Subgroups — C₂×#11722201 #13163848

Order 1728.

4620 conjugacy classes of subgroups.

The subgroup lattice has 176 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	C₂³×C₃⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
3	#11722201	C₂	1	4	yes	maximal, direct product factor
4	C₂×#63658	—	3	13	no	maximal
5	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
6	C₂²×C₃³×C₄	—	2	4	no	—
7	C₂²×C₃³×C₄	C₄	1	4	yes	complemented
8	C₂²×C₃³×C₄	C₂²	1	2	yes	—
9	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
10	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
11	C₂²×C₃⋊(C₃⋊Dic₃)	C₄	1	8	yes	complemented
12	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
13	C₂³×C₃²×C₄	D₃	1	13	yes	—
14	C₂³×C₃⋊Dic₃	—	3	26	no	—
15	#63658	—	3	52	no	—
16	C₂³×C₃³	C₂×C₄	1	4	yes	—
17	C₂³×C₃³	C₂×C₄	1	2	yes	—
18	C₂³×C₃³	D₄	1	2	yes	—
19	C₂³×C₃³	D₄	1	4	yes	—
20	C₂³×C₃³	Q₈	1	2	yes	—
21	C₂³×C₃³	C₂³	1	1	yes	characteristic
22	C₂×C₃³×C₄	—	2	8	no	—
23	C₂×C₃³×C₄	—	2	4	no	—
24	C₂×C₃³×C₄	C₂×C₄	1	4	yes	complemented
25	C₂×C₃⋊(C₃⋊Dic₃)	—	2	16	no	—
26	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
27	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
28	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	8	yes	complemented
29	C₂×C₃⋊#95	—	9	13	no	—
30	C₂⁴×C₃²	C₂×D₃	1	13	yes	—
31	C₂²×C₃²×C₄	—	2	52	no	—
32	C₂²×C₃²×C₄	Dic₃	1	52	yes	complemented
33	C₂²×C₃²×C₄	C₂×D₃	1	26	yes	—
34	C₂²×C₃⋊Dic₃	—	3	104	no	—
35	C₂²×C₃⋊Dic₃	—	3	52	no	—
36	C₂²×C₃⋊Dic₃	—	6	52	no	—
37	C₂²×C₃⋊Dic₃	—	6	52	no	—
38	C₂²×C₃³	C₄²	1	4	yes	—
39	C₂²×C₃³	C₂²⋊C₄	1	8	yes	—
40	C₂²×C₃³	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₃³	C₄⋊C₄	1	8	yes	—
42	C₂²×C₃³	C₄⋊C₄	1	4	yes	—
43	C₂²×C₃³	C₂²×C₄	1	2	yes	—
44	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₃³	C₂×D₄	1	2	yes	—
47	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
48	C₃³×C₄	—	2	4	no	—
49	C₃⋊(C₃⋊Dic₃)	—	2	8	no	—
50	C₂³×C₃×C₄	C₃⋊D₃	1	13	yes	—
51	C₂³×Dic₃	—	9	26	no	—
52	C₃⋊#95	—	9	52	no	—
53	C₂³×C₃²	Dic₆	1	26	yes	—
54	C₂³×C₃²	C₄×D₃	1	52	yes	—
55	C₂³×C₃²	D₁₂	1	26	yes	—
56	C₂³×C₃²	C₂×Dic₃	1	26	yes	—
57	C₂³×C₃²	#67	1	52	yes	—
58	C₂³×C₃²	C₂²×D₃	1	13	yes	—
59	C₂×C₃²×C₄	—	2	104	no	—
60	C₂×C₃²×C₄	—	2	52	no	—
61	C₂×C₃²×C₄	C₂×Dic₃	1	52	yes	complemented
62	C₂×C₃⋊Dic₃	—	3	104	no	—
63	C₂×C₃⋊Dic₃	—	6	208	no	—
64	C₂×C₃⋊Dic₃	—	6	52	no	—
65	C₂×C₃⋊Dic₃	—	6	52	no	—
66	C₂×#95	—	27	1	no	Sylow
67	C₂×C₃³	#95	1	8	yes	complemented
68	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₃³	C₂×C₂²⋊C₄	1	2	yes	—
70	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃³	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂⁴×C₃	C₂×C₃⋊D₃	1	13	yes	—
74	C₂²×C₃×C₄	—	2	52	no	—
75	C₂²×C₃×C₄	C₃⋊Dic₃	1	52	yes	complemented
76	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
77	C₂²×Dic₃	—	9	104	no	—
78	C₂²×Dic₃	—	9	52	no	—
79	C₂²×Dic₃	—	18	52	no	—
80	C₂²×Dic₃	—	18	52	no	—
81	C₂²×C₃²	C₄×Dic₃	1	52	yes	—
82	C₂²×C₃²	Dic₃⋊C₄	1	104	yes	—
83	C₂²×C₃²	C₄⋊Dic₃	1	52	yes	—
84	C₂²×C₃²	(C₂×D₃)⋊C₄	1	104	yes	—
85	C₂²×C₃²	#217	1	52	yes	—
86	C₂²×C₃²	C₂×Dic₆	1	13	yes	—
87	C₂²×C₃²	C₂×C₄×D₃	1	26	yes	—
88	C₂²×C₃²	C₂×D₁₂	1	13	yes	—
89	C₂²×C₃²	C₂²×Dic₃	1	13	yes	—
90	C₂²×C₃²	C₂×#67	1	26	yes	—
91	C₃²×C₄	—	2	52	no	—
92	C₃⋊Dic₃	—	6	104	no	—
93	C₂³×C₄	—	27	2	no	—
94	C₂³×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
95	#95	—	27	4	no	—
96	C₃³	C₂×#95	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₃	#634	1	26	yes	—
98	C₂³×C₃	C₄×C₃⋊D₃	1	52	yes	—
99	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	26	yes	—
100	C₂³×C₃	C₂×C₃⋊Dic₃	1	26	yes	—
101	C₂³×C₃	#638	1	52	yes	—
102	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
103	C₂×C₃×C₄	—	2	104	no	—
104	C₂×C₃×C₄	—	2	52	no	—
105	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	52	yes	complemented
