Subgroups — C₃×D₄×C₃⋊C₈ #10592107

Order 576.

315 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₃×#830	C₂	1	2	yes	maximal, complemented
4	C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₈×D₄	—	3	1	no	maximal
7	D₄×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×D₄	C₄	1	1	yes	characteristic
9	C₂²×C₃²×C₄	C₂²	1	2	yes	—
10	C₃²×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₃²×C₂²⋊C₄	C₄	1	2	yes	—
12	C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₃⋊C₈	—	2	4	no	—
14	C₂×C₃×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₈	—	3	2	no	—
18	C₃×C₄×D₄	—	2	1	no	—
19	C₃×C₄×D₄	D₃	1	1	yes	characteristic
20	C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₃⋊C₈	C₂×C₃	1	2	yes	complemented
22	C₃×C₄⋊C₈	—	3	1	no	—
23	C₃×C₂²⋊C₈	—	3	2	no	—
24	C₃×C₄×C₈	—	3	1	no	—
25	#830	C₂×C₃	1	2	yes	complemented
26	C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
27	C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×C₃²	C₂×C₄	1	2	yes	—
29	C₃²×D₄	C₈	1	4	yes	complemented
30	C₂×C₃²×C₄	—	2	2	no	—
31	C₂×C₃²×C₄	—	2	2	no	—
32	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
36	C₃×C₃⋊C₈	—	2	2	no	—
37	C₃×C₃⋊C₈	—	2	1	no	—
38	C₃×C₃⋊C₈	D₄	1	2	yes	direct product factor
39	C₈×D₄	—	3	1	no	Sylow
40	C₂×C₃×D₄	—	2	1	no	—
41	C₂×C₃×D₄	Dic₃	1	1	yes	characteristic
42	C₂×C₃×D₄	C₃×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
45	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
46	C₂×C₃×C₈	—	3	1	no	—
47	C₂×C₃×C₈	—	3	1	no	—
48	C₂×C₃×C₈	—	3	2	no	—
49	C₂×C₃×C₈	—	6	4	no	—
50	C₃×C₄⋊C₄	—	2	1	no	—
51	C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
52	C₃×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
53	C₃×C₂²⋊C₄	—	2	2	no	—
54	C₃×C₂²⋊C₄	Dic₃	1	2	yes	—
55	C₃×C₂²⋊C₄	C₃×C₄	1	2	yes	—
56	C₃×C₄²	—	2	1	no	—
57	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃⋊C₈	—	2	4	no	—
60	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃⋊C₈	C₂²×C₃	1	2	yes	—
62	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃²	—	2	2	no	—
64	C₂²×C₃²	—	2	2	no	—
65	C₂²×C₃²	C₂×C₈	1	4	yes	complemented
66	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
67	C₃²×C₄	—	2	2	no	—
68	C₃²×C₄	—	2	1	no	—
69	C₃²×C₄	C₂×C₈	1	2	yes	complemented
70	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₃²×C₄	#41	1	1	yes	characteristic
72	C₂²×C₈	—	3	2	no	—
73	C₄×D₄	C₃×D₃	1	1	yes	characteristic
74	C₄⋊C₈	—	3	1	no	—
75	C₂²⋊C₈	—	3	2	no	—
76	C₄×C₈	—	3	1	no	—
77	C₂³×C₃	—	2	2	no	—
78	C₂³×C₃	C₂×Dic₃	1	2	yes	—
79	C₂³×C₃	C₂×C₃×C₄	1	2	yes	—
80	C₃×D₄	—	2	4	no	—
81	C₃×D₄	C₃⋊C₈	1	4	yes	direct product factor
82	C₃×D₄	C₃×C₈	1	4	yes	complemented
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	1	no	—
88	C₂×C₃×C₄	—	2	1	no	—
89	C₂×C₃×C₄	—	2	2	no	—
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	4	2	no	—
92	C₂×C₃×C₄	—	4	2	no	—
93	C₂×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂×Dic₃	1	2	yes	—
95	C₂×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
98	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₃×C₈	—	3	2	no	—
102	C₃×C₈	—	6	2	no	—
103	C₃×C₈	—	6	1	no	—
104	C₃⋊C₈	—	2	2	no	—
105	C₃⋊C₈	—	2	1	no	—
106	C₃⋊C₈	C₃×D₄	1	2	yes	direct product factor
107	C₂×C₃²	—	2	4	no	—
108	C₂×C₃²	C₄×D₄	1	1	yes	characteristic
109	C₂×C₃²	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₃²	#131	1	1	yes	characteristic
111	C₂×D₄	C₃×Dic₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₃×D₃	1	2	yes	—
113	C₂×C₈	—	3	1	no	—
114	C₂×C₈	—	3	1	no	—
115	C₂×C₈	—	3	2	no	—
116	C₂×C₈	—	6	4	no	—
