Subgroups — C₃×#3214 #85078

Order 384.

434 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#496	C₂	1	2	yes	maximal, complemented
4	C₃×#489	C₂	1	2	yes	maximal, complemented
5	C₃×#461	C₂	1	2	yes	maximal, complemented
6	C₃×#459	C₂	1	2	yes	maximal, complemented
7	C₃×#449	C₂	1	4	yes	maximal, complemented
8	C₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	#3214	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	—	2	1	no	—
11	C₂×C₃×#41	C₂²	1	2	yes	—
12	C₂×C₃×SD₁₆	C₂²	1	4	yes	—
13	C₂×C₃×D₈	C₂²	1	4	yes	—
14	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃×#127	C₂²	1	1	yes	characteristic
17	C₃×#127	C₂²	1	1	yes	characteristic
18	C₃×#124	—	2	1	no	—
19	C₃×#124	C₂²	1	2	yes	—
20	C₃×#121	—	2	1	no	—
21	C₃×#121	—	2	1	no	—
22	C₃×#121	—	2	1	no	—
23	C₃×#121	C₂²	1	2	yes	—
24	C₃×C₄×D₄	—	2	1	no	—
25	C₃×C₄×D₄	—	2	1	no	—
26	C₃×C₄×D₄	—	2	1	no	—
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₃×#103	C₂²	1	4	yes	—
31	C₃×#102	C₂²	1	4	yes	—
32	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₃×#97	C₂²	1	2	yes	—
34	#532	C₂×C₃	1	1	yes	characteristic, complemented
35	#532	C₂×C₃	1	1	yes	characteristic, complemented
36	#496	C₂×C₃	1	2	yes	complemented
37	#489	C₂×C₃	1	2	yes	complemented
38	#461	C₂×C₃	1	2	yes	complemented
39	#459	C₂×C₃	1	2	yes	complemented
40	#449	C₂×C₃	1	4	yes	complemented
41	#432	C₂×C₃	1	1	yes	characteristic, complemented
42	C₃×#41	—	4	4	no	—
43	C₃×#41	—	4	4	no	—
44	C₂×C₃×Q₈	—	2	1	no	—
45	C₂×C₃×Q₈	C₂³	1	2	yes	—
46	C₂×C₃×D₄	—	2	2	no	—
47	C₂×C₃×D₄	—	2	2	no	—
48	C₂×C₃×D₄	—	2	1	no	—
49	C₂×C₃×D₄	—	2	1	no	—
50	C₂×C₃×D₄	—	2	1	no	—
51	C₂×C₃×D₄	—	2	1	no	—
52	C₂×C₃×D₄	C₂³	1	2	yes	—
53	C₂×C₃×D₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₃×SD₁₆	—	4	4	no	—
60	C₃×D₈	—	4	4	no	—
61	C₂×C₃×C₈	C₂³	1	4	yes	—
62	C₃×C₄⋊C₄	—	2	1	no	—
63	C₃×C₄⋊C₄	—	2	1	no	—
64	C₃×C₄⋊C₄	—	2	1	no	—
65	C₃×C₄⋊C₄	C₂³	1	2	yes	—
66	C₃×C₂²⋊C₄	—	2	2	no	—
67	C₃×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₂²⋊C₄	—	2	2	no	—
69	C₃×C₂²⋊C₄	—	2	2	no	—
70	C₃×C₄²	D₄	1	1	yes	characteristic
71	C₃×C₄²	D₄	1	1	yes	characteristic
72	C₃×C₄²	C₂³	1	1	yes	characteristic
73	C₃×C₄²	C₂³	1	1	yes	characteristic
74	C₂×#41	—	2	1	no	—
75	C₂×#41	C₂²×C₃	1	2	yes	—
76	C₂×SD₁₆	C₂²×C₃	1	4	yes	—
77	C₂×D₈	C₂²×C₃	1	4	yes	—
78	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
79	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
80	#127	C₂²×C₃	1	1	yes	characteristic
81	#127	C₂²×C₃	1	1	yes	characteristic
82	#124	—	2	1	no	—
83	#124	C₂²×C₃	1	2	yes	—
84	#121	—	2	1	no	—
85	#121	—	2	1	no	—
86	#121	—	2	1	no	—
87	#121	C₂²×C₃	1	2	yes	—
88	C₄×D₄	—	2	1	no	—
89	C₄×D₄	—	2	1	no	—
90	C₄×D₄	—	2	1	no	—
91	C₄×D₄	C₂²×C₃	1	2	yes	—
92	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
93	C₄⋊C₈	C₂²×C₃	1	2	yes	—
94	#103	C₂²×C₃	1	4	yes	—
95	#102	C₂²×C₃	1	4	yes	—
96	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	#97	C₂²×C₃	1	2	yes	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
101	C₃×Q₈	—	2	4	no	—
102	C₃×Q₈	—	4	2	no	—
103	C₃×D₄	—	2	2	no	—
104	C₃×D₄	—	2	4	no	—
105	C₃×D₄	—	2	2	no	—
