Subgroups — C₃×C₄×#441 #10639109

Order 768.

1086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#441	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄²×Q₈	C₂	1	2	yes	maximal
4	C₃×#1776	C₂	1	2	yes	maximal
5	C₃×C₄×#106	C₂	1	1	yes	maximal, characteristic
6	C₃×#1763	C₂	1	4	yes	maximal
7	C₃×C₄×#103	C₂	1	2	yes	maximal
8	C₃×C₄×#97	C₂	1	1	yes	maximal, characteristic
9	C₄×#441	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×Q₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Q₈	C₂²	1	4	yes	—
13	C₃×#441	C₄	1	16	yes	direct product factor
14	C₃×C₄×Q₁₆	C₄	1	8	yes	complemented
15	C₂×C₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃×#106	C₂²	1	2	yes	—
17	C₂×C₃×#103	C₂²	1	4	yes	—
18	C₂×C₃×#103	C₂²	1	2	yes	—
19	C₂×C₃×#97	C₂²	1	2	yes	—
20	C₂×C₃×#97	C₂²	1	2	yes	—
21	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₄⋊C₄	—	2	4	no	—
24	C₃×C₄×C₄⋊C₄	—	2	2	no	—
25	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
26	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
27	C₃×C₄³	—	2	2	no	—
28	C₃×C₄³	C₂²	1	1	yes	characteristic
29	C₃×#340	C₂²	1	4	yes	—
30	C₃×#340	C₂²	1	2	yes	—
31	C₂×#441	C₂×C₃	1	2	yes	—
32	C₂×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
33	C₄²×Q₈	C₂×C₃	1	2	yes	—
34	#1776	C₂×C₃	1	2	yes	—
35	C₄×#106	C₂×C₃	1	1	yes	characteristic
36	#1763	C₂×C₃	1	4	yes	—
37	C₄×#103	C₂×C₃	1	2	yes	—
38	C₄×#97	C₂×C₃	1	1	yes	characteristic
39	C₂²×C₃×Q₈	C₂³	1	2	yes	—
40	C₂×C₃×Q₁₆	C₂×C₄	1	4	yes	—
41	C₂×C₃×Q₁₆	C₂×C₄	1	8	yes	—
42	C₂²×C₃×C₈	C₂³	1	2	yes	—
43	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₄×Q₈	—	2	4	no	—
46	C₃×C₄×Q₈	—	2	8	no	—
47	C₃×C₄×Q₈	C₂×C₄	1	8	yes	—
48	C₃×C₄×Q₈	C₂×C₄	1	16	yes	—
49	C₂×C₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₃×C₄⋊C₄	—	2	4	no	—
51	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
52	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₃×C₄²	—	2	4	no	—
54	C₂×C₃×C₄²	—	2	2	no	—
55	C₂×C₃×C₄²	—	2	4	no	—
56	C₂×C₃×C₄²	—	2	2	no	—
57	C₂×C₃×C₄²	D₄	1	2	yes	—
58	C₂×C₃×C₄²	D₄	1	2	yes	—
59	C₂×C₃×C₄²	C₂³	1	2	yes	—
60	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
61	C₃×#107	C₂×C₄	1	4	yes	—
62	C₃×#106	C₂×C₄	1	8	yes	—
63	C₃×#103	C₂×C₄	1	8	yes	—
64	C₃×#103	C₂×C₄	1	16	yes	—
65	C₃×#97	—	2	4	no	—
66	C₃×#97	C₂×C₄	1	8	yes	—
67	C₃×C₄×C₈	—	2	2	no	—
68	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
69	C₃×#95	—	2	2	no	—
70	C₃×#95	—	2	4	no	—
71	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄×Q₈	C₂²×C₃	1	2	yes	—
73	C₂×C₄×Q₈	C₂²×C₃	1	4	yes	—
74	#441	C₃×C₄	1	16	yes	direct product factor
75	C₄×Q₁₆	C₃×C₄	1	8	yes	complemented
76	C₂×#107	C₂²×C₃	1	1	yes	characteristic
77	C₂×#106	C₂²×C₃	1	2	yes	—
78	C₂×#103	C₂²×C₃	1	2	yes	—
79	C₂×#103	C₂²×C₃	1	4	yes	—
80	C₂×#97	C₂²×C₃	1	2	yes	—
81	C₂×#97	C₂²×C₃	1	2	yes	—
82	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
84	C₄×C₄⋊C₄	—	2	2	no	—
85	C₄×C₄⋊C₄	—	2	4	no	—
