Subgroups — C₃×#1806 #72521

Order 384.

312 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#41	C₂	1	1	yes	maximal, characteristic
2	C₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#104	C₂	1	2	yes	maximal
4	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#339	C₂	1	2	yes	maximal, complemented
6	#1806	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#41	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	C₂²	1	2	yes	—
9	C₃×C₄×Q₈	C₄	1	1	yes	characteristic, complemented
10	C₃×C₄×Q₈	C₄	1	1	yes	characteristic, complemented
11	C₃×C₄×D₄	—	2	1	no	—
12	C₃×C₄×D₄	—	2	1	no	—
13	C₃×C₄×D₄	C₄	1	1	yes	characteristic, complemented
14	C₃×C₄×D₄	C₄	1	1	yes	characteristic, complemented
15	C₃×#117	—	2	1	no	—
16	C₃×#117	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	—	2	1	no	—
18	C₂×C₃×C₄²	—	2	1	no	—
19	C₂×C₃×C₄²	C₂²	1	2	yes	—
20	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊C₈	—	2	1	no	—
22	C₃×C₄⋊C₈	—	2	1	no	—
23	C₃×#104	—	2	4	no	—
24	C₃×#95	—	2	1	no	—
25	C₄×#41	C₂×C₃	1	1	yes	characteristic
26	#423	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#104	C₂×C₃	1	2	yes	—
28	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
29	#339	C₂×C₃	1	2	yes	complemented
30	C₃×#41	—	2	1	no	—
31	C₃×#41	—	2	1	no	—
32	C₃×#41	D₄	1	2	yes	—
33	C₃×#41	Q₈	1	2	yes	—
34	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×D₄	—	2	1	no	—
36	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄	—	2	1	no	—
38	C₂²×C₃×C₄	—	2	2	no	—
39	C₂²×C₃×C₄	—	2	1	no	—
40	C₂²×C₃×C₄	—	2	1	no	—
41	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₃×M₁₆	—	2	2	no	—
45	C₃×M₁₆	—	2	2	no	—
46	C₂×C₃×C₈	—	2	2	no	—
47	C₃×C₄⋊C₄	—	2	4	no	—
48	C₃×C₄⋊C₄	—	2	1	no	—
49	C₃×C₄⋊C₄	—	2	1	no	—
50	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₃×C₂²⋊C₄	—	2	1	no	—
53	C₃×C₂²⋊C₄	—	2	1	no	—
54	C₃×C₂²⋊C₄	—	2	1	no	—
55	C₃×C₄²	—	2	1	no	—
56	C₃×C₄²	—	2	1	no	—
57	C₃×C₄²	—	2	1	no	—
58	C₃×C₄²	—	2	2	no	—
59	C₃×C₄²	—	2	2	no	—
60	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
61	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
62	C₃×C₄²	D₄	1	1	yes	characteristic
63	C₃×C₄²	D₄	1	1	yes	characteristic
64	C₂×#41	C₂²×C₃	1	1	yes	characteristic
65	C₂×M₁₆	C₂²×C₃	1	2	yes	—
66	C₄×Q₈	C₃×C₄	1	1	yes	characteristic, complemented
67	C₄×Q₈	C₃×C₄	1	1	yes	characteristic, complemented
68	C₄×D₄	—	2	1	no	—
69	C₄×D₄	—	2	1	no	—
70	C₄×D₄	C₃×C₄	1	1	yes	characteristic, complemented
71	C₄×D₄	C₃×C₄	1	1	yes	characteristic, complemented
72	#117	—	2	1	no	—
73	#117	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄²	—	2	1	no	—
76	C₂×C₄²	C₂²×C₃	1	2	yes	—
77	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
78	C₄⋊C₈	—	2	1	no	—
79	C₄⋊C₈	—	2	1	no	—
80	#104	—	2	4	no	—
81	#95	—	2	1	no	—
82	C₂³×C₃	—	2	1	no	—
83	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
84	C₃×Q₈	—	2	1	no	—
85	C₃×Q₈	C₄⋊C₄	1	2	yes	complemented
86	C₃×D₄	—	2	2	no	—
87	C₃×D₄	—	2	2	no	—
88	C₃×D₄	—	2	1	no	—
89	C₃×D₄	C₄⋊C₄	1	2	yes	complemented
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	2	1	no	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	4	1	no	—
98	C₂×C₃×C₄	—	4	2	no	—
99	C₂×C₃×C₄	—	4	2	no	—
100	C₂×C₃×C₄	—	4	2	no	—
101	C₂×C₃×C₄	—	4	2	no	—
102	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₃×C₄	#41	1	1	yes	characteristic
111	C₂×C₃×C₄	#41	1	1	yes	characteristic
112	C₃×C₈	—	4	2	no	—
113	#41	—	2	1	no	—
114	#41	—	2	1	no	—
