Subgroups — C₃×#10279 #10628886

Order 768.

510 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#477	C₂	1	2	yes	maximal
3	C₃×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#333	C₂	1	2	yes	maximal
5	C₃×#1277	C₂	1	1	yes	maximal, characteristic
6	#10279	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
9	C₃×#500	—	2	4	no	—
10	C₃×#477	—	2	4	no	—
11	C₃×#477	—	2	4	no	—
12	C₂×C₃×#107	—	2	2	no	—
13	C₂×C₃×#107	—	2	2	no	—
14	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃×#103	—	2	2	no	—
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₃×#336	—	2	2	no	—
19	C₃×#333	—	2	4	no	—
20	C₂×#500	C₂×C₃	1	1	yes	characteristic
21	C₂×#477	C₂×C₃	1	2	yes	—
22	#1984	C₂×C₃	1	1	yes	characteristic
23	C₂×#333	C₂×C₃	1	2	yes	—
24	#1277	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₃×Q₈	—	2	2	no	—
26	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	—	2	2	no	—
28	C₂²×C₃×C₈	—	2	1	no	—
29	C₂²×C₃×C₈	D₄	1	2	yes	—
30	C₃×C₄⋊Q₈	—	2	2	no	—
31	C₃×C₄⋊Q₈	—	2	2	no	—
32	C₃×C₄⋊Q₈	—	2	2	no	—
33	C₃×C₄⋊Q₈	D₄	1	4	yes	—
34	C₃×C₄×Q₈	—	2	2	no	—
35	C₃×C₄×Q₈	—	2	2	no	—
36	C₂×C₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	—	2	1	no	—
41	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
42	C₃×#107	—	2	8	no	—
43	C₃×#107	—	2	8	no	—
44	C₃×C₄⋊C₈	—	2	2	no	—
45	C₃×C₄⋊C₈	—	2	2	no	—
46	C₃×C₄⋊C₈	D₄	1	4	yes	—
47	C₃×#103	—	2	8	no	—
48	C₃×C₄×C₈	—	2	2	no	—
49	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
50	C₂×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
51	#500	—	2	4	no	—
52	#477	—	2	4	no	—
53	#477	—	2	4	no	—
54	C₂×#107	—	2	2	no	—
55	C₂×#107	—	2	2	no	—
56	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
58	C₂×#103	—	2	2	no	—
59	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	#336	—	2	2	no	—
61	#333	—	2	4	no	—
62	C₂×C₃×Q₈	—	2	1	no	—
63	C₂×C₃×Q₈	—	2	2	no	—
64	C₂×C₃×Q₈	—	2	4	no	—
65	C₂×C₃×Q₈	—	2	2	no	—
66	C₂×C₃×Q₈	—	4	2	no	—
67	C₂×C₃×Q₈	—	8	4	no	—
68	C₂×C₃×Q₈	Q₁₆	1	2	yes	—
69	C₂²×C₃×C₄	—	2	1	no	—
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	—	4	2	no	—
72	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	#41	1	1	yes	characteristic
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	—	2	2	no	—
77	C₂×C₃×C₈	—	2	4	no	—
78	C₂×C₃×C₈	—	4	2	no	—
79	C₂×C₃×C₈	—	4	2	no	—
80	C₂×C₃×C₈	—	4	1	no	—
81	C₂×C₃×C₈	—	4	2	no	—
82	C₂×C₃×C₈	Q₁₆	1	4	yes	—
83	C₃×C₄⋊C₄	—	2	4	no	—
84	C₃×C₄⋊C₄	—	2	4	no	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	—	4	2	no	—
87	C₃×C₄⋊C₄	—	4	2	no	—
88	C₃×C₄⋊C₄	—	4	2	no	—
89	C₃×C₄²	—	4	1	no	—
90	C₃×C₄²	—	4	1	no	—
91	C₃×C₄²	C₂×D₄	1	2	yes	—
92	C₃×C₄²	C₂×D₄	1	2	yes	—
93	C₂²×Q₈	—	2	2	no	—
94	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
95	C₂²×C₈	—	2	2	no	—
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	C₃×D₄	1	2	yes	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄⋊Q₈	—	2	2	no	—
100	C₄⋊Q₈	—	2	2	no	—
101	C₄⋊Q₈	C₃×D₄	1	4	yes	—
102	C₄×Q₈	—	2	2	no	—
103	C₄×Q₈	—	2	2	no	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
108	C₂×C₄²	—	2	1	no	—
109	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
110	#107	—	2	8	no	—
111	#107	—	2	8	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	C₃×D₄	1	4	yes	—