86	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
88	C₄³	—	2	2	no	—
89	C₄³	C₂²×C₃	1	1	yes	characteristic
90	#340	C₂²×C₃	1	4	yes	—
91	#340	C₂²×C₃	1	2	yes	—
92	C₂×C₃×Q₈	—	2	2	no	—
93	C₂×C₃×Q₈	—	2	4	no	—
94	C₂×C₃×Q₈	C₂²×C₄	1	4	yes	—
95	C₂×C₃×Q₈	C₂²×C₄	1	8	yes	—
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	—	2	4	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
101	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
103	C₃×Q₁₆	C₄²	1	16	yes	complemented
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
109	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
111	C₃×C₄⋊C₄	—	2	16	no	—
112	C₃×C₄⋊C₄	—	2	8	no	—
113	C₃×C₄⋊C₄	—	2	4	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
116	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
117	C₃×C₄²	—	2	16	no	—
118	C₃×C₄²	—	2	8	no	—
119	C₃×C₄²	—	2	4	no	—
120	C₃×C₄²	—	4	2	no	—
121	C₃×C₄²	—	4	8	no	—
122	C₃×C₄²	—	4	4	no	—
123	C₃×C₄²	C₂²×C₄	1	2	yes	—
124	C₃×C₄²	C₂²×C₄	1	4	yes	—
125	C₃×C₄²	#41	1	2	yes	—
126	C₃×C₄²	#41	1	4	yes	—
127	C₂²×Q₈	C₂³×C₃	1	2	yes	—
128	C₂×Q₁₆	C₂×C₃×C₄	1	4	yes	—
129	C₂×Q₁₆	C₂×C₃×C₄	1	8	yes	—
130	C₂²×C₈	C₂³×C₃	1	2	yes	—
131	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
133	C₄×Q₈	—	2	4	no	—
134	C₄×Q₈	—	2	8	no	—
135	C₄×Q₈	C₂×C₃×C₄	1	8	yes	—
136	C₄×Q₈	C₂×C₃×C₄	1	16	yes	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
140	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	—	2	4	no	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₃×D₄	1	2	yes	—
146	C₂×C₄²	C₃×D₄	1	2	yes	—
147	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
148	C₂×C₄²	C₂³×C₃	1	2	yes	—
149	#107	C₂×C₃×C₄	1	4	yes	—
150	#106	C₂×C₃×C₄	1	8	yes	—
151	#103	C₂×C₃×C₄	1	8	yes	—
152	#103	C₂×C₃×C₄	1	16	yes	—
153	#97	—	2	4	no	—
154	#97	C₂×C₃×C₄	1	8	yes	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	C₂×C₃×C₄	1	4	yes	—
157	#95	—	2	2	no	—
158	#95	—	2	4	no	—
159	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
160	C₃×Q₈	—	2	4	no	—
161	C₃×Q₈	—	2	8	no	—
162	C₃×Q₈	C₂×C₄²	1	8	yes	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	8	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	8	no	—
170	C₂×C₃×C₄	—	4	4	no	—
171	C₂×C₃×C₄	—	4	4	no	—
172	C₂×C₃×C₄	—	4	4	no	—
173	C₂×C₃×C₄	C₄×D₄	1	8	yes	—
174	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
175	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
176	C₂×C₃×C₄	#137	1	4	yes	—
177	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
179	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×#41	1	2	yes	—
181	C₃×C₈	—	2	4	no	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	C₂×C₄²	1	4	yes	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	C₂²×C₃×C₄	1	4	yes	—
187	C₂×Q₈	C₂²×C₃×C₄	1	8	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
193	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
195	Q₁₆	C₃×C₄²	1	16	yes	complemented
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
