Subgroups — C₃²×C₈×SD₁₆ #11768625

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃²×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₈×SD₁₆	C₃	1	4	yes	maximal, direct product factor
9	C₂×C₃²×SD₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₃²×C₈	—	2	1	no	—
11	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
13	C₃²×#106	C₄	1	1	yes	characteristic
14	C₃²×C₄⋊C₈	—	2	1	no	—
15	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃²×#103	C₄	1	1	yes	characteristic
18	C₃²×#102	C₄	1	1	yes	characteristic
19	C₃²×C₂²⋊C₈	—	2	1	no	—
20	C₃²×C₄×C₈	—	2	1	no	—
21	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×C₈×Q₈	C₂×C₃	1	4	yes	complemented
25	C₃×C₄×SD₁₆	C₂×C₃	1	4	yes	—
26	C₃×C₈×D₄	C₂×C₃	1	4	yes	—
27	C₃×#334	C₂×C₃	1	4	yes	complemented
28	C₃×Q₈⋊C₈	C₂×C₃	1	4	yes	complemented
29	C₃×D₄⋊C₈	C₂×C₃	1	4	yes	—
30	C₃×C₈²	C₂×C₃	1	4	yes	complemented
31	C₂×C₃²×Q₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	—	2	1	no	—
34	C₃²×SD₁₆	C₈	1	4	yes	direct product factor
35	C₂×C₃²×C₈	—	2	1	no	—
36	C₂×C₃²×C₈	—	2	1	no	—
37	C₂×C₃²×C₈	—	2	1	no	—
38	C₂×C₃²×C₈	—	4	2	no	—
39	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
42	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
43	C₃²×C₄⋊C₄	—	2	1	no	—
44	C₃²×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₃²×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₃²×C₂²⋊C₄	—	2	1	no	—
47	C₃²×C₄²	—	2	1	no	—
48	C₃²×C₄²	C₂³	1	1	yes	characteristic
49	C₈×SD₁₆	C₃²	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₃×SD₁₆	C₃×C₄	1	4	yes	—
51	C₂²×C₃×C₈	—	2	4	no	—
52	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
53	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
54	C₃×#106	C₃×C₄	1	4	yes	—
55	C₃×C₄⋊C₈	—	2	4	no	—
56	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
57	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
58	C₃×#103	C₃×C₄	1	4	yes	—
59	C₃×#102	C₃×C₄	1	4	yes	—
60	C₃×C₂²⋊C₈	—	2	4	no	—
61	C₃×C₄×C₈	—	2	4	no	—
62	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
63	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
64	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
65	C₂³×C₃²	—	2	1	no	—
66	C₃²×Q₈	—	2	1	no	—
67	C₃²×Q₈	C₂×C₈	1	2	yes	complemented
68	C₃²×D₄	—	2	1	no	—
69	C₃²×D₄	C₂×C₈	1	2	yes	—
70	C₂×C₃²×C₄	—	2	1	no	—
71	C₂×C₃²×C₄	—	2	1	no	—
72	C₂×C₃²×C₄	—	2	1	no	—
73	C₂×C₃²×C₄	—	4	1	no	—
74	C₂×C₃²×C₄	—	4	1	no	—
75	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	#41	1	1	yes	characteristic
78	C₃²×C₈	—	2	2	no	—
79	C₃²×C₈	—	2	1	no	—
80	C₃²×C₈	—	2	1	no	—
81	C₃²×C₈	—	4	1	no	—
82	C₃²×C₈	—	4	1	no	—
83	C₃²×C₈	C₂×C₈	1	2	yes	complemented
84	C₃²×C₈	SD₁₆	1	2	yes	direct product factor
85	C₈×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
86	C₄×SD₁₆	C₂×C₃²	1	1	yes	characteristic
87	C₈×D₄	C₂×C₃²	1	1	yes	characteristic
88	#334	C₂×C₃²	1	1	yes	characteristic, complemented
89	Q₈⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
90	D₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
91	C₈²	C₂×C₃²	1	1	yes	characteristic, complemented
92	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
93	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₃×SD₁₆	C₃×C₈	1	16	yes	direct product factor
96	C₂×C₃×C₈	—	2	4	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	2	4	no	—
99	C₂×C₃×C₈	—	4	8	no	—
100	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
101	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
102	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
103	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
104	C₃×C₄⋊C₄	—	2	4	no	—
105	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
106	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
107	C₃×C₂²⋊C₄	—	2	4	no	—
108	C₃×C₄²	—	2	4	no	—
109	C₃×C₄²	C₂³×C₃	1	4	yes	—
110	C₂²×C₃²	—	2	2	no	—
111	C₂²×C₃²	—	4	1	no	—
112	C₂²×C₃²	—	4	1	no	—
113	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
114	C₃²×C₄	—	2	2	no	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	—	4	1	no	—
117	C₃²×C₄	—	4	1	no	—
118	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
119	C₃²×C₄	#131	1	1	yes	characteristic
120	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
121	C₃²×C₄	#135	1	1	yes	characteristic
122	C₂×SD₁₆	C₃²×C₄	1	1	yes	characteristic
123	C₂²×C₈	—	2	1	no	—
124	C₄×Q₈	C₂²×C₃²	1	1	yes	characteristic
