Subgroups — C₃×C₄×#444 #10639118

Order 768.

1022 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#444	C₂	1	2	yes	maximal
2	C₃×C₄×#135	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#131	C₂	1	2	yes	maximal, complemented
4	C₃×C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1773	C₂	1	2	yes	maximal, complemented
6	C₃×#1765	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×#104	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#444	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#135	C₂²	1	1	yes	characteristic
11	C₂×C₃×#131	C₂²	1	4	yes	—
12	C₃×C₄×#41	C₂²	1	2	yes	—
13	C₃×#444	C₄	1	16	yes	direct product factor
14	C₃×C₄×Q₁₆	C₄	1	2	yes	complemented
15	C₃×C₄×SD₁₆	C₄	1	4	yes	complemented
16	C₃×C₄×D₈	C₄	1	2	yes	complemented
17	C₂×C₃×#108	C₂²	1	2	yes	—
18	C₃×#427	C₂²	1	1	yes	characteristic
19	C₂×C₃×#104	C₂²	1	4	yes	—
20	C₃×#416	C₂²	1	2	yes	—
21	C₃×#405	—	2	4	no	—
22	C₃×#405	C₂²	1	4	yes	—
23	C₃×C₄×M₁₆	—	2	2	no	—
24	C₃×C₄×M₁₆	C₂²	1	2	yes	—
25	C₂×C₃×#97	C₂²	1	2	yes	—
26	C₂×C₃×#97	C₂²	1	2	yes	—
27	C₂×C₃×C₄×C₈	—	2	2	no	—
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₃×C₄³	C₂²	1	1	yes	characteristic
31	C₃×#341	C₂²	1	2	yes	—
32	C₃×#339	C₂²	1	4	yes	—
33	C₂×#444	C₂×C₃	1	2	yes	—
34	C₄×#135	C₂×C₃	1	1	yes	characteristic
35	C₄×#131	C₂×C₃	1	2	yes	complemented
36	C₄×#108	C₂×C₃	1	1	yes	characteristic, complemented
37	#1773	C₂×C₃	1	2	yes	complemented
38	#1765	C₂×C₃	1	4	yes	complemented
39	C₄×#104	C₂×C₃	1	2	yes	complemented
40	C₄×#97	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃×#41	C₂³	1	2	yes	—
42	C₃×#135	C₂×C₄	1	8	yes	—
43	C₂×C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×SD₁₆	C₂×C₄	1	2	yes	—
45	C₂×C₃×D₈	C₂×C₄	1	1	yes	characteristic
46	C₃×#131	—	2	8	no	—
47	C₃×#131	C₂×C₄	1	16	yes	complemented
48	C₂×C₃×M₁₆	—	2	4	no	—
49	C₂×C₃×M₁₆	C₂³	1	4	yes	—
50	C₂²×C₃×C₈	—	2	4	no	—
51	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂³	1	2	yes	—
54	C₃×C₄×Q₈	C₂×C₄	1	4	yes	—
55	C₃×C₄×D₄	—	2	4	no	—
56	C₃×C₄×D₄	C₂×C₄	1	4	yes	—
57	C₃×#117	—	2	2	no	—
58	C₃×#117	C₂³	1	2	yes	—
59	C₂×C₃×C₄²	—	2	2	no	—
60	C₂×C₃×C₄²	—	2	2	no	—
61	C₂×C₃×C₄²	C₂³	1	2	yes	—
62	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
63	C₃×#108	C₂×C₄	1	8	yes	complemented
64	C₃×#107	C₂×C₄	1	2	yes	—
65	C₃×#106	C₂×C₄	1	2	yes	—
66	C₃×#104	C₂×C₄	1	16	yes	complemented
67	C₃×#103	C₂×C₄	1	4	yes	—
68	C₃×#102	C₂×C₄	1	4	yes	—
69	C₃×#97	—	2	4	no	—
70	C₃×#97	—	2	4	no	—
71	C₃×#97	—	2	2	no	—
72	C₃×#97	C₂×C₄	1	8	yes	complemented
73	C₃×C₄×C₈	—	2	4	no	—
74	C₃×C₄×C₈	—	2	2	no	—
75	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
76	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
77	C₃×C₄×C₈	D₄	1	2	yes	—
78	C₃×C₄×C₈	D₄	1	2	yes	—
79	C₂×#135	C₂²×C₃	1	1	yes	characteristic
80	C₂×#131	C₂²×C₃	1	4	yes	—
81	C₄×#41	C₂²×C₃	1	2	yes	—
82	#444	C₃×C₄	1	16	yes	direct product factor
83	C₄×Q₁₆	C₃×C₄	1	2	yes	complemented
84	C₄×SD₁₆	C₃×C₄	1	4	yes	complemented
85	C₄×D₈	C₃×C₄	1	2	yes	complemented
86	C₂×#108	C₂²×C₃	1	2	yes	—
87	#427	C₂²×C₃	1	1	yes	characteristic
88	C₂×#104	C₂²×C₃	1	4	yes	—
89	#416	C₂²×C₃	1	2	yes	—
90	#405	—	2	4	no	—
91	#405	C₂²×C₃	1	4	yes	—
92	C₄×M₁₆	—	2	2	no	—
93	C₄×M₁₆	C₂²×C₃	1	2	yes	—
94	C₂×#97	C₂²×C₃	1	2	yes	—
95	C₂×#97	C₂²×C₃	1	2	yes	—
96	C₂×C₄×C₈	—	2	2	no	—
97	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	C₄³	C₂²×C₃	1	1	yes	characteristic
100	#341	C₂²×C₃	1	2	yes	—
101	#339	C₂²×C₃	1	4	yes	—
102	C₃×#41	—	2	4	no	—
103	C₃×#41	C₂²×C₄	1	8	yes	—
104	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
105	C₂×C₃×D₄	—	2	2	no	—