200	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
201	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
202	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
208	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
209	C₄²	—	2	16	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	8	no	—
214	C₄²	—	4	4	no	—
215	C₄²	C₂²×C₃×C₄	1	2	yes	—
216	C₄²	C₂²×C₃×C₄	1	4	yes	—
217	C₄²	C₃×#41	1	2	yes	—
218	C₄²	C₃×#41	1	4	yes	—
219	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₃	C₂×C₄×D₄	1	2	yes	—
221	C₂²×C₃	C₂×#137	1	2	yes	—
222	C₂²×C₃	#575	1	2	yes	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	8	no	—
225	C₃×C₄	—	2	8	no	—
226	C₃×C₄	—	4	4	no	—
227	C₃×C₄	—	4	8	no	—
228	C₃×C₄	#441	1	4	yes	direct product factor
229	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
230	C₃×C₄	C₄×#41	1	2	yes	—
231	C₃×C₄	C₄×#41	1	1	yes	characteristic
232	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	8	no	—
235	Q₈	C₂×C₃×C₄²	1	8	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₃×C₄×D₄	1	8	yes	—
247	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
248	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
249	C₂×C₄	C₃×#137	1	4	yes	—
250	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
251	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
252	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
253	C₂×C₄	C₂×C₃×#41	1	2	yes	—
254	C₈	—	2	4	no	—
255	C₈	—	2	2	no	—
256	C₈	C₂×C₃×C₄²	1	4	yes	—
257	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
258	C₂×C₃	C₂×#441	1	1	yes	characteristic
259	C₂×C₃	#2943	1	1	yes	characteristic
260	C₂×C₃	C₄×#137	1	4	yes	—
261	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
262	C₂²	C₂×C₃×C₄×D₄	1	2	yes	—
263	C₂²	C₂×C₃×#137	1	2	yes	—
264	C₂²	C₃×#575	1	2	yes	—
265	C₄	—	2	2	no	—
266	C₄	—	2	8	no	—
267	C₄	—	2	8	no	—
268	C₄	—	4	4	no	—
269	C₄	—	4	8	no	—
270	C₄	C₃×#441	1	4	yes	direct product factor
271	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
272	C₄	C₃×C₄×#41	1	2	yes	—
273	C₄	C₃×C₄×#41	1	1	yes	characteristic
274	C₃	C₄×#441	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
276	C₂	C₂×C₃×#441	1	1	yes	characteristic
277	C₂	C₃×#2943	1	1	yes	characteristic
278	C₂	C₃×C₄×#137	1	4	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×C₄³ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. C₄×#441	C₃	direct (N × H)
13. C₃×#441	C₄	direct (N × H)
14. C₃×C₄×Q₁₆	C₄	semidirect
74. #441	C₃×C₄	direct (N × H)
75. C₄×Q₁₆	C₃×C₄	semidirect
103. C₃×Q₁₆	C₄²	semidirect
195. Q₁₆	C₃×C₄²	semidirect
228. C₃×C₄	#441	direct (N × H)
270. C₄	C₃×#441	direct (N × H)
274. C₃	C₄×#441	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×#441	2
C₂×C₃×C₄×Q₁₆	2
C₃×C₄²×Q₈	2
C₃×#1776	2
C₃×C₄×#106	2
C₃×#1763	2
C₃×C₄×#103	2
C₃×C₄×#97	2
C₄×#441	3

Subgroups — C3×C4×#441 #10639109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#441 #10639109