Subgroups — C₂²×C₃×#488 #10679211

Order 768.

2310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#488	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×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#488	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₈	C₂²	1	2	yes	—
11	C₂×C₃×#125	—	2	6	no	—
12	C₂×C₃×#125	C₂²	1	12	yes	complemented
13	C₂×C₃×#124	—	2	6	no	—
14	C₂×C₃×#124	C₂²	1	12	yes	—
15	C₂²×C₃×C₄⋊C₄	—	2	2	no	—
16	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₃×C₂²⋊C₄	—	2	2	no	—
18	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₃×#488	C₂²	1	64	yes	direct product factor
20	C₂×C₃×#103	C₂²	1	24	yes	complemented
21	C₂×C₃×#102	C₂²	1	24	yes	—
22	C₂×C₃×C₄×C₈	C₂²	1	12	yes	complemented
23	C₂²×#125	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×#124	C₂×C₃	1	1	yes	characteristic
25	C₂×#488	C₂×C₃	1	24	yes	direct product factor
26	C₂²×#103	C₂×C₃	1	2	yes	complemented
27	C₂²×#102	C₂×C₃	1	2	yes	—
28	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂⁵×C₃	—	2	1	no	—
30	C₂²×C₃×Q₈	—	2	3	no	—
31	C₂²×C₃×Q₈	—	2	8	no	—
32	C₂²×C₃×Q₈	C₂³	1	6	yes	—
33	C₂²×C₃×D₄	—	2	3	no	—
34	C₂²×C₃×D₄	—	2	8	no	—
35	C₂²×C₃×D₄	C₂³	1	6	yes	—
36	C₂³×C₃×C₄	—	2	1	no	—
37	C₂³×C₃×C₄	—	2	2	no	—
38	C₂³×C₃×C₄	D₄	1	2	yes	—
39	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₈	—	2	8	no	—
41	C₂²×C₃×C₈	C₂³	1	12	yes	—
42	C₃×#125	—	2	24	no	—
43	C₃×#125	C₂³	1	16	yes	complemented
44	C₃×#124	—	2	24	no	—
45	C₃×#124	C₂³	1	16	yes	—
46	C₂×C₃×C₄⋊C₄	—	2	24	no	—
47	C₂×C₃×C₄⋊C₄	—	2	6	no	—
48	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
49	C₂×C₃×C₂²⋊C₄	—	2	24	no	—
50	C₂×C₃×C₄²	D₄	1	6	yes	—
51	C₂×C₃×C₄²	C₂³	1	6	yes	—
52	C₃×#103	C₂³	1	32	yes	complemented
53	C₃×#102	C₂³	1	32	yes	—
54	C₃×C₄×C₈	C₂³	1	16	yes	complemented
55	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂³×C₈	C₂²×C₃	1	2	yes	—
58	C₂×#125	—	2	6	no	—
59	C₂×#125	C₂²×C₃	1	12	yes	complemented
60	C₂×#124	—	2	6	no	—
61	C₂×#124	C₂²×C₃	1	12	yes	—
62	C₂²×C₄⋊C₄	—	2	2	no	—
63	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂²×C₂²⋊C₄	—	2	2	no	—
65	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
66	#488	C₂²×C₃	1	64	yes	direct product factor
67	C₂×#103	C₂²×C₃	1	24	yes	complemented
68	C₂×#102	C₂²×C₃	1	24	yes	—
69	C₂×C₄×C₈	C₂²×C₃	1	12	yes	complemented
70	C₂⁴×C₃	—	2	6	no	—
71	C₂⁴×C₃	—	4	4	no	—
72	C₂⁴×C₃	—	4	8	no	—
73	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×Q₈	—	2	6	no	—
75	C₂×C₃×Q₈	—	2	24	no	—
76	C₂×C₃×Q₈	—	4	12	no	—
77	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
78	C₂×C₃×D₄	—	2	6	no	—
79	C₂×C₃×D₄	—	2	24	no	—
80	C₂×C₃×D₄	—	4	12	no	—
81	C₂×C₃×D₄	C₂⁴	1	4	yes	—
82	C₂²×C₃×C₄	—	2	8	no	—
83	C₂²×C₃×C₄	—	2	6	no	—
84	C₂²×C₃×C₄	—	2	12	no	—
85	C₂²×C₃×C₄	—	4	4	no	—
86	C₂²×C₃×C₄	—	4	8	no	—
