Subgroups — C₃×#3217 #85081

Order 384.

490 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#496	C₂	1	2	yes	maximal, complemented
4	C₃×#490	C₂	1	2	yes	maximal, complemented
5	C₃×#460	C₂	1	2	yes	maximal, complemented
6	C₃×#459	C₂	1	2	yes	maximal, complemented
7	C₃×#450	C₂	1	2	yes	maximal, complemented
8	C₃×#447	C₂	1	2	yes	maximal, complemented
9	C₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	#3217	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×D₄	—	2	1	no	—
12	C₂²×C₃×D₄	—	2	1	no	—
13	C₂²×C₃×D₄	C₂²	1	2	yes	—
14	C₂×C₃×SD₁₆	C₂²	1	4	yes	complemented
15	C₂×C₃×D₈	C₂²	1	4	yes	complemented
16	C₃×#127	—	2	1	no	—
17	C₃×#127	—	2	1	no	—
18	C₃×#127	C₂²	1	2	yes	—
19	C₃×#127	C₂²	1	2	yes	—
20	C₃×#126	—	2	1	no	—
21	C₃×#125	C₂²	1	1	yes	characteristic, complemented
22	C₃×#124	C₂²	1	1	yes	characteristic, complemented
23	C₃×#123	—	2	1	no	—
24	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃×#121	C₂²	1	1	yes	characteristic
26	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
27	C₃×C₄×D₄	—	2	1	no	—
28	C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
29	C₃×#117	—	2	1	no	—
30	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
32	C₃×#102	C₂²	1	4	yes	complemented
33	C₃×#102	C₂²	1	4	yes	complemented
34	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₃×#97	C₂²	1	2	yes	complemented
36	C₂×#127	C₂×C₃	1	1	yes	characteristic, complemented
37	#529	C₂×C₃	1	1	yes	characteristic, complemented
38	#496	C₂×C₃	1	2	yes	complemented
39	#490	C₂×C₃	1	2	yes	complemented
40	#460	C₂×C₃	1	2	yes	complemented
41	#459	C₂×C₃	1	2	yes	complemented
42	#450	C₂×C₃	1	2	yes	complemented
43	#447	C₂×C₃	1	2	yes	complemented
44	#432	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂⁴×C₃	—	2	2	no	—
46	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
47	C₂×C₃×D₄	—	2	1	no	—
48	C₂×C₃×D₄	—	2	2	no	—
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	—	2	2	no	—
51	C₂×C₃×D₄	—	2	2	no	—
52	C₂×C₃×D₄	—	2	2	no	—
53	C₂×C₃×D₄	—	4	4	no	—
54	C₂×C₃×D₄	—	4	4	no	—
55	C₂×C₃×D₄	—	4	4	no	—
56	C₂×C₃×D₄	C₂³	1	4	yes	—
57	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₃×SD₁₆	—	4	4	no	—
63	C₃×D₈	—	4	4	no	—
64	C₂×C₃×C₈	C₂³	1	4	yes	—
65	C₃×C₄⋊C₄	—	2	1	no	—
66	C₃×C₄⋊C₄	—	2	1	no	—
67	C₃×C₄⋊C₄	—	2	1	no	—
68	C₃×C₄⋊C₄	—	2	1	no	—
69	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₃×C₂²⋊C₄	—	2	1	no	—
72	C₃×C₂²⋊C₄	—	2	1	no	—
73	C₃×C₂²⋊C₄	—	2	1	no	—
74	C₃×C₂²⋊C₄	—	2	1	no	—
75	C₃×C₂²⋊C₄	—	2	1	no	—
76	C₃×C₄²	—	2	1	no	—
77	C₃×C₄²	D₄	1	1	yes	characteristic
78	C₃×C₄²	D₄	1	1	yes	characteristic
79	C₃×C₄²	C₂³	1	1	yes	characteristic
80	C₃×C₄²	C₂³	1	1	yes	characteristic
81	C₂²×D₄	—	2	1	no	—
82	C₂²×D₄	—	2	1	no	—
83	C₂²×D₄	C₂²×C₃	1	2	yes	—
84	C₂×SD₁₆	C₂²×C₃	1	4	yes	complemented
85	C₂×D₈	C₂²×C₃	1	4	yes	complemented
86	#127	—	2	1	no	—
87	#127	—	2	1	no	—
88	#127	C₂²×C₃	1	2	yes	—
89	#127	C₂²×C₃	1	2	yes	—
90	#126	—	2	1	no	—
91	#125	C₂²×C₃	1	1	yes	characteristic, complemented
92	#124	C₂²×C₃	1	1	yes	characteristic, complemented
93	#123	—	2	1	no	—
94	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
95	#121	C₂²×C₃	1	1	yes	characteristic
96	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
97	C₄×D₄	—	2	1	no	—
98	C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
99	#117	—	2	1	no	—
100	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
101	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
102	#102	C₂²×C₃	1	4	yes	complemented
103	#102	C₂²×C₃	1	4	yes	complemented
104	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
105	#97	C₂²×C₃	1	2	yes	complemented
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	—	2	4	no	—
108	C₂³×C₃	—	4	4	no	—
109	C₂³×C₃	—	4	4	no	—
110	C₂³×C₃	—	4	4	no	—
111	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
112	C₃×Q₈	—	2	2	no	—
113	C₃×D₄	—	2	2	no	—
114	C₃×D₄	—	2	4	no	—
115	C₃×D₄	—	2	4	no	—
116	C₃×D₄	—	4	4	no	—
117	C₃×D₄	—	4	4	no	—
118	C₃×D₄	—	4	4	no	—
119	C₃×D₄	—	4	4	no	—
120	C₃×D₄	—	4	2	no	—
