Subgroups — C₃×#10276 #10628883

Order 768.

574 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#475	C₂	1	2	yes	maximal
3	C₃×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#332	C₂	1	2	yes	maximal
5	C₃×#1277	C₂	1	1	yes	maximal, characteristic
6	#10276	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₁₆	—	2	2	no	—
8	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₃×#494	—	2	4	no	—
11	C₃×#475	—	2	4	no	—
12	C₃×#475	—	2	4	no	—
13	C₂×C₃×#106	—	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×#332	—	2	4	no	—
20	C₂×#494	C₂×C₃	1	1	yes	characteristic
21	C₂×#475	C₂×C₃	1	2	yes	—
22	#1984	C₂×C₃	1	1	yes	characteristic
23	C₂×#332	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	C₂×C₃×Q₁₆	—	2	8	no	—
28	C₂×C₃×Q₁₆	—	4	4	no	—
29	C₂×C₃×Q₁₆	—	4	4	no	—
30	C₂²×C₃×C₈	—	2	2	no	—
31	C₂²×C₃×C₈	—	2	1	no	—
32	C₂²×C₃×C₈	D₄	1	2	yes	—
33	C₃×C₄⋊Q₈	—	2	2	no	—
34	C₃×C₄⋊Q₈	—	2	2	no	—
35	C₃×C₄⋊Q₈	—	2	2	no	—
36	C₃×C₄⋊Q₈	D₄	1	4	yes	—
37	C₃×C₄×Q₈	—	2	2	no	—
38	C₃×C₄×Q₈	—	2	2	no	—
39	C₂×C₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄²	—	2	1	no	—
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₃×#106	—	2	8	no	—
46	C₃×C₄⋊C₈	—	2	2	no	—
47	C₃×C₄⋊C₈	—	2	2	no	—
48	C₃×C₄⋊C₈	D₄	1	4	yes	—
49	C₃×#103	—	2	8	no	—
50	C₃×C₄×C₈	—	2	2	no	—
51	C₂²×Q₁₆	—	2	2	no	—
52	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
53	C₂×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
54	#494	—	2	4	no	—
55	#475	—	2	4	no	—
56	#475	—	2	4	no	—
57	C₂×#106	—	2	2	no	—
58	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
60	C₂×#103	—	2	2	no	—
61	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
62	#336	—	2	2	no	—
63	#332	—	2	4	no	—
64	C₂×C₃×Q₈	—	2	1	no	—
65	C₂×C₃×Q₈	—	2	2	no	—
66	C₂×C₃×Q₈	—	2	4	no	—
67	C₂×C₃×Q₈	—	2	2	no	—
68	C₂×C₃×Q₈	—	4	2	no	—
69	C₂×C₃×Q₈	—	4	4	no	—
70	C₂×C₃×Q₈	—	4	4	no	—
71	C₂×C₃×Q₈	Q₁₆	1	2	yes	complemented
72	C₂²×C₃×C₄	—	2	1	no	—
73	C₂²×C₃×C₄	—	2	1	no	—
74	C₂²×C₃×C₄	—	4	2	no	—
75	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	#41	1	1	yes	characteristic
78	C₃×Q₁₆	—	4	16	no	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	4	no	—
82	C₂×C₃×C₈	—	4	2	no	—
83	C₂×C₃×C₈	—	4	2	no	—
84	C₂×C₃×C₈	—	4	1	no	—
85	C₂×C₃×C₈	—	4	2	no	—
86	C₂×C₃×C₈	SD₁₆	1	4	yes	—
87	C₃×C₄⋊C₄	—	2	4	no	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	—	2	4	no	—
90	C₃×C₄⋊C₄	—	4	2	no	—
91	C₃×C₄⋊C₄	—	4	2	no	—
92	C₃×C₄⋊C₄	—	4	2	no	—
93	C₃×C₄²	—	4	1	no	—
94	C₃×C₄²	—	4	1	no	—
95	C₃×C₄²	C₂×D₄	1	2	yes	—
96	C₃×C₄²	C₂×D₄	1	2	yes	—
97	C₂²×Q₈	—	2	2	no	—
98	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
99	C₂×Q₁₆	—	2	8	no	—
100	C₂×Q₁₆	—	4	4	no	—
101	C₂×Q₁₆	—	4	4	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	C₃×D₄	1	2	yes	—
105	C₄⋊Q₈	—	2	2	no	—
106	C₄⋊Q₈	—	2	2	no	—
107	C₄⋊Q₈	—	2	2	no	—
108	C₄⋊Q₈	C₃×D₄	1	4	yes	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
115	C₂×C₄²	—	2	1	no	—
116	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
117	#106	—	2	8	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	C₃×D₄	1	4	yes	—
121	#103	—	2	8	no	—
122	C₄×C₈	—	2	2	no	—
123	C₂³×C₃	#121	1	1	yes	characteristic
124	C₃×Q₈	—	2	4	no	—
125	C₃×Q₈	—	4	2	no	—
126	C₃×Q₈	—	4	8	no	—
127	C₃×Q₈	—	8	2	no	—
