Subgroups — C₃×C₄×AGL(1,ℤ/8) #84808

Order 384.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#442	C₂	1	2	yes	maximal, complemented
5	C₃×#440	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×SD₁₆	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×D₈	C₂	1	2	yes	maximal, complemented
8	C₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#418	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×AGL(1,ℤ/8)	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	—	2	1	no	—
16	C₃×AGL(1,ℤ/8)	C₄	1	8	yes	direct product factor
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₃×D₈	C₂²	1	2	yes	—
19	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
21	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₃×C₄×D₄	—	2	1	no	—
24	C₃×C₄×D₄	—	2	1	no	—
25	C₃×C₄×D₄	—	2	1	no	—
26	C₃×C₄×D₄	—	2	1	no	—
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₃×C₄×D₄	C₂²	1	2	yes	—
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
30	C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
31	C₃×#117	—	2	1	no	—
32	C₃×#117	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₃×C₄²	—	2	1	no	—
36	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
37	C₃×#107	C₂²	1	2	yes	complemented
38	C₃×#106	C₂²	1	2	yes	complemented
39	C₃×#103	C₂²	1	2	yes	complemented
40	C₃×#102	C₂²	1	2	yes	—
41	C₃×#102	C₂²	1	2	yes	—
42	C₃×#102	C₂²	1	2	yes	complemented
43	C₃×#97	C₂²	1	2	yes	complemented
44	C₃×C₄×C₈	C₂²	1	2	yes	complemented
45	C₂×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic
46	C₄×#41	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
48	#442	C₂×C₃	1	2	yes	complemented
49	#440	C₂×C₃	1	2	yes	complemented
50	C₄×SD₁₆	C₂×C₃	1	2	yes	complemented
51	C₄×D₈	C₂×C₃	1	2	yes	complemented
52	#428	C₂×C₃	1	1	yes	characteristic, complemented
53	#418	C₂×C₃	1	1	yes	characteristic, complemented
54	#417	C₂×C₃	1	1	yes	characteristic, complemented
55	C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
56	C₂⁴×C₃	—	2	1	no	—
57	C₃×#41	—	2	2	no	—
58	C₃×#41	C₂×C₄	1	4	yes	complemented
59	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	2	1	no	—
62	C₂×C₃×D₄	—	2	2	no	—
63	C₂×C₃×D₄	C₂×C₄	1	4	yes	complemented
64	C₂×C₃×D₄	C₂³	1	2	yes	—
65	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	2	1	no	—
69	C₂²×C₃×C₄	—	2	2	no	—
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	—	2	2	no	—
72	C₂²×C₃×C₄	—	4	4	no	—
73	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
76	C₃×SD₁₆	C₂×C₄	1	8	yes	complemented
77	C₃×D₈	C₂×C₄	1	8	yes	complemented
78	C₃×M₁₆	—	2	2	no	—
79	C₃×M₁₆	C₂×C₄	1	4	yes	complemented
80	C₂×C₃×C₈	C₂³	1	2	yes	—
81	C₂×C₃×C₈	C₂³	1	2	yes	—
82	C₃×C₄⋊C₄	—	2	1	no	—
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
86	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	2	yes	—
88	C₃×C₂²⋊C₄	—	2	1	no	—
89	C₃×C₂²⋊C₄	—	2	2	no	—
90	C₃×C₂²⋊C₄	—	2	2	no	—
91	C₃×C₂²⋊C₄	—	2	1	no	—
92	C₃×C₂²⋊C₄	—	2	1	no	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	D₄	1	1	yes	characteristic
97	C₃×C₄²	D₄	1	1	yes	characteristic
98	C₃×C₄²	C₂³	1	1	yes	characteristic
99	C₃×C₄²	C₂³	1	1	yes	characteristic
100	C₂×#41	C₂²×C₃	1	1	yes	characteristic
101	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
102	C₂³×C₄	—	2	1	no	—
103	AGL(1,ℤ/8)	C₃×C₄	1	8	yes	direct product factor
