Subgroups — C₃×#2604 #84468

Order 384.

470 conjugacy classes of subgroups.

The subgroup lattice has 250 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	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#399	C₂	1	1	yes	maximal, characteristic
5	C₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#393	C₂	1	1	yes	maximal, characteristic
7	C₃×#389	C₂	1	2	yes	maximal, complemented
8	C₃×#388	C₂	1	2	yes	maximal
9	C₃×#388	C₂	1	2	yes	maximal
10	C₃×#387	C₂	1	2	yes	maximal, complemented
11	C₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
12	#2604	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₃×#126	—	2	4	no	—
15	C₃×C₄×Q₈	—	2	4	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	2	yes	—
29	C₂×C₃×C₄²	C₂²	1	2	yes	—
30	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₃×#95	C₂²	1	2	yes	—
32	C₃×#95	C₂²	1	1	yes	characteristic
33	C₃×#95	C₂²	1	1	yes	characteristic
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×#126	C₂×C₃	1	1	yes	characteristic
39	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
40	#400	C₂×C₃	1	1	yes	characteristic, complemented
41	#399	C₂×C₃	1	1	yes	characteristic
42	#395	C₂×C₃	1	1	yes	characteristic, complemented
43	#393	C₂×C₃	1	1	yes	characteristic
44	#389	C₂×C₃	1	2	yes	complemented
45	#388	C₂×C₃	1	2	yes	—
46	#388	C₂×C₃	1	2	yes	—
47	#387	C₂×C₃	1	2	yes	complemented
48	#376	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂⁴×C₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×Q₈	—	2	2	no	—
51	C₂×C₃×Q₈	—	2	4	no	—
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
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	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	4	no	—
68	C₃×C₄⋊C₄	—	2	4	no	—
69	C₃×C₂²⋊C₄	—	2	2	no	—
70	C₃×C₂²⋊C₄	—	2	2	no	—
71	C₃×C₂²⋊C₄	—	2	2	no	—
72	C₃×C₂²⋊C₄	—	2	4	no	—
73	C₃×C₂²⋊C₄	—	2	4	no	—
74	C₃×C₄²	—	2	4	no	—
75	C₃×C₄²	—	2	4	no	—
76	C₃×C₄²	D₄	1	4	yes	—
77	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
78	#126	—	2	4	no	—
79	C₄×Q₈	—	2	4	no	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
91	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
92	C₂×C₄²	C₂²×C₃	1	2	yes	—
93	C₂×C₄²	C₂²×C₃	1	2	yes	—
94	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
95	#95	C₂²×C₃	1	2	yes	—
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	C₂³×C₃	—	2	1	no	—
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	—	2	2	no	—
105	C₂³×C₃	—	2	1	no	—
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	C₂⁴	1	1	yes	characteristic
108	C₃×Q₈	—	2	8	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	1	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	2	no	—
130	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
131	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
133	C₂×C₃×C₄	#41	1	4	yes	—
134	C₂×C₃×C₄	#41	1	4	yes	—
135	C₂⁴	C₂³×C₃	1	1	yes	characteristic
136	C₂×Q₈	—	2	2	no	—
137	C₂×Q₈	—	2	4	no	—
138	C₂²×C₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
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	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	4	no	—
162	C₄²	C₃×D₄	1	4	yes	—
163	C₂²×C₃	—	4	1	no	—
164	C₂²×C₃	—	4	1	no	—
165	C₂²×C₃	—	4	1	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
169	C₂²×C₃	C₂×#41	1	2	yes	—
170	C₂²×C₃	C₂×#41	1	2	yes	—
171	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
172	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	4	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	4	1	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	4	1	no	—
181	C₃×C₄	—	4	1	no	—
182	C₃×C₄	—	4	2	no	—
183	C₂³	—	2	1	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	1	no	—
187	C₂³	—	2	1	no	—
188	C₂³	C₂⁴×C₃	1	1	yes	characteristic
189	Q₈	—	2	8	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	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	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂×C₄	C₃×#41	1	4	yes	—
215	C₂×C₄	C₃×#41	1	4	yes	—
216	C₂×C₃	—	8	1	no	—
217	C₂×C₃	#525	1	1	yes	characteristic
218	C₂×C₃	#532	1	1	yes	characteristic
219	C₂×C₃	#536	1	1	yes	characteristic
220	C₂×C₃	#552	1	2	yes	—
221	C₂×C₃	#556	1	2	yes	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
228	C₂²	C₂×C₃×#41	1	2	yes	—
229	C₂²	C₂×C₃×#41	1	2	yes	—
230	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
231	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	2	no	—
242	C₃	#2604	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	8	1	no	—
244	C₂	C₃×#525	1	1	yes	characteristic
245	C₂	C₃×#532	1	1	yes	characteristic
246	C₂	C₃×#536	1	1	yes	characteristic
247	C₂	C₃×#552	1	2	yes	—
248	C₂	C₃×#556	1	2	yes	—

	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×#395	C₂	semidirect
7. C₃×#389	C₂	semidirect
10. C₃×#387	C₂	semidirect
11. C₃×#376	C₂	semidirect
12. #2604	C₃	direct (N × H)
39. C₂×C₄×Q₈	C₂×C₃	semidirect
40. #400	C₂×C₃	semidirect
42. #395	C₂×C₃	semidirect
44. #389	C₂×C₃	semidirect
47. #387	C₂×C₃	semidirect
48. #376	C₂×C₃	semidirect
242. C₃	#2604	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
C₂×C₃×C₄×Q₈	2
C₃×#400	2
C₃×#399	2
C₃×#395	2
C₃×#393	2
C₃×#389	2
C₃×#388	2
C₃×#387	2
C₃×#376	2
#2604	3

Subgroups — C3×#2604 #84468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2604 #84468