Subgroups — C₂²×C₃×#489 #10679213

Order 768.

2486 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	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#489	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#103	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃×#102	C₂	1	2	yes	maximal
6	C₂²×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#489	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	—	2	1	no	—
9	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
13	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
14	C₂×C₃×#124	—	2	6	no	—
15	C₂×C₃×#124	C₂²	1	12	yes	—
16	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
17	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
19	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
20	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₃×#489	C₂²	1	64	yes	direct product factor
22	C₂×C₃×#103	C₂²	1	24	yes	complemented
23	C₂×C₃×#102	C₂²	1	24	yes	—
24	C₂×C₃×#97	C₂²	1	12	yes	complemented
25	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²×#124	C₂×C₃	1	1	yes	characteristic
27	C₂×#489	C₂×C₃	1	24	yes	direct product factor
28	C₂²×#103	C₂×C₃	1	2	yes	complemented
29	C₂²×#102	C₂×C₃	1	2	yes	—
30	C₂²×#97	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁵×C₃	—	2	1	no	—
32	C₂²×C₃×Q₈	—	2	12	no	—
33	C₂²×C₃×Q₈	—	2	3	no	—
34	C₂²×C₃×Q₈	—	2	8	no	—
35	C₂²×C₃×Q₈	—	4	8	no	—
36	C₂²×C₃×Q₈	C₂³	1	6	yes	—
37	C₂²×C₃×D₄	—	2	3	no	—
38	C₂²×C₃×D₄	—	2	8	no	—
39	C₂²×C₃×D₄	C₂³	1	6	yes	—
40	C₂³×C₃×C₄	—	2	2	no	—
41	C₂³×C₃×C₄	—	2	1	no	—
42	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
43	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
44	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₈	—	2	8	no	—
46	C₂²×C₃×C₈	C₂³	1	12	yes	—
47	C₃×C₄⋊Q₈	—	2	24	no	—
48	C₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
49	C₃×#124	—	2	24	no	—
50	C₃×#124	C₂³	1	16	yes	—
51	C₂×C₃×C₄⋊C₄	—	2	12	no	—
52	C₂×C₃×C₄⋊C₄	—	2	6	no	—
53	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
54	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
55	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
56	C₂×C₃×C₄²	D₄	1	6	yes	—
57	C₂×C₃×C₄²	C₂³	1	6	yes	—
58	C₃×#103	C₂³	1	32	yes	complemented
59	C₃×#102	C₂³	1	32	yes	—
60	C₃×#97	C₂³	1	16	yes	complemented
61	C₂³×Q₈	—	2	1	no	—
62	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
63	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
64	C₂³×C₈	C₂²×C₃	1	2	yes	—
65	C₂×C₄⋊Q₈	—	2	6	no	—
66	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	complemented
67	C₂×#124	—	2	6	no	—
68	C₂×#124	C₂²×C₃	1	12	yes	—
69	C₂²×C₄⋊C₄	—	2	1	no	—
70	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂²×C₂²⋊C₄	—	2	1	no	—
72	C₂²×C₂²⋊C₄	—	2	1	no	—
73	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
74	#489	C₂²×C₃	1	64	yes	direct product factor
75	C₂×#103	C₂²×C₃	1	24	yes	complemented
76	C₂×#102	C₂²×C₃	1	24	yes	—
77	C₂×#97	C₂²×C₃	1	12	yes	complemented
78	C₂⁴×C₃	—	2	6	no	—
79	C₂⁴×C₃	—	4	4	no	—
80	C₂⁴×C₃	—	4	4	no	—
81	C₂⁴×C₃	—	4	4	no	—
82	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×Q₈	—	2	6	no	—
84	C₂×C₃×Q₈	—	2	16	no	—
85	C₂×C₃×Q₈	—	2	24	no	—
86	C₂×C₃×Q₈	—	4	48	no	—
87	C₂×C₃×Q₈	—	4	12	no	—
88	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
89	C₂×C₃×D₄	—	2	6	no	—
90	C₂×C₃×D₄	—	2	24	no	—
91	C₂×C₃×D₄	—	4	12	no	—
92	C₂×C₃×D₄	C₂⁴	1	4	yes	—
