Subgroups — C₃×#2339 #84203

Order 384.

928 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
3	C₃×#391	C₂	1	2	yes	maximal, complemented
4	C₃×#386	C₂	1	4	yes	maximal, complemented
5	C₃×#381	C₂	1	2	yes	maximal, complemented
6	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#95	C₂	1	2	yes	maximal, complemented
9	#2339	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	—	2	8	no	—
11	C₂×C₃×#41	C₄	1	8	yes	complemented
12	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	4	yes	—
16	C₂³×C₃×C₄	C₂²	1	2	yes	—
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
18	C₃×C₄×D₄	—	2	8	no	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₃×#95	C₂²	1	1	yes	characteristic
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	4	yes	—
27	C₃×#95	C₂²	1	1	yes	characteristic
28	C₃×#95	C₂²	1	4	yes	—
29	C₂²×#41	C₂×C₃	1	1	yes	characteristic
30	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
31	#391	C₂×C₃	1	2	yes	complemented
32	#386	C₂×C₃	1	4	yes	complemented
33	#381	C₂×C₃	1	2	yes	complemented
34	#380	C₂×C₃	1	1	yes	characteristic, complemented
35	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#95	C₂×C₃	1	2	yes	complemented
37	C₂⁴×C₃	C₂³	1	1	yes	characteristic
38	C₂⁴×C₃	C₂³	1	2	yes	—
39	C₃×#41	—	2	32	no	—
40	C₂×C₃×Q₈	—	2	4	no	—
41	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
42	C₂×C₃×D₄	—	2	8	no	—
43	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
44	C₂×C₃×D₄	C₂×C₄	1	8	yes	complemented
45	C₂×C₃×D₄	D₄	1	8	yes	—
46	C₂²×C₃×C₄	—	2	8	no	—
47	C₂²×C₃×C₄	—	2	4	no	—
48	C₂²×C₃×C₄	—	2	8	no	—
49	C₂²×C₃×C₄	—	2	8	no	—
50	C₂²×C₃×C₄	—	4	4	no	—
51	C₂²×C₃×C₄	C₂×C₄	1	8	yes	complemented
52	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	4	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₃×C₄⋊C₄	—	2	4	no	—
60	C₃×C₂²⋊C₄	—	2	8	no	—
61	C₃×C₂²⋊C₄	—	2	4	no	—
62	C₃×C₂²⋊C₄	—	4	2	no	—
63	C₃×C₄²	—	2	4	no	—
64	C₂×#41	—	2	8	no	—
65	C₂×#41	C₃×C₄	1	8	yes	complemented
66	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂²×D₄	C₂²×C₃	1	2	yes	—
68	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₂³×C₄	C₂²×C₃	1	4	yes	—
70	C₂³×C₄	C₂²×C₃	1	2	yes	—
71	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
72	C₄×D₄	—	2	8	no	—
73	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
75	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₄²	C₂²×C₃	1	2	yes	—
78	#95	C₂²×C₃	1	1	yes	characteristic
79	#95	C₂²×C₃	1	2	yes	—
80	#95	C₂²×C₃	1	4	yes	—
81	#95	C₂²×C₃	1	1	yes	characteristic
82	#95	C₂²×C₃	1	4	yes	—
83	C₂³×C₃	—	2	4	no	—
84	C₂³×C₃	—	2	2	no	—
85	C₂³×C₃	—	2	2	no	—
86	C₂³×C₃	—	2	2	no	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	—	2	1	no	—
89	C₂³×C₃	—	2	1	no	—
90	C₂³×C₃	C₂²×C₄	1	2	yes	—
91	C₂³×C₃	C₂²×C₄	1	4	yes	—
92	C₂³×C₃	C₂×D₄	1	8	yes	—
93	C₂³×C₃	C₂⁴	1	1	yes	characteristic
94	C₃×Q₈	—	2	8	no	—
95	C₃×D₄	—	2	16	no	—
96	C₃×D₄	C₂²⋊C₄	1	16	yes	complemented
97	C₂×C₃×C₄	—	2	16	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	8	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	4	16	no	—
110	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
111	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
112	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
113	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
114	C₂×C₃×C₄	#41	1	4	yes	complemented
115	C₂⁴	C₂³×C₃	1	1	yes	characteristic
116	C₂⁴	C₂³×C₃	1	2	yes	—
117	#41	—	2	32	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
120	C₂×D₄	—	2	8	no	—
121	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
122	C₂×D₄	C₂×C₃×C₄	1	8	yes	complemented
123	C₂×D₄	C₃×D₄	1	8	yes	—
124	C₂²×C₄	—	2	8	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	8	no	—
