Subgroups — C₃×#2408 #84272

Order 384.

1232 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#121	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#120	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#394	C₂	1	2	yes	maximal, complemented
6	C₃×#392	C₂	1	2	yes	maximal, complemented
7	C₃×#390	C₂	1	2	yes	maximal, complemented
8	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#379	C₂	1	2	yes	maximal, complemented
10	#2408	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁵×C₃	C₂²	1	2	yes	—
12	C₂²×C₃×D₄	—	2	8	no	—
13	C₂²×C₃×D₄	—	2	4	no	—
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃×D₄	C₂²	1	4	yes	complemented
18	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
19	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
20	C₃×#121	—	2	8	no	—
21	C₃×#120	—	2	8	no	—
22	C₃×C₄×D₄	—	2	4	no	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
24	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
28	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
29	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
31	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
32	C₃×#95	C₂²	1	2	yes	complemented
33	C₂³×D₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#121	C₂×C₃	1	2	yes	complemented
35	C₂×#120	C₂×C₃	1	2	yes	complemented
36	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
37	#394	C₂×C₃	1	2	yes	complemented
38	#392	C₂×C₃	1	2	yes	complemented
39	#390	C₂×C₃	1	2	yes	complemented
40	#380	C₂×C₃	1	1	yes	characteristic, complemented
41	#379	C₂×C₃	1	2	yes	complemented
42	C₂⁴×C₃	—	2	8	no	—
43	C₂⁴×C₃	—	2	4	no	—
44	C₂⁴×C₃	—	2	4	no	—
45	C₂⁴×C₃	—	2	4	no	—
46	C₂⁴×C₃	—	4	4	no	—
47	C₂⁴×C₃	C₂³	1	4	yes	—
48	C₂⁴×C₃	C₂³	1	1	yes	characteristic
49	C₂⁴×C₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×D₄	—	2	32	no	—
51	C₂×C₃×D₄	—	2	2	no	—
52	C₂×C₃×D₄	—	2	8	no	—
53	C₂×C₃×D₄	—	2	2	no	—
54	C₂×C₃×D₄	—	2	8	no	—
55	C₂×C₃×D₄	—	2	8	no	—
56	C₂×C₃×D₄	—	2	4	no	—
57	C₂×C₃×D₄	—	2	4	no	—
58	C₂×C₃×D₄	—	2	4	no	—
59	C₂×C₃×D₄	—	4	4	no	—
60	C₂×C₃×D₄	D₄	1	8	yes	complemented
61	C₂×C₃×D₄	D₄	1	4	yes	complemented
62	C₂²×C₃×C₄	—	2	4	no	—
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₂²×C₃×C₄	—	2	4	no	—
65	C₂²×C₃×C₄	D₄	1	4	yes	complemented
66	C₂²×C₃×C₄	C₂³	1	4	yes	—
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	4	no	—
74	C₃×C₂²⋊C₄	—	2	2	no	—
75	C₃×C₂²⋊C₄	—	2	4	no	—
76	C₃×C₂²⋊C₄	—	2	8	no	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	2	8	no	—
79	C₃×C₂²⋊C₄	—	4	4	no	—
80	C₃×C₄²	—	2	2	no	—
81	C₂⁵	C₂²×C₃	1	2	yes	—
82	C₂²×D₄	—	2	8	no	—
83	C₂²×D₄	—	2	4	no	—
84	C₂²×D₄	C₂²×C₃	1	2	yes	—
85	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
86	C₂²×D₄	C₂²×C₃	1	2	yes	—
87	C₂²×D₄	C₂²×C₃	1	4	yes	complemented
88	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
90	#121	—	2	8	no	—
91	#120	—	2	8	no	—
92	C₄×D₄	—	2	4	no	—
93	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
94	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
95	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
98	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
99	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
100	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
101	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
