Subgroups — C₂²×C₃×#357 #10651981

Order 768.

1470 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#357	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#357	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×D₄	—	2	1	no	—
7	C₂²×C₃×D₈	—	2	3	no	—
8	C₂²×C₃×D₈	—	2	4	no	—
9	C₂²×C₃×D₈	C₄	1	8	yes	complemented
10	C₂²×C₃×D₈	C₂²	1	6	yes	—
11	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
13	C₂²×C₃×C₁₆	—	2	4	no	—
14	C₂²×C₃×C₁₆	C₂²	1	6	yes	complemented
15	C₂×C₃×#107	—	2	3	no	—
16	C₂×C₃×#107	C₂²	1	6	yes	complemented
17	C₃×#357	C₂²	1	16	yes	direct product factor
18	C₂³×D₈	C₂×C₃	1	1	yes	characteristic
19	C₂³×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂²×#107	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×#357	C₂×C₃	1	12	yes	direct product factor
22	C₂⁵×C₃	—	4	1	no	—
23	C₂²×C₃×D₄	—	2	8	no	—
24	C₂²×C₃×D₄	—	2	6	no	—
25	C₂²×C₃×D₄	—	4	3	no	—
26	C₂²×C₃×D₄	—	4	4	no	—
27	C₂³×C₃×C₄	—	4	1	no	—
28	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃×D₈	—	2	24	no	—
30	C₂×C₃×D₈	—	2	6	no	—
31	C₂×C₃×D₈	—	2	12	no	—
32	C₂×C₃×D₈	C₂×C₄	1	24	yes	complemented
33	C₂×C₃×D₈	C₂³	1	4	yes	—
34	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₃×C₈	D₄	1	3	yes	—
36	C₂²×C₃×C₈	D₄	1	4	yes	—
37	C₂²×C₃×C₈	C₂³	1	3	yes	—
38	C₂×C₃×C₄⋊C₄	—	2	6	no	—
39	C₂×C₃×C₄⋊C₄	—	4	3	no	—
40	C₂×C₃×C₁₆	—	2	12	no	—
41	C₂×C₃×C₁₆	C₂³	1	4	yes	complemented
42	C₃×#107	—	2	6	no	—
43	C₃×#107	C₂³	1	4	yes	complemented
44	C₂³×D₄	—	2	1	no	—
45	C₂²×D₈	—	2	3	no	—
46	C₂²×D₈	—	2	4	no	—
47	C₂²×D₈	C₃×C₄	1	8	yes	complemented
48	C₂²×D₈	C₂²×C₃	1	6	yes	—
49	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₄⋊C₄	—	2	1	no	—
51	C₂²×C₁₆	—	2	4	no	—
52	C₂²×C₁₆	C₂²×C₃	1	6	yes	complemented
53	C₂×#107	—	2	3	no	—
54	C₂×#107	C₂²×C₃	1	6	yes	complemented
55	#357	C₂²×C₃	1	16	yes	direct product factor
56	C₂⁴×C₃	—	4	8	no	—
57	C₂⁴×C₃	—	4	6	no	—
58	C₂⁴×C₃	—	8	4	no	—
59	C₂⁴×C₃	—	8	4	no	—
60	C₂⁴×C₃	D₈	1	1	yes	characteristic
61	C₂×C₃×D₄	—	2	24	no	—
62	C₂×C₃×D₄	—	2	4	no	—
63	C₂×C₃×D₄	—	4	24	no	—
64	C₂×C₃×D₄	—	4	6	no	—
65	C₂×C₃×D₄	—	4	12	no	—
66	C₂²×C₃×C₄	—	4	6	no	—
67	C₂²×C₃×C₄	—	8	4	no	—
68	C₂²×C₃×C₄	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₃×C₄	D₈	1	3	yes	—
70	C₂²×C₃×C₄	SD₁₆	1	4	yes	—
71	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
72	C₃×D₈	—	2	24	no	—
73	C₃×D₈	—	2	4	no	—
74	C₃×D₈	C₂²×C₄	1	8	yes	complemented
75	C₂×C₃×C₈	C₂²⋊C₄	1	12	yes	—
76	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
77	C₂×C₃×C₈	C₂×D₄	1	3	yes	—
78	C₂×C₃×C₈	C₂×D₄	1	6	yes	—
79	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
80	C₃×C₄⋊C₄	—	2	4	no	—
81	C₃×C₄⋊C₄	—	4	6	no	—
82	C₃×C₁₆	—	2	4	no	—
