Subgroups — C₃×D₄×SD₁₆ #85145

Order 384.

546 conjugacy classes of subgroups.

The subgroup lattice has 290 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	2	yes	maximal, complemented
2	C₃×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄²	C₂	1	1	yes	maximal, characteristic
4	C₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#465	C₂	1	2	yes	maximal, complemented
6	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#450	C₂	1	2	yes	maximal, complemented
9	C₃×#448	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	D₄×SD₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	2	yes	complemented
14	C₂²×C₃×D₄	—	2	1	no	—
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₂×C₃×SD₁₆	—	2	8	no	—
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
20	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
21	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
22	C₂²×C₃×C₈	C₂²	1	2	yes	complemented
23	C₃×C₄⋊Q₈	—	2	1	no	—
24	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
25	C₃×#127	C₂²	1	1	yes	characteristic
26	C₃×C₂²⋊Q₈	—	2	2	no	—
27	C₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
28	C₃×#121	—	2	1	no	—
29	C₃×#121	C₂²	1	2	yes	—
30	C₃×#120	—	2	2	no	—
31	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
32	C₃×C₄×D₄	—	2	1	no	—
33	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
34	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
35	C₃×#106	C₂²	1	1	yes	characteristic, complemented
36	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₃×#103	C₂²	1	1	yes	characteristic, complemented
38	C₃×#103	C₂²	1	2	yes	complemented
39	C₃×#102	C₂²	1	1	yes	characteristic
40	C₃×#102	C₂²	1	2	yes	complemented
41	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
42	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
43	C₂²×SD₁₆	C₂×C₃	1	2	yes	complemented
44	D₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
45	D₄²	C₂×C₃	1	1	yes	characteristic
46	#492	C₂×C₃	1	1	yes	characteristic, complemented
47	#465	C₂×C₃	1	2	yes	complemented
48	#461	C₂×C₃	1	1	yes	characteristic, complemented
49	#460	C₂×C₃	1	1	yes	characteristic, complemented
50	#450	C₂×C₃	1	2	yes	complemented
51	#448	C₂×C₃	1	2	yes	complemented
52	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
53	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂⁴×C₃	—	2	2	no	—
55	C₂×C₃×Q₈	—	2	2	no	—
56	C₂×C₃×Q₈	—	2	4	no	—
57	C₂×C₃×Q₈	—	4	2	no	—
58	C₂×C₃×Q₈	C₂³	1	2	yes	—
59	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×D₄	—	2	4	no	—
61	C₂×C₃×D₄	—	2	2	no	—
62	C₂×C₃×D₄	—	2	2	no	—
63	C₂×C₃×D₄	—	2	1	no	—
64	C₂×C₃×D₄	—	2	1	no	—
65	C₂×C₃×D₄	—	4	2	no	—
66	C₂×C₃×D₄	—	4	4	no	—
67	C₂×C₃×D₄	D₄	1	1	yes	characteristic
68	C₂×C₃×D₄	C₂³	1	2	yes	—
69	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄	—	2	2	no	—
71	C₂²×C₃×C₄	—	2	1	no	—
72	C₂²×C₃×C₄	C₂³	1	2	yes	—
73	C₃×SD₁₆	—	2	4	no	—
74	C₃×SD₁₆	—	2	4	no	—
75	C₃×SD₁₆	—	2	4	no	—
76	C₃×SD₁₆	—	4	2	no	—
77	C₃×SD₁₆	D₄	1	4	yes	direct product factor
78	C₂×C₃×C₈	—	2	4	no	—
79	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂³	1	2	yes	—
81	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
82	C₃×C₄⋊C₄	—	2	1	no	—
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₃×C₂²⋊C₄	—	2	2	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₂²⋊C₄	—	2	2	no	—
