Subgroups — C₂×#11722200 #13163846

Order 1728.

2828 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	2	yes	maximal, complemented
3	#11722200	C₂	1	4	yes	maximal, direct product factor
4	C₂×#63657	—	3	13	no	maximal
5	C₂⁴×C₃³	C₄	1	1	yes	characteristic
6	C₂²×C₃³×C₄	—	2	4	no	—
7	C₂²×C₃³×C₄	C₄	1	1	yes	characteristic
8	C₂²×C₃³×C₄	C₄	1	2	yes	—
9	C₂²×C₃³×C₄	C₂²	1	2	yes	—
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	4	no	—
12	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	4	yes	complemented
13	C₂³×C₃²×C₄	D₃	1	13	yes	—
14	C₂²×C₃⋊(C₃⋊C₈)	—	3	26	no	—
15	#63657	—	3	52	no	—
16	C₂³×C₃³	—	2	2	no	—
17	C₂³×C₃³	—	2	2	no	—
18	C₂³×C₃³	C₈	1	4	yes	complemented
19	C₂³×C₃³	C₂×C₄	1	2	yes	—
20	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃³×C₄	—	2	2	no	—
22	C₂×C₃³×C₄	—	2	4	no	—
23	C₂×C₃³×C₄	—	2	4	no	—
24	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	D₄	1	4	yes	—
27	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
28	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	4	no	—
29	C₂×#830	—	9	13	no	—
30	C₂⁴×C₃²	Dic₃	1	13	yes	—
31	C₂²×C₃²×C₄	—	2	52	no	—
32	C₂²×C₃²×C₄	Dic₃	1	13	yes	—
33	C₂²×C₃²×C₄	Dic₃	1	26	yes	—
34	C₂²×C₃²×C₄	C₂×D₃	1	26	yes	—
35	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
36	C₂×C₃⋊(C₃⋊C₈)	—	3	52	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	6	52	no	—
38	C₂²×C₃³	—	2	8	no	—
39	C₂²×C₃³	—	2	2	no	—
40	C₂²×C₃³	—	2	2	no	—
41	C₂²×C₃³	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₃³	C₂×C₈	1	2	yes	—
43	C₂²×C₃³	C₂×C₈	1	4	yes	complemented
44	C₂²×C₃³	M₁₆	1	2	yes	—
45	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
46	C₃³×C₄	—	2	2	no	—
47	C₃³×C₄	C₂²⋊C₄	1	2	yes	—
48	C₃³×C₄	C₂×D₄	1	2	yes	—
49	C₂³×C₃×C₄	C₃⋊D₃	1	13	yes	—
50	C₂²×C₃⋊C₈	—	9	26	no	—
51	#830	—	9	52	no	—
52	C₂³×C₃²	—	2	26	no	—
53	C₂³×C₃²	—	2	26	no	—
54	C₂³×C₃²	C₃⋊C₈	1	52	yes	complemented
55	C₂³×C₃²	C₂×Dic₃	1	26	yes	—
56	C₂³×C₃²	C₂×Dic₃	1	13	yes	—
57	C₂×C₃²×C₄	—	2	26	no	—
58	C₂×C₃²×C₄	—	2	52	no	—
59	C₂×C₃²×C₄	—	2	52	no	—
60	C₂×C₃²×C₄	C₂×Dic₃	1	26	yes	—
61	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
62	C₂×C₃²×C₄	#67	1	52	yes	—
63	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
64	C₃⋊(C₃⋊C₈)	—	6	52	no	—
65	C₂×C₂²⋊C₈	—	27	1	no	Sylow
66	C₂×C₃³	—	2	4	no	—
67	C₂×C₃³	C₂²⋊C₈	1	4	yes	complemented
68	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
71	C₂⁴×C₃	C₃⋊Dic₃	1	13	yes	—
72	C₂²×C₃×C₄	—	2	52	no	—
73	C₂²×C₃×C₄	C₃⋊Dic₃	1	13	yes	—
74	C₂²×C₃×C₄	C₃⋊Dic₃	1	26	yes	—
75	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
76	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
77	C₂×C₃⋊C₈	—	9	52	no	—
78	C₂×C₃⋊C₈	—	18	52	no	—
79	C₂²×C₃²	—	2	104	no	—
80	C₂²×C₃²	—	2	26	no	—
81	C₂²×C₃²	—	2	26	no	—
82	C₂²×C₃²	C₂×C₃⋊C₈	1	26	yes	—
83	C₂²×C₃²	C₂×C₃⋊C₈	1	52	yes	complemented
84	C₂²×C₃²	#208	1	26	yes	—
85	C₂²×C₃²	#217	1	26	yes	—
86	C₂²×C₃²	C₂²×Dic₃	1	13	yes	—
87	C₃²×C₄	—	2	26	no	—
88	C₃²×C₄	#217	1	26	yes	—
89	C₃²×C₄	C₂×#67	1	26	yes	—
90	C₂³×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
91	C₂²×C₈	—	27	2	no	—
92	C₂²⋊C₈	—	27	4	no	—
93	C₃³	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
94	C₂³×C₃	—	2	26	no	—
95	C₂³×C₃	—	2	26	no	—
96	C₂³×C₃	C₃⋊(C₃⋊C₈)	1	52	yes	complemented
97	C₂³×C₃	C₂×C₃⋊Dic₃	1	26	yes	—
98	C₂³×C₃	C₂×C₃⋊Dic₃	1	13	yes	—
99	C₂×C₃×C₄	—	2	26	no	—
100	C₂×C₃×C₄	—	2	52	no	—
101	C₂×C₃×C₄	—	2	52	no	—
102	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	26	yes	—