83	C₂⁵	—	4	1	no	—
84	C₂²×D₄	—	2	8	no	—
85	C₂²×D₄	—	2	6	no	—
86	C₂²×D₄	—	4	3	no	—
87	C₂²×D₄	—	4	4	no	—
88	C₂³×C₄	—	4	1	no	—
89	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
90	C₂×D₈	—	2	24	no	—
91	C₂×D₈	—	2	6	no	—
92	C₂×D₈	—	2	12	no	—
93	C₂×D₈	C₂×C₃×C₄	1	24	yes	complemented
94	C₂×D₈	C₂³×C₃	1	4	yes	—
95	C₂²×C₈	C₂×C₃×C₄	1	4	yes	—
96	C₂²×C₈	C₃×D₄	1	3	yes	—
97	C₂²×C₈	C₃×D₄	1	4	yes	—
98	C₂²×C₈	C₂³×C₃	1	3	yes	—
99	C₂×C₄⋊C₄	—	2	6	no	—
100	C₂×C₄⋊C₄	—	4	3	no	—
101	C₂×C₁₆	—	2	12	no	—
102	C₂×C₁₆	C₂³×C₃	1	4	yes	complemented
103	#107	—	2	6	no	—
104	#107	C₂³×C₃	1	4	yes	complemented
105	C₂³×C₃	—	4	4	no	—
106	C₂³×C₃	—	4	24	no	—
107	C₂³×C₃	—	8	32	no	—
108	C₂³×C₃	—	8	12	no	—
109	C₂³×C₃	—	8	12	no	—
110	C₂³×C₃	#102	1	4	yes	—
111	C₂³×C₃	D₁₆	1	4	yes	—
112	C₂³×C₃	SD₃₂	1	4	yes	—
113	C₂³×C₃	C₂×D₈	1	3	yes	—
114	C₃×D₄	—	2	8	no	—
115	C₃×D₄	—	4	4	no	—
116	C₃×D₄	—	4	24	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	—	8	12	no	—
119	C₂×C₃×C₄	#102	1	12	yes	—
120	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	6	yes	—
121	C₂×C₃×C₄	C₂×D₈	1	3	yes	—
122	C₂×C₃×C₄	C₂×SD₁₆	1	6	yes	—
123	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
124	C₃×C₈	C₂×C₂²⋊C₄	1	6	yes	—
125	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
126	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
127	C₂⁴	—	4	8	no	—
128	C₂⁴	—	4	6	no	—
129	C₂⁴	—	8	4	no	—
130	C₂⁴	—	8	4	no	—
131	C₂⁴	C₃×D₈	1	1	yes	characteristic
132	C₂×D₄	—	2	24	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	4	24	no	—
135	C₂×D₄	—	4	6	no	—
136	C₂×D₄	—	4	12	no	—
137	C₂²×C₄	—	4	6	no	—
138	C₂²×C₄	—	8	4	no	—
139	C₂²×C₄	C₃×C₂²⋊C₄	1	4	yes	—
140	C₂²×C₄	C₃×D₈	1	3	yes	—
141	C₂²×C₄	C₃×SD₁₆	1	4	yes	—
142	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
143	D₈	—	2	24	no	—
144	D₈	—	2	4	no	—
145	D₈	C₂²×C₃×C₄	1	8	yes	complemented
146	C₂×C₈	C₃×C₂²⋊C₄	1	12	yes	—
147	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
148	C₂×C₈	C₂×C₃×D₄	1	3	yes	—
149	C₂×C₈	C₂×C₃×D₄	1	6	yes	—
150	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	6	no	—
153	C₁₆	—	2	4	no	—
154	C₂²×C₃	—	4	8	no	—
155	C₂²×C₃	—	8	4	no	—
156	C₂²×C₃	—	8	4	no	—
157	C₂²×C₃	—	8	48	no	—
158	C₂²×C₃	#357	1	16	yes	direct product factor
159	C₂²×C₃	C₂×#102	1	6	yes	—
160	C₂²×C₃	C₂×D₁₆	1	6	yes	—
161	C₂²×C₃	C₂×SD₃₂	1	6	yes	—
162	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
163	C₃×C₄	—	8	4	no	—
164	C₃×C₄	C₂×#102	1	6	yes	—
165	C₃×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₃×C₄	C₂²×SD₁₆	1	1	yes	characteristic
167	C₂³	—	4	4	no	—
168	C₂³	—	4	24	no	—
169	C₂³	—	8	12	no	—
170	C₂³	—	8	12	no	—
171	C₂³	—	8	32	no	—
172	C₂³	C₃×#102	1	4	yes	—
