Subgroups — C₂×C₃×(C₂×D₃)⋊C₈ #10592205

Order 576.

442 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×(C₂×D₃)⋊C₈	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₂²⋊C₈	—	3	1	no	maximal
6	C₂×(C₂×D₃)⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃×D₃	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₃	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×D₃	—	2	2	no	—
11	C₂×C₃×C₄×D₃	—	2	2	no	—
12	C₂×C₃×C₄×D₃	C₄	1	2	yes	—
13	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
14	C₂×C₃²×C₈	—	2	2	no	—
15	C₂×C₃²×C₈	C₂²	1	2	yes	complemented
16	C₂×C₃×C₃⋊C₈	—	2	2	no	—
17	C₂×C₃×C₃⋊C₈	C₂²	1	2	yes	complemented
18	C₂³×C₃×C₄	—	3	1	no	—
19	C₂²×C₄×D₃	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₃×C₈	—	2	1	no	—
21	C₂²×C₃×C₈	—	3	1	no	—
22	C₂²×C₃×C₈	D₃	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×C₂²⋊C₈	—	3	4	no	—
26	(C₂×D₃)⋊C₈	C₂×C₃	1	4	yes	direct product factor
27	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×D₃	—	2	2	no	—
29	C₂²×C₃×D₃	—	2	2	no	—
30	C₂²×C₃×D₃	C₈	1	4	yes	complemented
31	C₂²×C₃×D₃	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₄	D₄	1	2	yes	—
34	C₂×C₃²×C₄	D₄	1	2	yes	—
35	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
36	C₂×C₃×Dic₃	—	2	2	no	—
37	C₂×C₃×Dic₃	C₂×C₄	1	2	yes	—
38	C₃×C₄×D₃	—	2	4	no	—
39	C₃×C₄×D₃	—	2	2	no	—
40	C₃×C₄×D₃	—	2	2	no	—
41	C₃²×C₈	—	2	2	no	—
42	C₃×C₃⋊C₈	—	2	2	no	—
43	C₂×C₂²⋊C₈	—	3	1	no	Sylow
44	C₂⁴×C₃	—	3	1	no	—
45	C₂³×D₃	C₃×C₄	1	1	yes	characteristic
46	C₂²×C₃×C₄	—	2	1	no	—
47	C₂²×C₃×C₄	—	3	1	no	—
48	C₂²×C₃×C₄	—	3	2	no	—
49	C₂²×C₃×C₄	—	3	2	no	—
50	C₂²×C₃×C₄	—	6	2	no	—
51	C₂²×C₃×C₄	—	6	2	no	—
52	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
54	C₂²×Dic₃	C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×D₃	—	2	2	no	—
56	C₂×C₄×D₃	—	2	2	no	—
57	C₂×C₄×D₃	C₃×C₄	1	2	yes	—
58	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
59	C₂×C₃×C₈	—	2	4	no	—
60	C₂×C₃×C₈	—	2	2	no	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	—	3	2	no	—
64	C₂×C₃×C₈	—	6	2	no	—
65	C₂×C₃×C₈	C₂×D₃	1	2	yes	complemented
66	C₂×C₃×C₈	C₂²×C₃	1	2	yes	complemented
67	C₂×C₃⋊C₈	—	2	2	no	—
68	C₂×C₃⋊C₈	C₂²×C₃	1	2	yes	complemented
69	C₂²×C₃²	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₃²	C₂×C₈	1	2	yes	—
71	C₂²×C₃²	M₁₆	1	2	yes	—
72	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×D₃	—	2	8	no	—
74	C₂×C₃×D₃	—	2	2	no	—
75	C₂×C₃×D₃	—	2	2	no	—
76	C₂×C₃×D₃	C₂×C₈	1	4	yes	complemented
77	C₃²×C₄	C₂²⋊C₄	1	2	yes	—
78	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₃×Dic₃	—	2	2	no	—
81	C₂³×C₄	—	3	1	no	—
82	C₂²×C₈	—	3	1	no	—
83	C₂²×C₈	C₃×D₃	1	1	yes	characteristic, complemented
84	C₂²⋊C₈	—	3	4	no	—
85	C₂³×C₃	—	2	1	no	—
86	C₂³×C₃	—	3	4	no	—
87	C₂³×C₃	—	3	2	no	—
88	C₂³×C₃	—	6	2	no	—
89	C₂³×C₃	—	6	2	no	—