87	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
88	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
89	C₂²×C₃×C₄	#41	1	8	yes	—
90	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
91	C₂×C₃×C₈	—	2	24	no	—
92	C₂×C₃×C₈	C₂⁴	1	8	yes	—
93	C₃×C₄⋊C₄	—	2	32	no	—
94	C₃×C₄⋊C₄	—	2	12	no	—
95	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
96	C₃×C₂²⋊C₄	—	2	32	no	—
97	C₃×C₄²	C₂×D₄	1	12	yes	—
98	C₃×C₄²	C₂⁴	1	4	yes	—
99	C₂⁵	—	2	1	no	—
100	C₂²×Q₈	—	2	3	no	—
101	C₂²×Q₈	—	2	8	no	—
102	C₂²×Q₈	C₂³×C₃	1	6	yes	—
103	C₂²×D₄	—	2	3	no	—
104	C₂²×D₄	—	2	8	no	—
105	C₂²×D₄	C₂³×C₃	1	6	yes	—
106	C₂³×C₄	—	2	1	no	—
107	C₂³×C₄	—	2	2	no	—
108	C₂³×C₄	C₃×D₄	1	2	yes	—
109	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂²×C₈	—	2	8	no	—
111	C₂²×C₈	C₂³×C₃	1	12	yes	—
112	#125	—	2	24	no	—
113	#125	C₂³×C₃	1	16	yes	complemented
114	#124	—	2	24	no	—
115	#124	C₂³×C₃	1	16	yes	—
116	C₂×C₄⋊C₄	—	2	24	no	—
117	C₂×C₄⋊C₄	—	2	6	no	—
118	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
119	C₂×C₂²⋊C₄	—	2	24	no	—
120	C₂×C₄²	C₃×D₄	1	6	yes	—
121	C₂×C₄²	C₂³×C₃	1	6	yes	—
122	#103	C₂³×C₃	1	32	yes	complemented
123	#102	C₂³×C₃	1	32	yes	—
124	C₄×C₈	C₂³×C₃	1	16	yes	complemented
125	C₂³×C₃	—	2	4	no	—
126	C₂³×C₃	—	4	12	no	—
127	C₂³×C₃	—	4	24	no	—
128	C₂³×C₃	—	8	16	no	—
129	C₂³×C₃	#124	1	4	yes	—
130	C₂³×C₃	#135	1	8	yes	—
131	C₂³×C₃	C₂²×D₄	1	3	yes	—
132	C₃×Q₈	—	2	8	no	—
133	C₃×Q₈	—	4	12	no	—
134	C₃×D₄	—	2	8	no	—
135	C₃×D₄	—	4	12	no	—
136	C₂×C₃×C₄	—	2	24	no	—
137	C₂×C₃×C₄	—	2	4	no	—
138	C₂×C₃×C₄	—	2	8	no	—
139	C₂×C₃×C₄	—	4	12	no	—
140	C₂×C₃×C₄	—	4	24	no	—
141	C₂×C₃×C₄	#124	1	12	yes	—
142	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
143	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
144	C₂×C₃×C₄	C₂×#41	1	12	yes	—
145	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
146	C₃×C₈	—	2	8	no	—
147	C₂⁴	—	2	6	no	—
148	C₂⁴	—	4	4	no	—
149	C₂⁴	—	4	8	no	—
150	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×Q₈	—	2	6	no	—
152	C₂×Q₈	—	2	24	no	—
153	C₂×Q₈	—	4	12	no	—
154	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
155	C₂×D₄	—	2	6	no	—
156	C₂×D₄	—	2	24	no	—
157	C₂×D₄	—	4	12	no	—
158	C₂×D₄	C₂⁴×C₃	1	4	yes	—
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	2	12	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	—	4	8	no	—
164	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
165	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
166	C₂²×C₄	C₃×#41	1	8	yes	—
167	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
168	C₂×C₈	—	2	24	no	—
169	C₂×C₈	C₂⁴×C₃	1	8	yes	—
170	C₄⋊C₄	—	2	32	no	—
171	C₄⋊C₄	—	2	12	no	—
172	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
173	C₂²⋊C₄	—	2	32	no	—
174	C₄²	C₂×C₃×D₄	1	12	yes	—
175	C₄²	C₂⁴×C₃	1	4	yes	—
176	C₂²×C₃	—	4	4	no	—
177	C₂²×C₃	—	4	8	no	—
