Subgroups — C₃×C₄×#442 #10639107

Order 768.

1326 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#442	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄²×D₄	C₂	1	2	yes	maximal, complemented
4	C₃×#1776	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1762	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×#102	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#442	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×D₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄×D₄	C₂²	1	4	yes	—
13	C₂²×C₃×C₄²	—	2	2	no	—
14	C₃×#442	C₄	1	16	yes	direct product factor
15	C₃×C₄×D₈	C₄	1	8	yes	complemented
16	C₂×C₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₃×#106	C₂²	1	2	yes	—
18	C₂×C₃×#102	C₂²	1	4	yes	—
19	C₂×C₃×#102	C₂²	1	2	yes	—
20	C₂×C₃×#97	C₂²	1	2	yes	—
21	C₂×C₃×#97	C₂²	1	2	yes	—
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
26	C₃×C₄×C₂²⋊C₄	—	2	4	no	—
27	C₃×C₄×C₂²⋊C₄	—	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×#442	C₂×C₃	1	2	yes	—
32	C₂×C₄×D₈	C₂×C₃	1	1	yes	characteristic
33	C₄²×D₄	C₂×C₃	1	2	yes	complemented
34	#1776	C₂×C₃	1	2	yes	complemented
35	C₄×#106	C₂×C₃	1	1	yes	characteristic, complemented
36	#1762	C₂×C₃	1	4	yes	complemented
37	C₄×#102	C₂×C₃	1	2	yes	complemented
38	C₄×#97	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂²×C₃×D₄	C₂³	1	2	yes	—
40	C₂³×C₃×C₄	—	2	4	no	—
41	C₂³×C₃×C₄	—	2	2	no	—
42	C₂×C₃×D₈	C₂×C₄	1	4	yes	—
43	C₂×C₃×D₈	C₂×C₄	1	8	yes	—
44	C₂²×C₃×C₈	C₂³	1	2	yes	—
45	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
47	C₃×C₄×D₄	—	2	4	no	—
48	C₃×C₄×D₄	—	2	8	no	—
49	C₃×C₄×D₄	C₂×C₄	1	8	yes	complemented
50	C₃×C₄×D₄	C₂×C₄	1	16	yes	complemented
51	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
52	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
55	C₂×C₃×C₄²	—	2	4	no	—
56	C₂×C₃×C₄²	—	2	2	no	—
57	C₂×C₃×C₄²	—	4	8	no	—
58	C₂×C₃×C₄²	D₄	1	2	yes	—
59	C₂×C₃×C₄²	D₄	1	2	yes	—
60	C₂×C₃×C₄²	C₂³	1	2	yes	—
61	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
62	C₃×#107	C₂×C₄	1	4	yes	complemented
63	C₃×#106	C₂×C₄	1	8	yes	complemented
64	C₃×#102	C₂×C₄	1	8	yes	complemented
65	C₃×#102	C₂×C₄	1	16	yes	complemented
66	C₃×#97	—	2	4	no	—
67	C₃×#97	C₂×C₄	1	8	yes	complemented
68	C₃×C₄×C₈	—	2	2	no	—
69	C₃×C₄×C₈	C₂×C₄	1	4	yes	complemented
70	C₃×#95	—	2	2	no	—
71	C₃×#95	—	2	4	no	—
72	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄×D₄	C₂²×C₃	1	2	yes	—
74	C₂×C₄×D₄	C₂²×C₃	1	4	yes	—
75	C₂²×C₄²	—	2	2	no	—
76	#442	C₃×C₄	1	16	yes	direct product factor
77	C₄×D₈	C₃×C₄	1	8	yes	complemented
78	C₂×#107	C₂²×C₃	1	1	yes	characteristic
79	C₂×#106	C₂²×C₃	1	2	yes	—
80	C₂×#102	C₂²×C₃	1	2	yes	—
81	C₂×#102	C₂²×C₃	1	4	yes	—
82	C₂×#97	C₂²×C₃	1	2	yes	—
83	C₂×#97	C₂²×C₃	1	2	yes	—
84	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
86	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
88	C₄×C₂²⋊C₄	—	2	2	no	—
89	C₄×C₂²⋊C₄	—	2	4	no	—
90	C₄³	C₂²×C₃	1	1	yes	characteristic
91	#340	C₂²×C₃	1	4	yes	—
92	#340	C₂²×C₃	1	2	yes	—
93	C₂⁴×C₃	—	2	2	no	—
94	C₂×C₃×D₄	—	2	2	no	—
95	C₂×C₃×D₄	—	2	4	no	—
96	C₂×C₃×D₄	C₂²×C₄	1	4	yes	—
97	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	4	no	—
100	C₂²×C₃×C₄	—	2	16	no	—
101	C₂²×C₃×C₄	—	2	8	no	—
102	C₂²×C₃×C₄	—	4	8	no	—
103	C₂²×C₃×C₄	—	4	8	no	—
104	C₂²×C₃×C₄	—	4	4	no	—
105	C₂²×C₃×C₄	—	4	4	no	—
106	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
108	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
110	C₃×D₈	C₄²	1	16	yes	complemented
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
116	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
117	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