90	C₂³×C₃	C₄×D₃	1	1	yes	characteristic
91	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂²×D₃	—	2	2	no	—
93	C₂²×D₃	—	2	2	no	—
94	C₂²×D₃	C₃×C₈	1	4	yes	complemented
95	C₂²×D₃	C₂×C₃×C₄	1	2	yes	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	3	2	no	—
101	C₂×C₃×C₄	—	6	2	no	—
102	C₂×C₃×C₄	—	6	4	no	—
103	C₂×C₃×C₄	—	6	2	no	—
104	C₂×C₃×C₄	—	6	2	no	—
105	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
106	C₂×C₃×C₄	D₁₂	1	2	yes	—
107	C₂×C₃×C₄	#67	1	2	yes	—
108	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₃×D₄	1	2	yes	—
110	C₂×C₃×C₄	C₃×D₄	1	2	yes	—
111	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂×Dic₃	—	2	2	no	—
114	C₂×Dic₃	C₂×C₃×C₄	1	2	yes	—
115	C₄×D₃	—	2	4	no	—
116	C₄×D₃	—	2	2	no	—
117	C₄×D₃	—	2	2	no	—
118	C₃×C₈	—	2	4	no	—
119	C₃×C₈	—	2	2	no	—
120	C₃×C₈	—	2	2	no	—
121	C₃×C₈	—	6	2	no	—
122	C₃⋊C₈	—	2	2	no	—
123	C₂×C₃²	C₂²⋊C₈	1	4	yes	complemented
124	C₂×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₃²	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₃²	C₂×M₁₆	1	1	yes	characteristic
127	C₃×D₃	—	2	4	no	—
128	C₂⁴	—	3	1	no	—
129	C₂²×C₄	—	3	1	no	—
130	C₂²×C₄	—	3	2	no	—
131	C₂²×C₄	—	3	2	no	—
132	C₂²×C₄	—	6	2	no	—
133	C₂²×C₄	—	6	2	no	—
134	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	3	2	no	—
137	C₂×C₈	—	6	2	no	—
138	C₂×C₈	C₂×C₃×D₃	1	2	yes	complemented
139	C₂²×C₃	—	2	2	no	—
140	C₂²×C₃	—	2	2	no	—
141	C₂²×C₃	—	2	2	no	—
142	C₂²×C₃	—	2	1	no	—
143	C₂²×C₃	—	3	4	no	—
144	C₂²×C₃	—	6	8	no	—
145	C₂²×C₃	—	6	2	no	—
146	C₂²×C₃	—	6	2	no	—
147	C₂²×C₃	D₃×C₈	1	2	yes	—
148	C₂²×C₃	ΓL(1,𝔽₂₅)	1	2	yes	—
149	C₂²×C₃	(C₂×D₃)⋊C₄	1	2	yes	—
150	C₂²×C₃	C₃×C₂²⋊C₄	1	2	yes	—
151	C₂²×C₃	C₂×C₃×C₈	1	2	yes	—
152	C₂²×C₃	C₃×M₁₆	1	2	yes	—
153	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
154	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂×D₃	—	2	8	no	—
156	C₂×D₃	—	2	2	no	—
157	C₂×D₃	—	2	2	no	—
158	C₂×D₃	C₂×C₃×C₈	1	4	yes	complemented
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	6	2	no	—
163	C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
164	C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
165	C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
166	C₃×C₄	C₂×#67	1	1	yes	characteristic
167	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
168	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
169	Dic₃	—	2	2	no	—
170	C₃²	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
171	C₂³	—	3	4	no	—
172	C₂³	—	3	2	no	—
173	C₂³	—	6	2	no	—
174	C₂³	—	6	2	no	—
175	C₂³	C₃×C₄×D₃	1	1	yes	characteristic
176	C₂×C₄	—	3	2	no	—
177	C₂×C₄	—	6	2	no	—
178	C₂×C₄	—	6	4	no	—
179	C₂×C₄	—	6	2	no	—
180	C₂×C₄	—	6	2	no	—
181	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
182	C₂×C₄	C₃×D₁₂	1	2	yes	—
183	C₂×C₄	C₃×#67	1	2	yes	—
184	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	6	2	no	—