104	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
105	C₂×D₈	C₂²×C₃	1	2	yes	—
106	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
107	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
108	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
109	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
110	C₄×D₄	—	2	1	no	—
111	C₄×D₄	—	2	1	no	—
112	C₄×D₄	—	2	1	no	—
113	C₄×D₄	—	2	1	no	—
114	C₄×D₄	C₂²×C₃	1	2	yes	—
115	C₄×D₄	C₂²×C₃	1	2	yes	—
116	C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
117	C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
118	#117	—	2	1	no	—
119	#117	C₂²×C₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	—	2	1	no	—
122	C₂×C₄²	—	2	1	no	—
123	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
124	#107	C₂²×C₃	1	2	yes	complemented
125	#106	C₂²×C₃	1	2	yes	complemented
126	#103	C₂²×C₃	1	2	yes	complemented
127	#102	C₂²×C₃	1	2	yes	—
128	#102	C₂²×C₃	1	2	yes	—
129	#102	C₂²×C₃	1	2	yes	complemented
130	#97	C₂²×C₃	1	2	yes	complemented
131	C₄×C₈	C₂²×C₃	1	2	yes	complemented
132	C₂³×C₃	—	2	4	no	—
133	C₂³×C₃	—	2	1	no	—
134	C₂³×C₃	—	2	2	no	—
135	C₂³×C₃	—	4	2	no	—
136	C₂³×C₃	—	4	2	no	—
137	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
138	C₃×Q₈	—	2	1	no	—
139	C₃×Q₈	C₂²×C₄	1	2	yes	complemented
140	C₃×D₄	—	2	4	no	—
141	C₃×D₄	—	2	4	no	—
142	C₃×D₄	—	2	2	no	—
143	C₃×D₄	—	2	1	no	—
144	C₃×D₄	C₂²×C₄	1	4	yes	—
145	C₃×D₄	C₂²×C₄	1	2	yes	complemented
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	4	1	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	4	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	4	no	—
158	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
159	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	#41	1	2	yes	—
165	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
166	C₃×C₈	—	2	2	no	—
167	C₃×C₈	C₂²×C₄	1	4	yes	complemented
168	C₂⁴	—	2	1	no	—
169	#41	—	2	2	no	—
170	#41	C₂×C₃×C₄	1	4	yes	complemented
171	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	C₂×C₃×C₄	1	4	yes	complemented
176	C₂×D₄	C₂³×C₃	1	2	yes	—
177	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
188	SD₁₆	C₂×C₃×C₄	1	8	yes	complemented
189	D₈	C₂×C₃×C₄	1	8	yes	complemented
190	M₁₆	—	2	2	no	—
191	M₁₆	C₂×C₃×C₄	1	4	yes	complemented
192	C₂×C₈	C₂³×C₃	1	2	yes	—
193	C₂×C₈	C₂³×C₃	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
199	C₄⋊C₄	C₂³×C₃	1	2	yes	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₃×D₄	1	1	yes	characteristic
209	C₄²	C₃×D₄	1	1	yes	characteristic
210	C₄²	C₂³×C₃	1	1	yes	characteristic
211	C₄²	C₂³×C₃	1	1	yes	characteristic
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	4	no	—
216	C₂²×C₃	—	4	8	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	—	4	1	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	C₄×D₄	1	2	yes	—
222	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	4	1	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	—	4	1	no	—
230	C₃×C₄	C₄×D₄	1	2	yes	—
231	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	direct product factor
232	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
233	C₃×C₄	C₂×#41	1	1	yes	characteristic
234	C₂³	—	2	4	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
240	Q₈	—	2	1	no	—
241	Q₈	C₂²×C₃×C₄	1	2	yes	complemented