106	C₂×C₃×D₄	C₂²×C₄	1	2	yes	—
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	—	2	4	no	—
110	C₂²×C₃×C₄	#41	1	2	yes	—
111	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
112	C₃×Q₁₆	C₄²	1	4	yes	complemented
113	C₃×SD₁₆	C₄²	1	8	yes	complemented
114	C₃×D₈	C₄²	1	4	yes	complemented
115	C₃×M₁₆	—	2	4	no	—
116	C₃×M₁₆	—	2	16	no	—
117	C₃×M₁₆	C₂²×C₄	1	8	yes	—
118	C₂×C₃×C₈	—	2	4	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	2	16	no	—
121	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
122	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
125	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
126	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
127	C₂×C₃×C₈	#41	1	4	yes	—
128	C₃×C₄⋊C₄	—	2	4	no	—
129	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
130	C₃×C₂²⋊C₄	—	2	2	no	—
131	C₃×C₂²⋊C₄	—	2	4	no	—
132	C₃×C₄²	—	2	4	no	—
133	C₃×C₄²	—	2	2	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	—	4	4	no	—
136	C₃×C₄²	C₂²×C₄	1	4	yes	—
137	C₃×C₄²	C₂²×C₄	1	2	yes	—
138	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
139	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
140	C₂×#41	C₂³×C₃	1	2	yes	—
141	#135	C₂×C₃×C₄	1	8	yes	—
142	C₂×Q₁₆	C₂×C₃×C₄	1	1	yes	characteristic
143	C₂×SD₁₆	C₂×C₃×C₄	1	2	yes	—
144	C₂×D₈	C₂×C₃×C₄	1	1	yes	characteristic
145	#131	—	2	8	no	—
146	#131	C₂×C₃×C₄	1	16	yes	complemented
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	C₂³×C₃	1	4	yes	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₃	1	2	yes	—
153	C₄×Q₈	C₂×C₃×C₄	1	4	yes	—
154	C₄×D₄	—	2	4	no	—
155	C₄×D₄	C₂×C₃×C₄	1	4	yes	—
156	#117	—	2	2	no	—
157	#117	C₂³×C₃	1	2	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	—	2	2	no	—
160	C₂×C₄²	C₂³×C₃	1	2	yes	—
161	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
162	#108	C₂×C₃×C₄	1	8	yes	complemented
163	#107	C₂×C₃×C₄	1	2	yes	—
164	#106	C₂×C₃×C₄	1	2	yes	—
165	#104	C₂×C₃×C₄	1	16	yes	complemented
166	#103	C₂×C₃×C₄	1	4	yes	—
167	#102	C₂×C₃×C₄	1	4	yes	—
168	#97	—	2	4	no	—
169	#97	—	2	4	no	—
170	#97	—	2	2	no	—
171	#97	C₂×C₃×C₄	1	8	yes	complemented
172	C₄×C₈	—	2	4	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
175	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
176	C₄×C₈	C₃×D₄	1	2	yes	—
177	C₄×C₈	C₃×D₄	1	2	yes	—
178	C₂³×C₃	—	2	2	no	—
179	C₂³×C₃	C₂×#41	1	1	yes	characteristic
180	C₃×Q₈	—	2	2	no	—
181	C₃×Q₈	C₂×C₄²	1	4	yes	—
182	C₃×D₄	—	2	8	no	—
183	C₃×D₄	—	2	2	no	—
184	C₃×D₄	C₂×C₄²	1	4	yes	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	8	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	4	4	no	—
191	C₂×C₃×C₄	—	4	8	no	—
192	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
193	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
194	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
195	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×#41	1	2	yes	—
198	C₃×C₈	—	2	8	no	—
199	C₃×C₈	—	2	2	no	—
200	C₃×C₈	C₂×C₄²	1	2	yes	—
201	C₃×C₈	C₂×C₄²	1	2	yes	—
202	C₃×C₈	C₄×D₄	1	8	yes	—
203	#41	—	2	4	no	—
204	#41	C₂²×C₃×C₄	1	8	yes	—
205	C₂×Q₈	C₂²×C₃×C₄	1	2	yes	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	C₂²×C₃×C₄	1	2	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	C₃×#41	1	2	yes	—
212	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
213	Q₁₆	C₃×C₄²	1	4	yes	complemented
214	SD₁₆	C₃×C₄²	1	8	yes	complemented
215	D₈	C₃×C₄²	1	4	yes	complemented
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	16	no	—
218	M₁₆	C₂²×C₃×C₄	1	8	yes	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	16	no	—