115	#41	C₃×D₄	1	2	yes	—
116	#41	C₃×Q₈	1	2	yes	—
117	C₂×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	C₂×C₃×C₄	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
125	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
134	C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
135	C₂²⋊C₄	—	2	1	no	—
136	C₂²⋊C₄	—	2	1	no	—
137	C₂²⋊C₄	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
144	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
145	C₄²	C₃×D₄	1	1	yes	characteristic
146	C₄²	C₃×D₄	1	1	yes	characteristic
147	C₂²×C₃	—	2	1	no	—
148	C₂²×C₃	—	2	1	no	—
149	C₂²×C₃	—	2	2	no	—
150	C₂²×C₃	—	4	1	no	—
151	C₂²×C₃	—	4	1	no	—
152	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₃	#121	1	1	yes	characteristic
154	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	—	2	1	no	—
157	C₃×C₄	—	2	1	no	—
158	C₃×C₄	—	2	1	no	—
159	C₃×C₄	—	4	4	no	—
160	C₃×C₄	—	4	1	no	—
161	C₃×C₄	—	4	1	no	—
162	C₃×C₄	#104	1	2	yes	complemented
163	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₃×C₄	#117	1	1	yes	characteristic
165	C₃×C₄	#121	1	1	yes	characteristic
166	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂³	—	2	1	no	—
168	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
169	Q₈	—	2	1	no	—
170	Q₈	C₃×C₄⋊C₄	1	2	yes	complemented
171	D₄	—	2	2	no	—
172	D₄	—	2	2	no	—
173	D₄	—	2	1	no	—
174	D₄	C₃×C₄⋊C₄	1	2	yes	complemented
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
192	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
195	C₂×C₄	C₃×#41	1	1	yes	characteristic
196	C₂×C₄	C₃×#41	1	1	yes	characteristic
197	C₈	—	4	2	no	—
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	—	2	1	no	—
200	C₂×C₃	—	4	2	no	—
201	C₂×C₃	#380	1	1	yes	characteristic
202	C₂×C₃	C₂×#104	1	1	yes	characteristic
203	C₂×C₃	#421	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	1	no	—
209	C₂²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²	C₃×#121	1	1	yes	characteristic
211	C₂²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	1	no	—
218	C₄	—	4	1	no	—
219	C₄	C₃×#104	1	2	yes	complemented
220	C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
221	C₄	C₃×#117	1	1	yes	characteristic
222	C₄	C₃×#121	1	1	yes	characteristic
223	C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
224	C₃	#1806	1	1	yes	characteristic, direct product factor, Sylow
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	—	4	2	no	—
228	C₂	C₃×#380	1	1	yes	characteristic
229	C₂	C₂×C₃×#104	1	1	yes	characteristic
230	C₂	C₃×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂×C₃×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×#423	C₂	semidirect
4. C₃×C₄×C₄⋊C₄	C₂	semidirect
5. C₃×#339	C₂	semidirect
6. #1806	C₃	direct (N × H)
9. C₃×C₄×Q₈	C₄	semidirect
10. C₃×C₄×Q₈	C₄	semidirect
13. C₃×C₄×D₄	C₄	semidirect
14. C₃×C₄×D₄	C₄	semidirect
26. #423	C₂×C₃	semidirect
28. C₄×C₄⋊C₄	C₂×C₃	semidirect
29. #339	C₂×C₃	semidirect
66. C₄×Q₈	C₃×C₄	semidirect
67. C₄×Q₈	C₃×C₄	semidirect
70. C₄×D₄	C₃×C₄	semidirect
71. C₄×D₄	C₃×C₄	semidirect
85. C₃×Q₈	C₄⋊C₄	semidirect
89. C₃×D₄	C₄⋊C₄	semidirect
162. C₃×C₄	#104	semidirect
170. Q₈	C₃×C₄⋊C₄	semidirect
174. D₄	C₃×C₄⋊C₄	semidirect
219. C₄	C₃×#104	semidirect
224. C₃	#1806	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×#41	2
C₃×#423	2
C₂×C₃×#104	2
C₃×C₄×C₄⋊C₄	2
C₃×#339	2
#1806	3

Subgroups — C3×#1806 #72521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1806 #72521