117	C₄⋊C₄	C₃×Dic₃	1	1	yes	characteristic
118	C₂²⋊C₄	C₃×Dic₃	1	2	yes	—
119	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	—	2	2	no	—
123	C₂²×C₃	—	2	2	no	—
124	C₂²×C₃	—	2	4	no	—
125	C₂²×C₃	—	2	1	no	—
126	C₂²×C₃	—	4	2	no	—
127	C₂²×C₃	—	4	2	no	—
128	C₂²×C₃	C₂×C₃⋊C₈	1	4	yes	complemented
129	C₂²×C₃	C₂×C₃×C₈	1	4	yes	complemented
130	C₂²×C₃	C₂²×Dic₃	1	1	yes	characteristic
131	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	—	2	1	no	—
134	C₃×C₄	—	2	2	no	—
135	C₃×C₄	—	2	1	no	—
136	C₃×C₄	—	2	2	no	—
137	C₃×C₄	—	2	1	no	—
138	C₃×C₄	—	2	1	no	—
139	C₃×C₄	—	4	2	no	—
140	C₃×C₄	—	4	1	no	—
141	C₃×C₄	C₂×C₃⋊C₈	1	2	yes	complemented
142	C₃×C₄	C₂×C₃×C₈	1	2	yes	complemented
143	C₃×C₄	D₃×D₄	1	1	yes	characteristic
144	C₃×C₄	#237	1	1	yes	characteristic
145	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
146	C₃×C₄	C₃×#41	1	1	yes	characteristic
147	C₃²	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
148	C₂³	C₂×C₃×Dic₃	1	2	yes	—
149	D₄	C₃×C₃⋊C₈	1	4	yes	direct product factor
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₃×Dic₃	1	2	yes	—
154	C₂×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
155	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
156	C₈	—	3	2	no	—
157	C₈	—	6	2	no	—
158	C₈	—	6	1	no	—
159	C₂×C₃	—	2	4	no	—
160	C₂×C₃	—	2	4	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	—	2	1	no	—
164	C₂×C₃	—	4	4	no	—
165	C₂×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
166	C₂×C₃	D₄×Dic₃	1	1	yes	characteristic
167	C₂×C₃	#948	1	1	yes	characteristic
168	C₂×C₃	C₃×C₄×D₄	1	1	yes	characteristic
169	C₂×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
170	C₂×C₃	C₃×#131	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	C₂×C₃×C₃⋊C₈	1	4	yes	complemented
174	C₂²	C₂²×C₃×Dic₃	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	1	no	—
177	C₄	C₂×C₃×C₃⋊C₈	1	2	yes	complemented
178	C₄	C₃×D₃×D₄	1	1	yes	characteristic
179	C₄	C₃×#237	1	1	yes	characteristic
180	C₃	—	2	1	no	—
181	C₃	D₄×C₃⋊C₈	1	1	yes	characteristic, direct product factor
182	C₃	C₃×C₈×D₄	1	1	yes	characteristic, complemented
183	C₂	—	2	4	no	—
184	C₂	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
185	C₂	C₃×D₄×Dic₃	1	1	yes	characteristic
186	C₂	C₃×#948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₄×D₄	C₂
socle	C₂²×C₃²	C₂²×C₄
prime radical	C₂×C₃²×D₄	C₄
cosocle	C₂×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₄×D₄	D₃
hypocenter	C₃	C₃×C₈×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₃⋊C₈	C₂	semidirect
3. C₃×#830	C₂	semidirect
4. C₃×C₄⋊(C₃⋊C₈)	C₂	semidirect
5. C₃×C₄×C₃⋊C₈	C₂	semidirect
7. D₄×C₃⋊C₈	C₃	direct (N × H)
21. C₂²×C₃⋊C₈	C₂×C₃	semidirect
25. #830	C₂×C₃	semidirect
26. C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
27. C₄×C₃⋊C₈	C₂×C₃	semidirect
29. C₃²×D₄	C₈	semidirect
38. C₃×C₃⋊C₈	D₄	direct (N × H)
65. C₂²×C₃²	C₂×C₈	semidirect
69. C₃²×C₄	C₂×C₈	semidirect
81. C₃×D₄	C₃⋊C₈	direct (N × H)
82. C₃×D₄	C₃×C₈	semidirect
106. C₃⋊C₈	C₃×D₄	direct (N × H)
128. C₂²×C₃	C₂×C₃⋊C₈	semidirect
129. C₂²×C₃	C₂×C₃×C₈	semidirect
141. C₃×C₄	C₂×C₃⋊C₈	semidirect
142. C₃×C₄	C₂×C₃×C₈	semidirect
147. C₃²	C₈×D₄	semidirect
149. D₄	C₃×C₃⋊C₈	direct (N × H)
173. C₂²	C₂×C₃×C₃⋊C₈	semidirect
177. C₄	C₂×C₃×C₃⋊C₈	semidirect
181. C₃	D₄×C₃⋊C₈	direct (N × H)
182. C₃	C₃×C₈×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₄	2
C₂²×C₃×C₃⋊C₈	2
C₃×#830	2
C₃×C₄⋊(C₃⋊C₈)	2
C₃×C₄×C₃⋊C₈	2
C₃×C₈×D₄	3
D₄×C₃⋊C₈	3

Subgroups — C3×D4×C3⋊C8 #10592107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×C₃⋊C₈ #10592107