115	#103	—	2	8	no	—
116	C₄×C₈	—	2	2	no	—
117	C₂³×C₃	#121	1	1	yes	characteristic
118	C₃×Q₈	—	2	4	no	—
119	C₃×Q₈	—	4	2	no	—
120	C₃×Q₈	—	8	4	no	—
121	C₃×Q₈	—	8	4	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	4	1	no	—
125	C₂×C₃×C₄	—	4	1	no	—
126	C₂×C₃×C₄	—	4	1	no	—
127	C₂×C₃×C₄	—	4	4	no	—
128	C₂×C₃×C₄	—	4	2	no	—
129	C₂×C₃×C₄	—	8	4	no	—
130	C₂×C₃×C₄	#120	1	1	yes	characteristic
131	C₂×C₃×C₄	#120	1	2	yes	—
132	C₂×C₃×C₄	#121	1	2	yes	—
133	C₂×C₃×C₄	#127	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
136	C₂×C₃×C₄	#135	1	1	yes	characteristic
137	C₂×C₃×C₄	#137	1	2	yes	—
138	C₂×C₃×C₄	#137	1	2	yes	—
139	C₃×C₈	—	2	4	no	—
140	C₃×C₈	—	4	4	no	—
141	C₃×C₈	—	8	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	8	4	no	—
148	C₂×Q₈	C₃×Q₁₆	1	2	yes	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂²×C₄	C₃×#41	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	C₃×Q₁₆	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂×C₃×D₄	1	2	yes	—
172	C₄²	C₂×C₃×D₄	1	2	yes	—
173	C₂²×C₃	#392	1	1	yes	characteristic
174	C₂²×C₃	#462	1	2	yes	—
175	C₂²×C₃	#467	1	1	yes	characteristic
176	C₂²×C₃	#470	1	1	yes	characteristic
177	C₂²×C₃	#472	1	2	yes	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	8	1	no	—
181	C₃×C₄	—	8	4	no	—
182	C₃×C₄	#451	1	2	yes	—
183	C₃×C₄	#451	1	2	yes	—
184	C₃×C₄	#452	1	2	yes	—
185	C₃×C₄	#494	1	1	yes	characteristic
186	C₃×C₄	#497	1	1	yes	characteristic
187	C₂³	C₃×#121	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	4	2	no	—
190	Q₈	—	8	4	no	—
191	Q₈	—	8	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	C₃×#120	1	1	yes	characteristic
201	C₂×C₄	C₃×#120	1	2	yes	—
202	C₂×C₄	C₃×#121	1	2	yes	—
203	C₂×C₄	C₃×#127	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
205	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
206	C₂×C₄	C₃×#135	1	1	yes	characteristic
207	C₂×C₄	C₃×#137	1	2	yes	—
208	C₂×C₄	C₃×#137	1	2	yes	—
209	C₈	—	2	4	no	—
210	C₈	—	4	4	no	—
211	C₈	—	8	1	no	—
212	C₂×C₃	#1676	1	2	yes	—
213	C₂×C₃	#1696	1	2	yes	—
214	C₂×C₃	#2001	1	1	yes	characteristic
215	C₂×C₃	#2016	1	1	yes	characteristic
216	C₂×C₃	#2018	1	1	yes	characteristic
217	C₂²	C₃×#392	1	1	yes	characteristic
218	C₂²	C₃×#462	1	2	yes	—
219	C₂²	C₃×#467	1	1	yes	characteristic
220	C₂²	C₃×#470	1	1	yes	characteristic
221	C₂²	C₃×#472	1	2	yes	—
222	C₄	—	2	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	1	no	—
225	C₄	—	8	4	no	—
226	C₄	C₃×#451	1	2	yes	—
227	C₄	C₃×#451	1	2	yes	—
228	C₄	C₃×#452	1	2	yes	—
229	C₄	C₃×#494	1	1	yes	characteristic
230	C₄	C₃×#497	1	1	yes	characteristic
231	C₃	#10279	1	1	yes	characteristic, direct product factor, Sylow
232	C₂	C₃×#1676	1	2	yes	—
233	C₂	C₃×#1696	1	2	yes	—
234	C₂	C₃×#2001	1	1	yes	characteristic
235	C₂	C₃×#2016	1	1	yes	characteristic
236	C₂	C₃×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
LCS	G ▷ C₂×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. #10279	C₃	direct (N × H)
231. C₃	#10279	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×#500	2
C₂×C₃×#477	2
C₃×#1984	2
C₂×C₃×#333	2
C₃×#1277	2
#10279	3

Subgroups — C3×#10279 #10628886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10279 #10628886