93	C₂²×C₃×C₄	—	2	4	no	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₂²×C₃×C₄	—	2	6	no	—
96	C₂²×C₃×C₄	—	2	12	no	—
97	C₂²×C₃×C₄	—	4	4	no	—
98	C₂²×C₃×C₄	—	4	8	no	—
99	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
100	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
101	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
102	C₂²×C₃×C₄	#41	1	8	yes	—
103	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
104	C₂×C₃×C₈	—	2	24	no	—
105	C₂×C₃×C₈	C₂⁴	1	8	yes	—
106	C₃×C₄⋊C₄	—	2	16	no	—
107	C₃×C₄⋊C₄	—	2	12	no	—
108	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
109	C₃×C₂²⋊C₄	—	2	16	no	—
110	C₃×C₂²⋊C₄	—	2	16	no	—
111	C₃×C₄²	C₂×D₄	1	12	yes	—
112	C₃×C₄²	C₂⁴	1	4	yes	—
113	C₂⁵	—	2	1	no	—
114	C₂²×Q₈	—	2	3	no	—
115	C₂²×Q₈	—	2	12	no	—
116	C₂²×Q₈	—	2	8	no	—
117	C₂²×Q₈	—	4	8	no	—
118	C₂²×Q₈	C₂³×C₃	1	6	yes	—
119	C₂²×D₄	—	2	3	no	—
120	C₂²×D₄	—	2	8	no	—
121	C₂²×D₄	C₂³×C₃	1	6	yes	—
122	C₂³×C₄	—	2	1	no	—
123	C₂³×C₄	—	2	2	no	—
124	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
125	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
126	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₂²×C₈	—	2	8	no	—
128	C₂²×C₈	C₂³×C₃	1	12	yes	—
129	C₄⋊Q₈	—	2	24	no	—
130	C₄⋊Q₈	C₂³×C₃	1	16	yes	complemented
131	#124	—	2	24	no	—
132	#124	C₂³×C₃	1	16	yes	—
133	C₂×C₄⋊C₄	—	2	12	no	—
134	C₂×C₄⋊C₄	—	2	6	no	—
135	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
136	C₂×C₂²⋊C₄	—	2	12	no	—
137	C₂×C₂²⋊C₄	—	2	12	no	—
138	C₂×C₄²	C₃×D₄	1	6	yes	—
139	C₂×C₄²	C₂³×C₃	1	6	yes	—
140	#103	C₂³×C₃	1	32	yes	complemented
141	#102	C₂³×C₃	1	32	yes	—
142	#97	C₂³×C₃	1	16	yes	complemented
143	C₂³×C₃	—	2	4	no	—
144	C₂³×C₃	—	4	12	no	—
145	C₂³×C₃	—	4	12	no	—
146	C₂³×C₃	—	4	12	no	—
147	C₂³×C₃	—	8	16	no	—
148	C₂³×C₃	#124	1	4	yes	—
149	C₂³×C₃	AGL(1,ℤ/8)	1	4	yes	—
150	C₂³×C₃	#137	1	4	yes	—
151	C₂³×C₃	C₂²×D₄	1	3	yes	—
152	C₃×Q₈	—	2	8	no	—
153	C₃×Q₈	—	4	32	no	—
154	C₃×Q₈	—	4	12	no	—
155	C₃×D₄	—	2	8	no	—
156	C₃×D₄	—	4	12	no	—
157	C₂×C₃×C₄	—	2	12	no	—
158	C₂×C₃×C₄	—	2	12	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	8	no	—
161	C₂×C₃×C₄	—	4	12	no	—
162	C₂×C₃×C₄	—	4	24	no	—
163	C₂×C₃×C₄	#124	1	12	yes	—
164	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
165	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
166	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
167	C₂×C₃×C₄	C₂×#41	1	12	yes	—
168	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
169	C₃×C₈	—	2	8	no	—
170	C₂⁴	—	2	6	no	—
171	C₂⁴	—	4	4	no	—
172	C₂⁴	—	4	4	no	—
173	C₂⁴	—	4	4	no	—
174	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×Q₈	—	2	6	no	—
176	C₂×Q₈	—	2	16	no	—
177	C₂×Q₈	—	2	24	no	—
178	C₂×Q₈	—	4	48	no	—
179	C₂×Q₈	—	4	12	no	—
180	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
181	C₂×D₄	—	2	6	no	—
182	C₂×D₄	—	2	24	no	—
183	C₂×D₄	—	4	12	no	—
184	C₂×D₄	C₂⁴×C₃	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	6	no	—
188	C₂²×C₄	—	2	12	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	4	8	no	—
191	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
192	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
193	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
194	C₂²×C₄	C₃×#41	1	8	yes	—
195	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
196	C₂×C₈	—	2	24	no	—
197	C₂×C₈	C₂⁴×C₃	1	8	yes	—
198	C₄⋊C₄	—	2	16	no	—
199	C₄⋊C₄	—	2	12	no	—
200	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
201	C₂²⋊C₄	—	2	16	no	—
