Subgroups — C₃×#69394 #11786179

Order 1152.

729 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	6	yes	maximal
3	C₃×#1842	—	3	1	no	maximal
4	#69394	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃²×C₄²	C₄	1	3	yes	—
6	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
7	C₂×C₃²×C₄²	C₂²	1	3	yes	—
8	C₂²×C₃×C₃⋊C₈	C₂²	1	4	yes	—
9	C₃×C₄⋊(C₃⋊C₈)	—	2	12	no	—
10	C₂×C₃×C₄⋊C₈	—	3	6	no	—
11	C₃×C₄³	—	2	1	no	—
12	C₃×C₄³	D₃	1	1	yes	characteristic
13	C₃×C₄³	C₂×C₃	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊C₈)	C₂×C₃	1	6	yes	—
15	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
16	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
17	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
18	C₃²×C₄²	—	2	6	no	—
19	C₃²×C₄²	C₈	1	4	yes	complemented
20	C₃²×C₄²	D₄	1	6	yes	—
21	C₃²×C₄²	Q₈	1	6	yes	—
22	C₂×C₃×C₃⋊C₈	—	2	12	no	—
23	#1842	—	3	1	no	Sylow
24	C₂²×C₃×C₈	—	3	4	no	—
25	C₂×C₃×C₄²	—	2	3	no	—
26	C₂×C₃×C₄²	—	2	1	no	—
27	C₂×C₃×C₄²	—	2	3	no	—
28	C₂×C₃×C₄²	Dic₃	1	3	yes	—
29	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₃×C₄	1	3	yes	—
31	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
33	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
34	C₂²×C₃⋊C₈	C₂²×C₃	1	4	yes	—
35	C₃×C₄⋊C₈	—	6	12	no	—
36	C₄⋊(C₃⋊C₈)	—	2	12	no	—
37	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₃²×C₄	—	2	6	no	—
39	C₂×C₃²×C₄	C₄⋊C₄	1	6	yes	—
40	C₂×C₃²×C₄	C₄⋊C₄	1	6	yes	—
41	C₂×C₃²×C₄	C₂×C₈	1	6	yes	—
42	C₂×C₃²×C₄	M₁₆	1	6	yes	—
43	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
44	C₂×C₃²×C₄	C₂×Q₈	1	3	yes	—
45	C₃×C₃⋊C₈	—	4	4	no	—
46	C₂×C₄⋊C₈	—	3	6	no	—
47	C₄³	C₃×D₃	1	1	yes	characteristic
48	C₂²×C₃×C₄	—	2	3	no	—
49	C₂²×C₃×C₄	—	2	3	no	—
50	C₂²×C₃×C₄	—	2	1	no	—
51	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
52	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
53	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
54	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
55	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
57	C₂×C₃×C₈	—	6	12	no	—
58	C₃×C₄²	—	2	12	no	—
59	C₃×C₄²	—	2	6	no	—
60	C₃×C₄²	—	2	6	no	—
61	C₃×C₄²	—	2	4	no	—
62	C₃×C₄²	—	2	6	no	—
63	C₃×C₄²	—	2	6	no	—
64	C₃×C₄²	C₃⋊C₈	1	4	yes	complemented
65	C₃×C₄²	C₃×C₈	1	4	yes	complemented
66	C₃×C₄²	Dic₆	1	6	yes	—
67	C₃×C₄²	D₁₂	1	6	yes	—
68	C₃×C₄²	C₃×D₄	1	6	yes	—
69	C₃×C₄²	C₃×Q₈	1	6	yes	—
70	C₂×C₃⋊C₈	—	2	12	no	—
71	C₂²×C₃²	C₂×C₄⋊C₄	1	3	yes	—
72	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
73	C₂²×C₃²	C₂×M₁₆	1	3	yes	—
74	C₃²×C₄	—	2	6	no	—
75	C₃²×C₄	C₄⋊C₈	1	12	yes	complemented
76	C₃²×C₄	#127	1	1	yes	characteristic
77	C₃²×C₄	C₄⋊Q₈	1	3	yes	—
78	C₂²×C₈	—	3	4	no	—
79	C₂×C₄²	C₃×Dic₃	1	3	yes	—
80	C₂×C₄²	C₃×Dic₃	1	1	yes	characteristic
81	C₂×C₄²	C₂×C₃×D₃	1	3	yes	—
82	C₄⋊C₈	—	6	12	no	—
83	C₂³×C₃	—	2	1	no	—
84	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
85	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄	—	2	12	no	—
87	C₂×C₃×C₄	—	2	6	no	—
88	C₂×C₃×C₄	—	2	6	no	—
89	C₂×C₃×C₄	—	2	6	no	—
90	C₂×C₃×C₄	—	2	6	no	—
91	C₂×C₃×C₄	—	2	6	no	—
92	C₂×C₃×C₄	—	2	6	no	—
93	C₂×C₃×C₄	—	2	3	no	—
94	C₂×C₃×C₄	—	2	3	no	—
95	C₂×C₃×C₄	C₂×C₃⋊C₈	1	6	yes	—
96	C₂×C₃×C₄	#208	1	6	yes	—
97	C₂×C₃×C₄	C₄⋊Dic₃	1	6	yes	—
98	C₂×C₃×C₄	C₄⋊Dic₃	1	6	yes	—
99	C₂×C₃×C₄	C₃×C₄⋊C₄	1	6	yes	—
