Subgroups — C₃×#1924 #72639

Order 384.

280 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	#1924	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	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
16	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊C₈	—	2	2	no	—
19	C₃×C₂²⋊C₈	—	2	2	no	—
20	C₃×C₂²⋊C₈	—	2	2	no	—
21	C₃×#97	—	2	2	no	—
22	C₃×#95	C₄	1	1	yes	characteristic
23	C₃×#95	C₄	1	1	yes	characteristic
24	C₂×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
26	C₂×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
27	C₂×#97	C₂×C₃	1	1	yes	characteristic, complemented
28	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
29	#336	C₂×C₃	1	1	yes	characteristic, complemented
30	#336	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₈	—	2	1	no	—
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	—	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	—	2	1	no	—
54	C₂×C₃×C₈	D₄	1	2	yes	—
55	C₂×C₃×C₈	D₄	1	2	yes	—
56	C₃×C₂²⋊C₄	—	2	2	no	—
57	C₃×C₂²⋊C₄	—	2	2	no	—
58	C₃×C₄²	—	2	1	no	—
59	C₃×C₄²	—	2	1	no	—
60	C₃×C₄²	—	2	1	no	—
61	C₃×C₄²	—	2	1	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	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
69	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	C₂²⋊C₈	—	2	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	#97	—	2	2	no	—
75	#95	C₃×C₄	1	1	yes	characteristic
76	#95	C₃×C₄	1	1	yes	characteristic
77	C₂³×C₃	—	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	M₁₆	1	2	yes	—
84	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	—	2	1	no	—
86	C₂×C₃×C₄	—	2	1	no	—
87	C₂×C₃×C₄	—	2	1	no	—
88	C₂×C₃×C₄	—	2	1	no	—
89	C₂×C₃×C₄	—	2	1	no	—
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	2	1	no	—
95	C₂×C₃×C₄	—	2	1	no	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	2	1	no	—
98	C₂×C₃×C₄	—	2	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	M₁₆	1	2	yes	—
104	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃×C₄	#41	1	1	yes	characteristic
107	C₂×C₃×C₄	#41	1	1	yes	characteristic
108	C₂×C₃×C₄	#41	1	1	yes	characteristic
109	C₂×C₃×C₄	#41	1	1	yes	characteristic
110	C₃×C₈	—	2	2	no	—
111	C₃×C₈	—	2	2	no	—
112	C₃×C₈	—	4	1	no	—
113	C₃×C₈	—	4	1	no	—
114	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
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	C₂×C₃×C₄	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
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	C₃×D₄	1	2	yes	—
138	C₂×C₈	C₃×D₄	1	2	yes	—
139	C₂²⋊C₄	—	2	2	no	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	—	2	2	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	4	1	no	—
149	C₂²×C₃	—	4	1	no	—
150	C₂²×C₃	—	4	1	no	—
151	C₂²×C₃	—	4	1	no	—
152	C₂²×C₃	#117	1	1	yes	characteristic
153	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
154	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
155	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
156	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
157	C₂²×C₃	#131	1	1	yes	characteristic
158	C₂²×C₃	#131	1	1	yes	characteristic
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	4	1	no	—
161	C₃×C₄	—	4	1	no	—
162	C₃×C₄	—	4	1	no	—
163	C₃×C₄	—	4	1	no	—
164	C₃×C₄	#121	1	1	yes	characteristic
165	C₃×C₄	#123	1	1	yes	characteristic
166	C₃×C₄	#124	1	1	yes	characteristic
167	C₃×C₄	#126	1	1	yes	characteristic
168	C₂³	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	1	no	—
172	C₂³	—	2	1	no	—
173	C₂³	—	2	1	no	—
174	C₂³	C₃×M₁₆	1	2	yes	—
175	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₃×M₁₆	1	2	yes	—
195	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
197	C₂×C₄	C₃×#41	1	1	yes	characteristic
198	C₂×C₄	C₃×#41	1	1	yes	characteristic
199	C₂×C₄	C₃×#41	1	1	yes	characteristic
200	C₂×C₄	C₃×#41	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	4	1	no	—
204	C₈	—	4	1	no	—
205	C₂×C₃	—	4	2	no	—
206	C₂×C₃	#388	1	1	yes	characteristic
207	C₂×C₃	#432	1	1	yes	characteristic
208	C₂×C₃	#433	1	1	yes	characteristic
209	C₂×C₃	#435	1	1	yes	characteristic
210	C₂×C₃	#435	1	1	yes	characteristic
211	C₂×C₃	#435	1	1	yes	characteristic
212	C₂×C₃	#436	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	4	1	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	C₃×#117	1	1	yes	characteristic
221	C₂²	C₃×C₄×D₄	1	1	yes	characteristic
222	C₂²	C₃×C₄×D₄	1	1	yes	characteristic
223	C₂²	C₂×C₃×M₁₆	1	1	yes	characteristic
224	C₂²	C₂×C₃×M₁₆	1	1	yes	characteristic
225	C₂²	C₃×#131	1	1	yes	characteristic
226	C₂²	C₃×#131	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	C₃×#121	1	1	yes	characteristic
233	C₄	C₃×#123	1	1	yes	characteristic
234	C₄	C₃×#124	1	1	yes	characteristic
235	C₄	C₃×#126	1	1	yes	characteristic
236	C₃	#1924	1	1	yes	characteristic, direct product factor, Sylow
237	C₂	—	4	2	no	—
238	C₂	C₃×#388	1	1	yes	characteristic
239	C₂	C₃×#432	1	1	yes	characteristic
240	C₂	C₃×#433	1	1	yes	characteristic
241	C₂	C₃×#435	1	1	yes	characteristic
242	C₂	C₃×#435	1	1	yes	characteristic
243	C₂	C₃×#435	1	1	yes	characteristic
244	C₂	C₃×#436	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄⋊C₈	C₂	semidirect
4. C₂×C₃×#97	C₂	semidirect
6. C₃×#336	C₂	semidirect
7. C₃×#336	C₂	semidirect
8. #1924	C₃	direct (N × H)
24. C₂×C₄⋊C₈	C₂×C₃	semidirect
27. C₂×#97	C₂×C₃	semidirect
29. #336	C₂×C₃	semidirect
30. #336	C₂×C₃	semidirect
236. C₃	#1924	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⋊C₈	2
C₂×C₃×C₂²⋊C₈	2
C₂×C₃×#97	2
C₃×C₄×C₂²⋊C₄	2
C₃×#336	2
#1924	3

Subgroups — C3×#1924 #72639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1924 #72639