Subgroups — C₂×C₃×C₄×SD₁₆ #84801

Order 384.

552 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×SD₁₆	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×SD₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	—	2	1	no	—
13	C₂×C₃×SD₁₆	C₄	1	8	yes	direct product factor
14	C₂×C₃×SD₁₆	C₂²	1	4	yes	—
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×Q₈	—	2	2	no	—
18	C₃×C₄×Q₈	C₂²	1	4	yes	complemented
19	C₃×C₄×D₄	—	2	2	no	—
20	C₃×C₄×D₄	C₂²	1	4	yes	—
21	C₂×C₃×C₄⋊C₄	—	2	1	no	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
25	C₂×C₃×C₄²	—	2	1	no	—
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₃×#106	C₂²	1	4	yes	complemented
28	C₃×#103	C₂²	1	4	yes	complemented
29	C₃×#102	C₂²	1	4	yes	—
30	C₃×C₄×C₈	C₂²	1	4	yes	complemented
31	C₂²×SD₁₆	C₂×C₃	1	1	yes	characteristic
32	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
34	C₄×SD₁₆	C₂×C₃	1	8	yes	direct product factor
35	C₂×#106	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#103	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×#102	C₂×C₃	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂⁴×C₃	—	2	1	no	—
40	C₂×C₃×Q₈	—	2	1	no	—
41	C₂×C₃×Q₈	—	2	2	no	—
42	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
43	C₂×C₃×Q₈	C₂³	1	2	yes	—
44	C₂×C₃×D₄	—	2	1	no	—
45	C₂×C₃×D₄	—	2	2	no	—
46	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
47	C₂×C₃×D₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	—	2	1	no	—
49	C₂²×C₃×C₄	—	2	1	no	—
50	C₂²×C₃×C₄	—	2	1	no	—
51	C₂²×C₃×C₄	—	2	4	no	—
52	C₂²×C₃×C₄	—	4	4	no	—
53	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₃×SD₁₆	C₂×C₄	1	16	yes	direct product factor
57	C₂×C₃×C₈	—	2	2	no	—
58	C₂×C₃×C₈	C₂×C₄	1	4	yes	complemented
59	C₂×C₃×C₈	C₂³	1	2	yes	—
60	C₂×C₃×C₈	C₂³	1	2	yes	—
61	C₃×C₄⋊C₄	—	2	4	no	—
62	C₃×C₄⋊C₄	—	2	1	no	—
63	C₃×C₄⋊C₄	—	2	1	no	—
64	C₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₃×C₄⋊C₄	C₂³	1	2	yes	—
66	C₃×C₂²⋊C₄	—	2	4	no	—
67	C₃×C₄²	—	2	4	no	—
68	C₃×C₄²	D₄	1	2	yes	—
69	C₃×C₄²	C₂³	1	2	yes	—
70	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
72	C₂³×C₄	—	2	1	no	—
73	C₂×SD₁₆	C₃×C₄	1	8	yes	direct product factor
74	C₂×SD₁₆	C₂²×C₃	1	4	yes	—
75	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	C₂²×C₃	1	4	yes	complemented
79	C₄×D₄	—	2	2	no	—
80	C₄×D₄	C₂²×C₃	1	4	yes	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	—	2	1	no	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
87	#106	C₂²×C₃	1	4	yes	complemented
88	#103	C₂²×C₃	1	4	yes	complemented
89	#102	C₂²×C₃	1	4	yes	—
90	C₄×C₈	C₂²×C₃	1	4	yes	complemented
91	C₂³×C₃	—	2	4	no	—
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	4	2	no	—
94	C₂³×C₃	—	4	2	no	—
95	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
96	C₃×Q₈	—	2	4	no	—
97	C₃×Q₈	—	2	2	no	—
98	C₃×Q₈	C₂²×C₄	1	4	yes	complemented
99	C₃×D₄	—	2	4	no	—
100	C₃×D₄	—	2	2	no	—
101	C₃×D₄	C₂²×C₄	1	4	yes	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	4	2	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	—	4	8	no	—
110	C₂×C₃×C₄	SD₁₆	1	4	yes	direct product factor
111	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
112	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
113	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
115	C₂×C₃×C₄	#41	1	2	yes	—
116	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
117	C₃×C₈	—	2	2	no	—
118	C₃×C₈	C₂²×C₄	1	4	yes	complemented
119	C₂⁴	—	2	1	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	2	2	no	—
122	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
123	C₂×Q₈	C₂³×C₃	1	2	yes	—
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
127	C₂×D₄	C₂³×C₃	1	2	yes	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	4	4	no	—
133	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
134	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	SD₁₆	C₂×C₃×C₄	1	16	yes	direct product factor
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	C₂×C₃×C₄	1	4	yes	complemented
