Subgroups — C₃×#2337 #84201

Order 384.

540 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₃×#391	C₂	1	2	yes	maximal, complemented
3	C₃×#388	C₂	1	2	yes	maximal, complemented
4	C₃×#388	C₂	1	2	yes	maximal, complemented
5	C₃×#386	C₂	1	2	yes	maximal
6	C₃×#385	C₂	1	2	yes	maximal, complemented
7	C₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
10	#2337	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₃×#124	C₄	1	8	yes	complemented
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
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×C₄²	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	1	yes	characteristic
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	1	yes	characteristic
32	C₃×#95	C₂²	1	2	yes	—
33	C₂×#124	C₂×C₃	1	1	yes	characteristic
34	#391	C₂×C₃	1	2	yes	complemented
35	#388	C₂×C₃	1	2	yes	complemented
36	#388	C₂×C₃	1	2	yes	complemented
37	#386	C₂×C₃	1	2	yes	—
38	#385	C₂×C₃	1	2	yes	complemented
39	#383	C₂×C₃	1	1	yes	characteristic, complemented
40	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
41	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
42	C₂⁴×C₃	C₂³	1	2	yes	—
43	C₂×C₃×Q₈	—	4	2	no	—
44	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
45	C₂×C₃×D₄	—	4	2	no	—
46	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
47	C₂²×C₃×C₄	—	2	4	no	—
48	C₂²×C₃×C₄	—	4	2	no	—
49	C₂²×C₃×C₄	—	4	2	no	—
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	1	yes	characteristic
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	C₃×C₄⋊C₄	—	2	4	no	—
59	C₃×C₄⋊C₄	—	4	2	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	C₂×C₄	1	8	yes	complemented
64	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
65	C₃×C₄²	—	4	1	no	—
66	C₃×C₄²	—	4	1	no	—
67	C₃×C₄²	—	4	2	no	—
68	C₃×C₄²	C₂×C₄	1	4	yes	complemented
69	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
70	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₂³×C₄	C₂²×C₃	1	2	yes	—
72	#124	C₃×C₄	1	8	yes	complemented
73	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₄⋊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×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₄²	C₂²×C₃	1	2	yes	—
82	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄²	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	2	yes	—
87	#95	C₂²×C₃	1	1	yes	characteristic
88	#95	C₂²×C₃	1	2	yes	—
89	#95	C₂²×C₃	1	1	yes	characteristic
90	#95	C₂²×C₃	1	2	yes	—
91	C₂³×C₃	—	2	2	no	—
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	C₂²×C₄	1	4	yes	—
98	C₂³×C₃	C₂⁴	1	1	yes	characteristic
99	C₃×Q₈	—	4	4	no	—
100	C₃×D₄	—	4	4	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	1	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	1	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	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	4	8	no	—
119	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
122	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
123	C₂×C₃×C₄	#41	1	4	yes	—
124	C₂⁴	C₂³×C₃	1	2	yes	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
127	C₂×D₄	—	4	2	no	—
128	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₂²⋊C₄	—	4	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
146	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
147	C₄²	—	4	1	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	2	no	—
150	C₄²	C₂×C₃×C₄	1	4	yes	complemented
151	C₂²×C₃	—	2	4	no	—
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	—	4	2	no	—
156	C₂²×C₃	—	4	2	no	—
157	C₂²×C₃	—	4	2	no	—
158	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
159	C₂²×C₃	C₂×#41	1	2	yes	—
160	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
161	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	—	4	2	no	—
169	C₃×C₄	—	4	2	no	—
170	C₃×C₄	—	4	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	C₂²×C₃×C₄	1	4	yes	—
178	C₂³	C₂⁴×C₃	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	D₄	—	4	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	1	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	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
200	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
201	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
202	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
203	C₂×C₄	C₃×#41	1	4	yes	—
204	C₂×C₃	—	4	4	no	—
205	C₂×C₃	C₄×#41	1	1	yes	characteristic
206	C₂×C₃	#518	1	1	yes	characteristic
207	C₂×C₃	#520	1	1	yes	characteristic
208	C₂×C₃	#538	1	2	yes	—
209	C₂×C₃	#542	1	2	yes	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
218	C₂²	C₂×C₃×#41	1	2	yes	—
219	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
220	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₃	#2337	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	4	4	no	—
232	C₂	C₃×C₄×#41	1	1	yes	characteristic
233	C₂	C₃×#518	1	1	yes	characteristic
234	C₂	C₃×#520	1	1	yes	characteristic
235	C₂	C₃×#538	1	2	yes	—
236	C₂	C₃×#542	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×D₄	C₂²
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. C₃×#391	C₂	semidirect
3. C₃×#388	C₂	semidirect
4. C₃×#388	C₂	semidirect
6. C₃×#385	C₂	semidirect
7. C₃×#383	C₂	semidirect
8. C₃×C₄×C₄⋊C₄	C₂	semidirect
9. C₃×C₄×C₂²⋊C₄	C₂	semidirect
10. #2337	C₃	direct (N × H)
14. C₃×#124	C₄	semidirect
34. #391	C₂×C₃	semidirect
35. #388	C₂×C₃	semidirect
36. #388	C₂×C₃	semidirect
38. #385	C₂×C₃	semidirect
39. #383	C₂×C₃	semidirect
40. C₄×C₄⋊C₄	C₂×C₃	semidirect
41. C₄×C₂²⋊C₄	C₂×C₃	semidirect
44. C₂×C₃×Q₈	C₂×C₄	semidirect
63. C₃×C₂²⋊C₄	C₂×C₄	semidirect
64. C₃×C₂²⋊C₄	C₂×C₄	semidirect
68. C₃×C₄²	C₂×C₄	semidirect
72. #124	C₃×C₄	semidirect
126. C₂×Q₈	C₂×C₃×C₄	semidirect
145. C₂²⋊C₄	C₂×C₃×C₄	semidirect
146. C₂²⋊C₄	C₂×C₃×C₄	semidirect
150. C₄²	C₂×C₃×C₄	semidirect
230. C₃	#2337	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×#124	2
C₃×#391	2
C₃×#388	2
C₃×#386	2
C₃×#385	2
C₃×#383	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2337	3

Subgroups — C3×#2337 #84201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2337 #84201