Subgroups — C₃×#1854 #72569

Order 384.

316 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×#380	C₂	1	1	yes	maximal, characteristic
6	C₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
8	#1854	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×#117	—	2	2	no	—
13	C₃×#117	—	2	2	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
19	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
20	C₂×C₃×C₄²	—	2	1	no	—
21	C₃×#107	—	2	2	no	—
22	C₃×#106	—	2	2	no	—
23	C₃×C₂²⋊C₈	C₄	1	4	yes	complemented
24	C₃×#95	—	2	1	no	—
25	C₂×#117	C₂×C₃	1	1	yes	characteristic
26	C₂×#107	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#106	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
29	#380	C₂×C₃	1	1	yes	characteristic
30	#340	C₂×C₃	1	1	yes	characteristic, complemented
31	#340	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂⁴×C₃	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	—	2	1	no	—
34	C₂²×C₃×C₄	—	2	1	no	—
35	C₂²×C₃×C₄	—	2	1	no	—
36	C₂²×C₃×C₄	—	2	1	no	—
37	C₂²×C₃×C₄	—	2	1	no	—
38	C₂²×C₃×C₄	—	2	1	no	—
39	C₂²×C₃×C₄	—	2	1	no	—
40	C₂²×C₃×C₄	—	2	1	no	—
41	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
42	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
43	C₂²×C₃×C₄	Q₈	1	2	yes	—
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₈	—	2	1	no	—
46	C₂×C₃×C₈	—	2	1	no	—
47	C₂×C₃×C₈	—	2	1	no	—
48	C₂×C₃×C₈	—	2	1	no	—
49	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
51	C₃×C₄⋊C₄	—	2	1	no	—
52	C₃×C₄⋊C₄	—	2	1	no	—
53	C₃×C₄⋊C₄	—	2	1	no	—
54	C₃×C₄⋊C₄	—	2	1	no	—
55	C₃×C₄⋊C₄	—	2	1	no	—
56	C₃×C₄⋊C₄	—	2	1	no	—
57	C₃×C₄⋊C₄	D₄	1	2	yes	—
58	C₃×C₄⋊C₄	D₄	1	2	yes	—
59	C₃×C₂²⋊C₄	—	2	4	no	—
60	C₃×C₂²⋊C₄	—	4	2	no	—
61	C₃×C₄²	—	2	4	no	—
62	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
65	#117	—	2	2	no	—
66	#117	—	2	2	no	—
67	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₂²⋊C₄	—	2	1	no	—
72	C₂×C₂²⋊C₄	—	2	1	no	—
73	C₂×C₄²	—	2	1	no	—
74	#107	—	2	2	no	—
75	#106	—	2	2	no	—
76	C₂²⋊C₈	C₃×C₄	1	4	yes	complemented
77	#95	—	2	1	no	—
78	C₂³×C₃	—	2	1	no	—
79	C₂³×C₃	—	2	1	no	—
80	C₂³×C₃	—	2	1	no	—
81	C₂³×C₃	—	2	1	no	—
82	C₂³×C₃	—	2	1	no	—
83	C₂³×C₃	—	2	1	no	—
84	C₂³×C₃	C₄⋊C₄	1	2	yes	—
85	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄	—	2	1	no	—
87	C₂×C₃×C₄	—	2	1	no	—
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	2	2	no	—
90	C₂×C₃×C₄	—	2	2	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	—	4	1	no	—
95	C₂×C₃×C₄	—	4	1	no	—
96	C₂×C₃×C₄	—	4	1	no	—
97	C₂×C₃×C₄	—	4	1	no	—
98	C₂×C₃×C₄	—	4	1	no	—
99	C₂×C₃×C₄	—	4	1	no	—
100	C₂×C₃×C₄	—	4	1	no	—
101	C₂×C₃×C₄	—	4	1	no	—
102	C₂×C₃×C₄	—	4	1	no	—
103	C₂×C₃×C₄	—	4	1	no	—
104	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
105	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₃×C₄	#41	1	1	yes	characteristic
110	C₂×C₃×C₄	#41	1	1	yes	characteristic
111	C₃×C₈	—	2	2	no	—
112	C₃×C₈	—	2	2	no	—
113	C₂⁴	C₃×D₄	1	1	yes	characteristic
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
123	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
124	C₂²×C₄	C₃×Q₈	1	2	yes	—
125	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
131	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	C₃×D₄	1	2	yes	—
139	C₄⋊C₄	C₃×D₄	1	2	yes	—
