂	1	2	yes	maximal
9	#4790	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₁₆	—	3	1	no	maximal, Sylow
13	C₂×D₃×Q₈	C₂²	1	2	yes	—
14	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₃×C₄×Q₈	C₂²	1	2	yes	—
16	Q₈×Dic₃	C₂²	1	2	yes	—
17	C₂×#216	C₂²	1	2	yes	—
18	D₃×Q₁₆	C₄	1	8	yes	direct product factor
19	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
20	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
21	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
22	D₃×C₄⋊C₄	—	2	2	no	—
23	D₃×C₄⋊C₄	C₂²	1	2	yes	—
24	#887	—	2	2	no	—
25	#887	C₂²	1	2	yes	—
26	C₄²×D₃	—	2	2	no	—
27	C₄²×D₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₆	—	2	2	no	—
29	C₄×Dic₆	C₂²	1	2	yes	—
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×#103	C₂²	1	2	yes	—
32	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
33	#835	C₂²	1	2	yes	—
34	#818	C₂²	1	1	yes	characteristic
35	#816	C₂²	1	2	yes	—
36	C₈×Dic₃	C₂²	1	1	yes	characteristic
37	C₈×Dic₃	C₂²	1	1	yes	characteristic
38	#810	C₂²	1	2	yes	—
39	#807	C₂²	1	2	yes	—
40	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
41	C₂²×Q₁₆	—	3	1	no	—
42	C₂×C₄×Q₈	—	3	2	no	—
43	C₄×Q₁₆	—	3	1	no	—
44	C₄×Q₁₆	—	3	2	no	—
45	C₄×Q₁₆	—	3	1	no	—
46	C₄×Q₁₆	—	3	2	no	—
47	C₄×Q₁₆	—	3	1	no	—
48	C₄×Q₁₆	D₃	1	1	yes	characteristic, direct product factor
49	C₂×#107	—	3	1	no	—
50	C₂×#103	—	3	2	no	—
51	C₂×C₄×C₈	—	3	1	no	—
52	C₂×C₃×Q₈	C₂³	1	2	yes	—
53	D₃×Q₈	—	2	4	no	—
54	D₃×Q₈	C₂×C₄	1	8	yes	—
55	C₂×C₄×D₃	—	2	2	no	—
56	C₂×C₄×D₃	—	2	2	no	—
57	C₂×C₄×D₃	D₄	1	1	yes	characteristic
58	C₂×C₄×D₃	D₄	1	1	yes	characteristic
59	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
60	C₂×Dic₆	—	2	2	no	—
61	C₂×Dic₆	C₂³	1	2	yes	—
62	C₃×Q₁₆	C₂×C₄	1	4	yes	complemented
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄⋊C₄	C₂³	1	2	yes	—
67	C₃×C₄²	—	2	2	no	—
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	#216	C₂×C₄	1	8	yes	complemented
70	C₄⋊Dic₃	—	2	2	no	—
71	C₄⋊Dic₃	C₂³	1	2	yes	—
72	Dic₃⋊C₄	—	2	2	no	—
73	Dic₃⋊C₄	—	2	2	no	—
74	Dic₃⋊C₄	—	2	2	no	—
75	C₄×Dic₃	—	2	2	no	—
76	C₄×Dic₃	—	2	2	no	—
77	C₄×Dic₃	—	2	2	no	—
78	C₄×Dic₃	D₄	1	1	yes	characteristic
79	C₄×Dic₃	D₄	1	1	yes	characteristic
80	C₄×Dic₃	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	Dic₁₂	C₂×C₄	1	4	yes	complemented
84	D₃×C₈	—	2	2	no	—
85	D₃×C₈	C₂×C₄	1	4	yes	—
86	C₂²×Q₈	—	3	2	no	—
87	C₂×Q₁₆	—	3	2	no	—