178	C₂²×C₃	—	8	24	no	—
179	C₂²×C₃	#488	1	16	yes	direct product factor
180	C₂²×C₃	C₂×#124	1	6	yes	—
181	C₂²×C₃	C₂×#135	1	12	yes	—
182	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
183	C₃×C₄	—	2	8	no	—
184	C₃×C₄	—	4	4	no	—
185	C₃×C₄	—	4	8	no	—
186	C₃×C₄	C₂×#124	1	6	yes	—
187	C₃×C₄	C₂²×#41	1	2	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	—	4	24	no	—
190	C₂³	—	4	12	no	—
191	C₂³	—	8	16	no	—
192	C₂³	C₃×#124	1	4	yes	—
193	C₂³	C₃×#135	1	8	yes	—
194	C₂³	C₂²×C₃×D₄	1	3	yes	—
195	Q₈	—	2	8	no	—
196	Q₈	—	4	12	no	—
197	D₄	—	2	8	no	—
198	D₄	—	4	12	no	—
199	C₂×C₄	—	2	24	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	4	24	no	—
203	C₂×C₄	—	4	12	no	—
204	C₂×C₄	C₃×#124	1	12	yes	—
205	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
206	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
207	C₂×C₄	C₂×C₃×#41	1	12	yes	—
208	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
209	C₈	—	2	8	no	—
210	C₂×C₃	—	8	4	no	—
211	C₂×C₃	C₂×#488	1	12	yes	direct product factor
212	C₂×C₃	C₂²×#124	1	1	yes	characteristic
213	C₂×C₃	C₂²×#135	1	2	yes	—
214	C₂²	—	4	4	no	—
215	C₂²	—	4	8	no	—
216	C₂²	—	8	24	no	—
217	C₂²	C₃×#488	1	16	yes	direct product factor
218	C₂²	C₂×C₃×#124	1	6	yes	—
219	C₂²	C₂×C₃×#135	1	12	yes	—
220	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
221	C₄	—	2	8	no	—
222	C₄	—	4	4	no	—
223	C₄	—	4	8	no	—
224	C₄	C₂×C₃×#124	1	6	yes	—
225	C₄	C₂²×C₃×#41	1	2	yes	—
226	C₃	C₂²×#488	1	1	yes	characteristic, direct product factor, Sylow
227	C₂	—	8	4	no	—
228	C₂	C₂×C₃×#488	1	12	yes	direct product factor
229	C₂	C₂²×C₃×#124	1	1	yes	characteristic
230	C₂	C₂²×C₃×#135	1	2	yes	—

	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×#125	C₂	semidirect
3. C₂×C₃×#488	C₂	direct (N × H)
4. C₂²×C₃×#103	C₂	semidirect
6. C₂²×C₃×C₄×C₈	C₂	semidirect
7. C₂²×#488	C₃	direct (N × H)
12. C₂×C₃×#125	C₂²	semidirect
19. C₃×#488	C₂²	direct (N × H)
20. C₂×C₃×#103	C₂²	semidirect
22. C₂×C₃×C₄×C₈	C₂²	semidirect
23. C₂²×#125	C₂×C₃	semidirect
25. C₂×#488	C₂×C₃	direct (N × H)
26. C₂²×#103	C₂×C₃	semidirect
28. C₂²×C₄×C₈	C₂×C₃	semidirect
43. C₃×#125	C₂³	semidirect
52. C₃×#103	C₂³	semidirect
54. C₃×C₄×C₈	C₂³	semidirect
59. C₂×#125	C₂²×C₃	semidirect
66. #488	C₂²×C₃	direct (N × H)
67. C₂×#103	C₂²×C₃	semidirect
69. C₂×C₄×C₈	C₂²×C₃	semidirect
113. #125	C₂³×C₃	semidirect
122. #103	C₂³×C₃	semidirect
124. C₄×C₈	C₂³×C₃	semidirect
179. C₂²×C₃	#488	direct (N × H)
211. C₂×C₃	C₂×#488	direct (N × H)
217. C₂²	C₃×#488	direct (N × H)
226. C₃	C₂²×#488	direct (N × H)
228. C₂	C₂×C₃×#488	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×#125	2
C₂²×C₃×#124	2
C₂×C₃×#488	2
C₂²×C₃×#103	2
C₂²×C₃×#102	2
C₂²×C₃×C₄×C₈	2
C₂²×#488	3

Subgroups — C22×C3×#488 #10679211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#488 #10679211