106	C₂×Dic₃	—	9	104	no	—
107	C₂×Dic₃	—	18	208	no	—
108	C₂×Dic₃	—	18	52	no	—
109	C₂×Dic₃	—	18	52	no	—
110	C₂×C₃²	C₃⋊#95	1	104	yes	complemented
111	C₂×C₃²	C₂×C₄×Dic₃	1	13	yes	—
112	C₂×C₃²	C₂×Dic₃⋊C₄	1	26	yes	—
113	C₂×C₃²	C₂×C₄⋊Dic₃	1	13	yes	—
114	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	26	yes	—
115	C₂×C₃²	C₂×#217	1	13	yes	—
116	C₂⁴	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	27	8	no	—
119	C₂²×C₄	—	27	4	no	—
120	C₂²×C₄	—	54	4	no	—
121	C₂²×C₄	—	54	4	no	—
122	C₂²×C₄	C₃⋊(C₃⋊Dic₃)	1	4	yes	complemented
123	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
124	C₂²×C₃	C₄×C₃⋊Dic₃	1	52	yes	—
125	C₂²×C₃	#3750	1	104	yes	—
126	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	52	yes	—
127	C₂²×C₃	#3752	1	104	yes	—
128	C₂²×C₃	#3757	1	52	yes	—
129	C₂²×C₃	C₂×#634	1	13	yes	—
130	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	26	yes	—
131	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	13	yes	—
132	C₂²×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
133	C₂²×C₃	C₂×#638	1	26	yes	—
134	C₃×C₄	—	2	52	no	—
135	Dic₃	—	18	104	no	—
136	C₃²	C₂×C₃⋊#95	1	13	yes	complemented
137	C₂³	#6308	1	2	yes	—
138	C₂³	C₄×C₃⋊(C₃⋊D₃)	1	4	yes	—
139	C₂³	C₄⋊(C₃⋊(C₃⋊D₃))	1	2	yes	—
140	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
141	C₂³	#6312	1	4	yes	—
142	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	27	8	no	—
146	C₂×C₄	—	54	16	no	—
147	C₂×C₄	—	54	4	no	—
148	C₂×C₄	—	54	4	no	—
149	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	4	yes	complemented
150	C₂×C₃	#63658	1	104	yes	complemented
151	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	13	yes	—
152	C₂×C₃	C₂×#3750	1	26	yes	—
153	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	13	yes	—
154	C₂×C₃	C₂×#3752	1	26	yes	—
155	C₂×C₃	C₂×#3757	1	13	yes	—
156	C₂²	C₄×C₃⋊(C₃⋊Dic₃)	1	4	yes	—
157	C₂²	#88857	1	8	yes	—
158	C₂²	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	4	yes	—
159	C₂²	#88859	1	8	yes	—
160	C₂²	#88864	1	4	yes	—
161	C₂²	C₂×#6308	1	1	yes	characteristic
162	C₂²	C₂×C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
163	C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
164	C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
165	C₂²	C₂×#6312	1	2	yes	—
166	C₄	—	2	4	no	—
167	C₄	—	54	8	no	—
168	C₃	C₂×#63658	1	13	yes	complemented
169	C₂	#11722201	1	8	yes	direct product factor
170	C₂	C₂×C₄×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
171	C₂	C₂×#88857	1	2	yes	—
172	C₂	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic
173	C₂	C₂×#88859	1	2	yes	—
174	C₂	C₂×#88864	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂×C₄²
Frattini	C₂³	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂³×C₃³×C₄	C₂
socle, prime radical	C₂⁴×C₃³	C₂²
cosocle	C₂³×C₃³	C₂³
hypercenter	C₂³×C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×#95
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. #11722201	C₂	direct (N × H)
7. C₂²×C₃³×C₄	C₄	semidirect
11. C₂²×C₃⋊(C₃⋊Dic₃)	C₄	semidirect
24. C₂×C₃³×C₄	C₂×C₄	semidirect
28. C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	semidirect
32. C₂²×C₃²×C₄	Dic₃	semidirect
61. C₂×C₃²×C₄	C₂×Dic₃	semidirect
67. C₂×C₃³	#95	semidirect
75. C₂²×C₃×C₄	C₃⋊Dic₃	semidirect
96. C₃³	C₂×#95	semidirect
105. C₂×C₃×C₄	C₂×C₃⋊Dic₃	semidirect
110. C₂×C₃²	C₃⋊#95	semidirect
122. C₂²×C₄	C₃⋊(C₃⋊Dic₃)	semidirect
136. C₃²	C₂×C₃⋊#95	semidirect
149. C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	semidirect
150. C₂×C₃	#63658	semidirect
168. C₃	C₂×#63658	semidirect
169. C₂	#11722201	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
C₂³×C₃⋊(C₃⋊Dic₃)	2
#11722201	2
C₂×#63658	3

Subgroups — C2×#11722201 #13163848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11722201 #13163848