125	C₄×D₄	C₂²×C₃²	1	1	yes	characteristic
126	#106	C₃²×C₄	1	1	yes	characteristic
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
129	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
130	#103	C₃²×C₄	1	1	yes	characteristic
131	#102	C₃²×C₄	1	1	yes	characteristic
132	C₂²⋊C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
136	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
137	C₂³×C₃	—	2	4	no	—
138	C₃×Q₈	—	2	4	no	—
139	C₃×Q₈	C₂×C₃×C₈	1	8	yes	complemented
140	C₃×D₄	—	2	4	no	—
141	C₃×D₄	C₂×C₃×C₈	1	8	yes	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	—	4	4	no	—
147	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
148	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
149	C₂×C₃×C₄	C₃×#41	1	4	yes	—
150	C₃×C₈	—	2	8	no	—
151	C₃×C₈	—	2	4	no	—
152	C₃×C₈	—	2	4	no	—
153	C₃×C₈	—	4	4	no	—
154	C₃×C₈	—	4	4	no	—
155	C₃×C₈	C₂×C₃×C₈	1	8	yes	complemented
156	C₃×C₈	C₃×SD₁₆	1	8	yes	direct product factor
157	C₂×C₃²	—	4	2	no	—
158	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
159	C₂×C₃²	C₄×SD₁₆	1	1	yes	characteristic
160	C₂×C₃²	#443	1	1	yes	characteristic
161	C₂×Q₈	C₂×C₃²×C₄	1	1	yes	characteristic
162	C₂×D₄	C₂×C₃²×C₄	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	SD₁₆	C₃²×C₈	1	4	yes	direct product factor
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
171	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
172	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	C₂×C₃²×C₄	1	1	yes	characteristic
175	C₄⋊C₄	C₂×C₃²×C₄	1	1	yes	characteristic
176	C₂²⋊C₄	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	C₂³×C₃²	1	1	yes	characteristic
179	C₂²×C₃	—	2	8	no	—
180	C₂²×C₃	—	4	4	no	—
181	C₂²×C₃	—	4	4	no	—
182	C₂²×C₃	C₃×C₄×D₄	1	4	yes	—
183	C₃×C₄	—	2	8	no	—
184	C₃×C₄	—	2	4	no	—
185	C₃×C₄	—	4	4	no	—
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
188	C₃×C₄	C₃×#131	1	4	yes	—
189	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
190	C₃×C₄	C₃×#135	1	4	yes	—
191	C₃²	C₈×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	—	2	1	no	—
193	Q₈	—	2	1	no	—
194	Q₈	C₂×C₃²×C₈	1	2	yes	complemented
195	D₄	—	2	1	no	—
196	D₄	C₂×C₃²×C₈	1	2	yes	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
204	C₂×C₄	C₃²×#41	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	1	no	—
209	C₈	—	4	1	no	—
210	C₈	C₂×C₃²×C₈	1	2	yes	complemented
211	C₈	C₃²×SD₁₆	1	2	yes	direct product factor
212	C₂×C₃	—	4	8	no	—
213	C₂×C₃	C₃×C₈×D₄	1	4	yes	—
214	C₂×C₃	C₃×C₄×SD₁₆	1	4	yes	—
215	C₂×C₃	C₃×#443	1	4	yes	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	4	1	no	—
219	C₂²	C₃²×C₄×D₄	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	1	no	—
224	C₄	C₂²×C₃²×C₈	1	1	yes	characteristic
225	C₄	C₃²×#131	1	1	yes	characteristic
226	C₄	C₂×C₃²×SD₁₆	1	1	yes	characteristic
227	C₄	C₃²×#135	1	1	yes	characteristic
228	C₃	C₃×C₈×SD₁₆	1	4	yes	direct product factor
229	C₂	—	4	2	no	—
230	C₂	C₃²×C₈×D₄	1	1	yes	characteristic
231	C₂	C₃²×C₄×SD₁₆	1	1	yes	characteristic
232	C₂	C₃²×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₈	D₄
commutator	C₄	C₂²×C₃²×C₈
Frattini, cosocle	C₄²	C₂³×C₃²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃²×D₄	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₈ ◁ C₃²×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×C₈×Q₈	C₂	semidirect
4. C₃²×#334	C₂	semidirect
5. C₃²×Q₈⋊C₈	C₂	semidirect
7. C₃²×C₈²	C₂	semidirect
8. C₃×C₈×SD₁₆	C₃	direct (N × H)
24. C₃×C₈×Q₈	C₂×C₃	semidirect
27. C₃×#334	C₂×C₃	semidirect
28. C₃×Q₈⋊C₈	C₂×C₃	semidirect
30. C₃×C₈²	C₂×C₃	semidirect
34. C₃²×SD₁₆	C₈	direct (N × H)
49. C₈×SD₁₆	C₃²	direct (N × H)
67. C₃²×Q₈	C₂×C₈	semidirect
83. C₃²×C₈	C₂×C₈	semidirect
84. C₃²×C₈	SD₁₆	direct (N × H)
85. C₈×Q₈	C₂×C₃²	semidirect
88. #334	C₂×C₃²	semidirect
89. Q₈⋊C₈	C₂×C₃²	semidirect
91. C₈²	C₂×C₃²	semidirect
95. C₃×SD₁₆	C₃×C₈	direct (N × H)
139. C₃×Q₈	C₂×C₃×C₈	semidirect
155. C₃×C₈	C₂×C₃×C₈	semidirect
156. C₃×C₈	C₃×SD₁₆	direct (N × H)
164. SD₁₆	C₃²×C₈	direct (N × H)
191. C₃²	C₈×SD₁₆	direct (N × H)
194. Q₈	C₂×C₃²×C₈	semidirect
210. C₈	C₂×C₃²×C₈	semidirect
211. C₈	C₃²×SD₁₆	direct (N × H)
228. C₃	C₃×C₈×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
C₃²×C₈×Q₈	2
C₃²×C₄×SD₁₆	2
C₃²×C₈×D₄	2
C₃²×#334	2
C₃²×Q₈⋊C₈	2
C₃²×D₄⋊C₈	2
C₃²×C₈²	2
C₃×C₈×SD₁₆	3

Subgroups — C32×C8×SD16 #11768625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₈×SD₁₆ #11768625