100	C₂×C₃×C₄	C₃×C₄⋊C₄	1	6	yes	—
101	C₂×C₃×C₄	C₂×C₃×C₈	1	6	yes	—
102	C₂×C₃×C₄	C₃×M₁₆	1	6	yes	—
103	C₂×C₃×C₄	C₂×Dic₆	1	3	yes	—
104	C₂×C₃×C₄	C₂×D₁₂	1	3	yes	—
105	C₂×C₃×C₄	C₂×C₃×D₄	1	3	yes	—
106	C₂×C₃×C₄	C₂×C₃×Q₈	1	3	yes	—
107	C₃×C₈	—	12	4	no	—
108	C₃⋊C₈	—	4	4	no	—
109	C₂×C₃²	#384	1	1	yes	characteristic
110	C₂×C₃²	C₂×C₄⋊C₈	1	3	yes	—
111	C₂×C₃²	#423	1	3	yes	—
112	C₂²×C₄	C₂×C₃×Dic₃	1	3	yes	—
113	C₂²×C₄	C₂×C₃×Dic₃	1	3	yes	—
114	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
115	C₂×C₈	—	6	12	no	—
116	C₄²	—	2	6	no	—
117	C₄²	C₃×C₃⋊C₈	1	4	yes	complemented
118	C₄²	C₃×Dic₆	1	6	yes	—
119	C₄²	C₃×D₁₂	1	6	yes	—
120	C₂²×C₃	—	2	1	no	—
121	C₂²×C₃	—	2	3	no	—
122	C₂²×C₃	—	2	3	no	—
123	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
124	C₂²×C₃	C₂×#208	1	3	yes	—
125	C₂²×C₃	C₂×C₄⋊Dic₃	1	3	yes	—
126	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	3	yes	—
127	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
128	C₂²×C₃	C₂×C₃×M₁₆	1	3	yes	—
129	C₃×C₄	—	2	12	no	—
130	C₃×C₄	—	2	6	no	—
131	C₃×C₄	—	2	6	no	—
132	C₃×C₄	—	2	12	no	—
133	C₃×C₄	—	2	3	no	—
134	C₃×C₄	—	2	1	no	—
135	C₃×C₄	C₄⋊(C₃⋊C₈)	1	12	yes	complemented
136	C₃×C₄	C₃×C₄⋊C₈	1	12	yes	complemented
137	C₃×C₄	#869	1	3	yes	—
138	C₃×C₄	#874	1	1	yes	characteristic
139	C₃×C₄	C₃×#127	1	1	yes	characteristic
140	C₃×C₄	C₃×C₄⋊Q₈	1	3	yes	—
141	C₃²	#1842	1	1	yes	characteristic, complemented, Sylow
142	C₂³	C₂²×C₃×Dic₃	1	1	yes	characteristic
143	C₂×C₄	—	2	6	no	—
144	C₂×C₄	C₂×C₃×C₃⋊C₈	1	6	yes	—
145	C₂×C₄	C₃×#208	1	6	yes	—
146	C₂×C₄	C₃×C₄⋊Dic₃	1	6	yes	—
147	C₂×C₄	C₃×C₄⋊Dic₃	1	6	yes	—
148	C₂×C₄	C₂×C₃×Dic₆	1	3	yes	—
149	C₂×C₄	C₂×C₃×D₁₂	1	3	yes	—
150	C₈	—	12	4	no	—
151	C₂×C₃	—	2	3	no	—
152	C₂×C₃	—	2	3	no	—
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	3	yes	—
155	C₂×C₃	#4924	1	3	yes	—
156	C₂×C₃	#4935	1	1	yes	characteristic
157	C₂×C₃	C₃×#384	1	1	yes	characteristic
158	C₂×C₃	C₂×C₃×C₄⋊C₈	1	3	yes	—
159	C₂×C₃	C₃×#423	1	3	yes	—
160	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
161	C₂²	C₂×C₃×#208	1	3	yes	—
162	C₂²	C₂×C₃×C₄⋊Dic₃	1	3	yes	—
163	C₄	—	2	6	no	—
164	C₄	C₃×C₄⋊(C₃⋊C₈)	1	12	yes	complemented
165	C₄	C₃×#869	1	3	yes	—
166	C₄	C₃×#874	1	1	yes	characteristic
167	C₃	—	2	1	no	—
168	C₃	#69394	1	1	yes	characteristic, direct product factor
169	C₃	C₃×#1842	1	1	yes	characteristic, complemented
170	C₂	C₂×C₃×C₄⋊(C₃⋊C₈)	1	3	yes	—
171	C₂	C₃×#4924	1	3	yes	—
172	C₂	C₃×#4935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₄³	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₄³	D₃
hypocenter	C₃	C₃×#1842
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×C₄³

Complements

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

N	H ≅ G/N	type
4. #69394	C₃	direct (N × H)
19. C₃²×C₄²	C₈	semidirect
64. C₃×C₄²	C₃⋊C₈	semidirect
65. C₃×C₄²	C₃×C₈	semidirect
75. C₃²×C₄	C₄⋊C₈	semidirect
117. C₄²	C₃×C₃⋊C₈	semidirect
135. C₃×C₄	C₄⋊(C₃⋊C₈)	semidirect
136. C₃×C₄	C₃×C₄⋊C₈	semidirect
141. C₃²	#1842	semidirect
164. C₄	C₃×C₄⋊(C₃⋊C₈)	semidirect
168. C₃	#69394	direct (N × H)
169. C₃	C₃×#1842	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄³	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₃×#1842	3
#69394	3

Subgroups — C3×#69394 #11786179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69394 #11786179