92	C₃×C₂²⋊C₄	D₄	1	2	yes	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	C₂²×Q₈	C₂²×C₃	1	2	yes	complemented
96	C₂²×D₄	—	2	1	no	—
97	C₂²×D₄	C₂²×C₃	1	2	yes	—
98	C₂×SD₁₆	—	2	8	no	—
99	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
100	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
101	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
102	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
103	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
104	C₂²×C₈	C₂²×C₃	1	2	yes	complemented
105	C₄⋊Q₈	—	2	1	no	—
106	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
107	#127	C₂²×C₃	1	1	yes	characteristic
108	C₂²⋊Q₈	—	2	2	no	—
109	C₂²⋊Q₈	C₂²×C₃	1	2	yes	complemented
110	#121	—	2	1	no	—
111	#121	C₂²×C₃	1	2	yes	—
112	#120	—	2	2	no	—
113	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
114	C₄×D₄	—	2	1	no	—
115	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
116	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
117	#106	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
119	#103	C₂²×C₃	1	1	yes	characteristic, complemented
120	#103	C₂²×C₃	1	2	yes	complemented
121	#102	C₂²×C₃	1	1	yes	characteristic
122	#102	C₂²×C₃	1	2	yes	complemented
123	C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
124	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	—	4	4	no	—
128	C₂³×C₃	—	4	4	no	—
129	C₂³×C₃	—	8	2	no	—
130	C₂³×C₃	C₂×D₄	1	2	yes	—
131	C₃×Q₈	—	2	1	no	—
132	C₃×Q₈	—	2	2	no	—
133	C₃×Q₈	—	2	2	no	—
134	C₃×Q₈	—	4	4	no	—
135	C₃×Q₈	C₂×D₄	1	2	yes	complemented
136	C₃×D₄	—	2	2	no	—
137	C₃×D₄	—	2	1	no	—
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	—	4	4	no	—
140	C₃×D₄	—	4	4	no	—
141	C₃×D₄	—	4	2	no	—
142	C₃×D₄	—	8	1	no	—
143	C₃×D₄	SD₁₆	1	4	yes	direct product factor
144	C₃×D₄	C₂×D₄	1	2	yes	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	4	4	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
155	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	C₂×D₄	1	2	yes	complemented
160	C₂⁴	—	2	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	C₂³×C₃	1	2	yes	—
165	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	C₃×D₄	1	1	yes	characteristic
174	C₂×D₄	C₂³×C₃	1	2	yes	—
175	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	C₂³×C₃	1	2	yes	—
179	SD₁₆	—	2	4	no	—
180	SD₁₆	—	2	4	no	—
181	SD₁₆	—	2	4	no	—
182	SD₁₆	—	4	2	no	—
183	SD₁₆	C₃×D₄	1	4	yes	direct product factor
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
186	C₂×C₈	C₂³×C₃	1	2	yes	—
187	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	C₃×D₄	1	2	yes	—
199	C₄²	—	2	1	no	—
200	C₄²	C₂³×C₃	1	1	yes	characteristic
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	—	4	2	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	4	2	no	—
209	C₂²×C₃	—	8	8	no	—
210	C₂²×C₃	C₂×SD₁₆	1	4	yes	complemented
211	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	4	2	no	—
216	C₃×C₄	—	4	1	no	—
217	C₃×C₄	—	4	1	no	—
218	C₃×C₄	C₂×SD₁₆	1	2	yes	complemented
219	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
220	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
221	C₂³	—	2	1	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	8	2	no	—
226	C₂³	C₂×C₃×D₄	1	2	yes	—
227	Q₈	—	2	1	no	—
228	Q₈	—	2	2	no	—
229	Q₈	—	2	2	no	—
230	Q₈	—	4	4	no	—
231	Q₈	C₂×C₃×D₄	1	2	yes	complemented