102	#95	C₂²×C₃	1	2	yes	complemented
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	—	2	2	no	—
105	C₂³×C₃	—	2	16	no	—
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	—	2	4	no	—
108	C₂³×C₃	—	2	2	no	—
109	C₂³×C₃	—	2	4	no	—
110	C₂³×C₃	—	2	4	no	—
111	C₂³×C₃	—	2	1	no	—
112	C₂³×C₃	—	2	4	no	—
113	C₂³×C₃	—	2	4	no	—
114	C₂³×C₃	—	2	2	no	—
115	C₂³×C₃	—	2	4	no	—
116	C₂³×C₃	—	4	16	no	—
117	C₂³×C₃	—	4	16	no	—
118	C₂³×C₃	—	4	8	no	—
119	C₂³×C₃	—	4	8	no	—
120	C₂³×C₃	C₂×D₄	1	8	yes	—
121	C₂³×C₃	C₂×D₄	1	2	yes	—
122	C₂³×C₃	C₂×D₄	1	2	yes	—
123	C₂³×C₃	C₂×D₄	1	2	yes	complemented
124	C₂³×C₃	#41	1	2	yes	—
125	C₂³×C₃	C₂⁴	1	1	yes	characteristic
126	C₃×D₄	—	2	16	no	—
127	C₃×D₄	—	2	8	no	—
128	C₃×D₄	—	2	16	no	—
129	C₃×D₄	—	4	16	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	8	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	4	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	4	4	no	—
142	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
143	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
144	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
145	C₂×C₃×C₄	C₂×D₄	1	2	yes	complemented
146	C₂⁴	—	2	8	no	—
147	C₂⁴	—	2	4	no	—
148	C₂⁴	—	2	4	no	—
149	C₂⁴	—	2	4	no	—
150	C₂⁴	—	4	4	no	—
151	C₂⁴	C₂³×C₃	1	4	yes	—
152	C₂⁴	C₂³×C₃	1	1	yes	characteristic
153	C₂⁴	C₂³×C₃	1	1	yes	characteristic
154	C₂×D₄	—	2	32	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	8	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	8	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	2	4	no	—
163	C₂×D₄	—	4	4	no	—
164	C₂×D₄	C₃×D₄	1	8	yes	complemented
165	C₂×D₄	C₃×D₄	1	4	yes	complemented
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	C₃×D₄	1	4	yes	complemented
170	C₂²×C₄	C₂³×C₃	1	4	yes	—
171	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	8	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	8	no	—
183	C₂²⋊C₄	—	4	4	no	—
184	C₄²	—	2	2	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	4	no	—
188	C₂²×C₃	—	2	4	no	—
189	C₂²×C₃	—	2	8	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	2	8	no	—
193	C₂²×C₃	—	2	8	no	—
194	C₂²×C₃	—	4	32	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	4	no	—
197	C₂²×C₃	—	4	1	no	—
198	C₂²×C₃	—	4	4	no	—
199	C₂²×C₃	—	4	4	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	—	4	4	no	—
202	C₂²×C₃	#121	1	4	yes	complemented
203	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
204	C₂²×C₃	C₂²×D₄	1	2	yes	—
205	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
206	C₂²×C₃	C₂×#41	1	1	yes	characteristic
207	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	4	1	no	—
211	C₃×C₄	—	4	1	no	—
212	C₃×C₄	—	4	4	no	—
213	C₃×C₄	#120	1	4	yes	complemented
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	16	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	4	16	no	—
228	C₂³	—	4	16	no	—
229	C₂³	—	4	8	no	—
230	C₂³	—	4	8	no	—
231	C₂³	C₂×C₃×D₄	1	8	yes	—
232	C₂³	C₂×C₃×D₄	1	2	yes	—
233	C₂³	C₂×C₃×D₄	1	2	yes	—
234	C₂³	C₂×C₃×D₄	1	2	yes	complemented
235	C₂³	C₃×#41	1	2	yes	—
236	C₂³	C₂⁴×C₃	1	1	yes	characteristic
237	D₄	—	2	16	no	—
238	D₄	—	2	8	no	—
239	D₄	—	2	16	no	—
240	D₄	—	4	16	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