118	C₃×C₄⋊C₄	—	2	4	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
121	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
122	C₃×C₂²⋊C₄	—	2	8	no	—
123	C₃×C₂²⋊C₄	—	2	16	no	—
124	C₃×C₄²	—	2	4	no	—
125	C₃×C₄²	—	4	2	no	—
126	C₃×C₄²	—	4	8	no	—
127	C₃×C₄²	—	4	4	no	—
128	C₃×C₄²	C₂²×C₄	1	2	yes	—
129	C₃×C₄²	C₂²×C₄	1	4	yes	—
130	C₃×C₄²	#41	1	2	yes	—
131	C₃×C₄²	#41	1	4	yes	—
132	C₂²×D₄	C₂³×C₃	1	2	yes	—
133	C₂³×C₄	—	2	4	no	—
134	C₂³×C₄	—	2	2	no	—
135	C₂×D₈	C₂×C₃×C₄	1	4	yes	—
136	C₂×D₈	C₂×C₃×C₄	1	8	yes	—
137	C₂²×C₈	C₂³×C₃	1	2	yes	—
138	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
140	C₄×D₄	—	2	4	no	—
141	C₄×D₄	—	2	8	no	—
142	C₄×D₄	C₂×C₃×C₄	1	8	yes	complemented
143	C₄×D₄	C₂×C₃×C₄	1	16	yes	complemented
144	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
145	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	2	4	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	—	2	2	no	—
150	C₂×C₄²	—	4	8	no	—
151	C₂×C₄²	C₃×D₄	1	2	yes	—
152	C₂×C₄²	C₃×D₄	1	2	yes	—
153	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₃	1	2	yes	—
155	#107	C₂×C₃×C₄	1	4	yes	complemented
156	#106	C₂×C₃×C₄	1	8	yes	complemented
157	#102	C₂×C₃×C₄	1	8	yes	complemented
158	#102	C₂×C₃×C₄	1	16	yes	complemented
159	#97	—	2	4	no	—
160	#97	C₂×C₃×C₄	1	8	yes	complemented
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	C₂×C₃×C₄	1	4	yes	complemented
163	#95	—	2	2	no	—
164	#95	—	2	4	no	—
165	C₂³×C₃	—	2	4	no	—
166	C₂³×C₃	—	2	8	no	—
167	C₂³×C₃	—	4	2	no	—
168	C₂³×C₃	—	4	2	no	—
169	C₂³×C₃	—	4	4	no	—
170	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
171	C₃×D₄	—	2	4	no	—
172	C₃×D₄	—	2	8	no	—
173	C₃×D₄	C₂×C₄²	1	8	yes	complemented
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	8	no	—
179	C₂×C₃×C₄	—	4	32	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	4	4	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	—	4	4	no	—
184	C₂×C₃×C₄	—	4	16	no	—
185	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
186	C₂×C₃×C₄	C₄×D₄	1	8	yes	—
187	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
188	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
189	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
191	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×#41	1	2	yes	—
193	C₃×C₈	—	2	4	no	—
194	C₃×C₈	—	2	2	no	—
195	C₃×C₈	C₂×C₄²	1	4	yes	complemented
196	C₂⁴	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	C₂²×C₃×C₄	1	4	yes	—
200	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	16	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	—	4	8	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
211	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
213	D₈	C₃×C₄²	1	16	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
218	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
219	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
220	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
224	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
225	C₂²⋊C₄	—	2	8	no	—
226	C₂²⋊C₄	—	2	16	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	4	2	no	—
229	C₄²	—	4	8	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₂²×C₃×C₄	1	2	yes	—
232	C₄²	C₂²×C₃×C₄	1	4	yes	—
233	C₄²	C₃×#41	1	2	yes	—
234	C₄²	C₃×#41	1	4	yes	—
235	C₂²×C₃	—	2	8	no	—
236	C₂²×C₃	—	4	4	no	—
237	C₂²×C₃	—	4	4	no	—
238	C₂²×C₃	—	4	8	no	—
239	C₂²×C₃	—	4	8	no	—
240	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
241	C₂²×C₃	C₂×C₄×D₄	1	2	yes	—
242	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
243	C₂²×C₃	#575	1	2	yes	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	8	no	—
246	C₃×C₄	—	4	4	no	—
247	C₃×C₄	—	4	8	no	—
248	C₃×C₄	#442	1	4	yes	direct product factor
