Subgroups — C₃×#1862 #72577

Order 384.

362 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
5	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	#1862	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₄	—	2	1	no	—
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×D₄	—	2	2	no	—
14	C₃×C₄×D₄	—	2	2	no	—
15	C₂×C₃×C₄⋊C₄	—	2	1	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
20	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₃×#107	—	2	2	no	—
22	C₃×#102	—	2	2	no	—
23	C₃×#102	C₄	1	4	yes	complemented
24	C₃×#95	—	2	1	no	—
25	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
26	C₂×#107	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#102	C₂×C₃	1	1	yes	characteristic
28	C₂×#102	C₂×C₃	1	1	yes	characteristic
29	#389	C₂×C₃	1	1	yes	characteristic, complemented
30	#340	C₂×C₃	1	1	yes	characteristic, complemented
31	#336	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂⁴×C₃	—	2	1	no	—
33	C₂×C₃×D₄	—	2	1	no	—
34	C₂×C₃×D₄	—	2	1	no	—
35	C₂×C₃×D₄	—	2	1	no	—
36	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
37	C₂×C₃×D₄	D₄	1	2	yes	—
38	C₂×C₃×D₄	Q₈	1	2	yes	—
39	C₂²×C₃×C₄	—	2	4	no	—
40	C₂²×C₃×C₄	—	2	1	no	—
41	C₂²×C₃×C₄	—	2	1	no	—
42	C₂²×C₃×C₄	—	2	1	no	—
43	C₂²×C₃×C₄	—	4	2	no	—
44	C₂²×C₃×C₄	—	4	2	no	—
45	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
46	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₈	—	2	1	no	—
49	C₂×C₃×C₈	—	2	1	no	—
50	C₂×C₃×C₈	—	2	1	no	—
51	C₂×C₃×C₈	—	2	1	no	—
52	C₂×C₃×C₈	—	2	1	no	—
53	C₂×C₃×C₈	C₂×C₄	1	2	yes	complemented
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	—	2	1	no	—
58	C₃×C₄⋊C₄	—	4	2	no	—
59	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	complemented
60	C₃×C₄⋊C₄	D₄	1	2	yes	—
61	C₃×C₂²⋊C₄	—	2	4	no	—
62	C₃×C₄²	—	2	1	no	—
63	C₃×C₄²	—	2	1	no	—
64	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
65	C₂³×C₄	—	2	1	no	—
66	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
68	C₄×D₄	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	2	1	no	—
71	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₂²⋊C₄	—	2	1	no	—
75	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
76	#107	—	2	2	no	—
77	#102	—	2	2	no	—
78	#102	C₃×C₄	1	4	yes	complemented
79	#95	—	2	1	no	—
80	C₂³×C₃	—	2	2	no	—
81	C₂³×C₃	—	2	2	no	—
82	C₂³×C₃	—	2	2	no	—
83	C₂³×C₃	—	4	1	no	—
84	C₂³×C₃	—	4	1	no	—
85	C₂³×C₃	—	4	1	no	—
86	C₂³×C₃	—	4	1	no	—
87	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
88	C₃×D₄	—	2	2	no	—
89	C₃×D₄	—	2	2	no	—
90	C₃×D₄	—	2	2	no	—
91	C₃×D₄	C₄⋊C₄	1	4	yes	complemented
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	1	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	1	no	—
98	C₂×C₃×C₄	—	2	1	no	—
99	C₂×C₃×C₄	—	4	8	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	—	4	1	no	—
105	C₂×C₃×C₄	—	4	1	no	—
106	C₂×C₃×C₄	—	4	1	no	—
107	C₂×C₃×C₄	D₈	1	2	yes	—
108	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₃×C₄	#41	1	1	yes	characteristic
113	C₂×C₃×C₄	#41	1	1	yes	characteristic
114	C₃×C₈	—	2	2	no	—
115	C₃×C₈	—	4	1	no	—
116	C₂⁴	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	—	2	1	no	—
120	C₂×D₄	C₂×C₃×C₄	1	2	yes	—
121	C₂×D₄	C₃×D₄	1	2	yes	—
122	C₂×D₄	C₃×Q₈	1	2	yes	—
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
130	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
131	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	C₂×C₃×C₄	1	2	yes	complemented
