Subgroups — C₃×#9841 #10628448

Order 768.

510 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#498	C₂	1	1	yes	maximal, characteristic
2	C₃×#1990	C₂	1	2	yes	maximal
3	C₃×#1956	C₂	1	2	yes	maximal
4	C₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
6	#9841	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#498	—	2	2	no	—
9	C₃×#498	C₄	1	4	yes	complemented
10	C₂×C₃×#107	—	2	2	no	—
11	C₂×C₃×#106	—	2	2	no	—
12	C₂×C₃×#103	—	2	4	no	—
13	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×#384	C₂²	1	1	yes	characteristic
16	C₃×#384	C₂²	1	1	yes	characteristic
17	C₃×#340	—	2	4	no	—
18	C₃×#322	—	2	2	no	—
19	C₂×#498	C₂×C₃	1	1	yes	characteristic
20	#1990	C₂×C₃	1	2	yes	—
21	#1956	C₂×C₃	1	2	yes	—
22	#1845	C₂×C₃	1	1	yes	characteristic
23	C₂×#322	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃×Q₈	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	4	yes	—
26	C₂²×C₃×C₈	D₄	1	2	yes	—
27	C₃×C₄⋊Q₈	—	2	2	no	—
28	C₃×C₄⋊Q₈	—	2	2	no	—
29	C₃×C₄⋊Q₈	—	2	2	no	—
30	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
38	C₃×#107	—	4	4	no	—
39	C₃×#106	—	2	8	no	—
40	C₃×#103	—	4	8	no	—
41	C₃×C₄×C₈	—	2	2	no	—
42	C₃×C₄×C₈	—	2	2	no	—
43	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
44	C₃×C₄×C₈	D₄	1	2	yes	—
45	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
46	#498	—	2	2	no	—
47	#498	C₃×C₄	1	4	yes	complemented
48	C₂×#107	—	2	2	no	—
49	C₂×#106	—	2	2	no	—
50	C₂×#103	—	2	4	no	—
51	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
52	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
53	#384	C₂²×C₃	1	1	yes	characteristic
54	#384	C₂²×C₃	1	1	yes	characteristic
55	#340	—	2	4	no	—
56	#322	—	2	2	no	—
57	C₂×C₃×Q₈	—	2	4	no	—
58	C₂×C₃×Q₈	—	4	2	no	—
59	C₂×C₃×Q₈	—	4	4	no	—
60	C₂×C₃×Q₈	—	4	4	no	—
61	C₂²×C₃×C₄	—	4	2	no	—
62	C₂²×C₃×C₄	—	4	1	no	—
63	C₂²×C₃×C₄	—	4	1	no	—
64	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₈	—	2	4	no	—
67	C₂×C₃×C₈	—	2	4	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
70	C₂×C₃×C₈	SD₁₆	1	4	yes	—
71	C₂×C₃×C₈	Q₁₆	1	4	yes	—
72	C₃×C₄⋊C₄	—	2	4	no	—
73	C₃×C₄⋊C₄	—	2	4	no	—
74	C₃×C₄⋊C₄	—	4	2	no	—
75	C₃×C₄⋊C₄	—	4	2	no	—
76	C₃×C₄⋊C₄	—	4	2	no	—
77	C₃×C₄⋊C₄	—	4	2	no	—
78	C₃×C₄⋊C₄	—	4	4	no	—
79	C₃×C₄⋊C₄	—	4	4	no	—
80	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄²	#41	1	2	yes	—
83	C₂²×Q₈	—	2	2	no	—
84	C₂²×C₈	C₃×D₄	1	4	yes	—
85	C₂²×C₈	C₃×D₄	1	2	yes	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	C₄⋊Q₈	—	2	2	no	—
89	C₄⋊Q₈	C₂×C₃×C₄	1	4	yes	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	2	2	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	2	1	no	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
97	#107	—	4	4	no	—
98	#106	—	2	8	no	—
99	#103	—	4	8	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
103	C₄×C₈	C₃×D₄	1	2	yes	—
104	C₂³×C₃	#127	1	1	yes	characteristic
105	C₃×Q₈	—	4	8	no	—
106	C₃×Q₈	—	8	4	no	—
107	C₂×C₃×C₄	—	4	4	no	—
108	C₂×C₃×C₄	—	8	2	no	—
109	C₂×C₃×C₄	—	8	2	no	—
