Subgroups — C₃×#10278 #10628885

Order 768.

562 conjugacy classes of subgroups.

The subgroup lattice has 264 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, complemented
2	C₂×C₃×#476	C₂	1	2	yes	maximal
3	C₃×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#333	C₂	1	2	yes	maximal, complemented
5	C₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	#10278	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic
9	C₃×#500	—	2	4	no	—
10	C₃×#476	—	2	4	no	—
11	C₃×#476	—	2	4	no	—
12	C₂×C₃×#107	—	2	2	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×#102	—	2	2	no	—
17	C₂×C₃×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₃×#333	—	2	4	no	—
20	C₂×#500	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×#476	C₂×C₃	1	2	yes	—
22	#1954	C₂×C₃	1	1	yes	characteristic
23	C₂×#333	C₂×C₃	1	2	yes	complemented
24	#1277	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²×C₃×Q₈	—	2	2	no	—
26	C₂²×C₃×D₄	D₄	1	1	yes	characteristic
27	C₂³×C₃×C₄	—	2	1	no	—
28	C₂²×C₃×C₈	—	2	2	no	—
29	C₂²×C₃×C₈	—	2	1	no	—
30	C₂²×C₃×C₈	D₄	1	2	yes	—
31	C₃×C₄⋊Q₈	—	2	2	no	—
32	C₃×C₄⋊Q₈	—	2	2	no	—
33	C₃×C₄⋊Q₈	—	2	2	no	—
34	C₃×C₄⋊Q₈	D₄	1	4	yes	—
35	C₃×C₄×D₄	—	2	2	no	—
36	C₃×C₄×D₄	—	2	2	no	—
37	C₂×C₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
41	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
42	C₃×#107	—	2	8	no	—
43	C₃×#106	—	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×#102	—	2	8	no	—
48	C₃×C₂²⋊C₈	—	4	4	no	—
49	C₃×C₄×C₈	—	2	2	no	—
50	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
51	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
52	#500	—	2	4	no	—
53	#476	—	2	4	no	—
54	#476	—	2	4	no	—
55	C₂×#107	—	2	2	no	—
56	C₂×#106	—	2	2	no	—
57	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
59	C₂×#102	—	2	2	no	—
60	C₂×C₂²⋊C₈	—	2	2	no	—
61	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
62	#333	—	2	4	no	—
63	C₂⁴×C₃	—	2	1	no	—
64	C₂×C₃×Q₈	—	2	4	no	—
65	C₂×C₃×Q₈	—	4	2	no	—
66	C₂×C₃×Q₈	—	8	4	no	—
67	C₂×C₃×D₄	—	2	2	no	—
68	C₂×C₃×D₄	—	2	1	no	—
69	C₂×C₃×D₄	—	2	2	no	—
70	C₂×C₃×D₄	Q₁₆	1	2	yes	—
71	C₂²×C₃×C₄	—	2	1	no	—
72	C₂²×C₃×C₄	—	4	2	no	—
73	C₂²×C₃×C₄	—	4	1	no	—
74	C₂²×C₃×C₄	—	4	1	no	—
75	C₂²×C₃×C₄	—	4	2	no	—
76	C₂²×C₃×C₄	—	4	2	no	—
77	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₃×C₄	#41	1	1	yes	characteristic
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	4	no	—
83	C₂×C₃×C₈	—	4	2	no	—
84	C₂×C₃×C₈	—	4	2	no	—
85	C₂×C₃×C₈	—	4	1	no	—
86	C₂×C₃×C₈	—	4	2	no	—
87	C₂×C₃×C₈	SD₁₆	1	4	yes	—
88	C₃×C₄⋊C₄	—	2	4	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	—	2	4	no	—
91	C₃×C₄⋊C₄	—	4	2	no	—
92	C₃×C₄⋊C₄	—	4	2	no	—
93	C₃×C₂²⋊C₄	—	4	2	no	—
94	C₃×C₄²	C₂×D₄	1	2	yes	—
95	C₃×C₄²	C₂×D₄	1	2	yes	—
96	C₂²×Q₈	—	2	2	no	—
97	C₂²×D₄	C₃×D₄	1	1	yes	characteristic
98	C₂³×C₄	—	2	1	no	—
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	C₃×D₄	1	2	yes	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	—	2	2	no	—
104	C₄⋊Q₈	—	2	2	no	—
105	C₄⋊Q₈	C₃×D₄	1	4	yes	—
106	C₄×D₄	—	2	2	no	—
107	C₄×D₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
113	#107	—	2	8	no	—
114	#106	—	2	8	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	C₃×D₄	1	4	yes	—
118	#102	—	2	8	no	—
119	C₂²⋊C₈	—	4	4	no	—
120	C₄×C₈	—	2	2	no	—
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	4	1	no	—
123	C₂³×C₃	—	4	2	no	—
124	C₂³×C₃	—	4	1	no	—
125	C₂³×C₃	—	4	2	no	—
126	C₂³×C₃	#121	1	1	yes	characteristic
127	C₃×Q₈	—	8	4	no	—
128	C₃×Q₈	—	8	4	no	—
129	C₃×D₄	—	2	4	no	—
130	C₃×D₄	—	4	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	4	1	no	—
