Subgroups — #13163623

Order 1728.

2100 conjugacy classes of subgroups.

The subgroup lattice has 252 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	1	yes	maximal, characteristic
4	C₂×#88857	C₂	1	1	yes	maximal, characteristic
5	#11722201	C₂	1	1	yes	maximal, characteristic
6	#11722201	C₂	1	1	yes	maximal, characteristic
7	#11722201	C₂	1	1	yes	maximal, characteristic
8	#10592681	—	3	13	no	maximal
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
16	C₃³×C₄⋊C₄	—	2	2	no	—
17	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	2	no	—
18	#88857	—	2	2	no	—
19	#88857	—	2	2	no	—
20	C₂×C₃²×C₄⋊C₄	D₃	1	13	yes	—
21	C₂×C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
22	C₂×#3750	—	3	13	no	—
23	C₂×#3750	—	3	13	no	—
24	#63658	—	3	13	no	—
25	#63658	—	3	13	no	—
26	#63658	—	3	13	no	—
27	C₂³×C₃³	C₂³	1	1	yes	characteristic
28	C₂×C₃³×C₄	—	2	1	no	—
29	C₂×C₃³×C₄	—	2	1	no	—
30	C₂×C₃³×C₄	—	2	1	no	—
31	C₂×C₃³×C₄	—	2	1	no	—
32	C₂×C₃³×C₄	—	2	1	no	—
33	C₂×C₃³×C₄	—	2	1	no	—
34	C₂×C₃³×C₄	—	2	1	no	—
35	C₂×C₃³×C₄	D₄	1	2	yes	—
36	C₂×C₃³×C₄	Q₈	1	2	yes	—
37	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
38	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
39	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
40	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
41	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
42	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
43	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
44	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
45	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
46	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
47	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
48	C₂×C₃⋊(C₃⋊Dic₃)	Q₈	1	2	yes	—
49	#4982	—	9	13	no	—
50	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
51	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
52	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
53	C₂²×C₃⋊Dic₃	—	3	13	no	—
54	C₂²×C₃⋊Dic₃	—	3	13	no	—
55	C₂²×C₃⋊Dic₃	—	3	13	no	—
56	C₂²×C₃⋊Dic₃	—	3	13	no	—
57	C₃²×C₄⋊C₄	—	2	26	no	—
58	C₄⋊(C₃⋊Dic₃)	—	6	26	no	—
59	#3750	—	6	26	no	—
60	#3750	—	6	26	no	—
61	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₃³	#41	1	1	yes	characteristic
66	C₂²×C₃³	#41	1	1	yes	characteristic
67	C₂²×C₃³	#41	1	1	yes	characteristic
68	C₃³×C₄	—	2	2	no	—
69	C₃³×C₄	—	2	2	no	—
70	C₃³×C₄	—	4	1	no	—
71	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
72	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
73	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
