Subgroups — C₃×#10274 #10628881

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 274 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, complemented
2	C₂×C₃×#474	C₂	1	2	yes	maximal
3	C₃×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#332	C₂	1	2	yes	maximal, complemented
5	C₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	#10274	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×D₄	C₂²	1	1	yes	characteristic
10	C₃×#494	—	2	4	no	—
11	C₃×#474	—	2	4	no	—
12	C₃×#474	—	2	4	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×#102	—	2	2	no	—
17	C₂×C₃×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₃×#332	—	2	4	no	—
20	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×#474	C₂×C₃	1	2	yes	—
22	#1954	C₂×C₃	1	1	yes	characteristic
23	C₂×#332	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	C₂×C₃×Q₁₆	—	2	8	no	—
29	C₂×C₃×Q₁₆	—	4	4	no	—
30	C₂×C₃×Q₁₆	—	4	4	no	—
31	C₂²×C₃×C₈	—	2	2	no	—
32	C₂²×C₃×C₈	—	2	1	no	—
33	C₂²×C₃×C₈	D₄	1	2	yes	—
34	C₃×C₄⋊Q₈	—	2	2	no	—
35	C₃×C₄⋊Q₈	—	2	2	no	—
36	C₃×C₄⋊Q₈	—	2	2	no	—
37	C₃×C₄⋊Q₈	D₄	1	4	yes	—
38	C₃×C₄×D₄	—	2	2	no	—
39	C₃×C₄×D₄	—	2	2	no	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₃×#107	—	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×#102	—	2	8	no	—
50	C₃×C₂²⋊C₈	—	4	4	no	—
51	C₃×C₄×C₈	—	2	2	no	—
52	C₂²×Q₁₆	—	2	2	no	—
53	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
54	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
55	#494	—	2	4	no	—
56	#474	—	2	4	no	—
57	#474	—	2	4	no	—
58	C₂×#107	—	2	2	no	—
59	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
61	C₂×#102	—	2	2	no	—
62	C₂×C₂²⋊C₈	—	2	2	no	—
63	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
64	#332	—	2	4	no	—
65	C₂⁴×C₃	—	2	1	no	—
66	C₂×C₃×Q₈	—	2	4	no	—
67	C₂×C₃×Q₈	—	4	2	no	—
68	C₂×C₃×Q₈	—	4	4	no	—
69	C₂×C₃×Q₈	—	4	4	no	—
70	C₂×C₃×D₄	—	2	2	no	—
71	C₂×C₃×D₄	—	2	1	no	—
72	C₂×C₃×D₄	—	2	2	no	—
73	C₂×C₃×D₄	Q₁₆	1	2	yes	complemented
74	C₂²×C₃×C₄	—	2	1	no	—
75	C₂²×C₃×C₄	—	4	2	no	—
76	C₂²×C₃×C₄	—	4	1	no	—
77	C₂²×C₃×C₄	—	4	1	no	—
78	C₂²×C₃×C₄	—	4	2	no	—
79	C₂²×C₃×C₄	—	4	2	no	—
80	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	#41	1	1	yes	characteristic
83	C₃×Q₁₆	—	4	16	no	—
84	C₂×C₃×C₈	—	2	2	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	—	2	4	no	—
87	C₂×C₃×C₈	—	4	2	no	—
88	C₂×C₃×C₈	—	4	2	no	—
89	C₂×C₃×C₈	—	4	1	no	—
90	C₂×C₃×C₈	—	4	2	no	—
91	C₂×C₃×C₈	D₈	1	4	yes	—
92	C₃×C₄⋊C₄	—	2	4	no	—
93	C₃×C₄⋊C₄	—	2	2	no	—
94	C₃×C₄⋊C₄	—	2	4	no	—
95	C₃×C₄⋊C₄	—	4	2	no	—
96	C₃×C₄⋊C₄	—	4	2	no	—
97	C₃×C₂²⋊C₄	—	4	2	no	—
98	C₃×C₄²	C₂×D₄	1	2	yes	—
99	C₃×C₄²	C₂×D₄	1	2	yes	—
100	C₂²×Q₈	—	2	2	no	—
101	C₂²×D₄	C₃×D₄	1	1	yes	characteristic
102	C₂³×C₄	—	2	1	no	—
103	C₂×Q₁₆	—	2	8	no	—
104	C₂×Q₁₆	—	4	4	no	—
105	C₂×Q₁₆	—	4	4	no	—
106	C₂²×C₈	—	2	2	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	C₃×D₄	1	2	yes	—
109	C₄⋊Q₈	—	2	2	no	—
110	C₄⋊Q₈	—	2	2	no	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	C₃×D₄	1	4	yes	—
113	C₄×D₄	—	2	2	no	—
114	C₄×D₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	—	2	1	no	—
119	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
120	#107	—	2	8	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	C₃×D₄	1	4	yes	—
124	#102	—	2	8	no	—
125	C₂²⋊C₈	—	4	4	no	—
126	C₄×C₈	—	2	2	no	—
127	C₂³×C₃	—	2	2	no	—
128	C₂³×C₃	—	4	1	no	—
129	C₂³×C₃	—	4	2	no	—
130	C₂³×C₃	—	4	1	no	—
131	C₂³×C₃	—	4	2	no	—
132	C₂³×C₃	#121	1	1	yes	characteristic
133	C₃×Q₈	—	4	8	no	—
134	C₃×Q₈	—	8	2	no	—
135	C₃×Q₈	—	8	2	no	—
136	C₃×D₄	—	2	4	no	—
137	C₃×D₄	—	4	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	4	1	no	—
140	C₂×C₃×C₄	—	4	4	no	—
141	C₂×C₃×C₄	—	4	2	no	—
142	C₂×C₃×C₄	—	8	4	no	—
143	C₂×C₃×C₄	—	8	2	no	—
