Subgroups — C₃×#3182 #85046

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#468	C₂	1	4	yes	maximal, complemented
8	C₃×#440	C₂	1	4	yes	maximal, complemented
9	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#3182	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#41	—	2	1	no	—
12	C₂×C₃×SD₁₆	C₂²	1	4	yes	—
13	C₂×C₃×M₁₆	C₂²	1	2	yes	—
14	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×#127	C₂²	1	1	yes	characteristic
18	C₃×#125	—	2	1	no	—
19	C₃×#124	—	2	1	no	—
20	C₃×C₂²⋊Q₈	—	2	1	no	—
21	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₃×#121	—	2	1	no	—
23	C₃×#121	C₂²	1	2	yes	—
24	C₃×C₄×Q₈	—	2	1	no	—
25	C₃×C₄×Q₈	C₂²	1	2	yes	—
26	C₃×C₄×D₄	—	2	1	no	—
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₃×#117	—	2	1	no	—
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₃×#107	C₂²	1	4	yes	—
31	C₃×#103	C₂²	1	4	yes	—
32	C₃×#102	C₂²	1	4	yes	—
33	C₃×#97	C₂²	1	2	yes	—
34	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
35	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
36	#533	C₂×C₃	1	1	yes	characteristic, complemented
37	#532	C₂×C₃	1	1	yes	characteristic
38	#500	C₂×C₃	1	1	yes	characteristic, complemented
39	#492	C₂×C₃	1	1	yes	characteristic, complemented
40	#487	C₂×C₃	1	1	yes	characteristic, complemented
41	#486	C₂×C₃	1	1	yes	characteristic, complemented
42	#468	C₂×C₃	1	4	yes	complemented
43	#440	C₂×C₃	1	4	yes	complemented
44	C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
45	C₃×#41	—	4	4	no	—
46	C₂×C₃×Q₈	—	2	1	no	—
47	C₂×C₃×Q₈	—	2	1	no	—
48	C₂×C₃×Q₈	C₂³	1	2	yes	—
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	—	2	1	no	—
51	C₂×C₃×D₄	—	2	1	no	—
52	C₂×C₃×D₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	—	2	2	no	—
54	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₃×SD₁₆	—	2	8	no	—
58	C₃×M₁₆	—	2	4	no	—
59	C₂×C₃×C₈	C₂³	1	2	yes	—
60	C₂×C₃×C₈	C₂³	1	2	yes	—
61	C₃×C₄⋊C₄	—	2	1	no	—
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	—	2	2	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	C₂³	1	2	yes	—
68	C₃×C₄⋊C₄	C₂³	1	2	yes	—
69	C₃×C₂²⋊C₄	—	2	2	no	—
70	C₃×C₂²⋊C₄	—	2	2	no	—
71	C₃×C₂²⋊C₄	—	2	2	no	—
72	C₃×C₄²	—	2	2	no	—
73	C₃×C₄²	D₄	1	1	yes	characteristic
74	C₃×C₄²	D₄	1	1	yes	characteristic
75	C₃×C₄²	C₂³	1	1	yes	characteristic
76	C₃×C₄²	C₂³	1	1	yes	characteristic
77	C₂×#41	—	2	1	no	—
78	C₂×SD₁₆	C₂²×C₃	1	4	yes	—
79	C₂×M₁₆	C₂²×C₃	1	2	yes	—
80	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
81	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
82	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
83	#127	C₂²×C₃	1	1	yes	characteristic
84	#125	—	2	1	no	—
85	#124	—	2	1	no	—
86	C₂²⋊Q₈	—	2	1	no	—
87	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
88	#121	—	2	1	no	—
89	#121	C₂²×C₃	1	2	yes	—
90	C₄×Q₈	—	2	1	no	—
91	C₄×Q₈	C₂²×C₃	1	2	yes	—
92	C₄×D₄	—	2	1	no	—
93	C₄×D₄	C₂²×C₃	1	2	yes	—
94	#117	—	2	1	no	—
95	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
96	#107	C₂²×C₃	1	4	yes	—
97	#103	C₂²×C₃	1	4	yes	—
98	#102	C₂²×C₃	1	4	yes	—
99	#97	C₂²×C₃	1	2	yes	—
100	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
102	C₂³×C₃	—	2	2	no	—
103	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
104	C₃×Q₈	—	2	4	no	—
105	C₃×Q₈	—	4	2	no	—
106	C₃×Q₈	—	4	2	no	—
107	C₃×D₄	—	2	4	no	—
108	C₃×D₄	—	4	4	no	—
109	C₃×D₄	—	4	2	no	—
110	C₃×D₄	—	4	2	no	—
111	C₂×C₃×C₄	—	2	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	—	2	2	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
124	C₃×C₈	—	2	2	no	—
125	C₃×C₈	#41	1	4	yes	complemented
126	#41	—	4	4	no	—
127	C₂×Q₈	—	2	1	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	1	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
136	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
138	SD₁₆	—	2	8	no	—
139	M₁₆	—	2	4	no	—
140	C₂×C₈	C₂³×C₃	1	2	yes	—
141	C₂×C₈	C₂³×C₃	1	2	yes	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	C₂³×C₃	1	2	yes	—
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	C₄²	—	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	C₂²×D₄	1	1	yes	characteristic
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	4	2	no	—
169	C₃×C₄	—	4	2	no	—
170	C₃×C₄	—	4	2	no	—
171	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
172	C₃×C₄	#137	1	2	yes	—
173	C₃×C₄	C₂×#41	1	2	yes	—
174	C₂³	—	2	2	no	—
175	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
176	Q₈	—	2	4	no	—
177	Q₈	—	4	2	no	—
178	Q₈	—	4	2	no	—
179	D₄	—	2	4	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	C₃×#41	1	4	yes	complemented
198	C₂×C₃	—	4	1	no	—
199	C₂×C₃	—	8	2	no	—
200	C₂×C₃	#532	1	1	yes	characteristic
201	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₂×C₃	C₂×#137	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
217	C₄	C₃×#137	1	2	yes	—
218	C₄	C₂×C₃×#41	1	2	yes	—
219	C₃	#3182	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	C₃×#532	1	1	yes	characteristic
223	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂	C₂×C₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#532	C₂
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×#533	C₂	semidirect
3. C₃×#500	C₂	semidirect
4. C₃×#492	C₂	semidirect
5. C₃×#487	C₂	semidirect
6. C₃×#486	C₂	semidirect
7. C₃×#468	C₂	semidirect
8. C₃×#440	C₂	semidirect
9. C₃×C₄×M₁₆	C₂	semidirect
10. #3182	C₃	direct (N × H)
36. #533	C₂×C₃	semidirect
38. #500	C₂×C₃	semidirect
39. #492	C₂×C₃	semidirect
40. #487	C₂×C₃	semidirect
41. #486	C₂×C₃	semidirect
42. #468	C₂×C₃	semidirect
43. #440	C₂×C₃	semidirect
44. C₄×M₁₆	C₂×C₃	semidirect
125. C₃×C₈	#41	semidirect
197. C₈	C₃×#41	semidirect
219. C₃	#3182	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×#533	2
C₃×#532	2
C₃×#500	2
C₃×#492	2
C₃×#487	2
C₃×#486	2
C₃×#468	2
C₃×#440	2
C₃×C₄×M₁₆	2
#3182	3

Subgroups — C3×#3182 #85046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3182 #85046