Subgroups — C₃×#2492 #84356

Order 384.

508 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#400	C₂	1	2	yes	maximal, complemented
4	C₃×#393	C₂	1	2	yes	maximal
5	C₃×#391	C₂	1	2	yes	maximal, complemented
6	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#388	C₂	1	2	yes	maximal
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	#2492	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Q₈	C₂²	1	2	yes	—
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	—	2	4	no	—
17	C₃×C₂²⋊Q₈	—	2	4	no	—
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	1	yes	characteristic
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	C₃×#95	C₂²	1	2	yes	—
38	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
40	#400	C₂×C₃	1	2	yes	complemented
41	#393	C₂×C₃	1	2	yes	—
42	#391	C₂×C₃	1	2	yes	complemented
43	#391	C₂×C₃	1	1	yes	characteristic, complemented
44	#391	C₂×C₃	1	1	yes	characteristic, complemented
45	#388	C₂×C₃	1	2	yes	—
46	#380	C₂×C₃	1	1	yes	characteristic
47	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
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	4	no	—
52	C₂×C₃×Q₈	—	2	4	no	—
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	2	no	—
55	C₂×C₃×Q₈	—	4	2	no	—
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	—	2	2	no	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂³	1	2	yes	—
61	C₂²×C₃×C₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	2	yes	—
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	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₃×C₄⋊C₄	—	2	4	no	—
70	C₃×C₄⋊C₄	—	2	4	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	D₄	1	4	yes	—
74	C₃×C₂²⋊C₄	—	2	4	no	—
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₂²⋊C₄	—	4	2	no	—
77	C₃×C₂²⋊C₄	Q₈	1	4	yes	complemented
78	C₃×C₄²	—	2	2	no	—
79	C₃×C₄²	—	2	2	no	—
80	C₃×C₄²	—	4	2	no	—
81	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
82	C₂²×Q₈	C₂²×C₃	1	2	yes	—
83	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
84	C₄⋊Q₈	—	2	4	no	—
85	C₂²⋊Q₈	—	2	4	no	—
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×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
92	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
97	C₂×C₄²	C₂²×C₃	1	2	yes	—
98	#95	C₂²×C₃	1	1	yes	characteristic
99	#95	C₂²×C₃	1	1	yes	characteristic
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	#95	C₂²×C₃	1	2	yes	—
105	#95	C₂²×C₃	1	2	yes	—
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	—	2	1	no	—
111	C₂³×C₃	—	2	1	no	—
112	C₂³×C₃	C₂×Q₈	1	2	yes	—
113	C₂³×C₃	C₂⁴	1	1	yes	characteristic
114	C₃×Q₈	—	4	8	no	—
115	C₃×Q₈	—	4	4	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	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	2	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	1	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	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	4	4	no	—
140	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
141	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
142	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
143	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
144	C₂×C₃×C₄	#41	1	2	yes	—
145	C₂×C₃×C₄	#41	1	2	yes	—
146	C₂×C₃×C₄	#41	1	2	yes	—
147	C₂⁴	C₂³×C₃	1	1	yes	characteristic
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₃	1	2	yes	—
163	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	C₃×D₄	1	4	yes	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	C₃×Q₈	1	4	yes	complemented
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	4	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	4	1	no	—
183	C₂²×C₃	—	4	1	no	—
184	C₂²×C₃	—	4	1	no	—
185	C₂²×C₃	—	4	1	no	—
186	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
187	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
188	C₂²×C₃	C₂×#41	1	1	yes	characteristic
189	C₂²×C₃	C₂×#41	1	1	yes	characteristic
190	C₂²×C₃	C₂×#41	1	1	yes	characteristic
191	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	4	1	no	—
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	—	4	2	no	—
200	C₃×C₄	—	4	2	no	—
201	C₃×C₄	—	4	1	no	—
202	C₃×C₄	#124	1	4	yes	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	C₂×C₃×Q₈	1	2	yes	—
210	C₂³	C₂⁴×C₃	1	1	yes	characteristic
211	Q₈	—	4	8	no	—
212	Q₈	—	4	4	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
238	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
239	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
240	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
241	C₂×C₄	C₃×#41	1	2	yes	—
242	C₂×C₄	C₃×#41	1	2	yes	—
243	C₂×C₄	C₃×#41	1	2	yes	—
244	C₂×C₃	—	4	2	no	—
245	C₂×C₃	C₂×#124	1	1	yes	characteristic
246	C₂×C₃	#533	1	1	yes	characteristic
247	C₂×C₃	#542	1	1	yes	characteristic
248	C₂×C₃	#546	1	1	yes	characteristic
249	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
250	C₂×C₃	#554	1	1	yes	characteristic
251	C₂×C₃	#556	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
260	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
261	C₂²	C₂×C₃×#41	1	1	yes	characteristic
262	C₂²	C₂×C₃×#41	1	1	yes	characteristic
263	C₂²	C₂×C₃×#41	1	1	yes	characteristic
264	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	C₃×#124	1	4	yes	—
276	C₃	#2492	1	1	yes	characteristic, direct product factor, Sylow
277	C₂	—	4	2	no	—
278	C₂	C₂×C₃×#124	1	1	yes	characteristic
279	C₂	C₃×#533	1	1	yes	characteristic
280	C₂	C₃×#542	1	1	yes	characteristic
281	C₂	C₃×#546	1	1	yes	characteristic
282	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
283	C₂	C₃×#554	1	1	yes	characteristic
284	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
1. C₂×C₃×C₄⋊Q₈	C₂	semidirect
3. C₃×#400	C₂	semidirect
5. C₃×#391	C₂	semidirect
6. C₃×#391	C₂	semidirect
7. C₃×#391	C₂	semidirect
10. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. #2492	C₃	direct (N × H)
38. C₂×C₄⋊Q₈	C₂×C₃	semidirect
40. #400	C₂×C₃	semidirect
42. #391	C₂×C₃	semidirect
43. #391	C₂×C₃	semidirect
44. #391	C₂×C₃	semidirect
47. C₄×C₄⋊C₄	C₂×C₃	semidirect
77. C₃×C₂²⋊C₄	Q₈	semidirect
175. C₂²⋊C₄	C₃×Q₈	semidirect
276. C₃	#2492	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⋊Q₈	2
C₂×C₃×C₂²⋊Q₈	2
C₃×#400	2
C₃×#393	2
C₃×#391	2
C₃×#388	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2492	3

Subgroups — C3×#2492 #84356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2492 #84356