222	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
223	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
224	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
226	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
227	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
228	C₂×C₈	C₃×#41	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	4	no	—
237	C₄²	C₂²×C₃×C₄	1	4	yes	—
238	C₄²	C₂²×C₃×C₄	1	2	yes	—
239	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
240	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	4	no	—
243	C₂²×C₃	—	4	4	no	—
244	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂²×C₃	C₄×#41	1	2	yes	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	4	4	no	—
249	C₃×C₄	—	4	4	no	—
250	C₃×C₄	#444	1	4	yes	direct product factor
251	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
252	C₃×C₄	C₂×C₄×D₄	1	2	yes	—
253	C₃×C₄	C₄×#41	1	1	yes	characteristic
254	C₂³	—	2	2	no	—
255	C₂³	C₂×C₃×#41	1	1	yes	characteristic
256	Q₈	—	2	2	no	—
257	Q₈	C₂×C₃×C₄²	1	4	yes	—
258	D₄	—	2	8	no	—
259	D₄	—	2	2	no	—
260	D₄	C₂×C₃×C₄²	1	4	yes	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
269	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
270	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
271	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₃×#41	1	2	yes	—
274	C₈	—	2	8	no	—
275	C₈	—	2	2	no	—
276	C₈	C₂×C₃×C₄²	1	2	yes	—
277	C₈	C₂×C₃×C₄²	1	2	yes	—
278	C₈	C₃×C₄×D₄	1	8	yes	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	4	4	no	—
281	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
282	C₂×C₃	C₂×#444	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
287	C₂²	C₃×C₄×#41	1	2	yes	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	4	no	—
292	C₄	C₃×#444	1	4	yes	direct product factor
293	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
294	C₄	C₂×C₃×C₄×D₄	1	2	yes	—
295	C₄	C₃×C₄×#41	1	1	yes	characteristic
296	C₃	C₄×#444	1	1	yes	characteristic, direct product factor, Sylow
297	C₂	—	2	2	no	—
298	C₂	—	4	4	no	—
299	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
300	C₂	C₂×C₃×#444	1	2	yes	—

	subgroup	G/H
center	C₃×C₄²	C₂×D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×#131	C₂	semidirect
4. C₃×C₄×#108	C₂	semidirect
5. C₃×#1773	C₂	semidirect
6. C₃×#1765	C₂	semidirect
7. C₃×C₄×#104	C₂	semidirect
8. C₃×C₄×#97	C₂	semidirect
9. C₄×#444	C₃	direct (N × H)
13. C₃×#444	C₄	direct (N × H)
14. C₃×C₄×Q₁₆	C₄	semidirect
15. C₃×C₄×SD₁₆	C₄	semidirect
16. C₃×C₄×D₈	C₄	semidirect
35. C₄×#131	C₂×C₃	semidirect
36. C₄×#108	C₂×C₃	semidirect
37. #1773	C₂×C₃	semidirect
38. #1765	C₂×C₃	semidirect
39. C₄×#104	C₂×C₃	semidirect
40. C₄×#97	C₂×C₃	semidirect
47. C₃×#131	C₂×C₄	semidirect
63. C₃×#108	C₂×C₄	semidirect
66. C₃×#104	C₂×C₄	semidirect
72. C₃×#97	C₂×C₄	semidirect
82. #444	C₃×C₄	direct (N × H)
83. C₄×Q₁₆	C₃×C₄	semidirect
84. C₄×SD₁₆	C₃×C₄	semidirect
85. C₄×D₈	C₃×C₄	semidirect
112. C₃×Q₁₆	C₄²	semidirect
113. C₃×SD₁₆	C₄²	semidirect
114. C₃×D₈	C₄²	semidirect
146. #131	C₂×C₃×C₄	semidirect
162. #108	C₂×C₃×C₄	semidirect
165. #104	C₂×C₃×C₄	semidirect
171. #97	C₂×C₃×C₄	semidirect
213. Q₁₆	C₃×C₄²	semidirect
214. SD₁₆	C₃×C₄²	semidirect
215. D₈	C₃×C₄²	semidirect
250. C₃×C₄	#444	direct (N × H)
292. C₄	C₃×#444	direct (N × H)
296. C₃	C₄×#444	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×#444	2
C₃×C₄×#135	2
C₃×C₄×#131	2
C₃×C₄×#108	2
C₃×#1773	2
C₃×#1765	2
C₃×C₄×#104	2
C₃×C₄×#97	2
C₄×#444	3

Subgroups — C3×C4×#444 #10639118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#444 #10639118