Subgroups — C₃×#3288 #85152

Order 384.

464 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#577	C₂	1	2	yes	maximal, complemented
2	C₃×#576	C₂	1	2	yes	maximal, complemented
3	C₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#472	C₂	1	2	yes	maximal, complemented
5	C₃×#471	C₂	1	2	yes	maximal, complemented
6	C₃×#455	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#454	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#453	C₂	1	2	yes	maximal, complemented
9	C₃×#444	C₂	1	2	yes	maximal, complemented
10	#3288	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic, complemented
12	C₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
13	C₃×H(5,𝔽₂)	C₂²	1	2	yes	complemented
14	C₃×#137	—	2	2	no	—
15	C₃×#137	C₂²	1	2	yes	complemented
16	C₃×AGL(1,ℤ/8)	—	2	2	no	—
17	C₃×AGL(1,ℤ/8)	—	2	2	no	—
18	C₃×AGL(1,ℤ/8)	—	2	2	no	—
19	C₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
20	C₃×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
21	C₃×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
22	C₃×#135	—	2	2	no	—
23	C₃×#135	—	2	2	no	—
24	C₃×#135	C₂²	1	2	yes	—
25	C₂×C₃×SD₁₆	—	2	2	no	—
26	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
27	C₂×C₃×D₈	—	2	2	no	—
28	C₂×C₃×D₈	C₂²	1	2	yes	complemented
29	C₃×#131	C₂²	1	2	yes	complemented
30	C₃×#131	C₂²	1	2	yes	complemented
31	C₃×#127	C₂²	1	1	yes	characteristic, complemented
32	C₃×#124	C₂²	1	1	yes	characteristic, complemented
33	C₃×#108	C₂²	1	2	yes	complemented
34	C₃×#104	C₂²	1	2	yes	complemented
35	C₃×#104	C₂²	1	2	yes	complemented
36	C₃×#101	C₂²	1	1	yes	characteristic, complemented
37	C₃×#100	C₂²	1	2	yes	complemented
38	C₃×#100	C₂²	1	1	yes	characteristic, complemented
39	C₃×#97	C₂²	1	1	yes	characteristic, complemented
40	#577	C₂×C₃	1	2	yes	complemented
41	#576	C₂×C₃	1	2	yes	complemented
42	#496	C₂×C₃	1	1	yes	characteristic, complemented
43	#472	C₂×C₃	1	2	yes	complemented
44	#471	C₂×C₃	1	2	yes	complemented
45	#455	C₂×C₃	1	1	yes	characteristic, complemented
46	#454	C₂×C₃	1	1	yes	characteristic, complemented
47	#453	C₂×C₃	1	2	yes	complemented
48	#444	C₂×C₃	1	2	yes	complemented
49	C₃×#41	—	2	2	no	—
50	C₃×#41	—	2	2	no	—
51	C₃×#41	—	2	2	no	—
52	C₃×#41	—	2	2	no	—
53	C₃×#41	—	4	1	no	—
54	C₃×#41	C₂³	1	2	yes	—
55	C₃×#41	C₂³	1	2	yes	—
56	C₂×C₃×Q₈	—	2	2	no	—
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	4	1	no	—
62	C₂×C₃×D₄	—	4	1	no	—
63	C₂×C₃×D₄	—	4	2	no	—
64	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
65	C₂×C₃×D₄	C₂³	1	2	yes	—
66	C₃×Q₁₆	—	2	2	no	—
67	C₃×Q₁₆	D₄	1	2	yes	complemented
68	C₃×SD₁₆	—	2	2	no	—
69	C₃×SD₁₆	—	2	2	no	—
70	C₃×SD₁₆	—	2	2	no	—
71	C₃×SD₁₆	—	2	2	no	—
72	C₃×SD₁₆	—	2	2	no	—
73	C₃×SD₁₆	—	2	2	no	—
74	C₃×SD₁₆	D₄	1	2	yes	complemented
75	C₃×SD₁₆	D₄	1	2	yes	complemented
76	C₃×D₈	—	2	2	no	—
77	C₃×D₈	—	2	2	no	—
78	C₃×D₈	—	2	2	no	—
79	C₃×D₈	—	2	2	no	—
80	C₃×D₈	—	2	2	no	—
81	C₃×D₈	D₄	1	2	yes	complemented
82	C₃×M₁₆	—	2	2	no	—
83	C₃×M₁₆	—	2	2	no	—
84	C₃×M₁₆	C₂³	1	2	yes	—
85	C₃×M₁₆	C₂³	1	2	yes	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	C₂³	1	2	yes	—