88	C₂×Q₁₆	—	3	8	no	—
89	C₂×Q₁₆	—	3	1	no	—
90	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
91	C₂²×C₈	—	3	1	no	—
92	C₂²×C₈	—	3	1	no	—
93	C₄×Q₈	—	3	2	no	—
94	C₄×Q₈	—	3	2	no	—
95	C₄×Q₈	—	3	2	no	—
96	C₄×Q₈	—	6	2	no	—
97	C₄×Q₈	—	6	2	no	—
98	C₄×Q₈	C₂×D₃	1	2	yes	—
99	C₂×C₄⋊C₄	—	3	2	no	—
100	C₂×C₄⋊C₄	—	6	2	no	—
101	C₂×C₄²	—	3	1	no	—
102	C₂×C₄²	—	6	2	no	—
103	#107	—	3	1	no	—
104	#107	—	3	2	no	—
105	#107	C₂×D₃	1	1	yes	characteristic
106	#103	—	3	2	no	—
107	#103	—	3	2	no	—
108	#103	—	3	2	no	—
109	#103	C₂×D₃	1	2	yes	—
110	C₄×C₈	—	3	1	no	—
111	C₄×C₈	—	3	1	no	—
112	C₄×C₈	—	3	1	no	—
113	C₄×C₈	C₂×D₃	1	1	yes	characteristic
114	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
115	C₃×Q₈	—	2	2	no	—
116	C₃×Q₈	C₂²×C₄	1	4	yes	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₃	—	2	2	no	—
123	C₂×Dic₃	—	2	2	no	—
124	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
127	C₄×D₃	—	2	8	no	—
128	C₄×D₃	—	2	2	no	—
129	C₄×D₃	—	4	4	no	—
130	C₄×D₃	Q₁₆	1	4	yes	direct product factor
131	C₄×D₃	C₂²×C₄	1	2	yes	—
132	C₄×D₃	#41	1	2	yes	—
133	Dic₆	—	2	2	no	—
134	Dic₆	—	2	8	no	—
135	Dic₆	C₂²×C₄	1	4	yes	—
136	C₃×C₈	—	2	1	no	—
137	C₃×C₈	C₂²×C₄	1	2	yes	—
138	C₃⋊C₈	—	2	1	no	—
139	C₃⋊C₈	C₂²×C₄	1	2	yes	—
140	C₂×Q₈	—	3	8	no	—
141	C₂×Q₈	—	3	2	no	—
142	C₂×Q₈	—	6	2	no	—
143	C₂×Q₈	—	6	4	no	—
144	C₂×Q₈	C₂²×D₃	1	2	yes	—
145	C₂²×C₄	—	3	1	no	—
146	C₂²×C₄	—	3	1	no	—
147	C₂²×C₄	—	3	1	no	—
148	C₂²×C₄	—	6	2	no	—
149	C₂²×C₄	—	6	2	no	—
150	Q₁₆	—	3	8	no	—
151	Q₁₆	—	3	4	no	—
152	Q₁₆	C₄×D₃	1	4	yes	direct product factor
153	C₂×C₈	—	3	1	no	—
154	C₂×C₈	—	3	4	no	—
155	C₂×C₈	—	3	1	no	—
156	C₂×C₈	—	6	2	no	—
157	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	3	2	no	—
161	C₄⋊C₄	—	6	2	no	—
162	C₄⋊C₄	—	6	2	no	—
163	C₄⋊C₄	—	6	2	no	—
164	C₄⋊C₄	—	6	2	no	—
165	C₄⋊C₄	C₂²×D₃	1	2	yes	—
166	C₄²	—	2	2	no	—
167	C₄²	—	3	1	no	—
168	C₄²	—	3	1	no	—
169	C₄²	—	3	1	no	—
170	C₄²	—	6	2	no	—
171	C₄²	—	6	2	no	—
172	C₄²	—	6	2	no	—
173	C₄²	C₂²×D₃	1	1	yes	characteristic
174	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×D₃	C₄×D₄	1	2	yes	—
176	C₂×D₃	C₂×Q₁₆	1	2	yes	—
177	C₂×D₃	#135	1	2	yes	—
178	C₃×C₄	—	2	4	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	C₂×Q₁₆	1	2	yes	complemented
182	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
183	C₃×C₄	C₂×#41	1	1	yes	characteristic
184	Dic₃	—	2	4	no	—
185	Dic₃	—	2	1	no	—
