Subgroups — C₃×#2841 #84705

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 222 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	2	yes	maximal, complemented
2	C₃×#399	C₂	1	1	yes	maximal, characteristic
3	C₃×#398	C₂	1	2	yes	maximal, complemented
4	C₃×#398	C₂	1	2	yes	maximal, complemented
5	C₃×#396	C₂	1	2	yes	maximal
6	C₃×#395	C₂	1	2	yes	maximal, complemented
7	C₃×#393	C₂	1	1	yes	maximal, characteristic
8	C₃×#381	C₂	1	1	yes	maximal, characteristic
9	C₃×#380	C₂	1	2	yes	maximal
10	#2841	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₃×#95	C₂²	1	2	yes	—
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	1	yes	characteristic
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	1	yes	characteristic
32	#400	C₂×C₃	1	2	yes	complemented
33	#399	C₂×C₃	1	1	yes	characteristic
34	#398	C₂×C₃	1	2	yes	complemented
35	#398	C₂×C₃	1	2	yes	complemented
36	#396	C₂×C₃	1	2	yes	—
37	#395	C₂×C₃	1	2	yes	complemented
38	#393	C₂×C₃	1	1	yes	characteristic
39	#381	C₂×C₃	1	1	yes	characteristic
40	#380	C₂×C₃	1	2	yes	—
41	C₂⁴×C₃	C₂³	1	1	yes	characteristic
42	C₂×C₃×Q₈	—	4	1	no	—
43	C₂×C₃×Q₈	—	4	2	no	—
44	C₂²×C₃×C₄	—	4	1	no	—
45	C₂²×C₃×C₄	—	4	1	no	—
46	C₂²×C₃×C₄	Q₈	1	4	yes	complemented
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	2	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	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₃×C₄⋊C₄	—	4	2	no	—
56	C₃×C₄⋊C₄	—	4	2	no	—
57	C₃×C₄⋊C₄	—	4	2	no	—
58	C₃×C₄⋊C₄	—	4	2	no	—
59	C₃×C₄⋊C₄	—	4	1	no	—
60	C₃×C₄⋊C₄	—	4	2	no	—
61	C₃×C₄⋊C₄	—	4	2	no	—
62	C₃×C₂²⋊C₄	—	4	2	no	—
63	C₃×C₂²⋊C₄	—	4	1	no	—
64	C₃×C₂²⋊C₄	—	4	2	no	—
65	C₃×C₂²⋊C₄	—	4	1	no	—
66	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
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	2	yes	—
72	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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	1	yes	characteristic
78	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	#95	C₂²×C₃	1	2	yes	—
80	#95	C₂²×C₃	1	2	yes	—
81	#95	C₂²×C₃	1	2	yes	—
82	#95	C₂²×C₃	1	2	yes	—
83	#95	C₂²×C₃	1	1	yes	characteristic
84	#95	C₂²×C₃	1	2	yes	—
85	#95	C₂²×C₃	1	2	yes	—
86	#95	C₂²×C₃	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
92	C₂³×C₃	C₂⁴	1	1	yes	characteristic
93	C₃×Q₈	—	4	4	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	1	no	—
101	C₂×C₃×C₄	—	2	1	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	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	4	4	no	—
116	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
117	C₂⁴	C₂³×C₃	1	1	yes	characteristic
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	C₃×Q₈	1	4	yes	complemented
123	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₃	1	2	yes	—
125	C₂²×C₄	C₂³×C₃	1	2	yes	—
126	C₂²×C₄	C₂³×C₃	1	2	yes	—
127	C₂²×C₄	C₂³×C₃	1	2	yes	—
128	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₂²×C₄	C₂³×C₃	1	2	yes	—
130	C₂²×C₄	C₂³×C₃	1	2	yes	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₂²⋊C₄	—	4	2	no	—
139	C₂²⋊C₄	—	4	1	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₂²⋊C₄	—	4	1	no	—
142	C₂²×C₃	—	2	2	no	—
143	C₂²×C₃	—	2	4	no	—
144	C₂²×C₃	—	4	1	no	—
145	C₂²×C₃	—	4	2	no	—
146	C₂²×C₃	—	4	1	no	—
147	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
148	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
151	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
152	C₃×C₄	—	2	4	no	—
153	C₃×C₄	—	4	2	no	—
154	C₃×C₄	—	4	1	no	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	—	4	2	no	—
157	C₃×C₄	—	4	2	no	—
158	C₃×C₄	—	4	2	no	—
159	C₃×C₄	—	4	1	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	1	no	—
162	C₂³	—	2	1	no	—
163	C₂³	—	2	2	no	—
164	C₂³	C₂×C₃×Q₈	1	2	yes	—
165	C₂³	C₂⁴×C₃	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	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	—	4	4	no	—
189	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
190	C₂×C₃	—	4	2	no	—
191	C₂×C₃	#543	1	1	yes	characteristic
192	C₂×C₃	#544	1	2	yes	—
193	C₂×C₃	#560	1	1	yes	characteristic
194	C₂×C₃	#562	1	2	yes	—
195	C₂×C₃	#563	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	4	1	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
202	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
203	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
205	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	1	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	1	no	—
214	C₃	#2841	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	4	2	no	—
216	C₂	C₃×#543	1	1	yes	characteristic
217	C₂	C₃×#544	1	2	yes	—
218	C₂	C₃×#560	1	1	yes	characteristic
219	C₂	C₃×#562	1	2	yes	—
220	C₂	C₃×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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
3. C₃×#398	C₂	semidirect
4. C₃×#398	C₂	semidirect
6. C₃×#395	C₂	semidirect
10. #2841	C₃	direct (N × H)
32. #400	C₂×C₃	semidirect
34. #398	C₂×C₃	semidirect
35. #398	C₂×C₃	semidirect
37. #395	C₂×C₃	semidirect
46. C₂²×C₃×C₄	Q₈	semidirect
122. C₂²×C₄	C₃×Q₈	semidirect
214. C₃	#2841	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×#399	2
C₃×#398	2
C₃×#396	2
C₃×#395	2
C₃×#393	2
C₃×#381	2
C₃×#380	2
#2841	3

Subgroups — C3×#2841 #84705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2841 #84705