128	C₃×Q₈	—	8	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	4	1	no	—
132	C₂×C₃×C₄	—	4	1	no	—
133	C₂×C₃×C₄	—	4	1	no	—
134	C₂×C₃×C₄	—	4	4	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	8	4	no	—
137	C₂×C₃×C₄	#120	1	1	yes	characteristic
138	C₂×C₃×C₄	#120	1	2	yes	—
139	C₂×C₃×C₄	#121	1	2	yes	—
140	C₂×C₃×C₄	#127	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
142	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
143	C₂×C₃×C₄	#135	1	1	yes	characteristic
144	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₃×C₄	#137	1	2	yes	—
146	C₃×C₈	—	2	4	no	—
147	C₃×C₈	—	4	4	no	—
148	C₃×C₈	—	8	1	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂×Q₈	—	4	4	no	—
156	C₂×Q₈	C₃×Q₁₆	1	2	yes	complemented
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₃×#41	1	1	yes	characteristic
163	Q₁₆	—	4	16	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	C₃×SD₁₆	1	4	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄²	—	4	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂×C₃×D₄	1	2	yes	—
181	C₄²	C₂×C₃×D₄	1	2	yes	—
182	C₂²×C₃	#392	1	1	yes	characteristic
183	C₂²×C₃	#460	1	2	yes	—
184	C₂²×C₃	#467	1	1	yes	characteristic
185	C₂²×C₃	#470	1	1	yes	characteristic
186	C₂²×C₃	#473	1	2	yes	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	4	2	no	—
189	C₃×C₄	—	8	4	no	—
190	C₃×C₄	—	8	1	no	—
191	C₃×C₄	#450	1	2	yes	—
192	C₃×C₄	#451	1	2	yes	—
193	C₃×C₄	#452	1	2	yes	—
194	C₃×C₄	#492	1	1	yes	characteristic
195	C₃×C₄	#496	1	1	yes	characteristic
196	C₂³	C₃×#121	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	4	8	no	—
200	Q₈	—	8	2	no	—
201	Q₈	—	8	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	4	no	—
210	C₂×C₄	C₃×#120	1	1	yes	characteristic
211	C₂×C₄	C₃×#120	1	2	yes	—
212	C₂×C₄	C₃×#121	1	2	yes	—
213	C₂×C₄	C₃×#127	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
215	C₂×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
216	C₂×C₄	C₃×#135	1	1	yes	characteristic
217	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
218	C₂×C₄	C₃×#137	1	2	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	4	4	no	—
221	C₈	—	8	1	no	—
222	C₂×C₃	#1670	1	2	yes	—
223	C₂×C₃	#1690	1	2	yes	—
224	C₂×C₃	#2001	1	1	yes	characteristic
225	C₂×C₃	#2014	1	1	yes	characteristic
226	C₂×C₃	#2020	1	1	yes	characteristic
227	C₂²	C₃×#392	1	1	yes	characteristic
228	C₂²	C₃×#460	1	2	yes	—
229	C₂²	C₃×#467	1	1	yes	characteristic
230	C₂²	C₃×#470	1	1	yes	characteristic
231	C₂²	C₃×#473	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	4	no	—
235	C₄	—	8	1	no	—
236	C₄	C₃×#450	1	2	yes	—
237	C₄	C₃×#451	1	2	yes	—
238	C₄	C₃×#452	1	2	yes	—
239	C₄	C₃×#492	1	1	yes	characteristic
240	C₄	C₃×#496	1	1	yes	characteristic
241	C₃	#10276	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	C₃×#1670	1	2	yes	—
243	C₂	C₃×#1690	1	2	yes	—
244	C₂	C₃×#2001	1	1	yes	characteristic
245	C₂	C₃×#2014	1	1	yes	characteristic
246	C₂	C₃×#2020	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. #10276	C₃	direct (N × H)
71. C₂×C₃×Q₈	Q₁₆	semidirect
156. C₂×Q₈	C₃×Q₁₆	semidirect
241. C₃	#10276	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×#494	2
C₂×C₃×#475	2
C₃×#1984	2
C₂×C₃×#332	2
C₃×#1277	2
#10276	3

Subgroups — C3×#10276 #10628883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10276 #10628883