Subgroups — C₃×#2795 #84659

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#396	C₂	1	2	yes	maximal, complemented
3	C₃×#394	C₂	1	1	yes	maximal, characteristic
4	C₃×#394	C₂	1	2	yes	maximal
5	C₃×#388	C₂	1	2	yes	maximal, complemented
6	C₃×#388	C₂	1	2	yes	maximal, complemented
7	C₃×#388	C₂	1	2	yes	maximal, complemented
8	C₃×#387	C₂	1	2	yes	maximal, complemented
9	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	#2795	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₂²⋊C₄	C₂²	1	2	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	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	1	yes	characteristic
27	C₃×#95	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	1	yes	characteristic
31	C₃×#95	C₂²	1	2	yes	—
32	#400	C₂×C₃	1	1	yes	characteristic, complemented
33	#396	C₂×C₃	1	2	yes	complemented
34	#394	C₂×C₃	1	1	yes	characteristic
35	#394	C₂×C₃	1	2	yes	—
36	#388	C₂×C₃	1	2	yes	complemented
37	#388	C₂×C₃	1	2	yes	complemented
38	#388	C₂×C₃	1	2	yes	complemented
39	#387	C₂×C₃	1	2	yes	complemented
40	#384	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂⁴×C₃	C₂³	1	2	yes	—
42	C₂×C₃×D₄	—	4	1	no	—
43	C₂×C₃×D₄	—	4	2	no	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	2	yes	—
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₃×C₄⋊C₄	—	2	4	no	—
53	C₃×C₄⋊C₄	—	4	2	no	—
54	C₃×C₄⋊C₄	—	4	1	no	—
55	C₃×C₄⋊C₄	—	4	2	no	—
56	C₃×C₄⋊C₄	—	4	1	no	—
57	C₃×C₄⋊C₄	—	4	1	no	—
58	C₃×C₂²⋊C₄	—	2	4	no	—
59	C₃×C₂²⋊C₄	—	2	4	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	4	2	no	—
62	C₃×C₂²⋊C₄	—	4	2	no	—
63	C₃×C₂²⋊C₄	—	4	2	no	—
64	C₃×C₄²	—	2	2	no	—
65	C₃×C₄²	—	4	2	no	—
66	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
69	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₄²	C₂²×C₃	1	2	yes	—
81	#95	C₂²×C₃	1	1	yes	characteristic
82	#95	C₂²×C₃	1	2	yes	—
83	#95	C₂²×C₃	1	2	yes	—
84	#95	C₂²×C₃	1	2	yes	—
85	#95	C₂²×C₃	1	1	yes	characteristic
86	#95	C₂²×C₃	1	2	yes	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	—	2	2	no	—
89	C₂³×C₃	—	2	2	no	—
90	C₂³×C₃	—	2	2	no	—
91	C₂³×C₃	—	2	2	no	—
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	C₂⁴	1	1	yes	characteristic
95	C₃×D₄	—	4	4	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	1	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	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	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	#41	1	2	yes	—
117	C₂×C₃×C₄	#41	1	4	yes	—
118	C₂⁴	C₂³×C₃	1	2	yes	—
119	C₂×D₄	—	4	1	no	—
120	C₂×D₄	—	4	2	no	—
121	C₂²×C₄	C₂³×C₃	1	2	yes	—
122	C₂²×C₄	C₂³×C₃	1	2	yes	—
123	C₂²×C₄	C₂³×C₃	1	2	yes	—
124	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₃	1	2	yes	—
126	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₃	1	2	yes	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₂²⋊C₄	—	2	4	no	—
138	C₂²⋊C₄	—	4	2	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	4	2	no	—
143	C₂²×C₃	—	4	2	no	—
144	C₂²×C₃	—	4	2	no	—
145	C₂²×C₃	—	4	2	no	—
146	C₂²×C₃	—	4	2	no	—
147	C₂²×C₃	—	4	2	no	—
148	C₂²×C₃	—	4	2	no	—
149	C₂²×C₃	—	4	2	no	—
150	C₂²×C₃	C₂×#41	1	1	yes	characteristic
151	C₂²×C₃	C₂×#41	1	2	yes	—
152	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
154	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	4	2	no	—
158	C₃×C₄	—	4	2	no	—
159	C₃×C₄	—	4	2	no	—
160	C₃×C₄	—	4	2	no	—
161	C₃×C₄	—	4	1	no	—
162	C₃×C₄	—	4	1	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	C₂⁴×C₃	1	1	yes	characteristic
171	D₄	—	4	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	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	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₃×#41	1	2	yes	—
193	C₂×C₄	C₃×#41	1	4	yes	—
194	C₂×C₃	—	8	2	no	—
195	C₂×C₃	#539	1	1	yes	characteristic
196	C₂×C₃	#542	1	2	yes	—
197	C₂×C₃	#553	1	2	yes	—
198	C₂×C₃	#555	1	1	yes	characteristic
199	C₂×C₃	#560	1	1	yes	characteristic
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	C₂×C₃×#41	1	1	yes	characteristic
208	C₂²	C₂×C₃×#41	1	2	yes	—
209	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
210	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
211	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	4	no	—
214	C₄	—	4	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₃	#2795	1	1	yes	characteristic, direct product factor, Sylow
221	C₂	—	8	2	no	—
222	C₂	C₃×#539	1	1	yes	characteristic
223	C₂	C₃×#542	1	2	yes	—
224	C₂	C₃×#553	1	2	yes	—
225	C₂	C₃×#555	1	1	yes	characteristic
226	C₂	C₃×#560	1	1	yes	characteristic

	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
1. C₃×#400	C₂	semidirect
2. C₃×#396	C₂	semidirect
5. C₃×#388	C₂	semidirect
6. C₃×#388	C₂	semidirect
7. C₃×#388	C₂	semidirect
8. C₃×#387	C₂	semidirect
9. C₃×#384	C₂	semidirect
10. #2795	C₃	direct (N × H)
32. #400	C₂×C₃	semidirect
33. #396	C₂×C₃	semidirect
36. #388	C₂×C₃	semidirect
37. #388	C₂×C₃	semidirect
38. #388	C₂×C₃	semidirect
39. #387	C₂×C₃	semidirect
40. #384	C₂×C₃	semidirect
220. C₃	#2795	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×#400	2
C₃×#396	2
C₃×#394	2
C₃×#388	2
C₃×#387	2
C₃×#384	2
#2795	3

Subgroups — C3×#2795 #84659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2795 #84659