133	C₂×C₃×C₄	—	4	4	no	—
134	C₂×C₃×C₄	—	4	2	no	—
135	C₂×C₃×C₄	—	8	4	no	—
136	C₂×C₃×C₄	—	8	2	no	—
137	C₂×C₃×C₄	—	8	1	no	—
138	C₂×C₃×C₄	—	8	1	no	—
139	C₂×C₃×C₄	#120	1	1	yes	characteristic
140	C₂×C₃×C₄	#120	1	2	yes	—
141	C₂×C₃×C₄	#121	1	2	yes	—
142	C₂×C₃×C₄	#127	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
144	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₃×C₄	#135	1	1	yes	characteristic
146	C₂×C₃×C₄	#137	1	2	yes	—
147	C₂×C₃×C₄	#137	1	2	yes	—
148	C₃×C₈	—	2	4	no	—
149	C₃×C₈	—	4	4	no	—
150	C₃×C₈	—	8	1	no	—
151	C₂⁴	—	2	1	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	C₃×Q₁₆	1	2	yes	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂²×C₄	C₃×#41	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	C₃×SD₁₆	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	C₂×C₃×D₄	1	2	yes	—
183	C₄²	C₂×C₃×D₄	1	2	yes	—
184	C₂²×C₃	—	4	4	no	—
185	C₂²×C₃	—	4	2	no	—
186	C₂²×C₃	—	8	2	no	—
187	C₂²×C₃	—	8	1	no	—
188	C₂²×C₃	—	8	1	no	—
189	C₂²×C₃	#392	1	1	yes	characteristic
190	C₂²×C₃	#461	1	2	yes	—
191	C₂²×C₃	#467	1	1	yes	characteristic
192	C₂²×C₃	#470	1	1	yes	characteristic
193	C₂²×C₃	#472	1	2	yes	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	8	1	no	—
196	C₃×C₄	—	8	4	no	—
197	C₃×C₄	#448	1	2	yes	—
198	C₃×C₄	#451	1	2	yes	—
199	C₃×C₄	#452	1	2	yes	—
200	C₃×C₄	#492	1	1	yes	characteristic
201	C₃×C₄	#497	1	1	yes	characteristic
202	C₂³	—	2	2	no	—
203	C₂³	—	4	1	no	—
204	C₂³	—	4	2	no	—
205	C₂³	—	4	1	no	—
206	C₂³	—	4	2	no	—
207	C₂³	C₃×#121	1	1	yes	characteristic
208	Q₈	—	8	4	no	—
209	Q₈	—	8	4	no	—
210	D₄	—	2	4	no	—
211	D₄	—	4	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	C₃×#120	1	1	yes	characteristic
221	C₂×C₄	C₃×#120	1	2	yes	—
222	C₂×C₄	C₃×#121	1	2	yes	—
223	C₂×C₄	C₃×#127	1	1	yes	characteristic
224	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
225	C₂×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
226	C₂×C₄	C₃×#135	1	1	yes	characteristic
227	C₂×C₄	C₃×#137	1	2	yes	—
228	C₂×C₄	C₃×#137	1	2	yes	—
229	C₈	—	2	4	no	—
230	C₈	—	4	4	no	—
231	C₈	—	8	1	no	—
232	C₂×C₃	—	8	2	no	—
233	C₂×C₃	#1675	1	2	yes	—
234	C₂×C₃	#1695	1	2	yes	—
235	C₂×C₃	#2001	1	1	yes	characteristic
236	C₂×C₃	#2013	1	1	yes	characteristic
237	C₂×C₃	#2018	1	1	yes	characteristic
238	C₂²	—	4	2	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	8	1	no	—
241	C₂²	—	8	1	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₃×#392	1	1	yes	characteristic
244	C₂²	C₃×#461	1	2	yes	—
245	C₂²	C₃×#467	1	1	yes	characteristic
246	C₂²	C₃×#470	1	1	yes	characteristic
247	C₂²	C₃×#472	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	8	1	no	—
250	C₄	—	8	4	no	—
251	C₄	C₃×#448	1	2	yes	—
252	C₄	C₃×#451	1	2	yes	—
253	C₄	C₃×#452	1	2	yes	—
254	C₄	C₃×#492	1	1	yes	characteristic
255	C₄	C₃×#497	1	1	yes	characteristic
256	C₃	#10278	1	1	yes	characteristic, direct product factor, Sylow
257	C₂	—	8	2	no	—
258	C₂	C₃×#1675	1	2	yes	—
259	C₂	C₃×#1695	1	2	yes	—
260	C₂	C₃×#2001	1	1	yes	characteristic
261	C₂	C₃×#2013	1	1	yes	characteristic
262	C₂	C₃×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×D₄	D₄
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
1. C₂×C₃×#500	C₂	semidirect
4. C₂×C₃×#333	C₂	semidirect
5. C₃×#1277	C₂	semidirect
6. #10278	C₃	direct (N × H)
20. C₂×#500	C₂×C₃	semidirect
23. C₂×#333	C₂×C₃	semidirect
24. #1277	C₂×C₃	semidirect
256. C₃	#10278	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×#476	2
C₃×#1954	2
C₂×C₃×#333	2
C₃×#1277	2
#10278	3

Subgroups — C3×#10278 #10628885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10278 #10628885