106	C₃×D₄	—	4	2	no	—
107	C₃×D₄	—	4	2	no	—
108	C₃×D₄	—	4	2	no	—
109	C₃×D₄	—	4	4	no	—
110	C₃×D₄	—	4	4	no	—
111	C₃×D₄	—	4	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	4	1	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	—	4	4	no	—
119	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
125	C₃×C₈	—	2	4	no	—
126	#41	—	4	4	no	—
127	#41	—	4	4	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	C₂³×C₃	1	2	yes	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	—	2	1	no	—
134	C₂×D₄	—	2	1	no	—
135	C₂×D₄	—	2	1	no	—
136	C₂×D₄	C₂³×C₃	1	2	yes	—
137	C₂×D₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
141	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
143	SD₁₆	—	4	4	no	—
144	D₈	—	4	4	no	—
145	C₂×C₈	C₂³×C₃	1	4	yes	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	C₂³×C₃	1	2	yes	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	C₃×D₄	1	1	yes	characteristic
155	C₄²	C₃×D₄	1	1	yes	characteristic
156	C₄²	C₂³×C₃	1	1	yes	characteristic
157	C₄²	C₂³×C₃	1	1	yes	characteristic
158	C₂²×C₃	—	2	1	no	—
159	C₂²×C₃	—	2	1	no	—
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	4	2	no	—
162	C₂²×C₃	—	4	2	no	—
163	C₂²×C₃	—	4	2	no	—
164	C₂²×C₃	—	4	2	no	—
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	4	2	no	—
173	C₃×C₄	—	4	2	no	—
174	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
175	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
179	Q₈	—	2	4	no	—
180	Q₈	—	4	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	2	4	no	—
183	D₄	—	2	2	no	—
184	D₄	—	4	2	no	—
185	D₄	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	4	no	—
188	D₄	—	4	4	no	—
189	D₄	—	4	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
198	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
203	C₈	—	2	4	no	—
204	C₂×C₃	—	4	1	no	—
205	C₂×C₃	—	8	2	no	—
206	C₂×C₃	—	8	2	no	—
207	C₂×C₃	#535	1	1	yes	characteristic
208	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₂×C₃	#575	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
227	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
228	C₃	#3214	1	1	yes	characteristic, direct product factor, Sylow
229	C₂	—	4	1	no	—
230	C₂	—	8	2	no	—
231	C₂	—	8	2	no	—
232	C₂	C₃×#535	1	1	yes	characteristic
233	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂	C₃×#575	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. C₃×#532	C₂	semidirect
2. C₃×#532	C₂	semidirect
3. C₃×#496	C₂	semidirect
4. C₃×#489	C₂	semidirect
5. C₃×#461	C₂	semidirect
6. C₃×#459	C₂	semidirect
7. C₃×#449	C₂	semidirect
8. C₃×#432	C₂	semidirect
9. #3214	C₃	direct (N × H)
34. #532	C₂×C₃	semidirect
35. #532	C₂×C₃	semidirect
36. #496	C₂×C₃	semidirect
37. #489	C₂×C₃	semidirect
38. #461	C₂×C₃	semidirect
39. #459	C₂×C₃	semidirect
40. #449	C₂×C₃	semidirect
41. #432	C₂×C₃	semidirect
228. C₃	#3214	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×#532	2
C₃×#496	2
C₃×#489	2
C₃×#461	2
C₃×#459	2
C₃×#449	2
C₃×#432	2
#3214	3

Subgroups — C3×#3214 #85078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3214 #85078