Subgroups — C₃×#3272 #85136

Order 384.

546 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#468	C₂	1	2	yes	maximal, complemented
6	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#450	C₂	1	2	yes	maximal, complemented
9	C₃×#449	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	#3272	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	complemented
14	C₂²×C₃×D₄	—	2	1	no	—
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₃×AGL(1,ℤ/8)	—	2	8	no	—
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
20	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×D₈	C₂²	1	2	yes	complemented
22	C₂×C₃×M₁₆	C₂²	1	2	yes	complemented
23	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₃×#127	C₂²	1	1	yes	characteristic
25	C₃×#123	—	2	2	no	—
26	C₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₃×#121	—	2	1	no	—
28	C₃×#121	—	2	1	no	—
29	C₃×#121	C₂²	1	2	yes	—
30	C₃×#120	—	2	2	no	—
31	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
32	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
33	C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₃×C₄⋊C₄	—	2	1	no	—
35	C₃×#107	C₂²	1	1	yes	characteristic, complemented
36	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₃×#103	C₂²	1	2	yes	complemented
38	C₃×#102	C₂²	1	1	yes	characteristic
39	C₃×#102	C₂²	1	1	yes	characteristic, complemented
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	C₂×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
44	#547	C₂×C₃	1	1	yes	characteristic, complemented
45	D₄²	C₂×C₃	1	1	yes	characteristic, complemented
46	#492	C₂×C₃	1	1	yes	characteristic, complemented
47	#468	C₂×C₃	1	2	yes	complemented
48	#461	C₂×C₃	1	1	yes	characteristic, complemented
49	#459	C₂×C₃	1	1	yes	characteristic, complemented
50	#450	C₂×C₃	1	2	yes	complemented
51	#449	C₂×C₃	1	2	yes	complemented
52	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
53	#436	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂⁴×C₃	—	2	2	no	—
55	C₃×#41	—	2	4	no	—
56	C₃×#41	—	4	2	no	—
57	C₂×C₃×Q₈	C₂³	1	2	yes	—
58	C₂×C₃×D₄	—	2	4	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	2	2	no	—
62	C₂×C₃×D₄	—	2	1	no	—
63	C₂×C₃×D₄	—	2	1	no	—
64	C₂×C₃×D₄	—	4	2	no	—
65	C₂×C₃×D₄	—	4	4	no	—
66	C₂×C₃×D₄	D₄	1	1	yes	characteristic
67	C₂×C₃×D₄	C₂³	1	2	yes	—
68	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
69	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	—	2	2	no	—
72	C₂²×C₃×C₄	—	2	1	no	—
73	C₂²×C₃×C₄	C₂³	1	2	yes	—
74	C₃×SD₁₆	—	2	4	no	—
75	C₃×SD₁₆	—	2	4	no	—
76	C₃×SD₁₆	—	4	2	no	—
77	C₃×D₈	—	2	4	no	—
78	C₃×D₈	D₄	1	4	yes	complemented
79	C₃×M₁₆	—	2	4	no	—
80	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂³	1	2	yes	—
82	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	—	2	2	no	—
85	C₃×C₄⋊C₄	—	2	1	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	1	no	—
92	C₃×C₂²⋊C₄	—	2	2	no	—
93	C₃×C₂²⋊C₄	D₄	1	2	yes	—
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	C₂×#41	C₂²×C₃	1	2	yes	complemented
96	C₂²×D₄	—	2	1	no	—
97	C₂²×D₄	C₂²×C₃	1	2	yes	—
98	AGL(1,ℤ/8)	—	2	8	no	—
99	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
100	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
101	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
102	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
103	C₂×D₈	C₂²×C₃	1	2	yes	complemented
104	C₂×M₁₆	C₂²×C₃	1	2	yes	complemented
105	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
106	#127	C₂²×C₃	1	1	yes	characteristic
107	#123	—	2	2	no	—
108	C₂²⋊Q₈	C₂²×C₃	1	2	yes	complemented
109	#121	—	2	1	no	—
110	#121	—	2	1	no	—
111	#121	C₂²×C₃	1	2	yes	—
112	#120	—	2	2	no	—
113	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
114	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
115	C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
116	C₂×C₄⋊C₄	—	2	1	no	—
117	#107	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
119	#103	C₂²×C₃	1	2	yes	complemented
120	#102	C₂²×C₃	1	1	yes	characteristic
121	#102	C₂²×C₃	1	1	yes	characteristic, complemented
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	1	no	—
127	C₂³×C₃	—	2	2	no	—
128	C₂³×C₃	—	4	4	no	—
129	C₂³×C₃	—	4	4	no	—
130	C₂³×C₃	—	8	2	no	—
131	C₂³×C₃	C₂×D₄	1	2	yes	—
132	C₃×Q₈	—	2	2	no	—
133	C₃×Q₈	—	2	2	no	—
134	C₃×D₄	—	2	2	no	—
135	C₃×D₄	—	2	2	no	—
136	C₃×D₄	—	2	1	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	4	no	—
142	C₃×D₄	—	4	2	no	—
143	C₃×D₄	—	8	1	no	—
144	C₃×D₄	C₂×D₄	1	2	yes	—
145	C₃×D₄	C₂×D₄	1	2	yes	complemented
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	—	4	2	no	—
154	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	C₂×D₄	1	2	yes	complemented
161	C₂⁴	—	2	2	no	—
