Subgroups — C₃×#2421 #84285

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#400	C₂	1	2	yes	maximal, complemented
4	C₃×#397	C₂	1	2	yes	maximal
5	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#388	C₂	1	2	yes	maximal
8	C₃×#387	C₂	1	2	yes	maximal, complemented
9	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#380	C₂	1	1	yes	maximal, characteristic
11	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	#2421	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×C₂²⋊Q₈	—	2	4	no	—
16	C₃×C₄×Q₈	—	2	4	no	—
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	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	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
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	C₃×#95	C₂²	1	2	yes	—
38	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
39	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
40	#400	C₂×C₃	1	2	yes	complemented
41	#397	C₂×C₃	1	2	yes	—
42	#391	C₂×C₃	1	1	yes	characteristic, complemented
43	#389	C₂×C₃	1	1	yes	characteristic, complemented
44	#388	C₂×C₃	1	2	yes	—
45	#387	C₂×C₃	1	2	yes	complemented
46	#382	C₂×C₃	1	1	yes	characteristic, complemented
47	#380	C₂×C₃	1	1	yes	characteristic
48	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
49	C₂⁴×C₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×Q₈	—	2	2	no	—
51	C₂×C₃×Q₈	—	2	2	no	—
52	C₂×C₃×Q₈	D₄	1	4	yes	—
53	C₂²×C₃×C₄	—	2	2	no	—
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₃×C₄⋊C₄	—	2	4	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	4	no	—
70	C₃×C₄⋊C₄	—	2	4	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₂²⋊C₄	—	2	2	no	—
73	C₃×C₂²⋊C₄	—	2	4	no	—
74	C₃×C₂²⋊C₄	—	2	4	no	—
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₄²	—	2	4	no	—
77	C₃×C₄²	—	2	2	no	—
78	C₃×C₄²	—	2	2	no	—
79	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
80	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
81	C₂²⋊Q₈	—	2	4	no	—
82	C₄×Q₈	—	2	4	no	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
92	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
93	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₃	1	2	yes	—
95	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
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	#95	C₂²×C₃	1	2	yes	—
103	#95	C₂²×C₃	1	2	yes	—
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	—	2	1	no	—
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	—	2	1	no	—
108	C₂³×C₃	—	2	1	no	—
109	C₂³×C₃	—	2	1	no	—
110	C₂³×C₃	#41	1	2	yes	—
111	C₂³×C₃	C₂⁴	1	1	yes	characteristic
112	C₃×Q₈	—	2	8	no	—
113	C₂×C₃×C₄	—	2	1	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	1	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	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	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	4	4	no	—
139	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
140	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
141	C₂×C₃×C₄	#41	1	2	yes	—
142	C₂×C₃×C₄	#41	1	2	yes	—
143	C₂×C₃×C₄	#41	1	2	yes	—
144	C₂⁴	C₂³×C₃	1	1	yes	characteristic
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	C₃×D₄	1	4	yes	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₃	1	2	yes	—
154	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₃	1	2	yes	—
157	C₂²×C₄	C₂³×C₃	1	2	yes	—
158	C₂²×C₄	C₂³×C₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	4	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	—	4	1	no	—
179	C₂²×C₃	—	4	1	no	—
180	C₂²×C₃	—	4	1	no	—
181	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
182	C₂²×C₃	C₂×#41	1	1	yes	characteristic
183	C₂²×C₃	C₂×#41	1	1	yes	characteristic
184	C₂²×C₃	C₂×#41	1	1	yes	characteristic
185	C₂²×C₃	C₂×#41	1	1	yes	characteristic
186	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
187	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	4	1	no	—
191	C₃×C₄	—	4	2	no	—
192	C₃×C₄	—	4	2	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	—	4	1	no	—
196	C₃×C₄	—	4	1	no	—
197	C₃×C₄	#123	1	4	yes	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	C₃×#41	1	2	yes	—
205	C₂³	C₂⁴×C₃	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	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	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
234	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
235	C₂×C₄	C₃×#41	1	2	yes	—
236	C₂×C₄	C₃×#41	1	2	yes	—
237	C₂×C₄	C₃×#41	1	2	yes	—
238	C₂×C₃	—	4	2	no	—
239	C₂×C₃	C₂×#123	1	1	yes	characteristic
240	C₂×C₃	#529	1	1	yes	characteristic
241	C₂×C₃	#541	1	1	yes	characteristic
242	C₂×C₃	#548	1	1	yes	characteristic
243	C₂×C₃	#548	1	1	yes	characteristic
244	C₂×C₃	#552	1	1	yes	characteristic
245	C₂×C₃	#556	1	1	yes	characteristic
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
254	C₂²	C₂×C₃×#41	1	1	yes	characteristic
255	C₂²	C₂×C₃×#41	1	1	yes	characteristic
256	C₂²	C₂×C₃×#41	1	1	yes	characteristic
257	C₂²	C₂×C₃×#41	1	1	yes	characteristic
258	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
259	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	1	no	—
269	C₄	C₃×#123	1	4	yes	—
270	C₃	#2421	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	4	2	no	—
272	C₂	C₂×C₃×#123	1	1	yes	characteristic
273	C₂	C₃×#529	1	1	yes	characteristic
274	C₂	C₃×#541	1	1	yes	characteristic
275	C₂	C₃×#548	1	1	yes	characteristic
276	C₂	C₃×#548	1	1	yes	characteristic
277	C₂	C₃×#552	1	1	yes	characteristic
278	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. C₂×C₃×C₄×Q₈	C₂	semidirect
3. C₃×#400	C₂	semidirect
5. C₃×#391	C₂	semidirect
6. C₃×#389	C₂	semidirect
8. C₃×#387	C₂	semidirect
9. C₃×#382	C₂	semidirect
12. #2421	C₃	direct (N × H)
39. C₂×C₄×Q₈	C₂×C₃	semidirect
40. #400	C₂×C₃	semidirect
42. #391	C₂×C₃	semidirect
43. #389	C₂×C₃	semidirect
45. #387	C₂×C₃	semidirect
46. #382	C₂×C₃	semidirect
270. C₃	#2421	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×C₂²⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₃×#400	2
C₃×#397	2
C₃×#391	2
C₃×#389	2
C₃×#388	2
C₃×#387	2
C₃×#382	2
C₃×#380	2
C₃×C₄×C₂²⋊C₄	2
#2421	3

Subgroups — C3×#2421 #84285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2421 #84285