Subgroups — #13163852

Order 1728.

3276 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	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#88864	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	2	yes	maximal, complemented
4	#11722201	C₂	1	2	yes	maximal, complemented
5	#10592911	—	3	13	no	maximal
6	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
7	C₂²×C₃³×C₄	—	2	4	no	—
8	C₂²×C₃³×C₄	C₄	1	4	yes	complemented
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
12	#88864	—	2	4	no	—
13	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	4	no	—
14	C₂³×C₃²×C₄	D₃	1	13	yes	—
15	C₂×#3757	—	3	26	no	—
16	C₂×C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
17	#63658	—	3	26	no	—
18	C₂³×C₃³	—	2	2	no	—
19	C₂³×C₃³	—	2	2	no	—
20	C₂³×C₃³	C₂×C₄	1	2	yes	—
21	C₂³×C₃³	D₄	1	2	yes	—
22	C₂³×C₃³	Q₈	1	2	yes	—
23	C₂³×C₃³	C₂³	1	1	yes	characteristic
24	C₂×C₃³×C₄	—	2	2	no	—
25	C₂×C₃³×C₄	—	2	8	no	—
26	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃³×C₄	D₄	1	4	yes	—
29	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
30	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
31	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
32	#5212	—	9	13	no	—
33	C₂⁴×C₃²	C₂×D₃	1	13	yes	—
34	C₂²×C₃²×C₄	—	2	52	no	—
35	C₂²×C₃²×C₄	Dic₃	1	52	yes	complemented
36	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
37	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
38	C₂²×C₃⋊Dic₃	—	3	52	no	—
39	#3757	—	6	52	no	—
40	C₄⋊(C₃⋊Dic₃)	—	6	52	no	—
41	C₂²×C₃³	—	2	2	no	—
42	C₂²×C₃³	—	2	2	no	—
43	C₂²×C₃³	—	2	4	no	—
44	C₂²×C₃³	—	2	4	no	—
45	C₂²×C₃³	C₄⋊C₄	1	4	yes	complemented
46	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₃³	C₂×D₄	1	2	yes	—
48	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₃³	#41	1	2	yes	—
51	C₃³×C₄	—	2	2	no	—
52	C₃³×C₄	C₂²⋊C₄	1	4	yes	complemented
53	C₃⋊(C₃⋊Dic₃)	—	4	4	no	—
54	C₂³×C₃×C₄	C₃⋊D₃	1	13	yes	—
55	C₂×#217	—	9	26	no	—
56	C₂×C₄⋊Dic₃	—	9	26	no	—
57	C₃⋊#95	—	9	26	no	—
58	C₂³×C₃²	—	2	26	no	—
59	C₂³×C₃²	—	2	26	no	—
60	C₂³×C₃²	Dic₆	1	26	yes	—
61	C₂³×C₃²	D₁₂	1	26	yes	—
62	C₂³×C₃²	C₂×Dic₃	1	26	yes	—
63	C₂³×C₃²	C₂²×D₃	1	13	yes	—
64	C₂×C₃²×C₄	—	2	26	no	—
65	C₂×C₃²×C₄	—	2	104	no	—
66	C₂×C₃²×C₄	C₂×Dic₃	1	26	yes	—
67	C₂×C₃²×C₄	C₂×Dic₃	1	26	yes	—
68	C₂×C₃²×C₄	#67	1	52	yes	—
69	C₂×C₃⋊Dic₃	—	6	52	no	—
70	C₂×C₃⋊Dic₃	—	6	52	no	—
71	C₂×C₃⋊Dic₃	—	6	52	no	—
72	#380	—	27	1	no	Sylow
73	C₂×C₃³	—	2	4	no	—
74	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃³	#117	1	1	yes	characteristic
77	C₂×C₃³	#121	1	2	yes	—
78	C₂×C₃³	C₂²⋊Q₈	1	2	yes	—
79	C₂⁴×C₃	C₂×C₃⋊D₃	1	13	yes	—
80	C₂²×C₃×C₄	—	2	52	no	—
81	C₂²×C₃×C₄	C₃⋊Dic₃	1	52	yes	complemented
82	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
83	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
84	C₂²×Dic₃	—	9	52	no	—
85	#217	—	18	52	no	—
86	C₄⋊Dic₃	—	18	52	no	—
87	C₂²×C₃²	—	2	52	no	—
88	C₂²×C₃²	—	2	26	no	—
89	C₂²×C₃²	—	2	52	no	—
90	C₂²×C₃²	—	2	26	no	—
91	C₂²×C₃²	C₄⋊Dic₃	1	52	yes	complemented
92	C₂²×C₃²	C₂×Dic₆	1	13	yes	—
93	C₂²×C₃²	C₂×D₁₂	1	13	yes	—
94	C₂²×C₃²	#235	1	26	yes	—
95	C₂²×C₃²	C₂²×Dic₃	1	13	yes	—
96	C₂²×C₃²	C₂×#67	1	26	yes	—
97	C₃²×C₄	—	2	26	no	—
98	C₃²×C₄	#217	1	52	yes	complemented
99	C₃⋊Dic₃	—	12	52	no	—
100	C₂³×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	27	2	no	—
102	C₂×C₂²⋊C₄	—	27	2	no	—
103	#95	—	27	2	no	—
104	C₃³	#380	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₃	—	2	26	no	—
106	C₂³×C₃	—	2	26	no	—
107	C₂³×C₃	#634	1	26	yes	—
108	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	26	yes	—
109	C₂³×C₃	C₂×C₃⋊Dic₃	1	26	yes	—
