Subgroups — C₃×#21952 #10640559

Order 768.

826 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#431	C₂	1	2	yes	maximal
2	C₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₃×#1939	C₂	1	8	yes	maximal, complemented
4	C₃×#1798	C₂	1	4	yes	maximal
5	#21952	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×#117	C₂²	1	2	yes	—
7	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₃×#431	—	2	8	no	—
9	C₂×C₃×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊C₈	C₂²	1	8	yes	—
11	C₂×C₃×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	4	yes	—
13	C₃×#384	C₄	1	2	yes	—
14	C₃×#382	C₄	1	2	yes	—
15	C₃×#380	C₄	1	4	yes	—
16	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×#336	C₂²	1	8	yes	—
18	C₂×#431	C₂×C₃	1	2	yes	—
19	#2285	C₂×C₃	1	1	yes	characteristic
20	#1939	C₂×C₃	1	8	yes	complemented
21	#1798	C₂×C₃	1	4	yes	—
22	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
23	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	8	yes	—
25	C₃×#117	—	4	4	no	—
26	C₂×C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
27	C₂×C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄²	D₄	1	4	yes	—
32	C₂×C₃×C₄²	Q₈	1	4	yes	—
33	C₂×C₃×C₄²	C₂³	1	4	yes	—
34	C₂×C₃×C₄²	C₂³	1	2	yes	—
35	C₃×C₄⋊C₈	—	2	8	no	—
36	C₃×C₄⋊C₈	—	4	8	no	—
37	C₃×C₂²⋊C₈	—	2	8	no	—
38	C₃×C₄×C₈	—	2	8	no	—
39	C₃×#95	C₂×C₄	1	8	yes	—
40	C₂×#117	C₂²×C₃	1	2	yes	—
41	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
42	#431	—	2	8	no	—
43	C₂×C₄⋊C₈	C₂²×C₃	1	4	yes	—
44	C₂×C₄⋊C₈	C₂²×C₃	1	8	yes	—
45	C₂×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
46	C₂×C₄×C₈	C₂²×C₃	1	4	yes	—
47	#384	C₃×C₄	1	2	yes	—
48	#382	C₃×C₄	1	2	yes	—
49	#380	C₃×C₄	1	4	yes	—
50	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
51	#336	C₂²×C₃	1	8	yes	—
52	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₃×C₄	—	2	8	no	—
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	C₄⋊C₄	1	8	yes	—
57	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
58	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
59	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
61	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
62	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
63	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
64	C₂×C₃×C₈	—	2	8	no	—
65	C₂×C₃×C₈	—	2	8	no	—
66	C₂×C₃×C₈	—	2	8	no	—
67	C₃×C₄⋊C₄	—	2	16	no	—
68	C₃×C₄⋊C₄	—	4	4	no	—
69	C₃×C₂²⋊C₄	—	4	4	no	—
70	C₃×C₄²	—	2	8	no	—
71	C₃×C₄²	C₄⋊C₄	1	4	yes	—
72	C₃×C₄²	C₄⋊C₄	1	4	yes	—
73	C₃×C₄²	C₂×D₄	1	4	yes	—
74	C₃×C₄²	C₂×Q₈	1	4	yes	—
75	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
76	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
77	C₂²×C₈	C₂³×C₃	1	8	yes	—
78	#117	—	4	4	no	—
79	C₂×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
80	C₂×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
81	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
82	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₄²	C₃×D₄	1	4	yes	—
85	C₂×C₄²	C₃×Q₈	1	4	yes	—
86	C₂×C₄²	C₂³×C₃	1	4	yes	—
87	C₂×C₄²	C₂³×C₃	1	2	yes	—
88	C₄⋊C₈	—	2	8	no	—
89	C₄⋊C₈	—	4	8	no	—
90	C₂²⋊C₈	—	2	8	no	—
91	C₄×C₈	—	2	8	no	—
92	#95	C₂×C₃×C₄	1	8	yes	—
93	C₂³×C₃	—	2	1	no	—
94	C₂³×C₃	—	2	1	no	—
95	C₂³×C₃	—	2	4	no	—
