Subgroups — C₃×#3223 #85087

Order 384.

364 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#536	C₂	1	1	yes	maximal, characteristic
3	C₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#464	C₂	1	2	yes	maximal, complemented
8	C₃×#462	C₂	1	2	yes	maximal, complemented
9	C₃×#452	C₂	1	2	yes	maximal
10	C₃×#451	C₂	1	2	yes	maximal, complemented
11	C₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	#3223	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₃×#126	—	2	2	no	—
23	C₃×#125	—	2	2	no	—
24	C₃×#125	C₂²	1	1	yes	characteristic
25	C₃×#124	C₂²	1	1	yes	characteristic
26	C₃×#124	C₂²	1	1	yes	characteristic
27	C₃×#123	—	2	2	no	—
28	C₃×C₂²⋊Q₈	—	2	2	no	—
29	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
30	C₃×C₄×Q₈	C₂²	1	2	yes	—
31	C₃×#117	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₃×#103	C₂²	1	2	yes	—
34	C₃×#103	C₂²	1	2	yes	—
35	C₃×#103	C₂²	1	2	yes	—
36	C₃×#102	C₂²	1	2	yes	—
37	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
38	C₃×#97	C₂²	1	1	yes	characteristic
39	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
40	#541	C₂×C₃	1	1	yes	characteristic, complemented
41	#536	C₂×C₃	1	1	yes	characteristic
42	#497	C₂×C₃	1	1	yes	characteristic, complemented
43	#494	C₂×C₃	1	1	yes	characteristic, complemented
44	#491	C₂×C₃	1	1	yes	characteristic, complemented
45	#488	C₂×C₃	1	1	yes	characteristic, complemented
46	#464	C₂×C₃	1	2	yes	complemented
47	#462	C₂×C₃	1	2	yes	complemented
48	#452	C₂×C₃	1	2	yes	—
49	#451	C₂×C₃	1	2	yes	complemented
50	#433	C₂×C₃	1	1	yes	characteristic, complemented
51	C₃×#41	—	4	2	no	—
52	C₂×C₃×Q₈	—	2	1	no	—
53	C₂×C₃×Q₈	—	4	2	no	—
54	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
55	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	2	yes	—
58	C₂×C₃×D₄	—	2	1	no	—
59	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	—	2	1	no	—
61	C₂²×C₃×C₄	—	2	1	no	—
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₃×Q₁₆	—	4	2	no	—
64	C₃×Q₁₆	—	4	2	no	—
65	C₃×Q₁₆	—	4	2	no	—
66	C₃×SD₁₆	—	4	2	no	—
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₂×C₃×C₈	C₂³	1	2	yes	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄⋊C₄	D₄	1	2	yes	—
76	C₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	2	2	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	D₄	1	2	yes	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	C₂³	1	1	yes	characteristic
83	C₃×C₄²	C₂³	1	1	yes	characteristic
84	C₂×#41	C₂²×C₃	1	1	yes	characteristic
85	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
86	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
87	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
88	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
89	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
90	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
91	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
92	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
93	#126	—	2	2	no	—
94	#125	—	2	2	no	—
95	#125	C₂²×C₃	1	1	yes	characteristic
96	#124	C₂²×C₃	1	1	yes	characteristic
97	#124	C₂²×C₃	1	1	yes	characteristic
98	#123	—	2	2	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
101	C₄×Q₈	C₂²×C₃	1	2	yes	—
102	#117	C₂²×C₃	1	1	yes	characteristic
103	C₄⋊C₈	C₂²×C₃	1	2	yes	—
104	#103	C₂²×C₃	1	2	yes	—
105	#103	C₂²×C₃	1	2	yes	—
106	#103	C₂²×C₃	1	2	yes	—
107	#102	C₂²×C₃	1	2	yes	—
108	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
109	#97	C₂²×C₃	1	1	yes	characteristic
110	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
111	C₂³×C₃	—	2	1	no	—
112	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
113	C₃×Q₈	—	2	2	no	—
114	C₃×Q₈	—	2	2	no	—
115	C₃×Q₈	—	2	2	no	—
116	C₃×Q₈	—	2	2	no	—
117	C₃×Q₈	—	2	2	no	—
118	C₃×Q₈	—	4	2	no	—
119	C₃×D₄	—	2	2	no	—
120	C₃×D₄	—	4	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	—	4	2	no	—
129	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
130	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
131	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
133	C₃×C₈	—	2	2	no	—
134	C₃×C₈	—	2	2	no	—
135	#41	—	4	2	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
139	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
140	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
141	C₂×Q₈	C₂³×C₃	1	2	yes	—
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
147	Q₁₆	—	4	2	no	—
148	Q₁₆	—	4	2	no	—
149	Q₁₆	—	4	2	no	—
150	SD₁₆	—	4	2	no	—
151	C₂×C₈	C₂³×C₃	1	2	yes	—
152	C₂×C₈	C₂³×C₃	1	2	yes	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	C₃×D₄	1	2	yes	—
160	C₄⋊C₄	C₂³×C₃	1	2	yes	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	C₃×D₄	1	2	yes	—
165	C₄²	—	2	2	no	—
166	C₄²	C₂³×C₃	1	1	yes	characteristic
167	C₄²	C₂³×C₃	1	1	yes	characteristic
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	—	4	2	no	—
172	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	4	1	no	—
177	C₃×C₄	—	4	1	no	—
178	C₃×C₄	—	4	1	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
182	C₂³	—	2	1	no	—
183	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
184	Q₈	—	2	2	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	2	2	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	4	2	no	—
190	D₄	—	2	2	no	—
191	D₄	—	4	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
201	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
202	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₂×C₃	—	4	1	no	—
207	C₂×C₃	—	8	1	no	—
208	C₂×C₃	#535	1	1	yes	characteristic
209	C₂×C₃	#578	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
224	C₃	#3223	1	1	yes	characteristic, direct product factor, Sylow
225	C₂	—	4	1	no	—
226	C₂	—	8	1	no	—
227	C₂	C₃×#535	1	1	yes	characteristic
228	C₂	C₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#541	C₂	semidirect
3. C₃×#497	C₂	semidirect
4. C₃×#494	C₂	semidirect
5. C₃×#491	C₂	semidirect
6. C₃×#488	C₂	semidirect
7. C₃×#464	C₂	semidirect
8. C₃×#462	C₂	semidirect
10. C₃×#451	C₂	semidirect
11. C₃×#433	C₂	semidirect
12. #3223	C₃	direct (N × H)
40. #541	C₂×C₃	semidirect
42. #497	C₂×C₃	semidirect
43. #494	C₂×C₃	semidirect
44. #491	C₂×C₃	semidirect
45. #488	C₂×C₃	semidirect
46. #464	C₂×C₃	semidirect
47. #462	C₂×C₃	semidirect
49. #451	C₂×C₃	semidirect
50. #433	C₂×C₃	semidirect
224. C₃	#3223	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×#541	2
C₃×#536	2
C₃×#497	2
C₃×#494	2
C₃×#491	2
C₃×#488	2
C₃×#464	2
C₃×#462	2
C₃×#452	2
C₃×#451	2
C₃×#433	2
#3223	3

Subgroups — C3×#3223 #85087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3223 #85087