110	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
111	C₂×C₃×C₄	—	2	26	no	—
112	C₂×C₃×C₄	—	2	104	no	—
113	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	26	yes	—
114	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	26	yes	—
115	C₂×C₃×C₄	#638	1	52	yes	—
116	C₂×Dic₃	—	18	52	no	—
117	C₂×Dic₃	—	18	52	no	—
118	C₂×Dic₃	—	18	52	no	—
119	C₂×C₃²	—	2	52	no	—
120	C₂×C₃²	#924	1	26	yes	—
121	C₂×C₃²	C₂×C₄⋊Dic₃	1	13	yes	—
122	C₂×C₃²	#926	1	13	yes	—
123	C₂×C₃²	#930	1	26	yes	—
124	C₂×C₃²	C₂×#217	1	13	yes	—
125	C₂⁴	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	27	4	no	—
128	C₂²×C₄	C₃⋊(C₃⋊Dic₃)	1	4	yes	complemented
129	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
131	C₄⋊C₄	—	54	4	no	—
132	C₂²⋊C₄	—	54	4	no	—
133	C₂²×C₃	—	2	52	no	—
134	C₂²×C₃	—	2	26	no	—
135	C₂²×C₃	—	2	52	no	—
136	C₂²×C₃	—	2	26	no	—
137	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	52	yes	complemented
138	C₂²×C₃	C₂×#634	1	13	yes	—
139	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	13	yes	—
140	C₂²×C₃	#3828	1	26	yes	—
141	C₂²×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
142	C₂²×C₃	C₂×#638	1	26	yes	—
143	C₃×C₄	—	2	26	no	—
144	C₃×C₄	#3757	1	52	yes	complemented
145	Dic₃	—	36	52	no	—
146	C₃²	#5212	1	13	yes	complemented
147	C₂³	—	2	2	no	—
148	C₂³	—	2	2	no	—
149	C₂³	#6308	1	2	yes	—
150	C₂³	C₄⋊(C₃⋊(C₃⋊D₃))	1	2	yes	—
151	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
152	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	54	4	no	—
156	C₂×C₄	—	54	4	no	—
157	C₂×C₄	—	54	4	no	—
158	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
159	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
160	C₂×C₄	#6312	1	4	yes	—
161	C₂×C₃	—	2	52	no	—
162	C₂×C₃	#64133	1	26	yes	—
163	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	13	yes	—
164	C₂×C₃	#64135	1	13	yes	—
165	C₂×C₃	#64139	1	26	yes	—
166	C₂×C₃	C₂×#3757	1	13	yes	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	2	no	—
171	C₂²	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	4	yes	complemented
172	C₂²	C₂×#6308	1	1	yes	characteristic
173	C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
174	C₂²	#89076	1	2	yes	—
175	C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
176	C₂²	C₂×#6312	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	108	4	no	—
179	C₄	#88864	1	4	yes	complemented
180	C₃	#10592911	1	13	yes	complemented
181	C₂	—	2	4	no	—
182	C₂	#11723965	1	2	yes	—
183	C₂	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic
184	C₂	#11723967	1	1	yes	characteristic
185	C₂	#11723971	1	2	yes	—
186	C₂	C₂×#88864	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₃⋊(C₃⋊D₃)
commutator	C₂²×C₃³	C₂²×C₄
Fitting	C₂³×C₃³×C₄	C₂
socle, cosocle	C₂³×C₃³	C₂³
prime radical	C₂⁴×C₃³	C₂²
hypercenter	C₂³×C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#380
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	semidirect
4. #11722201	C₂	semidirect
8. C₂²×C₃³×C₄	C₄	semidirect
35. C₂²×C₃²×C₄	Dic₃	semidirect
45. C₂²×C₃³	C₄⋊C₄	semidirect
52. C₃³×C₄	C₂²⋊C₄	semidirect
81. C₂²×C₃×C₄	C₃⋊Dic₃	semidirect
91. C₂²×C₃²	C₄⋊Dic₃	semidirect
98. C₃²×C₄	#217	semidirect
104. C₃³	#380	semidirect
128. C₂²×C₄	C₃⋊(C₃⋊Dic₃)	semidirect
137. C₂²×C₃	C₄⋊(C₃⋊Dic₃)	semidirect
144. C₃×C₄	#3757	semidirect
146. C₃²	#5212	semidirect
171. C₂²	C₄⋊(C₃⋊(C₃⋊Dic₃))	semidirect
179. C₄	#88864	semidirect
180. C₃	#10592911	semidirect

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
C₂×#88864	2
C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	2
#11722201	2
#10592911	3