96	C₂³×C₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₃×C₄	—	2	1	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	16	no	—
104	C₂×C₃×C₄	—	2	8	no	—
105	C₂×C₃×C₄	—	2	16	no	—
106	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₃×C₄	#131	1	8	yes	—
110	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₃×C₄	H(5,𝔽₂)	1	2	yes	—
113	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
114	C₃×C₈	—	4	8	no	—
115	C₂⁴	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	C₃×C₄⋊C₄	1	8	yes	—
120	C₂²×C₄	C₂²×C₃×C₄	1	8	yes	—
121	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
122	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
124	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
125	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
126	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
127	C₂×C₈	—	2	8	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	—	2	8	no	—
130	C₄⋊C₄	—	2	16	no	—
131	C₄⋊C₄	—	4	4	no	—
132	C₂²⋊C₄	—	4	4	no	—
133	C₄²	—	2	8	no	—
134	C₄²	C₃×C₄⋊C₄	1	4	yes	—
135	C₄²	C₃×C₄⋊C₄	1	4	yes	—
136	C₄²	C₂×C₃×D₄	1	4	yes	—
137	C₄²	C₂×C₃×Q₈	1	4	yes	—
138	C₂²×C₃	—	2	2	no	—
139	C₂²×C₃	—	2	4	no	—
140	C₂²×C₃	—	4	2	no	—
141	C₂²×C₃	—	4	2	no	—
142	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₃	#518	1	1	yes	characteristic
144	C₂²×C₃	#519	1	1	yes	characteristic
145	C₂²×C₃	C₂×#131	1	4	yes	—
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	—	2	4	no	—
148	C₃×C₄	—	4	8	no	—
149	C₃×C₄	#424	1	8	yes	—
150	C₃×C₄	#537	1	2	yes	—
151	C₃×C₄	#544	1	2	yes	—
152	C₂³	—	2	4	no	—
153	C₂³	—	2	1	no	—
154	C₂³	—	2	1	no	—
155	C₂³	C₂×C₃×C₄⋊C₄	1	2	yes	—
156	C₂³	C₂³×C₃×C₄	1	1	yes	characteristic
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	16	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	16	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
166	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
167	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
168	C₂×C₄	C₃×#131	1	8	yes	—
169	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
171	C₂×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
173	C₈	—	4	8	no	—
174	C₂×C₃	—	4	2	no	—
175	C₂×C₃	#2316	1	1	yes	characteristic
176	C₂×C₃	C₂×#424	1	2	yes	—
177	C₂×C₃	#2979	1	2	yes	—
178	C₂×C₃	#2986	1	2	yes	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
184	C₂²	C₃×#518	1	1	yes	characteristic
185	C₂²	C₃×#519	1	1	yes	characteristic
186	C₂²	C₂×C₃×#131	1	4	yes	—
187	C₄	—	2	4	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	8	no	—
190	C₄	C₃×#424	1	8	yes	—
191	C₄	C₃×#537	1	2	yes	—
192	C₄	C₃×#544	1	2	yes	—
193	C₃	#21952	1	1	yes	characteristic, direct product factor, Sylow
194	C₂	—	4	2	no	—
195	C₂	C₃×#2316	1	1	yes	characteristic
196	C₂	C₂×C₃×#424	1	2	yes	—
197	C₂	C₃×#2979	1	2	yes	—
198	C₂	C₃×#2986	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂⁴
commutator	C₂³	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂³×C₄
prime radical	C₂⁴×C₃	C₂²×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
3. C₃×#1939	C₂	semidirect
5. #21952	C₃	direct (N × H)
20. #1939	C₂×C₃	semidirect
193. C₃	#21952	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×#431	2
C₃×#2285	2
C₃×#1939	2
C₃×#1798	2
#21952	3

Subgroups — C3×#21952 #10640559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21952 #10640559