127	C₂²×C₄	—	2	8	no	—
128	C₂²×C₄	—	4	4	no	—
129	C₂²×C₄	C₂×C₃×C₄	1	8	yes	complemented
130	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
131	C₂²×C₄	C₂³×C₃	1	2	yes	—
132	C₂²×C₄	C₂³×C₃	1	4	yes	—
133	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₄⋊C₄	—	2	4	no	—
138	C₂²⋊C₄	—	2	8	no	—
139	C₂²⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	2	4	no	—
142	C₂²×C₃	—	2	4	no	—
143	C₂²×C₃	—	2	4	no	—
144	C₂²×C₃	—	2	8	no	—
145	C₂²×C₃	—	2	8	no	—
146	C₂²×C₃	—	2	2	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	4	2	no	—
150	C₂²×C₃	—	4	2	no	—
151	C₂²×C₃	C₂×C₂²⋊C₄	1	8	yes	—
152	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
153	C₂²×C₃	C₂²×D₄	1	2	yes	—
154	C₂²×C₃	C₂×#41	1	2	yes	—
155	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	2	4	no	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	—	4	4	no	—
161	C₃×C₄	—	4	2	no	—
162	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	—
163	C₂³	—	2	4	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	C₂²×C₃×C₄	1	2	yes	—
171	C₂³	C₂²×C₃×C₄	1	4	yes	—
172	C₂³	C₂×C₃×D₄	1	8	yes	—
173	C₂³	C₂⁴×C₃	1	1	yes	characteristic
174	Q₈	—	2	8	no	—
175	D₄	—	2	16	no	—
176	D₄	C₃×C₂²⋊C₄	1	16	yes	complemented
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	16	no	—
190	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
191	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
192	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
193	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
194	C₂×C₄	C₃×#41	1	4	yes	complemented
195	C₂×C₃	—	2	8	no	—
196	C₂×C₃	—	4	2	no	—
197	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₃	C₄×#41	1	1	yes	characteristic
199	C₂×C₃	#520	1	1	yes	characteristic
200	C₂×C₃	#546	1	2	yes	—
201	C₂×C₃	#547	1	2	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	8	no	—
205	C₂²	—	2	8	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	C₂×C₃×C₂²⋊C₄	1	8	yes	—
212	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
213	C₂²	C₂²×C₃×D₄	1	2	yes	—
214	C₂²	C₂×C₃×#41	1	2	yes	—
215	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	—	2	4	no	—
220	C₄	—	4	4	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
223	C₃	#2339	1	1	yes	characteristic, direct product factor, Sylow
224	C₂	—	2	8	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
227	C₂	C₃×C₄×#41	1	1	yes	characteristic
228	C₂	C₃×#520	1	1	yes	characteristic
229	C₂	C₃×#546	1	2	yes	—
230	C₂	C₃×#547	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×#41	C₂
LCS	G ▷ C₂² ▷ 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×C₄×D₄	C₂	semidirect
3. C₃×#391	C₂	semidirect
4. C₃×#386	C₂	semidirect
5. C₃×#381	C₂	semidirect
6. C₃×#380	C₂	semidirect
7. C₃×C₄×C₂²⋊C₄	C₂	semidirect
8. C₂×C₃×#95	C₂	semidirect
9. #2339	C₃	direct (N × H)
11. C₂×C₃×#41	C₄	semidirect
30. C₂×C₄×D₄	C₂×C₃	semidirect
31. #391	C₂×C₃	semidirect
32. #386	C₂×C₃	semidirect
33. #381	C₂×C₃	semidirect
34. #380	C₂×C₃	semidirect
35. C₄×C₂²⋊C₄	C₂×C₃	semidirect
36. C₂×#95	C₂×C₃	semidirect
41. C₂×C₃×Q₈	C₂×C₄	semidirect
44. C₂×C₃×D₄	C₂×C₄	semidirect
51. C₂²×C₃×C₄	C₂×C₄	semidirect
52. C₂²×C₃×C₄	C₂×C₄	semidirect
65. C₂×#41	C₃×C₄	semidirect
96. C₃×D₄	C₂²⋊C₄	semidirect
114. C₂×C₃×C₄	#41	semidirect
119. C₂×Q₈	C₂×C₃×C₄	semidirect
122. C₂×D₄	C₂×C₃×C₄	semidirect
129. C₂²×C₄	C₂×C₃×C₄	semidirect
130. C₂²×C₄	C₂×C₃×C₄	semidirect
176. D₄	C₃×C₂²⋊C₄	semidirect
194. C₂×C₄	C₃×#41	semidirect
223. C₃	#2339	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×#41	2
C₂×C₃×C₄×D₄	2
C₃×#391	2
C₃×#386	2
C₃×#381	2
C₃×#380	2
C₃×C₄×C₂²⋊C₄	2
C₂×C₃×#95	2
#2339	3

Subgroups — C3×#2339 #84203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2339 #84203