Subgroups — C₃×#3273 #85137

Order 384.

520 conjugacy classes of subgroups.

The subgroup lattice has 250 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×#546	C₂	1	2	yes	maximal, complemented
3	C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#468	C₂	1	2	yes	maximal, complemented
5	C₃×#463	C₂	1	2	yes	maximal, complemented
6	C₃×#452	C₂	1	2	yes	maximal, complemented
7	C₃×#447	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
10	#3273	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#41	C₂²	1	2	yes	—
12	C₂²×C₃×D₄	C₂²	1	2	yes	complemented
13	C₃×AGL(1,ℤ/8)	—	2	8	no	—
14	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
16	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
17	C₂×C₃×D₈	C₂²	1	2	yes	—
18	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×M₁₆	C₂²	1	2	yes	complemented
21	C₃×#124	C₂²	1	2	yes	complemented
22	C₃×#123	—	2	2	no	—
23	C₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
24	C₃×#121	—	2	2	no	—
25	C₃×#121	C₂²	1	2	yes	—
26	C₃×#120	—	2	2	no	—
27	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
28	C₃×C₄×D₄	C₂²	1	2	yes	complemented
29	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
30	C₃×#107	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₃×#103	C₂²	1	2	yes	complemented
33	C₃×#102	C₂²	1	2	yes	complemented
34	C₃×#102	C₂²	1	2	yes	complemented
35	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
36	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
37	C₂×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
38	#546	C₂×C₃	1	2	yes	complemented
39	#495	C₂×C₃	1	1	yes	characteristic, complemented
40	#468	C₂×C₃	1	2	yes	complemented
41	#463	C₂×C₃	1	2	yes	complemented
42	#452	C₂×C₃	1	2	yes	complemented
43	#447	C₂×C₃	1	2	yes	complemented
44	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
45	#436	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂⁴×C₃	—	2	2	no	—
47	C₃×#41	—	2	4	no	—
48	C₃×#41	—	4	2	no	—
49	C₂×C₃×Q₈	C₂³	1	2	yes	—
50	C₂×C₃×D₄	—	2	4	no	—
51	C₂×C₃×D₄	—	2	2	no	—
52	C₂×C₃×D₄	—	2	2	no	—
53	C₂×C₃×D₄	—	2	2	no	—
54	C₂×C₃×D₄	—	4	2	no	—
55	C₂×C₃×D₄	D₄	1	1	yes	characteristic
56	C₂×C₃×D₄	C₂³	1	2	yes	—
57	C₂×C₃×D₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	C₂³	1	2	yes	—
61	C₃×Q₁₆	—	4	1	no	—
62	C₃×SD₁₆	—	2	4	no	—
63	C₃×SD₁₆	—	2	4	no	—
64	C₃×D₈	—	2	4	no	—
65	C₃×D₈	—	4	1	no	—
66	C₃×D₈	D₄	1	4	yes	complemented
67	C₃×M₁₆	—	2	4	no	—
68	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂³	1	2	yes	—
70	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
73	C₃×C₄⋊C₄	C₂³	1	2	yes	—
74	C₃×C₂²⋊C₄	—	2	2	no	—
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₂²⋊C₄	—	2	2	no	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	2	2	no	—
79	C₃×C₂²⋊C₄	D₄	1	2	yes	—
80	C₃×C₄²	C₂³	1	1	yes	characteristic
81	C₂×#41	C₂²×C₃	1	2	yes	—
82	C₂²×D₄	C₂²×C₃	1	2	yes	complemented
83	AGL(1,ℤ/8)	—	2	8	no	—
84	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
85	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
86	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
87	C₂×D₈	C₂²×C₃	1	2	yes	—
88	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
89	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
90	C₂×M₁₆	C₂²×C₃	1	2	yes	complemented
91	#124	C₂²×C₃	1	2	yes	complemented
92	#123	—	2	2	no	—
93	C₂²⋊Q₈	C₂²×C₃	1	2	yes	complemented
94	#121	—	2	2	no	—
95	#121	C₂²×C₃	1	2	yes	—
96	#120	—	2	2	no	—
97	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
98	C₄×D₄	C₂²×C₃	1	2	yes	complemented
99	C₂×C₂²⋊C₄	—	2	2	no	—
100	#107	C₂²×C₃	1	1	yes	characteristic, complemented
101	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
102	#103	C₂²×C₃	1	2	yes	complemented
103	#102	C₂²×C₃	1	2	yes	complemented
104	#102	C₂²×C₃	1	2	yes	complemented
105	C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
106	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
107	C₂³×C₃	—	2	2	no	—
108	C₂³×C₃	—	2	2	no	—
109	C₂³×C₃	—	4	4	no	—
110	C₂³×C₃	—	4	4	no	—
111	C₂³×C₃	—	8	2	no	—
112	C₂³×C₃	C₂×D₄	1	2	yes	—
113	C₃×Q₈	—	2	2	no	—
114	C₃×Q₈	—	2	2	no	—