186	Dic₃	—	4	2	no	—
187	Dic₃	C₄×D₄	1	2	yes	—
188	Dic₃	C₂×Q₁₆	1	2	yes	complemented
189	C₂³	—	3	1	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	3	4	no	—
192	Q₈	—	6	8	no	—
193	Q₈	—	6	2	no	—
194	Q₈	C₂×C₄×D₃	1	4	yes	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	3	1	no	—
198	C₂×C₄	—	3	1	no	—
199	C₂×C₄	—	3	4	no	—
200	C₂×C₄	—	3	2	no	—
201	C₂×C₄	—	3	2	no	—
202	C₂×C₄	—	3	1	no	—
203	C₂×C₄	—	6	2	no	—
204	C₂×C₄	—	6	8	no	—
205	C₂×C₄	—	6	2	no	—
206	C₂×C₄	—	6	2	no	—
207	C₂×C₄	—	12	4	no	—
208	C₂×C₄	D₃×D₄	1	1	yes	characteristic
209	C₂×C₄	D₃×D₄	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	3	2	no	—
213	C₈	—	6	1	no	—
214	C₈	C₂×C₄×D₃	1	2	yes	—
215	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
216	C₂×C₃	C₂²×Q₁₆	1	1	yes	characteristic
217	C₂×C₃	C₂×#135	1	1	yes	characteristic
218	D₃	C₄×Q₁₆	1	4	yes	direct product factor
219	C₂²	—	3	2	no	—
220	C₂²	—	3	2	no	—
221	C₂²	—	3	2	no	—
222	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	4	no	—
225	C₄	—	3	2	no	—
226	C₄	—	3	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	6	1	no	—
229	C₄	—	6	4	no	—
230	C₄	—	12	2	no	—
231	C₄	D₃×Q₁₆	1	2	yes	direct product factor
232	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
233	C₄	D₃×#41	1	1	yes	characteristic
234	C₃	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
235	C₂	—	3	4	no	—
236	C₂	C₄×D₃×D₄	1	1	yes	characteristic
237	C₂	C₂×D₃×Q₁₆	1	1	yes	characteristic
238	C₂	D₃×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×Q₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	C₂×C₄×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×Q₁₆	C₂	semidirect
4. Dic₃×Q₁₆	C₂	semidirect
5. C₄×#216	C₂	semidirect
7. #4873	C₂	semidirect
9. #4790	C₂	semidirect
10. C₄×Dic₁₂	C₂	semidirect
18. D₃×Q₁₆	C₄	direct (N × H)
48. C₄×Q₁₆	D₃	direct (N × H)
62. C₃×Q₁₆	C₂×C₄	semidirect
69. #216	C₂×C₄	semidirect
83. Dic₁₂	C₂×C₄	semidirect
130. C₄×D₃	Q₁₆	direct (N × H)
152. Q₁₆	C₄×D₃	direct (N × H)
181. C₃×C₄	C₂×Q₁₆	semidirect
188. Dic₃	C₂×Q₁₆	semidirect
218. D₃	C₄×Q₁₆	direct (N × H)
231. C₄	D₃×Q₁₆	direct (N × H)
234. C₃	C₂×C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×Q₁₆	2
C₄×D₃×Q₈	2
C₃×C₄×Q₁₆	2
Dic₃×Q₁₆	2
C₄×#216	2
D₃×#107	2
#4873	2
D₃×#103	2
#4790	2
C₄×Dic₁₂	2
C₄×D₃×C₈	2
C₂×C₄×Q₁₆	3

Subgroups — C4×D3×Q16 #79716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₃×Q₁₆ #79716