202	C₂²⋊C₄	—	2	16	no	—
203	C₄²	C₂×C₃×D₄	1	12	yes	—
204	C₄²	C₂⁴×C₃	1	4	yes	—
205	C₂²×C₃	—	4	4	no	—
206	C₂²×C₃	—	4	4	no	—
207	C₂²×C₃	—	4	4	no	—
208	C₂²×C₃	—	8	24	no	—
209	C₂²×C₃	#489	1	16	yes	direct product factor
210	C₂²×C₃	C₂×#124	1	6	yes	—
211	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
212	C₂²×C₃	C₂×#137	1	6	yes	—
213	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	4	4	no	—
217	C₃×C₄	—	4	8	no	—
218	C₃×C₄	C₂×#124	1	6	yes	—
219	C₃×C₄	C₂²×#41	1	2	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	12	no	—
222	C₂³	—	4	12	no	—
223	C₂³	—	4	12	no	—
224	C₂³	—	8	16	no	—
225	C₂³	C₃×#124	1	4	yes	—
226	C₂³	C₃×AGL(1,ℤ/8)	1	4	yes	—
227	C₂³	C₃×#137	1	4	yes	—
228	C₂³	C₂²×C₃×D₄	1	3	yes	—
229	Q₈	—	2	8	no	—
230	Q₈	—	4	32	no	—
231	Q₈	—	4	12	no	—
232	D₄	—	2	8	no	—
233	D₄	—	4	12	no	—
234	C₂×C₄	—	2	12	no	—
235	C₂×C₄	—	2	12	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	4	24	no	—
239	C₂×C₄	—	4	12	no	—
240	C₂×C₄	C₃×#124	1	12	yes	—
241	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
242	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
243	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
244	C₂×C₄	C₂×C₃×#41	1	12	yes	—
245	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
246	C₈	—	2	8	no	—
247	C₂×C₃	—	8	4	no	—
248	C₂×C₃	C₂×#489	1	12	yes	direct product factor
249	C₂×C₃	C₂²×#124	1	1	yes	characteristic
250	C₂×C₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂×C₃	C₂²×#137	1	1	yes	characteristic
252	C₂²	—	4	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	8	24	no	—
256	C₂²	C₃×#489	1	16	yes	direct product factor
257	C₂²	C₂×C₃×#124	1	6	yes	—
258	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	6	yes	—
259	C₂²	C₂×C₃×#137	1	6	yes	—
260	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	2	4	no	—
263	C₄	—	4	4	no	—
264	C₄	—	4	8	no	—
265	C₄	C₂×C₃×#124	1	6	yes	—
266	C₄	C₂²×C₃×#41	1	2	yes	—
267	C₃	C₂²×#489	1	1	yes	characteristic, direct product factor, Sylow
268	C₂	—	8	4	no	—
269	C₂	C₂×C₃×#489	1	12	yes	direct product factor
270	C₂	C₂²×C₃×#124	1	1	yes	characteristic
271	C₂	C₂²×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂	C₂²×C₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂³×C₃×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄⋊Q₈	C₂	semidirect
3. C₂×C₃×#489	C₂	direct (N × H)
4. C₂²×C₃×#103	C₂	semidirect
6. C₂²×C₃×#97	C₂	semidirect
7. C₂²×#489	C₃	direct (N × H)
13. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
21. C₃×#489	C₂²	direct (N × H)
22. C₂×C₃×#103	C₂²	semidirect
24. C₂×C₃×#97	C₂²	semidirect
25. C₂²×C₄⋊Q₈	C₂×C₃	semidirect
27. C₂×#489	C₂×C₃	direct (N × H)
28. C₂²×#103	C₂×C₃	semidirect
30. C₂²×#97	C₂×C₃	semidirect
48. C₃×C₄⋊Q₈	C₂³	semidirect
58. C₃×#103	C₂³	semidirect
60. C₃×#97	C₂³	semidirect
66. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
74. #489	C₂²×C₃	direct (N × H)
75. C₂×#103	C₂²×C₃	semidirect
77. C₂×#97	C₂²×C₃	semidirect
130. C₄⋊Q₈	C₂³×C₃	semidirect
140. #103	C₂³×C₃	semidirect
142. #97	C₂³×C₃	semidirect
209. C₂²×C₃	#489	direct (N × H)
248. C₂×C₃	C₂×#489	direct (N × H)
256. C₂²	C₃×#489	direct (N × H)
267. C₃	C₂²×#489	direct (N × H)
269. C₂	C₂×C₃×#489	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₄⋊Q₈	2
C₂²×C₃×#124	2
C₂×C₃×#489	2
C₂²×C₃×#103	2
C₂²×C₃×#102	2
C₂²×C₃×#97	2
C₂²×#489	3

Subgroups — C22×C3×#489 #10679213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#489 #10679213