Subgroups — #13163613

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	1	yes	maximal, characteristic
3	C₂×#88857	C₂	1	2	yes	maximal
4	C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
5	#11722201	C₂	1	2	yes	maximal
6	#10592671	—	3	13	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	2	yes	—
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
12	C₃³×C₄⋊C₄	—	2	2	no	—
13	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₄	1	4	yes	complemented
14	#88857	—	2	4	no	—
15	C₄×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
16	C₂×C₃²×C₄⋊C₄	D₃	1	13	yes	—
17	C₂×C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
18	C₂×#3750	—	3	26	no	—
19	C₂×C₄×C₃⋊Dic₃	—	3	13	no	—
20	#63658	—	3	26	no	—
21	C₂³×C₃³	C₂³	1	1	yes	characteristic
22	C₂×C₃³×C₄	—	2	2	no	—
23	C₂×C₃³×C₄	—	2	2	no	—
24	C₂×C₃³×C₄	—	2	2	no	—
25	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃³×C₄	D₄	1	2	yes	—
27	C₂×C₃³×C₄	Q₈	1	2	yes	—
28	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
29	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
30	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
31	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
32	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
33	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
34	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
35	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
36	C₂×C₃⋊(C₃⋊Dic₃)	Q₈	1	2	yes	—
37	#4972	—	9	13	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	26	yes	—
39	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
40	C₂²×C₃⋊Dic₃	—	3	13	no	—
41	C₂²×C₃⋊Dic₃	—	3	13	no	—
42	C₂²×C₃⋊Dic₃	—	3	26	no	—
43	C₃²×C₄⋊C₄	—	2	26	no	—
44	C₄⋊(C₃⋊Dic₃)	—	3	52	no	—
45	#3750	—	6	52	no	—
46	C₄×C₃⋊Dic₃	—	6	26	no	—
47	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₃³	#41	1	1	yes	characteristic
53	C₂²×C₃³	#41	1	1	yes	characteristic
54	C₃³×C₄	—	4	2	no	—
55	C₃³×C₄	C₄⋊C₄	1	4	yes	complemented
56	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
57	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
58	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
59	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	13	yes	—
60	C₂×C₄⋊Dic₃	—	9	13	no	—
61	C₂×Dic₃⋊C₄	—	9	26	no	—
62	C₂×C₄×Dic₃	—	9	13	no	—
63	C₃⋊#95	—	9	26	no	—
64	C₂³×C₃²	C₂²×D₃	1	13	yes	—
65	C₂×C₃²×C₄	—	2	26	no	—
66	C₂×C₃²×C₄	—	2	26	no	—
67	C₂×C₃²×C₄	—	2	26	no	—
68	C₂×C₃²×C₄	Dic₆	1	26	yes	—
69	C₂×C₃²×C₄	C₄×D₃	1	26	yes	—
70	C₂×C₃²×C₄	#67	1	26	yes	—
71	C₂×C₃⋊Dic₃	—	3	52	no	—
72	C₂×C₃⋊Dic₃	—	3	26	no	—
73	C₂×C₃⋊Dic₃	—	3	26	no	—
74	C₂×C₃⋊Dic₃	—	6	26	no	—
75	C₂×C₃⋊Dic₃	—	6	13	no	—
76	C₂×C₃⋊Dic₃	—	6	26	no	—
77	C₂×C₃⋊Dic₃	—	6	13	no	—
78	C₂×C₃⋊Dic₃	—	6	13	no	—
79	C₂×C₃⋊Dic₃	—	6	13	no	—
80	#389	—	27	1	no	Sylow
81	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
83	C₂×C₃³	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₃³	#121	1	1	yes	characteristic
85	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₃³	#125	1	1	yes	characteristic
87	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
89	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
90	C₂²×Dic₃	—	9	13	no	—
91	C₂²×Dic₃	—	9	26	no	—
92	C₂²×Dic₃	—	9	13	no	—
93	C₃×C₄⋊C₄	—	2	26	no	—
94	C₄⋊Dic₃	—	9	52	no	—
95	Dic₃⋊C₄	—	18	52	no	—
96	C₄×Dic₃	—	18	26	no	—
97	C₂²×C₃²	C₂×Dic₆	1	13	yes	—
98	C₂²×C₃²	C₂×C₄×D₃	1	13	yes	—
99	C₂²×C₃²	D₃×D₄	1	13	yes	—
100	C₂²×C₃²	#237	1	13	yes	—
101	C₂²×C₃²	D₃×Q₈	1	13	yes	—
102	C₂²×C₃²	#239	1	13	yes	—
103	C₂²×C₃²	C₂×#67	1	13	yes	—
104	C₃²×C₄	—	4	26	no	—
105	C₃²×C₄	Dic₃⋊C₄	1	52	yes	complemented
106	C₃⋊Dic₃	—	6	52	no	—