173	C₂³	C₃×D₁₆	1	4	yes	—
174	C₂³	C₃×SD₃₂	1	4	yes	—
175	C₂³	C₂×C₃×D₈	1	3	yes	—
176	D₄	—	2	8	no	—
177	D₄	—	4	4	no	—
178	D₄	—	4	24	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	12	no	—
181	C₂×C₄	C₃×#102	1	12	yes	—
182	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	6	yes	—
183	C₂×C₄	C₂×C₃×D₈	1	3	yes	—
184	C₂×C₄	C₂×C₃×SD₁₆	1	6	yes	—
185	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
186	C₈	C₂×C₃×C₂²⋊C₄	1	6	yes	—
187	C₈	C₂³×C₃×C₄	1	1	yes	characteristic
188	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₂×C₃	—	8	8	no	—
190	C₂×C₃	C₂×#357	1	12	yes	direct product factor
191	C₂×C₃	C₂²×#102	1	1	yes	characteristic
192	C₂×C₃	C₂²×D₁₆	1	1	yes	characteristic
193	C₂×C₃	C₂²×SD₃₂	1	1	yes	characteristic
194	C₂²	—	4	8	no	—
195	C₂²	—	8	4	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	8	48	no	—
198	C₂²	C₃×#357	1	16	yes	direct product factor
199	C₂²	C₂×C₃×#102	1	6	yes	—
200	C₂²	C₂×C₃×D₁₆	1	6	yes	—
201	C₂²	C₂×C₃×SD₃₂	1	6	yes	—
202	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
203	C₄	—	8	4	no	—
204	C₄	C₂×C₃×#102	1	6	yes	—
205	C₄	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₄	C₂²×C₃×SD₁₆	1	1	yes	characteristic
207	C₃	C₂²×#357	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	—	8	8	no	—
209	C₂	C₂×C₃×#357	1	12	yes	direct product factor
210	C₂	C₂²×C₃×#102	1	1	yes	characteristic
211	C₂	C₂²×C₃×D₁₆	1	1	yes	characteristic
212	C₂	C₂²×C₃×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	D₈
commutator	C₈	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂³×C₃×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×C₁₆	C₂	semidirect
3. C₂²×C₃×#107	C₂	semidirect
4. C₂×C₃×#357	C₂	direct (N × H)
5. C₂²×#357	C₃	direct (N × H)
9. C₂²×C₃×D₈	C₄	semidirect
14. C₂²×C₃×C₁₆	C₂²	semidirect
16. C₂×C₃×#107	C₂²	semidirect
17. C₃×#357	C₂²	direct (N × H)
19. C₂³×C₁₆	C₂×C₃	semidirect
20. C₂²×#107	C₂×C₃	semidirect
21. C₂×#357	C₂×C₃	direct (N × H)
32. C₂×C₃×D₈	C₂×C₄	semidirect
41. C₂×C₃×C₁₆	C₂³	semidirect
43. C₃×#107	C₂³	semidirect
47. C₂²×D₈	C₃×C₄	semidirect
52. C₂²×C₁₆	C₂²×C₃	semidirect
54. C₂×#107	C₂²×C₃	semidirect
55. #357	C₂²×C₃	direct (N × H)
74. C₃×D₈	C₂²×C₄	semidirect
93. C₂×D₈	C₂×C₃×C₄	semidirect
102. C₂×C₁₆	C₂³×C₃	semidirect
104. #107	C₂³×C₃	semidirect
145. D₈	C₂²×C₃×C₄	semidirect
158. C₂²×C₃	#357	direct (N × H)
190. C₂×C₃	C₂×#357	direct (N × H)
198. C₂²	C₃×#357	direct (N × H)
207. C₃	C₂²×#357	direct (N × H)
209. C₂	C₂×C₃×#357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₈	2
C₂³×C₃×C₁₆	2
C₂²×C₃×#107	2
C₂×C₃×#357	2
C₂²×#357	3

Subgroups — C22×C3×#357 #10651981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#357 #10651981