74	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
75	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	13	yes	—
76	C₂×C₄⋊Dic₃	—	9	13	no	—
77	C₂×Dic₃⋊C₄	—	9	13	no	—
78	C₂×Dic₃⋊C₄	—	9	13	no	—
79	C₃⋊#95	—	9	13	no	—
80	C₃⋊#95	—	9	13	no	—
81	C₃⋊#95	—	9	13	no	—
82	C₂³×C₃²	C₂²×D₃	1	13	yes	—
83	C₂×C₃²×C₄	—	2	13	no	—
84	C₂×C₃²×C₄	—	2	13	no	—
85	C₂×C₃²×C₄	—	2	13	no	—
86	C₂×C₃²×C₄	—	2	13	no	—
87	C₂×C₃²×C₄	—	2	13	no	—
88	C₂×C₃²×C₄	—	2	13	no	—
89	C₂×C₃²×C₄	—	2	13	no	—
90	C₂×C₃²×C₄	Dic₆	1	26	yes	—
91	C₂×C₃²×C₄	#67	1	26	yes	—
92	C₂×C₃⋊Dic₃	—	3	26	no	—
93	C₂×C₃⋊Dic₃	—	3	26	no	—
94	C₂×C₃⋊Dic₃	—	6	13	no	—
95	C₂×C₃⋊Dic₃	—	6	13	no	—
96	C₂×C₃⋊Dic₃	—	6	13	no	—
97	C₂×C₃⋊Dic₃	—	6	13	no	—
98	C₂×C₃⋊Dic₃	—	6	13	no	—
99	C₂×C₃⋊Dic₃	—	6	13	no	—
100	C₂×C₃⋊Dic₃	—	6	13	no	—
101	C₂×C₃⋊Dic₃	—	6	13	no	—
102	C₂×C₃⋊Dic₃	—	6	13	no	—
103	C₂×C₃⋊Dic₃	—	6	13	no	—
104	#398	—	27	1	no	Sylow
105	C₂×C₃³	#121	1	1	yes	characteristic
106	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₃³	#123	1	1	yes	characteristic
109	C₂×C₃³	#125	1	1	yes	characteristic
110	C₂×C₃³	#125	1	1	yes	characteristic
111	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
113	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
114	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
115	C₂²×Dic₃	—	9	13	no	—
116	C₂²×Dic₃	—	9	13	no	—
117	C₂²×Dic₃	—	9	13	no	—
118	C₂²×Dic₃	—	9	13	no	—
119	C₃×C₄⋊C₄	—	2	26	no	—
120	C₄⋊Dic₃	—	18	26	no	—
121	Dic₃⋊C₄	—	18	26	no	—
122	Dic₃⋊C₄	—	18	26	no	—
123	C₂²×C₃²	C₂×Dic₆	1	13	yes	—
124	C₂²×C₃²	#235	1	13	yes	—
125	C₂²×C₃²	D₃×D₄	1	13	yes	—
126	C₂²×C₃²	#237	1	13	yes	—
127	C₂²×C₃²	D₃×Q₈	1	13	yes	—
128	C₂²×C₃²	#239	1	13	yes	—
129	C₂²×C₃²	C₂×#67	1	13	yes	—
130	C₃²×C₄	—	2	26	no	—
131	C₃²×C₄	—	2	26	no	—
132	C₃²×C₄	—	4	13	no	—
133	C₃⋊Dic₃	—	6	26	no	—
134	C₃⋊Dic₃	—	6	26	no	—
135	C₃⋊Dic₃	—	12	13	no	—
136	C₃⋊Dic₃	—	12	13	no	—
137	C₂×C₄⋊C₄	—	27	1	no	—
138	C₂×C₄⋊C₄	—	27	1	no	—
139	C₂×C₄⋊C₄	—	27	1	no	—
140	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
141	#95	—	27	1	no	—
142	#95	—	27	1	no	—
143	#95	—	27	1	no	—
144	C₃³	#398	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
146	C₂×C₃×C₄	—	2	13	no	—
147	C₂×C₃×C₄	—	2	13	no	—
148	C₂×C₃×C₄	—	2	13	no	—
149	C₂×C₃×C₄	—	2	13	no	—
150	C₂×C₃×C₄	—	2	13	no	—
151	C₂×C₃×C₄	—	2	13	no	—
152	C₂×C₃×C₄	—	2	13	no	—
153	C₂×C₃×C₄	#634	1	26	yes	—
154	C₂×C₃×C₄	#638	1	26	yes	—
155	C₂×Dic₃	—	9	26	no	—
156	C₂×Dic₃	—	9	26	no	—
157	C₂×Dic₃	—	18	13	no	—
158	C₂×Dic₃	—	18	13	no	—
159	C₂×Dic₃	—	18	13	no	—
160	C₂×Dic₃	—	18	13	no	—
161	C₂×Dic₃	—	18	13	no	—
162	C₂×Dic₃	—	18	13	no	—
163	C₂×Dic₃	—	18	13	no	—
164	C₂×Dic₃	—	18	13	no	—