232	D₄	—	2	2	no	—
233	D₄	—	2	1	no	—
234	D₄	—	2	2	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	2	no	—
238	D₄	—	8	1	no	—
239	D₄	C₃×SD₁₆	1	4	yes	direct product factor
240	D₄	C₂×C₃×D₄	1	2	yes	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
251	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	1	no	—
255	C₈	C₂×C₃×D₄	1	2	yes	complemented
256	C₂×C₃	—	2	4	no	—
257	C₂×C₃	—	4	2	no	—
258	C₂×C₃	—	8	2	no	—
259	C₂×C₃	D₄²	1	1	yes	characteristic
260	C₂×C₃	C₂²×SD₁₆	1	1	yes	characteristic
261	C₂×C₃	#577	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	8	8	no	—
271	C₂²	C₂×C₃×SD₁₆	1	4	yes	complemented
272	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	2	no	—
279	C₄	C₂×C₃×SD₁₆	1	2	yes	complemented
280	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
281	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
282	C₃	D₄×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
283	C₂	—	2	4	no	—
284	C₂	—	4	2	no	—
285	C₂	—	8	2	no	—
286	C₂	C₃×D₄²	1	1	yes	characteristic
287	C₂	C₂²×C₃×SD₁₆	1	1	yes	characteristic
288	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×D₄²	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×SD₁₆	C₂	semidirect
2. C₃×D₄×Q₈	C₂	semidirect
4. C₃×#492	C₂	semidirect
5. C₃×#465	C₂	semidirect
6. C₃×#461	C₂	semidirect
7. C₃×#460	C₂	semidirect
8. C₃×#450	C₂	semidirect
9. C₃×#448	C₂	semidirect
10. C₃×C₄×SD₁₆	C₂	semidirect
11. C₃×C₈×D₄	C₂	semidirect
12. D₄×SD₁₆	C₃	direct (N × H)
13. C₂²×C₃×Q₈	C₂²	semidirect
19. C₂×C₃×SD₁₆	C₂²	semidirect
20. C₂×C₃×SD₁₆	C₂²	semidirect
21. C₂×C₃×SD₁₆	C₂²	semidirect
22. C₂²×C₃×C₈	C₂²	semidirect
24. C₃×C₄⋊Q₈	C₂²	semidirect
27. C₃×C₂²⋊Q₈	C₂²	semidirect
31. C₃×C₄×Q₈	C₂²	semidirect
35. C₃×#106	C₂²	semidirect
36. C₃×C₄⋊C₈	C₂²	semidirect
37. C₃×#103	C₂²	semidirect
38. C₃×#103	C₂²	semidirect
40. C₃×#102	C₂²	semidirect
41. C₃×C₂²⋊C₈	C₂²	semidirect
42. C₃×C₄×C₈	C₂²	semidirect
43. C₂²×SD₁₆	C₂×C₃	semidirect
44. D₄×Q₈	C₂×C₃	semidirect
46. #492	C₂×C₃	semidirect
47. #465	C₂×C₃	semidirect
48. #461	C₂×C₃	semidirect
49. #460	C₂×C₃	semidirect
50. #450	C₂×C₃	semidirect
51. #448	C₂×C₃	semidirect
52. C₄×SD₁₆	C₂×C₃	semidirect
53. C₈×D₄	C₂×C₃	semidirect
77. C₃×SD₁₆	D₄	direct (N × H)
95. C₂²×Q₈	C₂²×C₃	semidirect
101. C₂×SD₁₆	C₂²×C₃	semidirect
102. C₂×SD₁₆	C₂²×C₃	semidirect
103. C₂×SD₁₆	C₂²×C₃	semidirect
104. C₂²×C₈	C₂²×C₃	semidirect
106. C₄⋊Q₈	C₂²×C₃	semidirect
109. C₂²⋊Q₈	C₂²×C₃	semidirect
113. C₄×Q₈	C₂²×C₃	semidirect
117. #106	C₂²×C₃	semidirect
118. C₄⋊C₈	C₂²×C₃	semidirect
119. #103	C₂²×C₃	semidirect
120. #103	C₂²×C₃	semidirect
122. #102	C₂²×C₃	semidirect
123. C₂²⋊C₈	C₂²×C₃	semidirect
124. C₄×C₈	C₂²×C₃	semidirect
135. C₃×Q₈	C₂×D₄	semidirect
143. C₃×D₄	SD₁₆	direct (N × H)
159. C₃×C₈	C₂×D₄	semidirect
183. SD₁₆	C₃×D₄	direct (N × H)
210. C₂²×C₃	C₂×SD₁₆	semidirect
218. C₃×C₄	C₂×SD₁₆	semidirect
231. Q₈	C₂×C₃×D₄	semidirect
239. D₄	C₃×SD₁₆	direct (N × H)
255. C₈	C₂×C₃×D₄	semidirect
271. C₂²	C₂×C₃×SD₁₆	semidirect
279. C₄	C₂×C₃×SD₁₆	semidirect
282. C₃	D₄×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
C₃×D₄×Q₈	2
C₃×D₄²	2
C₃×#492	2
C₃×#465	2
C₃×#461	2
C₃×#460	2
C₃×#450	2
C₃×#448	2
C₃×C₄×SD₁₆	2
C₃×C₈×D₄	2
D₄×SD₁₆	3

Subgroups — C3×D4×SD16 #85145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×SD₁₆ #85145