Subgroups — C₃×#2500 #84364

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#394	C₂	1	4	yes	maximal, complemented
4	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#388	C₂	1	4	yes	maximal, complemented
7	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#2500	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	2	yes	—
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₃×#121	—	2	4	no	—
15	C₃×#117	—	2	4	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄²	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	4	yes	—
30	C₂×#121	C₂×C₃	1	1	yes	characteristic
31	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
32	#394	C₂×C₃	1	4	yes	complemented
33	#390	C₂×C₃	1	1	yes	characteristic, complemented
34	#390	C₂×C₃	1	1	yes	characteristic, complemented
35	#388	C₂×C₃	1	4	yes	complemented
36	#384	C₂×C₃	1	1	yes	characteristic, complemented
37	#382	C₂×C₃	1	1	yes	characteristic, complemented
38	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂⁴×C₃	C₂³	1	2	yes	—
40	C₂⁴×C₃	C₂³	1	1	yes	characteristic
41	C₂×C₃×D₄	—	2	4	no	—
42	C₂×C₃×D₄	—	2	2	no	—
43	C₂×C₃×D₄	—	2	2	no	—
44	C₂×C₃×D₄	—	2	2	no	—
45	C₂×C₃×D₄	—	4	4	no	—
46	C₂²×C₃×C₄	—	2	2	no	—
47	C₂²×C₃×C₄	—	2	2	no	—
48	C₂²×C₃×C₄	D₄	1	4	yes	—
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	2	2	no	—
59	C₃×C₄⋊C₄	—	4	4	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	2	4	no	—
62	C₃×C₂²⋊C₄	—	2	4	no	—
63	C₃×C₂²⋊C₄	—	2	8	no	—
64	C₃×C₂²⋊C₄	—	4	4	no	—
65	C₃×C₄²	—	2	4	no	—
66	C₃×C₄²	—	2	2	no	—
67	C₃×C₄²	—	2	2	no	—
68	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₂²×D₄	C₂²×C₃	1	2	yes	—
70	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
71	#121	—	2	4	no	—
72	#117	—	2	4	no	—
73	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
77	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
82	C₂×C₄²	C₂²×C₃	1	2	yes	—
83	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
85	#95	C₂²×C₃	1	2	yes	—
86	#95	C₂²×C₃	1	4	yes	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	—	2	1	no	—
89	C₂³×C₃	—	2	1	no	—
90	C₂³×C₃	—	2	2	no	—
91	C₂³×C₃	—	2	2	no	—
92	C₂³×C₃	—	2	4	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	—	2	4	no	—
96	C₂³×C₃	C₂×D₄	1	2	yes	—
97	C₂³×C₃	C₂⁴	1	1	yes	characteristic
98	C₃×D₄	—	4	8	no	—
99	C₃×D₄	—	4	4	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	1	no	—
102	C₂×C₃×C₄	—	2	1	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	8	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	2	4	no	—
114	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
115	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
116	C₂×C₃×C₄	#41	1	4	yes	—
117	C₂×C₃×C₄	#41	1	2	yes	—
118	C₂×C₃×C₄	#41	1	2	yes	—
119	C₂⁴	C₂³×C₃	1	2	yes	—
120	C₂⁴	C₂³×C₃	1	1	yes	characteristic
121	C₂×D₄	—	2	4	no	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	—	4	4	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	C₃×D₄	1	4	yes	—
129	C₂²×C₄	C₂³×C₃	1	2	yes	—
130	C₂²×C₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₃	1	2	yes	—
132	C₂²×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₂²⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₂²⋊C₄	—	2	8	no	—
144	C₂²⋊C₄	—	4	4	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	2	no	—
150	C₂²×C₃	—	2	2	no	—
151	C₂²×C₃	—	4	2	no	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	—	4	2	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	—	4	2	no	—
156	C₂²×C₃	—	4	4	no	—
157	C₂²×C₃	—	4	4	no	—
158	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²×C₃	C₂×#41	1	1	yes	characteristic
160	C₂²×C₃	C₂×#41	1	1	yes	characteristic
161	C₂²×C₃	C₂×#41	1	2	yes	—
162	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	2	4	no	—
165	C₃×C₄	—	4	2	no	—
166	C₃×C₄	—	4	2	no	—
167	C₃×C₄	—	4	2	no	—
168	C₃×C₄	—	4	2	no	—
169	C₃×C₄	#123	1	4	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	1	no	—
172	C₂³	—	2	1	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	4	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	4	no	—
179	C₂³	C₂×C₃×D₄	1	2	yes	—
180	C₂³	C₂⁴×C₃	1	1	yes	characteristic
181	D₄	—	4	8	no	—
182	D₄	—	4	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
199	C₂×C₄	C₃×#41	1	4	yes	—
200	C₂×C₄	C₃×#41	1	2	yes	—
201	C₂×C₄	C₃×#41	1	2	yes	—
202	C₂×C₃	—	4	2	no	—
203	C₂×C₃	—	8	2	no	—
204	C₂×C₃	C₂×#123	1	1	yes	characteristic
205	C₂×C₃	#532	1	1	yes	characteristic
206	C₂×C₃	#535	1	1	yes	characteristic
207	C₂×C₃	#553	1	2	yes	—
208	C₂×C₃	#555	1	2	yes	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	4	no	—
219	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
220	C₂²	C₂×C₃×#41	1	1	yes	characteristic
221	C₂²	C₂×C₃×#41	1	1	yes	characteristic
222	C₂²	C₂×C₃×#41	1	2	yes	—
223	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	4	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	C₃×#123	1	4	yes	—
231	C₃	#2500	1	1	yes	characteristic, direct product factor, Sylow
232	C₂	—	4	2	no	—
233	C₂	—	8	2	no	—
234	C₂	C₂×C₃×#123	1	1	yes	characteristic
235	C₂	C₃×#532	1	1	yes	characteristic
236	C₂	C₃×#535	1	1	yes	characteristic
237	C₂	C₃×#553	1	2	yes	—
238	C₂	C₃×#555	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#117	C₂	semidirect
3. C₃×#394	C₂	semidirect
4. C₃×#390	C₂	semidirect
5. C₃×#390	C₂	semidirect
6. C₃×#388	C₂	semidirect
7. C₃×#384	C₂	semidirect
8. C₃×#382	C₂	semidirect
9. C₃×C₄×C₂²⋊C₄	C₂	semidirect
10. #2500	C₃	direct (N × H)
31. C₂×#117	C₂×C₃	semidirect
32. #394	C₂×C₃	semidirect
33. #390	C₂×C₃	semidirect
34. #390	C₂×C₃	semidirect
35. #388	C₂×C₃	semidirect
36. #384	C₂×C₃	semidirect
37. #382	C₂×C₃	semidirect
38. C₄×C₂²⋊C₄	C₂×C₃	semidirect
231. C₃	#2500	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×#121	2
C₂×C₃×#117	2
C₃×#394	2
C₃×#390	2
C₃×#388	2
C₃×#384	2
C₃×#382	2
C₃×C₄×C₂²⋊C₄	2
#2500	3

Subgroups — C3×#2500 #84364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2500 #84364