Subgroups — C₃×#2727 #84591

Order 384.

476 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#126	C₂	1	2	yes	maximal, complemented
2	C₃×#399	C₂	1	2	yes	maximal, complemented
3	C₃×#396	C₂	1	2	yes	maximal, complemented
4	C₃×#394	C₂	1	2	yes	maximal
5	C₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₃×#389	C₂	1	2	yes	maximal, complemented
7	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#388	C₂	1	2	yes	maximal, complemented
10	#2727	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₃×#126	—	2	8	no	—
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	1	yes	characteristic
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	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄²	C₂²	1	2	yes	—
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	2	yes	—
32	C₃×#95	C₂²	1	2	yes	—
33	C₂×#126	C₂×C₃	1	2	yes	complemented
34	#399	C₂×C₃	1	2	yes	complemented
35	#396	C₂×C₃	1	2	yes	complemented
36	#394	C₂×C₃	1	2	yes	—
37	#390	C₂×C₃	1	1	yes	characteristic
38	#389	C₂×C₃	1	2	yes	complemented
39	#389	C₂×C₃	1	1	yes	characteristic, complemented
40	#389	C₂×C₃	1	1	yes	characteristic, complemented
41	#388	C₂×C₃	1	2	yes	complemented
42	C₂⁴×C₃	C₂³	1	2	yes	—
43	C₂×C₃×D₄	—	2	2	no	—
44	C₂×C₃×D₄	—	4	1	no	—
45	C₂×C₃×D₄	—	4	1	no	—
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
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	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	4	no	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	4	1	no	—
59	C₃×C₄⋊C₄	—	4	1	no	—
60	C₃×C₄⋊C₄	—	4	1	no	—
61	C₃×C₄⋊C₄	—	4	2	no	—
62	C₃×C₄⋊C₄	D₄	1	4	yes	complemented
63	C₃×C₂²⋊C₄	—	2	4	no	—
64	C₃×C₂²⋊C₄	—	2	4	no	—
65	C₃×C₂²⋊C₄	—	2	4	no	—
66	C₃×C₂²⋊C₄	—	2	4	no	—
67	C₃×C₂²⋊C₄	—	4	2	no	—
68	C₃×C₂²⋊C₄	—	4	2	no	—
69	C₃×C₄²	—	2	4	no	—
70	C₃×C₄²	—	4	1	no	—
71	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
72	#126	—	2	8	no	—
73	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₄²	C₂²×C₃	1	2	yes	—
89	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
90	#95	C₂²×C₃	1	2	yes	—
91	#95	C₂²×C₃	1	2	yes	—
92	#95	C₂²×C₃	1	2	yes	—
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	—	2	2	no	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	C₂⁴	1	1	yes	characteristic
101	C₃×D₄	—	4	4	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	1	no	—
104	C₂×C₃×C₄	—	2	1	no	—
105	C₂×C₃×C₄	—	2	1	no	—
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	2	1	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	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
123	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₄	#41	1	2	yes	—
125	C₂×C₃×C₄	#41	1	2	yes	—
126	C₂⁴	C₂³×C₃	1	2	yes	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	4	1	no	—
129	C₂×D₄	—	4	1	no	—
130	C₂²×C₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₃	1	2	yes	—
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	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	C₃×D₄	1	4	yes	complemented
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	4	2	no	—
152	C₂²⋊C₄	—	4	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	1	no	—
155	C₂²×C₃	—	4	2	no	—
156	C₂²×C₃	—	4	2	no	—
157	C₂²×C₃	—	4	2	no	—
158	C₂²×C₃	—	4	2	no	—
159	C₂²×C₃	—	4	2	no	—
160	C₂²×C₃	—	4	2	no	—
161	C₂²×C₃	—	4	2	no	—
162	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
163	C₂²×C₃	C₂×#41	1	1	yes	characteristic
164	C₂²×C₃	C₂×#41	1	1	yes	characteristic
165	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	4	2	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	—	4	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	C₂⁴×C₃	1	1	yes	characteristic
185	D₄	—	4	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	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	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
207	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
208	C₂×C₄	C₃×#41	1	2	yes	—
209	C₂×C₄	C₃×#41	1	2	yes	—
210	C₂×C₃	—	8	2	no	—
211	C₂×C₃	#537	1	1	yes	characteristic
212	C₂×C₃	#538	1	1	yes	characteristic
213	C₂×C₃	#539	1	1	yes	characteristic
214	C₂×C₃	#547	1	1	yes	characteristic
215	C₂×C₃	#550	1	1	yes	characteristic
216	C₂×C₃	#556	1	1	yes	characteristic
217	C₂×C₃	#563	1	1	yes	characteristic
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
226	C₂²	C₂×C₃×#41	1	1	yes	characteristic
227	C₂²	C₂×C₃×#41	1	1	yes	characteristic
228	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
230	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
231	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₃	#2727	1	1	yes	characteristic, direct product factor, Sylow
241	C₂	—	8	2	no	—
242	C₂	C₃×#537	1	1	yes	characteristic
243	C₂	C₃×#538	1	1	yes	characteristic
244	C₂	C₃×#539	1	1	yes	characteristic
245	C₂	C₃×#547	1	1	yes	characteristic
246	C₂	C₃×#550	1	1	yes	characteristic
247	C₂	C₃×#556	1	1	yes	characteristic
248	C₂	C₃×#563	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. C₂×C₃×#126	C₂	semidirect
2. C₃×#399	C₂	semidirect
3. C₃×#396	C₂	semidirect
6. C₃×#389	C₂	semidirect
7. C₃×#389	C₂	semidirect
8. C₃×#389	C₂	semidirect
9. C₃×#388	C₂	semidirect
10. #2727	C₃	direct (N × H)
33. C₂×#126	C₂×C₃	semidirect
34. #399	C₂×C₃	semidirect
35. #396	C₂×C₃	semidirect
38. #389	C₂×C₃	semidirect
39. #389	C₂×C₃	semidirect
40. #389	C₂×C₃	semidirect
41. #388	C₂×C₃	semidirect
62. C₃×C₄⋊C₄	D₄	semidirect
146. C₄⋊C₄	C₃×D₄	semidirect
240. C₃	#2727	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×#126	2
C₃×#399	2
C₃×#396	2
C₃×#394	2
C₃×#390	2
C₃×#389	2
C₃×#388	2
#2727	3

Subgroups — C3×#2727 #84591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2727 #84591