Subgroups — C₃×#2623 #84487

Order 384.

526 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#123	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#400	C₂	1	2	yes	maximal, complemented
4	C₃×#399	C₂	1	2	yes	maximal, complemented
5	C₃×#398	C₂	1	2	yes	maximal, complemented
6	C₃×#397	C₂	1	2	yes	maximal, complemented
7	C₃×#386	C₂	1	2	yes	maximal
8	C₃×#385	C₂	1	2	yes	maximal, complemented
9	C₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
10	#2623	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₃×#123	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	—	2	4	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	4	yes	—
29	C₃×#95	C₂²	1	4	yes	—
30	C₂×#123	C₂×C₃	1	1	yes	characteristic
31	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
32	#400	C₂×C₃	1	2	yes	complemented
33	#399	C₂×C₃	1	2	yes	complemented
34	#398	C₂×C₃	1	2	yes	complemented
35	#397	C₂×C₃	1	2	yes	complemented
36	#386	C₂×C₃	1	2	yes	—
37	#385	C₂×C₃	1	2	yes	complemented
38	#381	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂⁴×C₃	C₂³	1	1	yes	characteristic
40	C₂⁴×C₃	C₂³	1	1	yes	characteristic
41	C₂×C₃×D₄	—	2	2	no	—
42	C₂×C₃×D₄	—	2	4	no	—
43	C₂²×C₃×C₄	—	2	4	no	—
44	C₂²×C₃×C₄	—	2	4	no	—
45	C₂²×C₃×C₄	—	4	4	no	—
46	C₂²×C₃×C₄	D₄	1	4	yes	—
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	4	yes	—
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂³	1	4	yes	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₃×C₄⋊C₄	—	2	4	no	—
55	C₃×C₄⋊C₄	—	2	4	no	—
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	4	4	no	—
59	C₃×C₂²⋊C₄	—	2	2	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	2	2	no	—
62	C₃×C₂²⋊C₄	—	2	2	no	—
63	C₃×C₂²⋊C₄	—	4	4	no	—
64	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
65	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂³×C₄	C₂²×C₃	1	2	yes	—
67	#123	—	2	4	no	—
68	C₂×C₄⋊C₄	—	2	4	no	—
69	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
79	#95	C₂²×C₃	1	2	yes	—
80	#95	C₂²×C₃	1	2	yes	—
81	#95	C₂²×C₃	1	4	yes	—
82	#95	C₂²×C₃	1	4	yes	—
83	C₂³×C₃	—	2	2	no	—
84	C₂³×C₃	—	2	2	no	—
85	C₂³×C₃	—	2	1	no	—
86	C₂³×C₃	—	2	2	no	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	—	2	2	no	—
89	C₂³×C₃	C₂×D₄	1	2	yes	—
90	C₂³×C₃	#41	1	4	yes	—
91	C₂³×C₃	C₂⁴	1	1	yes	characteristic
92	C₃×D₄	—	4	4	no	—
93	C₂×C₃×C₄	—	2	8	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	2	1	no	—
98	C₂×C₃×C₄	—	2	4	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	4	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	—	4	8	no	—
109	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
110	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
111	C₂⁴	C₂³×C₃	1	1	yes	characteristic
112	C₂⁴	C₂³×C₃	1	1	yes	characteristic
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	2	4	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	4	4	no	—
118	C₂²×C₄	C₃×D₄	1	4	yes	—
119	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂²×C₄	C₂³×C₃	1	4	yes	—
121	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂²×C₄	C₂³×C₃	1	4	yes	—
123	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	4	no	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	4	4	no	—
136	C₂²×C₃	—	2	2	no	—
137	C₂²×C₃	—	2	2	no	—
138	C₂²×C₃	—	2	8	no	—
139	C₂²×C₃	—	4	1	no	—
140	C₂²×C₃	—	4	1	no	—
141	C₂²×C₃	—	4	1	no	—
142	C₂²×C₃	—	4	4	no	—
143	C₂²×C₃	#123	1	4	yes	—
144	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
145	C₂²×C₃	C₂×#41	1	2	yes	—
146	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
147	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
148	C₃×C₄	—	2	2	no	—
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	4	1	no	—
151	C₃×C₄	—	4	1	no	—
152	C₃×C₄	—	4	4	no	—
153	C₃×C₄	—	4	1	no	—
154	C₃×C₄	—	4	4	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	1	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	C₂×C₃×D₄	1	2	yes	—
162	C₂³	C₃×#41	1	4	yes	—
163	C₂³	C₂⁴×C₃	1	1	yes	characteristic
164	D₄	—	4	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₃	—	2	4	no	—
184	C₂×C₃	—	8	1	no	—
185	C₂×C₃	C₂×#123	1	1	yes	characteristic
186	C₂×C₃	#535	1	1	yes	characteristic
187	C₂×C₃	#536	1	1	yes	characteristic
188	C₂×C₃	#539	1	4	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	8	no	—
192	C₂²	—	4	1	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	4	4	no	—
196	C₂²	C₃×#123	1	4	yes	—
197	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₂²	C₂×C₃×#41	1	2	yes	—
199	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
200	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	4	no	—
205	C₄	—	4	1	no	—
206	C₄	—	4	1	no	—
207	C₄	—	4	4	no	—
208	C₃	#2623	1	1	yes	characteristic, direct product factor, Sylow
209	C₂	—	2	4	no	—
210	C₂	—	8	1	no	—
211	C₂	C₂×C₃×#123	1	1	yes	characteristic
212	C₂	C₃×#535	1	1	yes	characteristic
213	C₂	C₃×#536	1	1	yes	characteristic
214	C₂	C₃×#539	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²
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. C₂²×C₃×C₄⋊C₄	C₂	semidirect
3. C₃×#400	C₂	semidirect
4. C₃×#399	C₂	semidirect
5. C₃×#398	C₂	semidirect
6. C₃×#397	C₂	semidirect
8. C₃×#385	C₂	semidirect
9. C₃×#381	C₂	semidirect
10. #2623	C₃	direct (N × H)
31. C₂²×C₄⋊C₄	C₂×C₃	semidirect
32. #400	C₂×C₃	semidirect
33. #399	C₂×C₃	semidirect
34. #398	C₂×C₃	semidirect
35. #397	C₂×C₃	semidirect
37. #385	C₂×C₃	semidirect
38. #381	C₂×C₃	semidirect
208. C₃	#2623	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×#123	2
C₂²×C₃×C₄⋊C₄	2
C₃×#400	2
C₃×#399	2
C₃×#398	2
C₃×#397	2
C₃×#386	2
C₃×#385	2
C₃×#381	2
#2623	3

Subgroups — C3×#2623 #84487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2623 #84487