Subgroups — C₃×#2620 #84484

Order 384.

612 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#396	C₂	1	4	yes	maximal, complemented
6	C₃×#394	C₂	1	4	yes	maximal, complemented
7	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	#2620	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×D₄	C₂²	1	2	yes	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×#124	—	2	4	no	—
16	C₃×C₂²⋊Q₈	—	2	4	no	—
17	C₃×#121	—	2	4	no	—
18	C₃×#117	—	2	4	no	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
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	4	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₃×#95	C₂²	1	4	yes	—
32	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#121	C₂×C₃	1	1	yes	characteristic
35	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
36	#396	C₂×C₃	1	4	yes	complemented
37	#394	C₂×C₃	1	4	yes	complemented
38	#391	C₂×C₃	1	1	yes	characteristic, complemented
39	#390	C₂×C₃	1	1	yes	characteristic, complemented
40	#384	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂⁴×C₃	C₂³	1	2	yes	—
42	C₂⁴×C₃	C₂³	1	1	yes	characteristic
43	C₂×C₃×Q₈	—	2	2	no	—
44	C₂×C₃×Q₈	—	2	2	no	—
45	C₂×C₃×Q₈	—	2	2	no	—
46	C₂×C₃×D₄	—	2	4	no	—
47	C₂×C₃×D₄	—	2	2	no	—
48	C₂×C₃×D₄	—	2	2	no	—
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	—	4	4	no	—
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	—	2	2	no	—
53	C₂²×C₃×C₄	D₄	1	4	yes	—
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	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₃×C₄⋊C₄	—	2	4	no	—
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	2	4	no	—
65	C₃×C₄⋊C₄	—	4	4	no	—
66	C₃×C₂²⋊C₄	—	2	4	no	—
67	C₃×C₂²⋊C₄	—	2	4	no	—
68	C₃×C₂²⋊C₄	—	2	4	no	—
69	C₃×C₂²⋊C₄	—	2	8	no	—
70	C₃×C₂²⋊C₄	—	4	4	no	—
71	C₃×C₄²	—	2	2	no	—
72	C₃×C₄²	D₄	1	4	yes	complemented
73	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
75	C₂²×D₄	C₂²×C₃	1	2	yes	—
76	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
77	#124	—	2	4	no	—
78	C₂²⋊Q₈	—	2	4	no	—
79	#121	—	2	4	no	—
80	#117	—	2	4	no	—
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	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
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	4	yes	—
90	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
91	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
93	#95	C₂²×C₃	1	4	yes	—
94	C₂³×C₃	—	2	1	no	—
95	C₂³×C₃	—	2	1	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	—	2	2	no	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	2	4	no	—
102	C₂³×C₃	—	2	4	no	—
103	C₂³×C₃	C₂×D₄	1	2	yes	—
104	C₂³×C₃	C₂⁴	1	1	yes	characteristic
105	C₃×Q₈	—	4	4	no	—
106	C₃×D₄	—	4	8	no	—
107	C₃×D₄	—	4	4	no	—
108	C₂×C₃×C₄	—	2	1	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	8	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	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	2	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
124	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
126	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
127	C₂×C₃×C₄	#41	1	2	yes	—
128	C₂⁴	C₂³×C₃	1	2	yes	—
129	C₂⁴	C₂³×C₃	1	1	yes	characteristic
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	4	4	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₃×D₄	1	4	yes	—
141	C₂²×C₄	C₂³×C₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₃	1	2	yes	—
144	C₂²×C₄	C₂³×C₃	1	2	yes	—
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₂²⋊C₄	—	2	4	no	—
154	C₂²⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	—	2	8	no	—
157	C₂²⋊C₄	—	4	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	C₃×D₄	1	4	yes	complemented
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	—	4	2	no	—
164	C₂²×C₃	—	4	2	no	—
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	—	4	4	no	—
169	C₂²×C₃	—	4	4	no	—
170	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
171	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
172	C₂²×C₃	C₂×#41	1	1	yes	characteristic
173	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
174	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	—	4	2	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	#121	1	4	yes	—
182	C₂³	—	2	1	no	—
183	C₂³	—	2	1	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×C₃×D₄	1	2	yes	—
192	C₂³	C₂⁴×C₃	1	1	yes	characteristic
193	Q₈	—	4	4	no	—
194	D₄	—	4	8	no	—
195	D₄	—	4	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
215	C₂×C₄	C₃×#41	1	2	yes	—
216	C₂×C₃	—	4	2	no	—
217	C₂×C₃	—	8	2	no	—
218	C₂×C₃	C₂×#121	1	1	yes	characteristic
219	C₂×C₃	#535	1	1	yes	characteristic
220	C₂×C₃	#536	1	1	yes	characteristic
221	C₂×C₃	#537	1	2	yes	—
222	C₂×C₃	#542	1	2	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	4	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
235	C₂²	C₂×C₃×#41	1	1	yes	characteristic
236	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
237	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
238	C₄	—	2	2	no	—
239	C₄	—	2	4	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	C₃×#121	1	4	yes	—
245	C₃	#2620	1	1	yes	characteristic, direct product factor, Sylow
246	C₂	—	4	2	no	—
247	C₂	—	8	2	no	—
248	C₂	C₂×C₃×#121	1	1	yes	characteristic
249	C₂	C₃×#535	1	1	yes	characteristic
250	C₂	C₃×#536	1	1	yes	characteristic
251	C₂	C₃×#537	1	2	yes	—
252	C₂	C₃×#542	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#121	C₂
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×#124	C₂	semidirect
2. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
4. C₂×C₃×#117	C₂	semidirect
5. C₃×#396	C₂	semidirect
6. C₃×#394	C₂	semidirect
7. C₃×#391	C₂	semidirect
8. C₃×#390	C₂	semidirect
9. C₃×#384	C₂	semidirect
10. #2620	C₃	direct (N × H)
32. C₂×#124	C₂×C₃	semidirect
33. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
35. C₂×#117	C₂×C₃	semidirect
36. #396	C₂×C₃	semidirect
37. #394	C₂×C₃	semidirect
38. #391	C₂×C₃	semidirect
39. #390	C₂×C₃	semidirect
40. #384	C₂×C₃	semidirect
72. C₃×C₄²	D₄	semidirect
159. C₄²	C₃×D₄	semidirect
245. C₃	#2620	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×#124	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×#121	2
C₂×C₃×#117	2
C₃×#396	2
C₃×#394	2
C₃×#391	2
C₃×#390	2
C₃×#384	2
#2620	3

Subgroups — C3×#2620 #84484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2620 #84484