Subgroups — C₃×#2318 #84182

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#388	C₂	1	2	yes	maximal, complemented
3	C₃×#387	C₂	1	2	yes	maximal, complemented
4	C₃×#386	C₂	1	2	yes	maximal
5	C₃×#385	C₂	1	2	yes	maximal, complemented
6	C₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
12	#2318	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	2	yes	—
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₄	C₄	1	8	yes	complemented
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂²	1	2	yes	—
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₃×#95	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
33	C₃×#95	C₂²	1	2	yes	—
34	C₃×#95	C₂²	1	2	yes	—
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
38	#388	C₂×C₃	1	2	yes	complemented
39	#387	C₂×C₃	1	2	yes	complemented
40	#386	C₂×C₃	1	2	yes	—
41	#385	C₂×C₃	1	2	yes	complemented
42	#383	C₂×C₃	1	1	yes	characteristic, complemented
43	#382	C₂×C₃	1	1	yes	characteristic, complemented
44	#381	C₂×C₃	1	1	yes	characteristic, complemented
45	#380	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×#95	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂⁴×C₃	C₂³	1	1	yes	characteristic
49	C₂⁴×C₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×D₄	—	2	4	no	—
51	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
52	C₂²×C₃×C₄	—	2	4	no	—
53	C₂²×C₃×C₄	—	4	2	no	—
54	C₂²×C₃×C₄	—	4	2	no	—
55	C₂²×C₃×C₄	—	4	2	no	—
56	C₂²×C₃×C₄	—	4	2	no	—
57	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
58	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	C₂³	1	2	yes	—
61	C₂²×C₃×C₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₃×C₄⋊C₄	—	4	2	no	—
69	C₃×C₄⋊C₄	—	4	2	no	—
70	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
71	C₃×C₂²⋊C₄	—	2	4	no	—
72	C₃×C₂²⋊C₄	—	4	2	no	—
73	C₃×C₂²⋊C₄	—	4	2	no	—
74	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
75	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	complemented
76	C₃×C₄²	—	4	2	no	—
77	C₃×C₄²	C₂×C₄	1	4	yes	complemented
78	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₂³×C₄	C₂²×C₃	1	2	yes	—
80	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
81	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
82	C₄×D₄	C₃×C₄	1	8	yes	complemented
83	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₃	1	2	yes	—
92	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
93	#95	C₂²×C₃	1	1	yes	characteristic
94	#95	C₂²×C₃	1	1	yes	characteristic
95	#95	C₂²×C₃	1	2	yes	—
96	#95	C₂²×C₃	1	1	yes	characteristic
97	#95	C₂²×C₃	1	1	yes	characteristic
98	#95	C₂²×C₃	1	2	yes	—
99	#95	C₂²×C₃	1	2	yes	—
100	#95	C₂²×C₃	1	2	yes	—
101	#95	C₂²×C₃	1	2	yes	—
102	C₂³×C₃	—	2	2	no	—
103	C₂³×C₃	—	2	1	no	—
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	—	2	2	no	—
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	—	2	2	no	—
108	C₂³×C₃	—	2	1	no	—
109	C₂³×C₃	C₂²×C₄	1	2	yes	—
110	C₂³×C₃	C₂²×C₄	1	2	yes	—
111	C₂³×C₃	#41	1	4	yes	—
112	C₂³×C₃	C₂⁴	1	1	yes	characteristic
113	C₃×D₄	—	2	8	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	4	4	no	—
133	C₂×C₃×C₄	—	4	4	no	—
134	C₂×C₃×C₄	—	4	8	no	—
135	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
136	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
137	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
138	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
139	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
140	C₂⁴	C₂³×C₃	1	1	yes	characteristic