121	C₃×D₄	—	4	4	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	4	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	1	yes	characteristic
132	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
135	C₃×C₈	—	2	4	no	—
136	C₂⁴	—	2	2	no	—
137	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
138	C₂×D₄	—	2	1	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	—	4	4	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	—	4	4	no	—
147	C₂×D₄	C₂³×C₃	1	4	yes	—
148	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
151	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
153	SD₁₆	—	4	4	no	—
154	D₈	—	4	4	no	—
155	C₂×C₈	C₂³×C₃	1	4	yes	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
161	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
162	C₂²⋊C₄	—	2	1	no	—
163	C₂²⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₂²⋊C₄	—	2	1	no	—
166	C₂²⋊C₄	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₃×D₄	1	1	yes	characteristic
169	C₄²	C₃×D₄	1	1	yes	characteristic
170	C₄²	C₂³×C₃	1	1	yes	characteristic
171	C₄²	C₂³×C₃	1	1	yes	characteristic
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	4	4	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	—	4	1	no	—
177	C₂²×C₃	—	4	4	no	—
178	C₂²×C₃	—	4	4	no	—
179	C₂²×C₃	—	4	1	no	—
180	C₂²×C₃	—	8	8	no	—
181	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
182	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	4	1	no	—
188	C₃×C₄	—	4	1	no	—
189	C₃×C₄	—	4	1	no	—
190	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
191	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	4	4	no	—
195	C₂³	—	4	4	no	—
196	C₂³	—	4	4	no	—
197	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
198	Q₈	—	2	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	2	4	no	—
201	D₄	—	2	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	4	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	2	no	—
207	D₄	—	4	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
219	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	—	8	1	no	—
224	C₂×C₃	—	8	4	no	—
225	C₂×C₃	#535	1	1	yes	characteristic
226	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	4	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	8	8	no	—
237	C₂²	C₃×AGL(1,ℤ/8)	1	2	yes	—
238	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
247	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
248	C₃	#3217	1	1	yes	characteristic, direct product factor, Sylow
249	C₂	—	2	2	no	—
250	C₂	—	8	4	no	—
251	C₂	—	8	1	no	—
252	C₂	C₃×#535	1	1	yes	characteristic
253	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₂	C₂×C₃×AGL(1,ℤ/8)	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 ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#127	C₂	semidirect
2. C₃×#529	C₂	semidirect
3. C₃×#496	C₂	semidirect
4. C₃×#490	C₂	semidirect
5. C₃×#460	C₂	semidirect
6. C₃×#459	C₂	semidirect
7. C₃×#450	C₂	semidirect
8. C₃×#447	C₂	semidirect
9. C₃×#432	C₂	semidirect
10. #3217	C₃	direct (N × H)
14. C₂×C₃×SD₁₆	C₂²	semidirect
15. C₂×C₃×D₈	C₂²	semidirect
21. C₃×#125	C₂²	semidirect
22. C₃×#124	C₂²	semidirect
26. C₃×C₄×Q₈	C₂²	semidirect
28. C₃×C₄×D₄	C₂²	semidirect
31. C₃×C₄⋊C₈	C₂²	semidirect
32. C₃×#102	C₂²	semidirect
33. C₃×#102	C₂²	semidirect
35. C₃×#97	C₂²	semidirect
36. C₂×#127	C₂×C₃	semidirect
37. #529	C₂×C₃	semidirect
38. #496	C₂×C₃	semidirect
39. #490	C₂×C₃	semidirect
40. #460	C₂×C₃	semidirect
41. #459	C₂×C₃	semidirect
42. #450	C₂×C₃	semidirect
43. #447	C₂×C₃	semidirect
44. #432	C₂×C₃	semidirect
84. C₂×SD₁₆	C₂²×C₃	semidirect
85. C₂×D₈	C₂²×C₃	semidirect
91. #125	C₂²×C₃	semidirect
92. #124	C₂²×C₃	semidirect
96. C₄×Q₈	C₂²×C₃	semidirect
98. C₄×D₄	C₂²×C₃	semidirect
101. C₄⋊C₈	C₂²×C₃	semidirect
102. #102	C₂²×C₃	semidirect
103. #102	C₂²×C₃	semidirect
105. #97	C₂²×C₃	semidirect
248. C₃	#3217	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×#127	2
C₃×#529	2
C₃×#496	2
C₃×#490	2
C₃×#460	2
C₃×#459	2
C₃×#450	2
C₃×#447	2
C₃×#432	2
#3217	3

Subgroups — C3×#3217 #85081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3217 #85081