89	C₃×C₂²⋊C₄	—	4	1	no	—
90	C₃×C₄²	C₂³	1	1	yes	characteristic
91	2₋¹⁺⁴	C₂²×C₃	1	1	yes	characteristic, complemented
92	H(5,𝔽₂)	C₂²×C₃	1	1	yes	characteristic
93	H(5,𝔽₂)	C₂²×C₃	1	2	yes	complemented
94	#137	—	2	2	no	—
95	#137	C₂²×C₃	1	2	yes	complemented
96	AGL(1,ℤ/8)	—	2	2	no	—
97	AGL(1,ℤ/8)	—	2	2	no	—
98	AGL(1,ℤ/8)	—	2	2	no	—
99	AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	—
100	AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	complemented
101	AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	complemented
102	#135	—	2	2	no	—
103	#135	—	2	2	no	—
104	#135	C₂²×C₃	1	2	yes	—
105	C₂×SD₁₆	—	2	2	no	—
106	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
107	C₂×D₈	—	2	2	no	—
108	C₂×D₈	C₂²×C₃	1	2	yes	complemented
109	#131	C₂²×C₃	1	2	yes	complemented
110	#131	C₂²×C₃	1	2	yes	complemented
111	#127	C₂²×C₃	1	1	yes	characteristic, complemented
112	#124	C₂²×C₃	1	1	yes	characteristic, complemented
113	#108	C₂²×C₃	1	2	yes	complemented
114	#104	C₂²×C₃	1	2	yes	complemented
115	#104	C₂²×C₃	1	2	yes	complemented
116	#101	C₂²×C₃	1	1	yes	characteristic, complemented
117	#100	C₂²×C₃	1	2	yes	complemented
118	#100	C₂²×C₃	1	1	yes	characteristic, complemented
119	#97	C₂²×C₃	1	1	yes	characteristic, complemented
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	4	2	no	—
123	C₂³×C₃	—	4	1	no	—
124	C₃×Q₈	—	2	2	no	—
125	C₃×Q₈	—	4	1	no	—
126	C₃×Q₈	C₂×D₄	1	2	yes	—
127	C₃×Q₈	C₂×D₄	1	2	yes	—
128	C₃×D₄	—	2	2	no	—
129	C₃×D₄	—	2	2	no	—
130	C₃×D₄	—	2	2	no	—
131	C₃×D₄	—	2	2	no	—
132	C₃×D₄	—	2	2	no	—
133	C₃×D₄	—	4	2	no	—
134	C₃×D₄	—	4	2	no	—
135	C₃×D₄	—	4	2	no	—
136	C₃×D₄	—	4	1	no	—
137	C₃×D₄	—	8	2	no	—
138	C₃×D₄	C₂×D₄	1	2	yes	—
139	C₃×D₄	C₂×D₄	1	2	yes	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	4	1	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	—	2	2	no	—
150	C₃×C₈	C₂×D₄	1	2	yes	—
151	C₃×C₈	C₂×D₄	1	2	yes	—
152	#41	—	2	2	no	—
153	#41	—	2	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	#41	—	4	1	no	—
157	#41	C₂³×C₃	1	2	yes	—
158	#41	C₂³×C₃	1	2	yes	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	2	no	—
164	C₂×D₄	—	4	1	no	—
165	C₂×D₄	—	4	1	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
168	C₂×D₄	C₂³×C₃	1	2	yes	—
169	Q₁₆	—	2	2	no	—
170	Q₁₆	C₃×D₄	1	2	yes	complemented
171	SD₁₆	—	2	2	no	—
172	SD₁₆	—	2	2	no	—
173	SD₁₆	—	2	2	no	—
174	SD₁₆	—	2	2	no	—
175	SD₁₆	—	2	2	no	—
176	SD₁₆	—	2	2	no	—
177	SD₁₆	C₃×D₄	1	2	yes	complemented
178	SD₁₆	C₃×D₄	1	2	yes	complemented
179	D₈	—	2	2	no	—
180	D₈	—	2	2	no	—
181	D₈	—	2	2	no	—
182	D₈	—	2	2	no	—
183	D₈	—	2	2	no	—
184	D₈	C₃×D₄	1	2	yes	complemented
185	M₁₆	—	2	2	no	—
186	M₁₆	—	2	2	no	—
187	M₁₆	C₂³×C₃	1	2	yes	—
188	M₁₆	C₂³×C₃	1	2	yes	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂³×C₃	1	2	yes	—
192	C₂²⋊C₄	—	4	1	no	—
193	C₄²	C₂³×C₃	1	1	yes	characteristic
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	—	8	2	no	—
199	C₂²×C₃	—	8	1	no	—
200	C₂²×C₃	—	16	2	no	—
201	C₂²×C₃	—	16	1	no	—
202	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	4	1	no	—
206	C₃×C₄	—	4	1	no	—