254	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
255	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
256	C₂×C₄	C₂×C₃×D₄	1	2	yes	complemented
257	C₂×C₃	—	2	4	no	—
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	—	4	8	no	—
260	C₂×C₃	C₂×#120	1	1	yes	characteristic
261	C₂×C₃	C₂×#121	1	1	yes	characteristic
262	C₂×C₃	#535	1	1	yes	characteristic
263	C₂×C₃	D₄²	1	2	yes	—
264	C₂×C₃	#546	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	4	32	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₃×#121	1	4	yes	complemented
283	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
284	C₂²	C₂²×C₃×D₄	1	2	yes	—
285	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
286	C₂²	C₂×C₃×#41	1	1	yes	characteristic
287	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	4	no	—
293	C₄	C₃×#120	1	4	yes	complemented
294	C₃	#2408	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	4	no	—
296	C₂	—	4	8	no	—
297	C₂	—	4	2	no	—
298	C₂	C₂×C₃×#120	1	1	yes	characteristic
299	C₂	C₂×C₃×#121	1	1	yes	characteristic
300	C₂	C₃×#535	1	1	yes	characteristic
301	C₂	C₃×D₄²	1	2	yes	—
302	C₂	C₃×#546	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
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. C₂³×C₃×D₄	C₂	semidirect
2. C₂×C₃×#121	C₂	semidirect
3. C₂×C₃×#120	C₂	semidirect
4. C₂×C₃×C₄×D₄	C₂	semidirect
5. C₃×#394	C₂	semidirect
6. C₃×#392	C₂	semidirect
7. C₃×#390	C₂	semidirect
8. C₃×#380	C₂	semidirect
9. C₃×#379	C₂	semidirect
10. #2408	C₃	direct (N × H)
17. C₂²×C₃×D₄	C₂²	semidirect
19. C₂³×C₃×C₄	C₂²	semidirect
23. C₂×C₃×C₄⋊C₄	C₂²	semidirect
24. C₂×C₃×C₄⋊C₄	C₂²	semidirect
27. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
28. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
29. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
30. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
31. C₂×C₃×C₄²	C₂²	semidirect
32. C₃×#95	C₂²	semidirect
33. C₂³×D₄	C₂×C₃	semidirect
34. C₂×#121	C₂×C₃	semidirect
35. C₂×#120	C₂×C₃	semidirect
36. C₂×C₄×D₄	C₂×C₃	semidirect
37. #394	C₂×C₃	semidirect
38. #392	C₂×C₃	semidirect
39. #390	C₂×C₃	semidirect
40. #380	C₂×C₃	semidirect
41. #379	C₂×C₃	semidirect
60. C₂×C₃×D₄	D₄	semidirect
61. C₂×C₃×D₄	D₄	semidirect
65. C₂²×C₃×C₄	D₄	semidirect
87. C₂²×D₄	C₂²×C₃	semidirect
89. C₂³×C₄	C₂²×C₃	semidirect
93. C₂×C₄⋊C₄	C₂²×C₃	semidirect
94. C₂×C₄⋊C₄	C₂²×C₃	semidirect
97. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
98. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
99. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
100. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
101. C₂×C₄²	C₂²×C₃	semidirect
102. #95	C₂²×C₃	semidirect
123. C₂³×C₃	C₂×D₄	semidirect
145. C₂×C₃×C₄	C₂×D₄	semidirect
164. C₂×D₄	C₃×D₄	semidirect
165. C₂×D₄	C₃×D₄	semidirect
169. C₂²×C₄	C₃×D₄	semidirect
202. C₂²×C₃	#121	semidirect
213. C₃×C₄	#120	semidirect
234. C₂³	C₂×C₃×D₄	semidirect
256. C₂×C₄	C₂×C₃×D₄	semidirect
282. C₂²	C₃×#121	semidirect
293. C₄	C₃×#120	semidirect
294. C₃	#2408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₄	2
C₂×C₃×#121	2
C₂×C₃×#120	2
C₂×C₃×C₄×D₄	2
C₃×#394	2
C₃×#392	2
C₃×#390	2
C₃×#380	2
C₃×#379	2
#2408	3

Subgroups — C3×#2408 #84272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2408 #84272