Subgroups — C₃×#3339 #85203

Order 384.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#553	C₂	1	2	yes	maximal, complemented
2	C₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#482	C₂	1	2	yes	maximal, complemented
4	C₃×#481	C₂	1	2	yes	maximal, complemented
5	C₃×#478	C₂	1	2	yes	maximal, complemented
6	C₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
12	#3339	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₃×#126	—	2	2	no	—
17	C₃×#125	C₂²	1	2	yes	—
18	C₃×#123	—	2	2	no	—
19	C₃×#121	—	2	2	no	—
20	C₃×#121	C₂²	1	2	yes	—
21	C₃×#121	C₂²	1	2	yes	—
22	C₃×C₄×D₄	—	2	2	no	—
23	C₃×C₄×D₄	C₂²	1	2	yes	—
24	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
25	C₃×#117	C₂²	1	2	yes	—
26	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₃×#107	C₂²	1	1	yes	characteristic
28	C₃×#107	C₂²	1	1	yes	characteristic
29	C₃×#107	C₂²	1	2	yes	—
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×#106	C₂²	1	2	yes	—
32	C₃×#106	C₂²	1	2	yes	—
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃×#102	C₂²	1	2	yes	—
35	C₃×#102	C₂²	1	2	yes	—
36	C₃×#102	C₂²	1	2	yes	—
37	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
40	#553	C₂×C₃	1	2	yes	complemented
41	#499	C₂×C₃	1	1	yes	characteristic, complemented
42	#482	C₂×C₃	1	2	yes	complemented
43	#481	C₂×C₃	1	2	yes	complemented
44	#478	C₂×C₃	1	2	yes	complemented
45	#466	C₂×C₃	1	1	yes	characteristic, complemented
46	#466	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
48	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
49	#427	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#107	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×C₃×D₄	—	2	2	no	—
52	C₂×C₃×D₄	—	2	2	no	—
53	C₂×C₃×D₄	D₄	1	1	yes	characteristic
54	C₂×C₃×D₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₃×D₈	—	2	2	no	—
60	C₂×C₃×C₈	—	2	2	no	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
70	C₃×C₄⋊C₄	C₂³	1	2	yes	—
71	C₃×C₄⋊C₄	C₂³	1	2	yes	—
72	C₃×C₄⋊C₄	C₂³	1	2	yes	—
73	C₃×C₂²⋊C₄	—	2	2	no	—
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	D₄	1	1	yes	characteristic
78	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
79	C₃×C₄²	—	2	2	no	—
80	C₃×C₄²	C₂³	1	1	yes	characteristic
81	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
84	#126	—	2	2	no	—
85	#125	C₂²×C₃	1	2	yes	—
86	#123	—	2	2	no	—
87	#121	—	2	2	no	—
88	#121	C₂²×C₃	1	2	yes	—
89	#121	C₂²×C₃	1	2	yes	—
90	C₄×D₄	—	2	2	no	—
91	C₄×D₄	C₂²×C₃	1	2	yes	—
92	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
93	#117	C₂²×C₃	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
95	#107	C₂²×C₃	1	1	yes	characteristic
96	#107	C₂²×C₃	1	1	yes	characteristic
97	#107	C₂²×C₃	1	2	yes	—
98	#107	C₂²×C₃	1	1	yes	characteristic
99	#106	C₂²×C₃	1	2	yes	—
100	#106	C₂²×C₃	1	2	yes	—
101	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
102	#102	C₂²×C₃	1	2	yes	—
103	#102	C₂²×C₃	1	2	yes	—
104	#102	C₂²×C₃	1	2	yes	—
105	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
106	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
108	C₂³×C₃	—	2	2	no	—
109	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
111	C₃×D₄	—	2	2	no	—
112	C₃×D₄	—	2	2	no	—
113	C₃×D₄	—	2	2	no	—
114	C₃×D₄	—	4	4	no	—
115	C₃×D₄	—	8	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	8	2	no	—
124	C₂×C₃×C₄	—	8	2	no	—
125	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
130	C₃×C₈	—	2	1	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	#41	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	C₃×D₄	1	1	yes	characteristic
137	C₂×D₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
142	D₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
146	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
148	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
153	C₄⋊C₄	C₂³×C₃	1	2	yes	—
154	C₄⋊C₄	C₂³×C₃	1	2	yes	—
155	C₄⋊C₄	C₂³×C₃	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
161	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
162	C₄²	—	2	2	no	—
163	C₄²	C₂³×C₃	1	1	yes	characteristic
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	—	4	2	no	—
171	C₂²×C₃	—	4	2	no	—
172	C₂²×C₃	#135	1	2	yes	—
173	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	C₂×#41	1	1	yes	characteristic
182	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
186	D₄	—	2	2	no	—
187	D₄	—	2	2	no	—
188	D₄	—	2	2	no	—
189	D₄	—	4	4	no	—
190	D₄	—	8	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	C₃×#41	1	2	yes	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	4	1	no	—
211	C₂×C₃	—	8	2	no	—
212	C₂×C₃	#547	1	1	yes	characteristic
213	C₂×C₃	C₂×#135	1	1	yes	characteristic
214	C₂×C₃	#576	1	1	yes	characteristic
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	C₃×#135	1	2	yes	—
224	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₂×C₃×#41	1	1	yes	characteristic
233	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
234	C₃	#3339	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	8	2	no	—
238	C₂	C₃×#547	1	1	yes	characteristic
239	C₂	C₂×C₃×#135	1	1	yes	characteristic
240	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. C₃×#553	C₂	semidirect
2. C₃×#499	C₂	semidirect
3. C₃×#482	C₂	semidirect
4. C₃×#481	C₂	semidirect
5. C₃×#478	C₂	semidirect
6. C₃×#466	C₂	semidirect
7. C₃×#466	C₂	semidirect
8. C₃×C₄×D₈	C₂	semidirect
9. C₃×C₈×D₄	C₂	semidirect
10. C₃×#427	C₂	semidirect
11. C₂×C₃×#107	C₂	semidirect
12. #3339	C₃	direct (N × H)
40. #553	C₂×C₃	semidirect
41. #499	C₂×C₃	semidirect
42. #482	C₂×C₃	semidirect
43. #481	C₂×C₃	semidirect
44. #478	C₂×C₃	semidirect
45. #466	C₂×C₃	semidirect
46. #466	C₂×C₃	semidirect
47. C₄×D₈	C₂×C₃	semidirect
48. C₈×D₄	C₂×C₃	semidirect
49. #427	C₂×C₃	semidirect
50. C₂×#107	C₂×C₃	semidirect
234. C₃	#3339	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×#553	2
C₃×#499	2
C₃×#482	2
C₃×#481	2
C₃×#478	2
C₃×#466	2
C₃×C₄×D₈	2
C₃×C₈×D₄	2
C₃×#427	2
C₂×C₃×#107	2
#3339	3

Subgroups — C3×#3339 #85203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3339 #85203