207	C₃×C₄	C₂²×D₄	1	2	yes	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	4	1	no	—
212	Q₈	—	2	2	no	—
213	Q₈	—	4	1	no	—
214	Q₈	C₂×C₃×D₄	1	2	yes	—
215	Q₈	C₂×C₃×D₄	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	2	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	D₄	—	8	2	no	—
226	D₄	C₂×C₃×D₄	1	2	yes	—
227	D₄	C₂×C₃×D₄	1	2	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	C₂×C₃×D₄	1	2	yes	—
239	C₈	C₂×C₃×D₄	1	2	yes	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	4	2	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	—	8	2	no	—
244	C₂×C₃	—	8	1	no	—
245	C₂×C₃	D₄²	1	1	yes	characteristic
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	1	no	—
252	C₂²	—	16	1	no	—
253	C₂²	—	16	2	no	—
254	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	1	no	—
259	C₄	C₂²×C₃×D₄	1	2	yes	—
260	C₃	#3288	1	1	yes	characteristic, direct product factor, Sylow
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	4	2	no	—
264	C₂	—	8	2	no	—
265	C₂	—	8	1	no	—
266	C₂	C₃×D₄²	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃×#577	C₂	semidirect
2. C₃×#576	C₂	semidirect
3. C₃×#496	C₂	semidirect
4. C₃×#472	C₂	semidirect
5. C₃×#471	C₂	semidirect
6. C₃×#455	C₂	semidirect
7. C₃×#454	C₂	semidirect
8. C₃×#453	C₂	semidirect
9. C₃×#444	C₂	semidirect
10. #3288	C₃	direct (N × H)
11. C₃×2₋¹⁺⁴	C₂²	semidirect
13. C₃×H(5,𝔽₂)	C₂²	semidirect
15. C₃×#137	C₂²	semidirect
20. C₃×AGL(1,ℤ/8)	C₂²	semidirect
21. C₃×AGL(1,ℤ/8)	C₂²	semidirect
26. C₂×C₃×SD₁₆	C₂²	semidirect
28. C₂×C₃×D₈	C₂²	semidirect
29. C₃×#131	C₂²	semidirect
30. C₃×#131	C₂²	semidirect
31. C₃×#127	C₂²	semidirect
32. C₃×#124	C₂²	semidirect
33. C₃×#108	C₂²	semidirect
34. C₃×#104	C₂²	semidirect
35. C₃×#104	C₂²	semidirect
36. C₃×#101	C₂²	semidirect
37. C₃×#100	C₂²	semidirect
38. C₃×#100	C₂²	semidirect
39. C₃×#97	C₂²	semidirect
40. #577	C₂×C₃	semidirect
41. #576	C₂×C₃	semidirect
42. #496	C₂×C₃	semidirect
43. #472	C₂×C₃	semidirect
44. #471	C₂×C₃	semidirect
45. #455	C₂×C₃	semidirect
46. #454	C₂×C₃	semidirect
47. #453	C₂×C₃	semidirect
48. #444	C₂×C₃	semidirect
67. C₃×Q₁₆	D₄	semidirect
74. C₃×SD₁₆	D₄	semidirect
75. C₃×SD₁₆	D₄	semidirect
81. C₃×D₈	D₄	semidirect
91. 2₋¹⁺⁴	C₂²×C₃	semidirect
93. H(5,𝔽₂)	C₂²×C₃	semidirect
95. #137	C₂²×C₃	semidirect
100. AGL(1,ℤ/8)	C₂²×C₃	semidirect
101. AGL(1,ℤ/8)	C₂²×C₃	semidirect
106. C₂×SD₁₆	C₂²×C₃	semidirect
108. C₂×D₈	C₂²×C₃	semidirect
109. #131	C₂²×C₃	semidirect
110. #131	C₂²×C₃	semidirect
111. #127	C₂²×C₃	semidirect
112. #124	C₂²×C₃	semidirect
113. #108	C₂²×C₃	semidirect
114. #104	C₂²×C₃	semidirect
115. #104	C₂²×C₃	semidirect
116. #101	C₂²×C₃	semidirect
117. #100	C₂²×C₃	semidirect
118. #100	C₂²×C₃	semidirect
119. #97	C₂²×C₃	semidirect
170. Q₁₆	C₃×D₄	semidirect
177. SD₁₆	C₃×D₄	semidirect
178. SD₁₆	C₃×D₄	semidirect
184. D₈	C₃×D₄	semidirect
260. C₃	#3288	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#577	2
C₃×#576	2
C₃×#496	2
C₃×#472	2
C₃×#471	2
C₃×#455	2
C₃×#454	2
C₃×#453	2
C₃×#444	2
#3288	3

Subgroups — C3×#3288 #85152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3288 #85152