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	1	no	—
246	D₄	C₂²×C₃×C₄	1	4	yes	—
247	D₄	C₂²×C₃×C₄	1	2	yes	complemented
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
261	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
262	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
266	C₂×C₄	C₃×#41	1	2	yes	—
267	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	C₂²×C₃×C₄	1	4	yes	complemented
270	C₂×C₃	—	2	2	no	—
271	C₂×C₃	—	4	2	no	—
272	C₂×C₃	—	4	4	no	—
273	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
274	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂×C₃	#575	1	1	yes	characteristic
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₃×C₄×D₄	1	2	yes	—
286	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	C₃×C₄×D₄	1	2	yes	—
295	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	direct product factor
296	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
297	C₄	C₂×C₃×#41	1	1	yes	characteristic
298	C₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	—	4	2	no	—
302	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
303	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×#41	C₂	semidirect
3. C₂×C₃×C₄×D₄	C₂	semidirect
4. C₃×#442	C₂	semidirect
5. C₃×#440	C₂	semidirect
6. C₃×C₄×SD₁₆	C₂	semidirect
7. C₃×C₄×D₈	C₂	semidirect
8. C₃×#428	C₂	semidirect
9. C₃×#418	C₂	semidirect
10. C₃×#417	C₂	semidirect
11. C₃×C₄×M₁₆	C₂	semidirect
12. C₄×AGL(1,ℤ/8)	C₃	direct (N × H)
16. C₃×AGL(1,ℤ/8)	C₄	direct (N × H)
21. C₃×C₄×Q₈	C₂²	semidirect
22. C₃×C₄×Q₈	C₂²	semidirect
29. C₃×C₄×D₄	C₂²	semidirect
30. C₃×C₄×D₄	C₂²	semidirect
37. C₃×#107	C₂²	semidirect
38. C₃×#106	C₂²	semidirect
39. C₃×#103	C₂²	semidirect
42. C₃×#102	C₂²	semidirect
43. C₃×#97	C₂²	semidirect
44. C₃×C₄×C₈	C₂²	semidirect
46. C₄×#41	C₂×C₃	semidirect
47. C₂×C₄×D₄	C₂×C₃	semidirect
48. #442	C₂×C₃	semidirect
49. #440	C₂×C₃	semidirect
50. C₄×SD₁₆	C₂×C₃	semidirect
51. C₄×D₈	C₂×C₃	semidirect
52. #428	C₂×C₃	semidirect
53. #418	C₂×C₃	semidirect
54. #417	C₂×C₃	semidirect
55. C₄×M₁₆	C₂×C₃	semidirect
58. C₃×#41	C₂×C₄	semidirect
63. C₂×C₃×D₄	C₂×C₄	semidirect
76. C₃×SD₁₆	C₂×C₄	semidirect
77. C₃×D₈	C₂×C₄	semidirect
79. C₃×M₁₆	C₂×C₄	semidirect
103. AGL(1,ℤ/8)	C₃×C₄	direct (N × H)
108. C₄×Q₈	C₂²×C₃	semidirect
109. C₄×Q₈	C₂²×C₃	semidirect
116. C₄×D₄	C₂²×C₃	semidirect
117. C₄×D₄	C₂²×C₃	semidirect
124. #107	C₂²×C₃	semidirect
125. #106	C₂²×C₃	semidirect
126. #103	C₂²×C₃	semidirect
129. #102	C₂²×C₃	semidirect
130. #97	C₂²×C₃	semidirect
131. C₄×C₈	C₂²×C₃	semidirect
139. C₃×Q₈	C₂²×C₄	semidirect
145. C₃×D₄	C₂²×C₄	semidirect
167. C₃×C₈	C₂²×C₄	semidirect
170. #41	C₂×C₃×C₄	semidirect
175. C₂×D₄	C₂×C₃×C₄	semidirect
188. SD₁₆	C₂×C₃×C₄	semidirect
189. D₈	C₂×C₃×C₄	semidirect
191. M₁₆	C₂×C₃×C₄	semidirect
231. C₃×C₄	AGL(1,ℤ/8)	direct (N × H)
241. Q₈	C₂²×C₃×C₄	semidirect
247. D₄	C₂²×C₃×C₄	semidirect
269. C₈	C₂²×C₃×C₄	semidirect
295. C₄	C₃×AGL(1,ℤ/8)	direct (N × H)
298. C₃	C₄×AGL(1,ℤ/8)	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×AGL(1,ℤ/8)	2
C₃×C₄×#41	2
C₂×C₃×C₄×D₄	2
C₃×#442	2
C₃×#440	2
C₃×C₄×SD₁₆	2
C₃×C₄×D₈	2
C₃×#428	2
C₃×#418	2
C₃×#417	2
C₃×C₄×M₁₆	2
C₄×AGL(1,ℤ/8)	3

Subgroups — C3×C4×AGL(1,ℤ/8) #84808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×AGL(1,ℤ/8) #84808