110	C₂×C₃×C₄	—	8	1	no	—
111	C₂×C₃×C₄	—	8	1	no	—
112	C₂×C₃×C₄	—	8	2	no	—
113	C₂×C₃×C₄	—	8	2	no	—
114	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
116	C₂×C₃×C₄	#121	1	2	yes	—
117	C₂×C₃×C₄	#124	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
119	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
120	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
121	C₂×C₃×C₄	#137	1	4	yes	—
122	C₃×C₈	—	4	4	no	—
123	C₃×C₈	#103	1	8	yes	complemented
124	C₂×Q₈	—	2	4	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	4	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	4	1	no	—
130	C₂²×C₄	—	4	1	no	—
131	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	C₃×C₂²⋊C₄	1	4	yes	—
137	C₂×C₈	C₃×SD₁₆	1	4	yes	—
138	C₂×C₈	C₃×Q₁₆	1	4	yes	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
149	C₄²	C₃×#41	1	2	yes	—
150	C₂²×C₃	#390	1	1	yes	characteristic
151	C₂²×C₃	#492	1	1	yes	characteristic
152	C₂²×C₃	#494	1	1	yes	characteristic
153	C₂²×C₃	#496	1	2	yes	—
154	C₂²×C₃	#497	1	2	yes	—
155	C₃×C₄	—	8	4	no	—
156	C₃×C₄	—	16	1	no	—
157	C₃×C₄	—	16	1	no	—
158	C₃×C₄	C₂×#103	1	2	yes	—
159	C₃×C₄	#440	1	4	yes	—
160	C₃×C₄	#468	1	4	yes	—
161	C₃×C₄	#487	1	2	yes	—
162	C₂³	C₃×#127	1	1	yes	characteristic
163	Q₈	—	4	8	no	—
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	1	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
174	C₂×C₄	C₃×#121	1	2	yes	—
175	C₂×C₄	C₃×#124	1	1	yes	characteristic
176	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
177	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
178	C₂×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
179	C₂×C₄	C₃×#137	1	4	yes	—
180	C₈	—	4	4	no	—
181	C₈	C₃×#103	1	8	yes	complemented
182	C₂×C₃	#1715	1	1	yes	characteristic
183	C₂×C₃	#1716	1	1	yes	characteristic
184	C₂×C₃	#1723	1	2	yes	—
185	C₂×C₃	#1956	1	1	yes	characteristic
186	C₂×C₃	#1973	1	2	yes	—
187	C₂²	C₃×#390	1	1	yes	characteristic
188	C₂²	C₃×#492	1	1	yes	characteristic
189	C₂²	C₃×#494	1	1	yes	characteristic
190	C₂²	C₃×#496	1	2	yes	—
191	C₂²	C₃×#497	1	2	yes	—
192	C₄	—	8	4	no	—
193	C₄	—	16	1	no	—
194	C₄	—	16	1	no	—
195	C₄	C₂×C₃×#103	1	2	yes	—
196	C₄	C₃×#440	1	4	yes	—
197	C₄	C₃×#468	1	4	yes	—
198	C₄	C₃×#487	1	2	yes	—
199	C₃	#9841	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	C₃×#1715	1	1	yes	characteristic
201	C₂	C₃×#1716	1	1	yes	characteristic
202	C₂	C₃×#1723	1	2	yes	—
203	C₂	C₃×#1956	1	1	yes	characteristic
204	C₂	C₃×#1973	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
LCS	G ▷ C₄² ▷ C₂² ▷ 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
6. #9841	C₃	direct (N × H)
9. C₃×#498	C₄	semidirect
47. #498	C₃×C₄	semidirect
123. C₃×C₈	#103	semidirect
181. C₈	C₃×#103	semidirect
199. C₃	#9841	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×#498	2
C₃×#1990	2
C₃×#1956	2
C₃×#1845	2
C₂×C₃×#322	2
#9841	3

Subgroups — C3×#9841 #10628448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9841 #10628448