165	C₂×Dic₃	—	18	13	no	—
166	C₂×Dic₃	—	18	13	no	—
167	C₂×C₃²	#888	1	13	yes	—
168	C₂×C₃²	#889	1	13	yes	—
169	C₂×C₃²	#890	1	13	yes	—
170	C₂×C₃²	#894	1	13	yes	—
171	C₂×C₃²	#924	1	13	yes	—
172	C₂×C₃²	#939	1	13	yes	—
173	C₂×C₃²	#946	1	13	yes	—
174	C₂²×C₄	—	27	1	no	—
175	C₂²×C₄	—	27	1	no	—
176	C₂²×C₄	—	27	1	no	—
177	C₂²×C₄	—	27	1	no	—
178	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	54	2	no	—
183	C₄⋊C₄	—	54	2	no	—
184	C₄⋊C₄	—	54	2	no	—
185	C₂²×C₃	C₂×#634	1	13	yes	—
186	C₂²×C₃	#3828	1	13	yes	—
187	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
188	C₂²×C₃	#3830	1	13	yes	—
189	C₂²×C₃	Q₈×C₃⋊D₃	1	13	yes	—
190	C₂²×C₃	#3832	1	13	yes	—
191	C₂²×C₃	C₂×#638	1	13	yes	—
192	C₃×C₄	—	2	26	no	—
193	C₃×C₄	—	2	26	no	—
194	C₃×C₄	—	4	13	no	—
195	Dic₃	—	18	26	no	—
196	Dic₃	—	18	26	no	—
197	Dic₃	—	36	13	no	—
198	Dic₃	—	36	13	no	—
199	C₃²	#4982	1	13	yes	complemented
200	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	27	2	no	—
209	C₂×C₄	—	27	2	no	—
210	C₂×C₄	—	54	1	no	—
211	C₂×C₄	—	54	1	no	—
212	C₂×C₄	—	54	1	no	—
213	C₂×C₄	—	54	1	no	—
214	C₂×C₄	—	54	1	no	—
215	C₂×C₄	—	54	1	no	—
216	C₂×C₄	—	54	1	no	—
217	C₂×C₄	—	54	1	no	—
218	C₂×C₄	—	54	1	no	—
219	C₂×C₄	—	54	1	no	—
220	C₂×C₄	#6308	1	2	yes	—
221	C₂×C₄	#6312	1	2	yes	—
222	C₂×C₃	#64097	1	13	yes	—
223	C₂×C₃	#64098	1	13	yes	—
224	C₂×C₃	#64099	1	13	yes	—
225	C₂×C₃	#64103	1	13	yes	—
226	C₂×C₃	#64133	1	13	yes	—
227	C₂×C₃	#64148	1	13	yes	—
228	C₂×C₃	#64155	1	13	yes	—
229	C₂²	C₂×#6308	1	1	yes	characteristic
230	C₂²	#89076	1	1	yes	characteristic
231	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
232	C₂²	#89078	1	1	yes	characteristic
233	C₂²	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
234	C₂²	#89080	1	1	yes	characteristic
235	C₂²	C₂×#6312	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	54	2	no	—
240	C₄	—	54	2	no	—
241	C₄	—	108	1	no	—
242	C₄	—	108	1	no	—
243	C₃	#10592681	1	13	yes	complemented
244	C₂	#11723929	1	1	yes	characteristic
245	C₂	#11723930	1	1	yes	characteristic
246	C₂	#11723931	1	1	yes	characteristic
247	C₂	#11723935	1	1	yes	characteristic
248	C₂	#11723965	1	1	yes	characteristic
249	C₂	#11723980	1	1	yes	characteristic
250	C₂	#11723987	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, prime radical, cosocle	C₂³×C₃³	C₂³
Fitting	C₂×C₃³×C₄⋊C₄	C₂
hypercenter	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#398
LCS	G ▷ C₂³×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
144. C₃³	#398	semidirect
199. C₃²	#4982	semidirect
243. C₃	#10592681	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
#11722201	2
#10592681	3