107	C₃⋊Dic₃	—	6	26	no	—
108	C₃⋊Dic₃	—	6	26	no	—
109	C₂×C₄⋊C₄	—	27	1	no	—
110	C₂×C₄⋊C₄	—	27	2	no	—
111	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
112	C₂×C₄²	—	27	1	no	—
113	#95	—	27	2	no	—
114	C₃³	#389	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
116	C₂×C₃×C₄	—	2	26	no	—
117	C₂×C₃×C₄	—	2	26	no	—
118	C₂×C₃×C₄	—	2	26	no	—
119	C₂×C₃×C₄	#634	1	26	yes	—
120	C₂×C₃×C₄	C₄×C₃⋊D₃	1	26	yes	—
121	C₂×C₃×C₄	#638	1	26	yes	—
122	C₂×Dic₃	—	9	52	no	—
123	C₂×Dic₃	—	9	26	no	—
124	C₂×Dic₃	—	9	26	no	—
125	C₂×Dic₃	—	18	26	no	—
126	C₂×Dic₃	—	18	13	no	—
127	C₂×Dic₃	—	18	26	no	—
128	C₂×Dic₃	—	18	13	no	—
129	C₂×Dic₃	—	18	13	no	—
130	C₂×Dic₃	—	18	13	no	—
131	C₂×C₃²	#887	1	13	yes	—
132	C₂×C₃²	#888	1	13	yes	—
133	C₂×C₃²	#890	1	13	yes	—
134	C₂×C₃²	#893	1	13	yes	—
135	C₂×C₃²	C₂×Dic₃⋊C₄	1	13	yes	—
136	C₂×C₃²	#938	1	13	yes	—
137	C₂×C₃²	#946	1	13	yes	—
138	C₂²×C₄	—	27	1	no	—
139	C₂²×C₄	—	27	2	no	—
140	C₂²×C₄	—	27	1	no	—
141	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
142	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	27	4	no	—
145	C₄⋊C₄	—	54	4	no	—
146	C₄²	—	54	2	no	—
147	C₂²×C₃	C₂×#634	1	13	yes	—
148	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	13	yes	—
149	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
150	C₂²×C₃	#3830	1	13	yes	—
151	C₂²×C₃	Q₈×C₃⋊D₃	1	13	yes	—
152	C₂²×C₃	#3832	1	13	yes	—
153	C₂²×C₃	C₂×#638	1	13	yes	—
154	C₃×C₄	—	4	26	no	—
155	C₃×C₄	#3750	1	52	yes	complemented
156	Dic₃	—	18	52	no	—
157	Dic₃	—	18	26	no	—
158	Dic₃	—	18	26	no	—
159	C₃²	#4972	1	13	yes	complemented
160	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	27	4	no	—
165	C₂×C₄	—	27	2	no	—
166	C₂×C₄	—	27	2	no	—
167	C₂×C₄	—	54	2	no	—
168	C₂×C₄	—	54	1	no	—
169	C₂×C₄	—	54	2	no	—
170	C₂×C₄	—	54	1	no	—
171	C₂×C₄	—	54	1	no	—
172	C₂×C₄	—	54	1	no	—
173	C₂×C₄	#6308	1	2	yes	—
174	C₂×C₄	C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
175	C₂×C₄	#6312	1	2	yes	—
176	C₂×C₃	#64096	1	13	yes	—
177	C₂×C₃	#64097	1	13	yes	—
178	C₂×C₃	#64099	1	13	yes	—
179	C₂×C₃	#64102	1	13	yes	—
180	C₂×C₃	C₂×#3750	1	13	yes	—
181	C₂×C₃	#64147	1	13	yes	—
182	C₂×C₃	#64155	1	13	yes	—
183	C₂²	C₂×#6308	1	1	yes	characteristic
184	C₂²	C₂×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
185	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
186	C₂²	#89078	1	1	yes	characteristic
187	C₂²	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
188	C₂²	#89080	1	1	yes	characteristic
189	C₂²	C₂×#6312	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	54	4	no	—
192	C₄	—	54	2	no	—
193	C₄	—	54	2	no	—
194	C₄	#88857	1	4	yes	complemented
195	C₃	#10592671	1	13	yes	complemented
196	C₂	#11723928	1	1	yes	characteristic
197	C₂	#11723929	1	1	yes	characteristic
198	C₂	#11723931	1	1	yes	characteristic
199	C₂	#11723934	1	1	yes	characteristic
200	C₂	C₂×#88857	1	1	yes	characteristic
201	C₂	#11723979	1	1	yes	characteristic
202	C₂	#11723987	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₃⋊(C₃⋊D₃)
commutator	C₂²×C₃³	C₂²×C₄
Fitting	C₂×C₃³×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃³	C₂³
hypercenter	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#389
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. C₄⋊(C₃⋊(C₃⋊Dic₃))	C₄	semidirect
55. C₃³×C₄	C₄⋊C₄	semidirect
105. C₃²×C₄	Dic₃⋊C₄	semidirect
114. C₃³	#389	semidirect
155. C₃×C₄	#3750	semidirect
159. C₃²	#4972	semidirect
194. C₄	#88857	semidirect
195. C₃	#10592671	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×C₄⋊C₄	2
C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	2
C₂×#88857	2
C₂×C₄×C₃⋊(C₃⋊Dic₃)	2
#11722201	2
#10592671	3