115	C₃×D₄	—	2	2	no	—
116	C₃×D₄	—	2	2	no	—
117	C₃×D₄	—	2	2	no	—
118	C₃×D₄	—	4	1	no	—
119	C₃×D₄	—	4	4	no	—
120	C₃×D₄	—	4	4	no	—
121	C₃×D₄	—	8	2	no	—
122	C₃×D₄	C₂×D₄	1	4	yes	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	4	4	no	—
128	C₂×C₃×C₄	—	8	2	no	—
129	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
133	C₃×C₈	—	2	2	no	—
134	C₃×C₈	—	2	1	no	—
135	C₃×C₈	C₂×D₄	1	2	yes	—
136	C₂⁴	—	2	2	no	—
137	#41	—	2	4	no	—
138	#41	—	4	2	no	—
139	C₂×Q₈	C₂³×C₃	1	2	yes	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	C₃×D₄	1	1	yes	characteristic
146	C₂×D₄	C₂³×C₃	1	2	yes	—
147	C₂×D₄	C₂³×C₃	1	2	yes	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂³×C₃	1	2	yes	—
151	Q₁₆	—	4	1	no	—
152	SD₁₆	—	2	4	no	—
153	SD₁₆	—	2	4	no	—
154	D₈	—	2	4	no	—
155	D₈	—	4	1	no	—
156	D₈	C₃×D₄	1	4	yes	complemented
157	M₁₆	—	2	4	no	—
158	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₃	1	2	yes	—
160	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
163	C₄⋊C₄	C₂³×C₃	1	2	yes	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	C₃×D₄	1	2	yes	—
170	C₄²	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	4	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	4	2	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	—	4	2	no	—
179	C₂²×C₃	—	4	2	no	—
180	C₂²×C₃	—	8	8	no	—
181	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
188	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	4	4	no	—
192	C₂³	—	4	4	no	—
193	C₂³	—	8	2	no	—
194	C₂³	C₂×C₃×D₄	1	2	yes	—
195	Q₈	—	2	2	no	—
196	Q₈	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	1	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	8	2	no	—
204	D₄	C₂×C₃×D₄	1	4	yes	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	C₂×C₃×D₄	1	2	yes	—
218	C₂×C₃	—	4	2	no	—
219	C₂×C₃	—	4	4	no	—
220	C₂×C₃	—	8	2	no	—
221	C₂×C₃	D₄²	1	1	yes	characteristic
222	C₂×C₃	#575	1	1	yes	characteristic
223	C₂×C₃	#576	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	8	8	no	—
234	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
241	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
242	C₃	#3273	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	4	2	no	—
244	C₂	—	4	4	no	—
245	C₂	—	8	2	no	—
246	C₂	C₃×D₄²	1	1	yes	characteristic
247	C₂	C₃×#575	1	1	yes	characteristic
248	C₂	C₃×#576	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×#546	C₂	semidirect
3. C₃×#495	C₂	semidirect
4. C₃×#468	C₂	semidirect
5. C₃×#463	C₂	semidirect
6. C₃×#452	C₂	semidirect
7. C₃×#447	C₂	semidirect
8. C₃×C₄×D₈	C₂	semidirect
9. C₃×#436	C₂	semidirect
10. #3273	C₃	direct (N × H)
12. C₂²×C₃×D₄	C₂²	semidirect
14. C₂×C₃×Q₁₆	C₂²	semidirect
15. C₂×C₃×SD₁₆	C₂²	semidirect
16. C₂×C₃×SD₁₆	C₂²	semidirect
20. C₂×C₃×M₁₆	C₂²	semidirect
21. C₃×#124	C₂²	semidirect
23. C₃×C₂²⋊Q₈	C₂²	semidirect
28. C₃×C₄×D₄	C₂²	semidirect
30. C₃×#107	C₂²	semidirect
31. C₃×C₄⋊C₈	C₂²	semidirect
32. C₃×#103	C₂²	semidirect
33. C₃×#102	C₂²	semidirect
34. C₃×#102	C₂²	semidirect
35. C₃×C₂²⋊C₈	C₂²	semidirect
36. C₃×C₄×C₈	C₂²	semidirect
37. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
38. #546	C₂×C₃	semidirect
39. #495	C₂×C₃	semidirect
40. #468	C₂×C₃	semidirect
41. #463	C₂×C₃	semidirect
42. #452	C₂×C₃	semidirect
43. #447	C₂×C₃	semidirect
44. C₄×D₈	C₂×C₃	semidirect
45. #436	C₂×C₃	semidirect
66. C₃×D₈	D₄	semidirect
82. C₂²×D₄	C₂²×C₃	semidirect
84. C₂×Q₁₆	C₂²×C₃	semidirect
85. C₂×SD₁₆	C₂²×C₃	semidirect
86. C₂×SD₁₆	C₂²×C₃	semidirect
90. C₂×M₁₆	C₂²×C₃	semidirect
91. #124	C₂²×C₃	semidirect
93. C₂²⋊Q₈	C₂²×C₃	semidirect
98. C₄×D₄	C₂²×C₃	semidirect
100. #107	C₂²×C₃	semidirect
101. C₄⋊C₈	C₂²×C₃	semidirect
102. #103	C₂²×C₃	semidirect
103. #102	C₂²×C₃	semidirect
104. #102	C₂²×C₃	semidirect
105. C₂²⋊C₈	C₂²×C₃	semidirect
106. C₄×C₈	C₂²×C₃	semidirect
156. D₈	C₃×D₄	semidirect
242. C₃	#3273	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×#546	2
C₃×#495	2
C₃×#468	2
C₃×#463	2
C₃×#452	2
C₃×#447	2
C₃×C₄×D₈	2
C₃×#436	2
#3273	3

Subgroups — C3×#3273 #85137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3273 #85137