140	C₂²⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	4	2	no	—
142	C₄²	—	2	4	no	—
143	C₂²×C₃	—	2	2	no	—
144	C₂²×C₃	—	2	2	no	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	—	4	1	no	—
147	C₂²×C₃	—	4	1	no	—
148	C₂²×C₃	—	4	1	no	—
149	C₂²×C₃	—	4	1	no	—
150	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₃	#120	1	1	yes	characteristic
152	C₂²×C₃	#123	1	1	yes	characteristic
153	C₂²×C₃	#135	1	1	yes	characteristic
154	C₂²×C₃	#135	1	1	yes	characteristic
155	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂²×C₃	#137	1	1	yes	characteristic
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	4	2	no	—
159	C₃×C₄	—	4	2	no	—
160	C₃×C₄	—	8	1	no	—
161	C₃×C₄	—	8	1	no	—
162	C₃×C₄	C₄×D₄	1	1	yes	characteristic
163	C₃×C₄	C₄×D₄	1	1	yes	characteristic
164	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
165	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
166	C₂³	—	2	1	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	1	no	—
172	C₂³	C₃×C₄⋊C₄	1	2	yes	—
173	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
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	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
193	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
197	C₂×C₄	C₃×#41	1	1	yes	characteristic
198	C₂×C₄	C₃×#41	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	2	no	—
201	C₂×C₃	—	4	2	no	—
202	C₂×C₃	#385	1	1	yes	characteristic
203	C₂×C₃	#427	1	1	yes	characteristic
204	C₂×C₃	#428	1	1	yes	characteristic
205	C₂×C₃	#449	1	1	yes	characteristic
206	C₂×C₃	#452	1	1	yes	characteristic
207	C₂×C₃	#482	1	1	yes	characteristic
208	C₂×C₃	#485	1	1	yes	characteristic
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂²	C₃×#120	1	1	yes	characteristic
218	C₂²	C₃×#123	1	1	yes	characteristic
219	C₂²	C₃×#135	1	1	yes	characteristic
220	C₂²	C₃×#135	1	1	yes	characteristic
221	C₂²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂²	C₃×#137	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	8	1	no	—
227	C₄	—	8	1	no	—
228	C₄	C₃×C₄×D₄	1	1	yes	characteristic
229	C₄	C₃×C₄×D₄	1	1	yes	characteristic
230	C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
231	C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
232	C₃	#1854	1	1	yes	characteristic, direct product factor, Sylow
233	C₂	—	4	2	no	—
234	C₂	C₃×#385	1	1	yes	characteristic
235	C₂	C₃×#427	1	1	yes	characteristic
236	C₂	C₃×#428	1	1	yes	characteristic
237	C₂	C₃×#449	1	1	yes	characteristic
238	C₂	C₃×#452	1	1	yes	characteristic
239	C₂	C₃×#482	1	1	yes	characteristic
240	C₂	C₃×#485	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃
prime radical	C₂⁴×C₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#107	C₂	semidirect
3. C₂×C₃×#106	C₂	semidirect
6. C₃×#340	C₂	semidirect
7. C₃×#340	C₂	semidirect
8. #1854	C₃	direct (N × H)
23. C₃×C₂²⋊C₈	C₄	semidirect
26. C₂×#107	C₂×C₃	semidirect
27. C₂×#106	C₂×C₃	semidirect
30. #340	C₂×C₃	semidirect
31. #340	C₂×C₃	semidirect
76. C₂²⋊C₈	C₃×C₄	semidirect
232. C₃	#1854	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×#117	2
C₂×C₃×#107	2
C₂×C₃×#106	2
C₂×C₃×C₂²⋊C₈	2
C₃×#380	2
C₃×#340	2
#1854	3

Subgroups — C3×#1854 #72569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1854 #72569