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	complemented
144	C₄⋊C₄	C₃×D₄	1	2	yes	—
145	C₂²⋊C₄	—	2	4	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₂²×C₃	—	2	4	no	—
149	C₂²×C₃	—	4	2	no	—
150	C₂²×C₃	—	4	2	no	—
151	C₂²×C₃	—	4	2	no	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	—	4	2	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
156	C₂²×C₃	#120	1	1	yes	characteristic
157	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
159	C₂²×C₃	#135	1	1	yes	characteristic
160	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂²×C₃	#137	1	1	yes	characteristic
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	4	2	no	—
164	C₃×C₄	—	4	2	no	—
165	C₃×C₄	—	4	1	no	—
166	C₃×C₄	—	8	1	no	—
167	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₃×C₄	C₄×D₄	1	1	yes	characteristic
169	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
170	C₃×C₄	#123	1	1	yes	characteristic
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	4	1	no	—
175	C₂³	—	4	1	no	—
176	C₂³	—	4	1	no	—
177	C₂³	—	4	1	no	—
178	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	C₃×C₄⋊C₄	1	4	yes	complemented
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	C₃×D₈	1	2	yes	—
199	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
203	C₂×C₄	C₃×#41	1	1	yes	characteristic
204	C₂×C₄	C₃×#41	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	4	1	no	—
207	C₂×C₃	—	4	4	no	—
208	C₂×C₃	#385	1	1	yes	characteristic
209	C₂×C₃	C₄×D₈	1	1	yes	characteristic
210	C₂×C₃	#440	1	1	yes	characteristic
211	C₂×C₃	#447	1	1	yes	characteristic
212	C₂×C₃	#452	1	1	yes	characteristic
213	C₂×C₃	#474	1	1	yes	characteristic
214	C₂×C₃	#478	1	1	yes	characteristic
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₃×C₄×D₄	1	1	yes	characteristic
223	C₂²	C₃×#120	1	1	yes	characteristic
224	C₂²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
225	C₂²	C₂×C₃×D₈	1	1	yes	characteristic
226	C₂²	C₃×#135	1	1	yes	characteristic
227	C₂²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₂²	C₃×#137	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	8	1	no	—
234	C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
235	C₄	C₃×C₄×D₄	1	1	yes	characteristic
236	C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
237	C₄	C₃×#123	1	1	yes	characteristic
238	C₃	#1862	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	4	4	no	—
240	C₂	C₃×#385	1	1	yes	characteristic
241	C₂	C₃×C₄×D₈	1	1	yes	characteristic
242	C₂	C₃×#440	1	1	yes	characteristic
243	C₂	C₃×#447	1	1	yes	characteristic
244	C₂	C₃×#452	1	1	yes	characteristic
245	C₂	C₃×#474	1	1	yes	characteristic
246	C₂	C₃×#478	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	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#107	C₂	semidirect
5. C₃×#389	C₂	semidirect
6. C₃×#340	C₂	semidirect
7. C₃×#336	C₂	semidirect
8. #1862	C₃	direct (N × H)
23. C₃×#102	C₄	semidirect
26. C₂×#107	C₂×C₃	semidirect
29. #389	C₂×C₃	semidirect
30. #340	C₂×C₃	semidirect
31. #336	C₂×C₃	semidirect
53. C₂×C₃×C₈	C₂×C₄	semidirect
59. C₃×C₄⋊C₄	C₂×C₄	semidirect
78. #102	C₃×C₄	semidirect
91. C₃×D₄	C₄⋊C₄	semidirect
137. C₂×C₈	C₂×C₃×C₄	semidirect
143. C₄⋊C₄	C₂×C₃×C₄	semidirect
182. D₄	C₃×C₄⋊C₄	semidirect
238. C₃	#1862	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×D₄	2
C₂×C₃×#107	2
C₂×C₃×#102	2
C₃×#389	2
C₃×#340	2
C₃×#336	2
#1862	3

Subgroups — C3×#1862 #72577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1862 #72577