103	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
104	C₂×C₃×C₄	#638	1	52	yes	—
105	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
106	C₃⋊C₈	—	18	52	no	—
107	C₂×C₃²	—	2	52	no	—
108	C₂×C₃²	#830	1	52	yes	complemented
109	C₂×C₃²	C₂²×C₃⋊C₈	1	13	yes	—
110	C₂×C₃²	C₂×#208	1	13	yes	—
111	C₂×C₃²	C₂×#217	1	13	yes	—
112	C₂⁴	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
115	C₂²×C₄	C₃⋊(C₃⋊Dic₃)	1	2	yes	—
116	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
117	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
118	C₂×C₈	—	27	4	no	—
119	C₂×C₈	—	54	4	no	—
120	C₂²×C₃	—	2	26	no	—
121	C₂²×C₃	—	2	26	no	—
122	C₂²×C₃	—	2	104	no	—
123	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	26	yes	—
124	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	52	yes	complemented
125	C₂²×C₃	#3748	1	26	yes	—
126	C₂²×C₃	#3757	1	26	yes	—
127	C₂²×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
128	C₃×C₄	—	2	26	no	—
129	C₃×C₄	#3757	1	26	yes	—
130	C₃×C₄	C₂×#638	1	26	yes	—
131	C₃²	C₂×#830	1	13	yes	complemented
132	C₂³	—	2	2	no	—
133	C₂³	—	2	2	no	—
134	C₂³	C₃⋊(C₃⋊(C₃⋊C₈))	1	4	yes	complemented
135	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
136	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
141	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
142	C₂×C₄	#6312	1	4	yes	—
143	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
144	C₈	—	54	4	no	—
145	C₂×C₃	—	2	52	no	—
146	C₂×C₃	#63657	1	52	yes	complemented
147	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	13	yes	—
148	C₂×C₃	C₂×#3748	1	13	yes	—
149	C₂×C₃	C₂×#3757	1	13	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	8	no	—
152	C₂²	—	2	2	no	—
153	C₂²	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	1	2	yes	—
154	C₂²	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	1	4	yes	complemented
155	C₂²	#88855	1	2	yes	—
156	C₂²	#88864	1	2	yes	—
157	C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	#88864	1	2	yes	—
160	C₄	C₂×#6312	1	2	yes	—
161	C₃	C₂×#63657	1	13	yes	complemented
162	C₂	—	2	4	no	—
163	C₂	#11722200	1	4	yes	direct product factor
164	C₂	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
165	C₂	C₂×#88855	1	1	yes	characteristic
166	C₂	C₂×#88864	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂²×C₈
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂³×C₃³×C₄	C₂
socle	C₂³×C₃³	C₂×C₄
prime radical	C₂⁴×C₃³	C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂³×C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
3. #11722200	C₂	direct (N × H)
12. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	semidirect
18. C₂³×C₃³	C₈	semidirect
43. C₂²×C₃³	C₂×C₈	semidirect
54. C₂³×C₃²	C₃⋊C₈	semidirect
67. C₂×C₃³	C₂²⋊C₈	semidirect
83. C₂²×C₃²	C₂×C₃⋊C₈	semidirect
93. C₃³	C₂×C₂²⋊C₈	semidirect
96. C₂³×C₃	C₃⋊(C₃⋊C₈)	semidirect
108. C₂×C₃²	#830	semidirect
124. C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	semidirect
131. C₃²	C₂×#830	semidirect
134. C₂³	C₃⋊(C₃⋊(C₃⋊C₈))	semidirect
146. C₂×C₃	#63657	semidirect
154. C₂²	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	semidirect
161. C₃	C₂×#63657	semidirect
163. C₂	#11722200	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×C₃³×C₄	2
C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	2
#11722200	2
C₂×#63657	3

Subgroups — C2×#11722200 #13163846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11722200 #13163846