141	C₂⁴	C₂³×C₃	1	1	yes	characteristic
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
150	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
151	C₂²×C₄	C₂³×C₃	1	2	yes	—
152	C₂²×C₄	C₂³×C₃	1	2	yes	—
153	C₂²×C₄	C₂³×C₃	1	2	yes	—
154	C₂²×C₄	C₂³×C₃	1	2	yes	—
155	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	4	2	no	—
165	C₂²⋊C₄	—	4	2	no	—
166	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
167	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
168	C₄²	—	4	2	no	—
169	C₄²	C₂×C₃×C₄	1	4	yes	complemented
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	4	no	—
173	C₂²×C₃	—	2	4	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	—	4	2	no	—
179	C₂²×C₃	#117	1	4	yes	—
180	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
181	C₂²×C₃	C₂×#41	1	2	yes	—
182	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
183	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	4	2	no	—
191	C₃×C₄	—	4	2	no	—
192	C₃×C₄	—	4	2	no	—
193	C₃×C₄	—	4	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₂²×C₃×C₄	1	2	yes	—
202	C₂³	C₂²×C₃×C₄	1	2	yes	—
203	C₂³	C₃×#41	1	4	yes	—
204	C₂³	C₂⁴×C₃	1	1	yes	characteristic
205	D₄	—	2	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
228	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
229	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
230	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
231	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
232	C₂×C₃	—	2	4	no	—
233	C₂×C₃	—	4	2	no	—
234	C₂×C₃	C₂×#117	1	1	yes	characteristic
235	C₂×C₃	#518	1	1	yes	characteristic
236	C₂×C₃	#520	1	1	yes	characteristic
237	C₂×C₃	#538	1	2	yes	—
238	C₂×C₃	#539	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	C₃×#117	1	4	yes	—
249	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
250	C₂²	C₂×C₃×#41	1	2	yes	—
251	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
252	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
253	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	2	no	—
263	C₃	#2318	1	1	yes	characteristic, direct product factor, Sylow
264	C₂	—	2	4	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×C₃×#117	1	1	yes	characteristic
267	C₂	C₃×#518	1	1	yes	characteristic
268	C₂	C₃×#520	1	1	yes	characteristic
269	C₂	C₃×#538	1	2	yes	—
270	C₂	C₃×#539	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×#388	C₂	semidirect
3. C₃×#387	C₂	semidirect
5. C₃×#385	C₂	semidirect
6. C₃×#383	C₂	semidirect
7. C₃×#382	C₂	semidirect
8. C₃×#381	C₂	semidirect
9. C₃×#380	C₂	semidirect
10. C₃×C₄×C₂²⋊C₄	C₂	semidirect
11. C₂×C₃×#95	C₂	semidirect
12. #2318	C₃	direct (N × H)
17. C₃×C₄×D₄	C₄	semidirect
38. #388	C₂×C₃	semidirect
39. #387	C₂×C₃	semidirect
41. #385	C₂×C₃	semidirect
42. #383	C₂×C₃	semidirect
43. #382	C₂×C₃	semidirect
44. #381	C₂×C₃	semidirect
45. #380	C₂×C₃	semidirect
46. C₄×C₂²⋊C₄	C₂×C₃	semidirect
47. C₂×#95	C₂×C₃	semidirect
58. C₂²×C₃×C₄	C₂×C₄	semidirect
70. C₃×C₄⋊C₄	C₂×C₄	semidirect
75. C₃×C₂²⋊C₄	C₂×C₄	semidirect
77. C₃×C₄²	C₂×C₄	semidirect
82. C₄×D₄	C₃×C₄	semidirect
150. C₂²×C₄	C₂×C₃×C₄	semidirect
162. C₄⋊C₄	C₂×C₃×C₄	semidirect
167. C₂²⋊C₄	C₂×C₃×C₄	semidirect
169. C₄²	C₂×C₃×C₄	semidirect
263. C₃	#2318	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×C₄×D₄	2
C₃×#388	2
C₃×#387	2
C₃×#386	2
C₃×#385	2
C₃×#383	2
C₃×#382	2
C₃×#381	2
C₃×#380	2
C₃×C₄×C₂²⋊C₄	2
C₂×C₃×#95	2
#2318	3

Subgroups — C3×#2318 #84182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2318 #84182