139	C₂×C₈	C₂³×C₃	1	2	yes	—
140	C₂×C₈	C₂³×C₃	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	C₂³×C₃	1	2	yes	—
145	C₄⋊C₄	C₂³×C₃	1	2	yes	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₄²	—	2	4	no	—
148	C₄²	C₃×D₄	1	2	yes	—
149	C₄²	C₂³×C₃	1	2	yes	—
150	C₂²×C₃	—	2	4	no	—
151	C₂²×C₃	—	4	2	no	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	—	4	8	no	—
154	C₂²×C₃	C₄×D₄	1	2	yes	—
155	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
156	C₂²×C₃	#135	1	2	yes	—
157	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
158	C₃×C₄	—	2	4	no	—
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	4	2	no	—
161	C₃×C₄	—	4	2	no	—
162	C₃×C₄	C₄×D₄	1	2	yes	—
163	C₃×C₄	C₂×SD₁₆	1	4	yes	direct product factor
164	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
165	C₃×C₄	C₂×#41	1	1	yes	characteristic
166	C₂³	—	2	4	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	4	2	no	—
169	C₂³	—	4	2	no	—
170	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
171	Q₈	—	2	4	no	—
172	Q₈	—	2	2	no	—
173	Q₈	C₂²×C₃×C₄	1	4	yes	complemented
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	D₄	C₂²×C₃×C₄	1	4	yes	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	C₃×SD₁₆	1	4	yes	direct product factor
186	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
187	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
188	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
190	C₂×C₄	C₃×#41	1	2	yes	—
191	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₂²×C₃×C₄	1	4	yes	complemented
194	C₂×C₃	—	4	4	no	—
195	C₂×C₃	C₄×SD₁₆	1	4	yes	direct product factor
196	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂×C₃	C₂²×SD₁₆	1	1	yes	characteristic
198	C₂×C₃	C₂×#135	1	1	yes	characteristic
199	C₂²	—	2	4	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	8	no	—
203	C₂²	C₃×C₄×D₄	1	2	yes	—
204	C₂²	C₂×C₃×SD₁₆	1	2	yes	—
205	C₂²	C₃×#135	1	2	yes	—
206	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	C₃×C₄×D₄	1	2	yes	—
212	C₄	C₂×C₃×SD₁₆	1	4	yes	direct product factor
213	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
214	C₄	C₂×C₃×#41	1	1	yes	characteristic
215	C₃	C₂×C₄×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
216	C₂	—	4	4	no	—
217	C₂	C₃×C₄×SD₁₆	1	4	yes	direct product factor
218	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
219	C₂	C₂²×C₃×SD₁₆	1	1	yes	characteristic
220	C₂	C₂×C₃×#135	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×Q₈	C₂	semidirect
4. C₃×C₄×SD₁₆	C₂	direct (N × H)
5. C₂×C₃×#106	C₂	semidirect
6. C₂×C₃×#103	C₂	semidirect
8. C₂×C₃×C₄×C₈	C₂	semidirect
9. C₂×C₄×SD₁₆	C₃	direct (N × H)
13. C₂×C₃×SD₁₆	C₄	direct (N × H)
18. C₃×C₄×Q₈	C₂²	semidirect
27. C₃×#106	C₂²	semidirect
28. C₃×#103	C₂²	semidirect
30. C₃×C₄×C₈	C₂²	semidirect
32. C₂×C₄×Q₈	C₂×C₃	semidirect
34. C₄×SD₁₆	C₂×C₃	direct (N × H)
35. C₂×#106	C₂×C₃	semidirect
36. C₂×#103	C₂×C₃	semidirect
38. C₂×C₄×C₈	C₂×C₃	semidirect
42. C₂×C₃×Q₈	C₂×C₄	semidirect
56. C₃×SD₁₆	C₂×C₄	direct (N × H)
58. C₂×C₃×C₈	C₂×C₄	semidirect
73. C₂×SD₁₆	C₃×C₄	direct (N × H)
78. C₄×Q₈	C₂²×C₃	semidirect
87. #106	C₂²×C₃	semidirect
88. #103	C₂²×C₃	semidirect
90. C₄×C₈	C₂²×C₃	semidirect
98. C₃×Q₈	C₂²×C₄	semidirect
110. C₂×C₃×C₄	SD₁₆	direct (N × H)
118. C₃×C₈	C₂²×C₄	semidirect
122. C₂×Q₈	C₂×C₃×C₄	semidirect
136. SD₁₆	C₂×C₃×C₄	direct (N × H)
138. C₂×C₈	C₂×C₃×C₄	semidirect
163. C₃×C₄	C₂×SD₁₆	direct (N × H)
173. Q₈	C₂²×C₃×C₄	semidirect
185. C₂×C₄	C₃×SD₁₆	direct (N × H)
193. C₈	C₂²×C₃×C₄	semidirect
195. C₂×C₃	C₄×SD₁₆	direct (N × H)
212. C₄	C₂×C₃×SD₁₆	direct (N × H)
215. C₃	C₂×C₄×SD₁₆	direct (N × H)
217. C₂	C₃×C₄×SD₁₆	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×SD₁₆	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄×D₄	2
C₃×C₄×SD₁₆	2
C₂×C₃×#106	2
C₂×C₃×#103	2
C₂×C₃×#102	2
C₂×C₃×C₄×C₈	2
C₂×C₄×SD₁₆	3

Subgroups — C2×C3×C4×SD16 #84801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×SD₁₆ #84801