187	C₂×C₃	—	2	1	no	—
188	C₂×C₃	—	2	1	no	—
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	4	no	—
191	C₂×C₃	—	6	4	no	—
192	C₂×C₃	(C₂×D₃)⋊C₈	1	4	yes	direct product factor
193	C₂×C₃	C₃×C₂²⋊C₈	1	4	yes	complemented
194	C₂×C₃	C₂×D₃×C₈	1	1	yes	characteristic
195	C₂×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
196	C₂×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₂×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
199	C₂×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
200	D₃	—	2	4	no	—
201	C₂²	—	3	4	no	—
202	C₂²	—	6	2	no	—
203	C₂²	—	6	2	no	—
204	C₂²	—	6	8	no	—
205	C₂²	C₃×D₃×C₈	1	2	yes	—
206	C₂²	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
207	C₂²	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
208	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
209	C₄	—	6	2	no	—
210	C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
211	C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
212	C₄	C₂×C₃×#67	1	1	yes	characteristic
213	C₃	—	2	1	no	—
214	C₃	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic, direct product factor
215	C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic, complemented
216	C₂	—	6	4	no	—
217	C₂	C₃×(C₂×D₃)⋊C₈	1	4	yes	direct product factor
218	C₂	C₂×C₃×D₃×C₈	1	1	yes	characteristic
219	C₂	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
220	C₂	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₃
commutator	C₂×C₃	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃²×C₈	C₂
socle	C₂³×C₃²	C₂×C₄
prime radical	C₂³×C₃×D₃	C₄
cosocle	C₂×C₃×C₄	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₃
hypocenter	C₃	C₂×C₃×C₂²⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₈	C₂	semidirect
3. C₂²×C₃×C₃⋊C₈	C₂	semidirect
4. C₃×(C₂×D₃)⋊C₈	C₂	direct (N × H)
6. C₂×(C₂×D₃)⋊C₈	C₃	direct (N × H)
15. C₂×C₃²×C₈	C₂²	semidirect
17. C₂×C₃×C₃⋊C₈	C₂²	semidirect
22. C₂²×C₃×C₈	D₃	semidirect
23. C₂²×C₃×C₈	C₂×C₃	semidirect
24. C₂²×C₃⋊C₈	C₂×C₃	semidirect
26. (C₂×D₃)⋊C₈	C₂×C₃	direct (N × H)
30. C₂²×C₃×D₃	C₈	semidirect
65. C₂×C₃×C₈	C₂×D₃	semidirect
66. C₂×C₃×C₈	C₂²×C₃	semidirect
68. C₂×C₃⋊C₈	C₂²×C₃	semidirect
76. C₂×C₃×D₃	C₂×C₈	semidirect
83. C₂²×C₈	C₃×D₃	semidirect
94. C₂²×D₃	C₃×C₈	semidirect
123. C₂×C₃²	C₂²⋊C₈	semidirect
138. C₂×C₈	C₂×C₃×D₃	semidirect
158. C₂×D₃	C₂×C₃×C₈	semidirect
170. C₃²	C₂×C₂²⋊C₈	semidirect
192. C₂×C₃	(C₂×D₃)⋊C₈	direct (N × H)
193. C₂×C₃	C₃×C₂²⋊C₈	semidirect
214. C₃	C₂×(C₂×D₃)⋊C₈	direct (N × H)
215. C₃	C₂×C₃×C₂²⋊C₈	semidirect
217. C₂	C₃×(C₂×D₃)⋊C₈	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×D₃	2
C₂²×C₃²×C₈	2
C₂²×C₃×C₃⋊C₈	2
C₃×(C₂×D₃)⋊C₈	2
C₂×C₃×C₂²⋊C₈	3
C₂×(C₂×D₃)⋊C₈	3

Subgroups — C2×C3×(C2×D3)⋊C8 #10592205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×(C₂×D₃)⋊C₈ #10592205