249	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
250	C₃×C₄	C₄×#41	1	2	yes	—
251	C₃×C₄	C₄×#41	1	1	yes	characteristic
252	C₂³	—	2	4	no	—
253	C₂³	—	2	8	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	4	no	—
257	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
258	D₄	—	2	4	no	—
259	D₄	—	2	8	no	—
260	D₄	C₂×C₃×C₄²	1	8	yes	complemented
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	4	32	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	16	no	—
272	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
273	C₂×C₄	C₃×C₄×D₄	1	8	yes	—
274	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
275	C₂×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
276	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
277	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
278	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
279	C₂×C₄	C₂×C₃×#41	1	2	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	2	no	—
282	C₈	C₂×C₃×C₄²	1	4	yes	complemented
283	C₂×C₃	—	4	8	no	—
284	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
285	C₂×C₃	C₂×#442	1	1	yes	characteristic
286	C₂×C₃	#2943	1	1	yes	characteristic
287	C₂×C₃	C₄×AGL(1,ℤ/8)	1	4	yes	—
288	C₂²	—	2	8	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	8	no	—
292	C₂²	—	4	8	no	—
293	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
294	C₂²	C₂×C₃×C₄×D₄	1	2	yes	—
295	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
296	C₂²	C₃×#575	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	2	8	no	—
299	C₄	—	4	4	no	—
300	C₄	—	4	8	no	—
301	C₄	C₃×#442	1	4	yes	direct product factor
302	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
303	C₄	C₃×C₄×#41	1	2	yes	—
304	C₄	C₃×C₄×#41	1	1	yes	characteristic
305	C₃	C₄×#442	1	1	yes	characteristic, direct product factor, Sylow
306	C₂	—	4	8	no	—
307	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
308	C₂	C₂×C₃×#442	1	1	yes	characteristic
309	C₂	C₃×#2943	1	1	yes	characteristic
310	C₂	C₃×C₄×AGL(1,ℤ/8)	1	4	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂²×D₄
prime radical	C₂²×C₃×D₈	C₂²
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
3. C₃×C₄²×D₄	C₂	semidirect
4. C₃×#1776	C₂	semidirect
5. C₃×C₄×#106	C₂	semidirect
6. C₃×#1762	C₂	semidirect
7. C₃×C₄×#102	C₂	semidirect
8. C₃×C₄×#97	C₂	semidirect
9. C₄×#442	C₃	direct (N × H)
14. C₃×#442	C₄	direct (N × H)
15. C₃×C₄×D₈	C₄	semidirect
33. C₄²×D₄	C₂×C₃	semidirect
34. #1776	C₂×C₃	semidirect
35. C₄×#106	C₂×C₃	semidirect
36. #1762	C₂×C₃	semidirect
37. C₄×#102	C₂×C₃	semidirect
38. C₄×#97	C₂×C₃	semidirect
49. C₃×C₄×D₄	C₂×C₄	semidirect
50. C₃×C₄×D₄	C₂×C₄	semidirect
62. C₃×#107	C₂×C₄	semidirect
63. C₃×#106	C₂×C₄	semidirect
64. C₃×#102	C₂×C₄	semidirect
65. C₃×#102	C₂×C₄	semidirect
67. C₃×#97	C₂×C₄	semidirect
69. C₃×C₄×C₈	C₂×C₄	semidirect
76. #442	C₃×C₄	direct (N × H)
77. C₄×D₈	C₃×C₄	semidirect
110. C₃×D₈	C₄²	semidirect
142. C₄×D₄	C₂×C₃×C₄	semidirect
143. C₄×D₄	C₂×C₃×C₄	semidirect
155. #107	C₂×C₃×C₄	semidirect
156. #106	C₂×C₃×C₄	semidirect
157. #102	C₂×C₃×C₄	semidirect
158. #102	C₂×C₃×C₄	semidirect
160. #97	C₂×C₃×C₄	semidirect
162. C₄×C₈	C₂×C₃×C₄	semidirect
173. C₃×D₄	C₂×C₄²	semidirect
195. C₃×C₈	C₂×C₄²	semidirect
213. D₈	C₃×C₄²	semidirect
248. C₃×C₄	#442	direct (N × H)
260. D₄	C₂×C₃×C₄²	semidirect
282. C₈	C₂×C₃×C₄²	semidirect
301. C₄	C₃×#442	direct (N × H)
305. C₃	C₄×#442	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×#442	2
C₂×C₃×C₄×D₈	2
C₃×C₄²×D₄	2
C₃×#1776	2
C₃×C₄×#106	2
C₃×#1762	2
C₃×C₄×#102	2
C₃×C₄×#97	2
C₄×#442	3

Subgroups — C3×C4×#442 #10639107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#442 #10639107