144	C₂×C₃×C₄	—	8	1	no	—
145	C₂×C₃×C₄	—	8	1	no	—
146	C₂×C₃×C₄	#120	1	1	yes	characteristic
147	C₂×C₃×C₄	#120	1	2	yes	—
148	C₂×C₃×C₄	#121	1	2	yes	—
149	C₂×C₃×C₄	#127	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
151	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
152	C₂×C₃×C₄	#135	1	1	yes	characteristic
153	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₃×C₄	#137	1	2	yes	—
155	C₃×C₈	—	2	4	no	—
156	C₃×C₈	—	4	4	no	—
157	C₃×C₈	—	8	1	no	—
158	C₂⁴	—	2	1	no	—
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂×D₄	—	2	2	no	—
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	C₃×Q₁₆	1	2	yes	complemented
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₃×#41	1	1	yes	characteristic
176	Q₁₆	—	4	16	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	C₃×D₈	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₄²	C₂×C₃×D₄	1	2	yes	—
192	C₄²	C₂×C₃×D₄	1	2	yes	—
193	C₂²×C₃	—	4	4	no	—
194	C₂²×C₃	—	4	2	no	—
195	C₂²×C₃	—	8	2	no	—
196	C₂²×C₃	—	8	1	no	—
197	C₂²×C₃	—	8	1	no	—
198	C₂²×C₃	#392	1	1	yes	characteristic
199	C₂²×C₃	#459	1	2	yes	—
200	C₂²×C₃	#467	1	1	yes	characteristic
201	C₂²×C₃	#470	1	1	yes	characteristic
202	C₂²×C₃	#473	1	2	yes	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	8	1	no	—
205	C₃×C₄	—	8	4	no	—
206	C₃×C₄	#447	1	2	yes	—
207	C₃×C₄	#451	1	2	yes	—
208	C₃×C₄	#452	1	2	yes	—
209	C₃×C₄	#493	1	1	yes	characteristic
210	C₃×C₄	#496	1	1	yes	characteristic
211	C₂³	—	2	2	no	—
212	C₂³	—	4	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	4	1	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₃×#121	1	1	yes	characteristic
217	Q₈	—	4	8	no	—
218	Q₈	—	8	2	no	—
219	Q₈	—	8	2	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	1	no	—
229	C₂×C₄	—	8	1	no	—
230	C₂×C₄	C₃×#120	1	1	yes	characteristic
231	C₂×C₄	C₃×#120	1	2	yes	—
232	C₂×C₄	C₃×#121	1	2	yes	—
233	C₂×C₄	C₃×#127	1	1	yes	characteristic
234	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
235	C₂×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
236	C₂×C₄	C₃×#135	1	1	yes	characteristic
237	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
238	C₂×C₄	C₃×#137	1	2	yes	—
239	C₈	—	2	4	no	—
240	C₈	—	4	4	no	—
241	C₈	—	8	1	no	—
242	C₂×C₃	—	8	2	no	—
243	C₂×C₃	#1669	1	2	yes	—
244	C₂×C₃	#1689	1	2	yes	—
245	C₂×C₃	#2001	1	1	yes	characteristic
246	C₂×C₃	#2011	1	1	yes	characteristic
247	C₂×C₃	#2020	1	1	yes	characteristic
248	C₂²	—	4	2	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	8	1	no	—
251	C₂²	—	8	1	no	—
252	C₂²	—	8	2	no	—
253	C₂²	C₃×#392	1	1	yes	characteristic
254	C₂²	C₃×#459	1	2	yes	—
255	C₂²	C₃×#467	1	1	yes	characteristic
256	C₂²	C₃×#470	1	1	yes	characteristic
257	C₂²	C₃×#473	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	8	1	no	—
260	C₄	—	8	4	no	—
261	C₄	C₃×#447	1	2	yes	—
262	C₄	C₃×#451	1	2	yes	—
263	C₄	C₃×#452	1	2	yes	—
264	C₄	C₃×#493	1	1	yes	characteristic
265	C₄	C₃×#496	1	1	yes	characteristic
266	C₃	#10274	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	8	2	no	—
268	C₂	C₃×#1669	1	2	yes	—
269	C₂	C₃×#1689	1	2	yes	—
270	C₂	C₃×#2001	1	1	yes	characteristic
271	C₂	C₃×#2011	1	1	yes	characteristic
272	C₂	C₃×#2020	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×#494	C₂	semidirect
4. C₂×C₃×#332	C₂	semidirect
5. C₃×#1277	C₂	semidirect
6. #10274	C₃	direct (N × H)
20. C₂×#494	C₂×C₃	semidirect
23. C₂×#332	C₂×C₃	semidirect
24. #1277	C₂×C₃	semidirect
73. C₂×C₃×D₄	Q₁₆	semidirect
166. C₂×D₄	C₃×Q₁₆	semidirect
266. C₃	#10274	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×#474	2
C₃×#1954	2
C₂×C₃×#332	2
C₃×#1277	2
#10274	3

Subgroups — C3×#10274 #10628881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10274 #10628881