Subgroups — C₃×#2496 #84360

Order 384.

468 conjugacy classes of subgroups.

The subgroup lattice has 274 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
2	C₃×#400	C₂	1	2	yes	maximal, complemented
3	C₃×#397	C₂	1	2	yes	maximal
4	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#387	C₂	1	2	yes	maximal, complemented
7	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₃×#380	C₂	1	1	yes	maximal, characteristic
10	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	#2496	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₃×#126	—	2	8	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	1	yes	characteristic
31	C₃×#95	C₂²	1	1	yes	characteristic
32	C₃×#95	C₂²	1	2	yes	—
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×#126	C₂×C₃	1	2	yes	—
38	#400	C₂×C₃	1	2	yes	complemented
39	#397	C₂×C₃	1	2	yes	—
40	#389	C₂×C₃	1	1	yes	characteristic, complemented
41	#389	C₂×C₃	1	1	yes	characteristic, complemented
42	#387	C₂×C₃	1	2	yes	complemented
43	#384	C₂×C₃	1	1	yes	characteristic, complemented
44	#380	C₂×C₃	1	1	yes	characteristic
45	#380	C₂×C₃	1	1	yes	characteristic
46	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
48	C₂⁴×C₃	C₂³	1	1	yes	characteristic
49	C₂²×C₃×C₄	—	2	2	no	—
50	C₂²×C₃×C₄	—	2	2	no	—
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
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	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₄⋊C₄	—	2	4	no	—
64	C₃×C₄⋊C₄	—	2	4	no	—
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₂²⋊C₄	—	2	4	no	—
71	C₃×C₂²⋊C₄	—	2	4	no	—
72	C₃×C₂²⋊C₄	—	2	2	no	—
73	C₃×C₂²⋊C₄	D₄	1	4	yes	—
74	C₃×C₄²	—	2	4	no	—
75	C₃×C₄²	—	2	2	no	—
76	C₃×C₄²	—	2	2	no	—
77	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
78	#126	—	2	8	no	—
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄⋊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	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
94	#95	C₂²×C₃	1	1	yes	characteristic
95	#95	C₂²×C₃	1	1	yes	characteristic
96	#95	C₂²×C₃	1	2	yes	—
97	#95	C₂²×C₃	1	2	yes	—
98	#95	C₂²×C₃	1	2	yes	—
99	#95	C₂²×C₃	1	2	yes	—
100	#95	C₂²×C₃	1	2	yes	—
101	C₂³×C₃	—	2	1	no	—
102	C₂³×C₃	—	2	1	no	—
103	C₂³×C₃	—	2	1	no	—
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	—	2	1	no	—
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	C₂×D₄	1	2	yes	—
108	C₂³×C₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	1	no	—
111	C₂×C₃×C₄	—	2	1	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	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	1	no	—
120	C₂×C₃×C₄	—	2	1	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	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	4	4	no	—
134	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
136	C₂×C₃×C₄	#41	1	2	yes	—
137	C₂×C₃×C₄	#41	1	2	yes	—
138	C₂×C₃×C₄	#41	1	2	yes	—
139	C₂×C₃×C₄	#41	1	2	yes	—
140	C₂⁴	C₂³×C₃	1	1	yes	characteristic
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₃	1	2	yes	—
149	C₂²×C₄	C₂³×C₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	C₃×D₄	1	4	yes	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	4	1	no	—
173	C₂²×C₃	—	4	1	no	—
174	C₂²×C₃	—	4	1	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
177	C₂²×C₃	C₂×#41	1	1	yes	characteristic
178	C₂²×C₃	C₂×#41	1	1	yes	characteristic
179	C₂²×C₃	C₂×#41	1	1	yes	characteristic
180	C₂²×C₃	C₂×#41	1	1	yes	characteristic
181	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
182	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	4	1	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	—	4	2	no	—
189	C₃×C₄	—	4	2	no	—
190	C₃×C₄	—	4	2	no	—
191	C₃×C₄	—	4	1	no	—
192	C₃×C₄	#126	1	4	yes	complemented
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	C₂³	C₂×C₃×D₄	1	2	yes	—
200	C₂³	C₂⁴×C₃	1	1	yes	characteristic
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	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	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
227	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
228	C₂×C₄	C₃×#41	1	2	yes	—
229	C₂×C₄	C₃×#41	1	2	yes	—
230	C₂×C₄	C₃×#41	1	2	yes	—
231	C₂×C₄	C₃×#41	1	2	yes	—
232	C₂×C₃	—	4	2	no	—
233	C₂×C₃	C₂×#126	1	1	yes	characteristic
234	C₂×C₃	#532	1	1	yes	characteristic
235	C₂×C₃	#541	1	1	yes	characteristic
236	C₂×C₃	#547	1	1	yes	characteristic
237	C₂×C₃	#547	1	1	yes	characteristic
238	C₂×C₃	#552	1	1	yes	characteristic
239	C₂×C₃	#556	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
248	C₂²	C₂×C₃×#41	1	1	yes	characteristic
249	C₂²	C₂×C₃×#41	1	1	yes	characteristic
250	C₂²	C₂×C₃×#41	1	1	yes	characteristic
251	C₂²	C₂×C₃×#41	1	1	yes	characteristic
252	C₂²	C₃×2₋¹⁺⁴	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	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	1	no	—
263	C₄	C₃×#126	1	4	yes	complemented
264	C₃	#2496	1	1	yes	characteristic, direct product factor, Sylow
265	C₂	—	4	2	no	—
266	C₂	C₂×C₃×#126	1	1	yes	characteristic
267	C₂	C₃×#532	1	1	yes	characteristic
268	C₂	C₃×#541	1	1	yes	characteristic
269	C₂	C₃×#547	1	1	yes	characteristic
270	C₂	C₃×#547	1	1	yes	characteristic
271	C₂	C₃×#552	1	1	yes	characteristic
272	C₂	C₃×#556	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
2. C₃×#400	C₂	semidirect
4. C₃×#389	C₂	semidirect
5. C₃×#389	C₂	semidirect
6. C₃×#387	C₂	semidirect
7. C₃×#384	C₂	semidirect
10. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. #2496	C₃	direct (N × H)
38. #400	C₂×C₃	semidirect
40. #389	C₂×C₃	semidirect
41. #389	C₂×C₃	semidirect
42. #387	C₂×C₃	semidirect
43. #384	C₂×C₃	semidirect
46. C₄×C₄⋊C₄	C₂×C₃	semidirect
192. C₃×C₄	#126	semidirect
263. C₄	C₃×#126	semidirect
264. C₃	#2496	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×#400	2
C₃×#397	2
C₃×#389	2
C₃×#387	2
C₃×#384	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2496	3

Subgroups — C3×#2496 #84360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2496 #84360