162	#41	—	2	4	no	—
163	#41	—	4	2	no	—
164	C₂×Q₈	C₂³×C₃	1	2	yes	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	2	2	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	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	C₂³×C₃	1	2	yes	—
181	SD₁₆	—	2	4	no	—
182	SD₁₆	—	2	4	no	—
183	SD₁₆	—	4	2	no	—
184	D₈	—	2	4	no	—
185	D₈	C₃×D₄	1	4	yes	complemented
186	M₁₆	—	2	4	no	—
187	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
188	C₂×C₈	C₂³×C₃	1	2	yes	—
189	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
195	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	C₃×D₄	1	2	yes	—
201	C₄²	C₂³×C₃	1	1	yes	characteristic
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	—	4	1	no	—
208	C₂²×C₃	—	4	1	no	—
209	C₂²×C₃	—	4	1	no	—
210	C₂²×C₃	—	4	1	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	—	4	2	no	—
214	C₂²×C₃	—	8	8	no	—
215	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	4	2	no	—
219	C₃×C₄	—	4	1	no	—
220	C₃×C₄	—	4	1	no	—
221	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	complemented
222	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
223	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	8	2	no	—
230	C₂³	C₂×C₃×D₄	1	2	yes	—
231	Q₈	—	2	2	no	—
232	Q₈	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	2	no	—
235	D₄	—	2	1	no	—
236	D₄	—	2	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	4	no	—
239	D₄	—	4	4	no	—
240	D₄	—	4	4	no	—
241	D₄	—	4	2	no	—
242	D₄	—	8	1	no	—
243	D₄	C₂×C₃×D₄	1	2	yes	—
244	D₄	C₂×C₃×D₄	1	2	yes	complemented
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
255	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
256	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	1	no	—
259	C₈	C₂×C₃×D₄	1	2	yes	complemented
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	—	4	2	no	—
262	C₂×C₃	—	4	2	no	—
263	C₂×C₃	—	8	2	no	—
264	C₂×C₃	D₄²	1	1	yes	characteristic
265	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₂×C₃	#577	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	8	8	no	—
280	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	2	no	—
286	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	complemented
287	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
288	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
289	C₃	#3272	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	4	2	no	—
291	C₂	—	4	2	no	—
292	C₂	—	4	2	no	—
293	C₂	—	8	2	no	—
294	C₂	C₃×D₄²	1	1	yes	characteristic
295	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
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. C₂×C₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₃×#547	C₂	semidirect
3. C₃×D₄²	C₂	semidirect
4. C₃×#492	C₂	semidirect
5. C₃×#468	C₂	semidirect
6. C₃×#461	C₂	semidirect
7. C₃×#459	C₂	semidirect
8. C₃×#450	C₂	semidirect
9. C₃×#449	C₂	semidirect
10. C₃×C₄×D₈	C₂	semidirect
11. C₃×#436	C₂	semidirect
12. #3272	C₃	direct (N × H)
13. C₂×C₃×#41	C₂²	semidirect
18. C₂×C₃×SD₁₆	C₂²	semidirect
19. C₂×C₃×SD₁₆	C₂²	semidirect
21. C₂×C₃×D₈	C₂²	semidirect
22. C₂×C₃×M₁₆	C₂²	semidirect
23. C₃×C₄⋊Q₈	C₂²	semidirect
26. C₃×C₂²⋊Q₈	C₂²	semidirect
33. C₃×C₄×D₄	C₂²	semidirect
35. C₃×#107	C₂²	semidirect
36. C₃×C₄⋊C₈	C₂²	semidirect
37. C₃×#103	C₂²	semidirect
39. C₃×#102	C₂²	semidirect
40. C₃×#102	C₂²	semidirect
41. C₃×C₂²⋊C₈	C₂²	semidirect
42. C₃×C₄×C₈	C₂²	semidirect
43. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
44. #547	C₂×C₃	semidirect
45. D₄²	C₂×C₃	semidirect
46. #492	C₂×C₃	semidirect
47. #468	C₂×C₃	semidirect
48. #461	C₂×C₃	semidirect
49. #459	C₂×C₃	semidirect
50. #450	C₂×C₃	semidirect
51. #449	C₂×C₃	semidirect
52. C₄×D₈	C₂×C₃	semidirect
53. #436	C₂×C₃	semidirect
78. C₃×D₈	D₄	semidirect
95. C₂×#41	C₂²×C₃	semidirect
100. C₂×SD₁₆	C₂²×C₃	semidirect
101. C₂×SD₁₆	C₂²×C₃	semidirect
103. C₂×D₈	C₂²×C₃	semidirect
104. C₂×M₁₆	C₂²×C₃	semidirect
105. C₄⋊Q₈	C₂²×C₃	semidirect
108. C₂²⋊Q₈	C₂²×C₃	semidirect
115. C₄×D₄	C₂²×C₃	semidirect
117. #107	C₂²×C₃	semidirect
118. C₄⋊C₈	C₂²×C₃	semidirect
119. #103	C₂²×C₃	semidirect
121. #102	C₂²×C₃	semidirect
122. #102	C₂²×C₃	semidirect
123. C₂²⋊C₈	C₂²×C₃	semidirect
124. C₄×C₈	C₂²×C₃	semidirect
145. C₃×D₄	C₂×D₄	semidirect
160. C₃×C₈	C₂×D₄	semidirect
185. D₈	C₃×D₄	semidirect
221. C₃×C₄	AGL(1,ℤ/8)	semidirect
244. D₄	C₂×C₃×D₄	semidirect
259. C₈	C₂×C₃×D₄	semidirect
286. C₄	C₃×AGL(1,ℤ/8)	semidirect
289. C₃	#3272	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×AGL(1,ℤ/8)	2
C₃×#547	2
C₃×D₄²	2
C₃×#492	2
C₃×#468	2
C₃×#461	2
C₃×#459	2
C₃×#450	2
C₃×#449	2
C₃×C₄×D₈	2
C₃×#436	2
#3272	3

Subgroups — C3×#3272 #85136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3272 #85136