Subgroups — #13163610

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 256 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×#88857	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	C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
6	#11722201	C₂	1	1	yes	maximal, characteristic
7	#11722201	C₂	1	1	yes	maximal, characteristic
8	#10592675	—	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	#88857	—	2	2	no	—
18	#88857	—	2	2	no	—
19	#88857	C₄	1	4	yes	complemented
20	C₄×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
21	C₂×C₃²×C₄⋊C₄	D₃	1	13	yes	—
22	C₂×#3750	—	3	13	no	—
23	C₂×#3750	—	3	13	no	—
24	C₂×#3750	—	3	13	no	—
25	C₂×C₄×C₃⋊Dic₃	—	3	13	no	—
26	#63658	—	3	13	no	—
27	#63658	—	3	13	no	—
28	C₂³×C₃³	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
36	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
37	C₂×C₃³×C₄	D₄	1	2	yes	—
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)	C₂×C₄	1	2	yes	—
46	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
47	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
48	C₂×C₃⋊(C₃⋊Dic₃)	Q₈	1	2	yes	—
49	C₂×C₃⋊(C₃⋊Dic₃)	Q₈	1	2	yes	—
50	#4976	—	9	13	no	—
51	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
52	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
53	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
54	C₂²×C₃⋊Dic₃	—	3	13	no	—
55	C₂²×C₃⋊Dic₃	—	3	13	no	—
56	C₂²×C₃⋊Dic₃	—	3	13	no	—
57	C₂²×C₃⋊Dic₃	—	3	13	no	—
58	C₃²×C₄⋊C₄	—	2	26	no	—
59	#3750	—	3	52	no	—
60	#3750	—	6	26	no	—
61	#3750	—	6	26	no	—
62	C₄×C₃⋊Dic₃	—	6	26	no	—
63	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₃³	#41	1	1	yes	characteristic
69	C₂²×C₃³	#41	1	1	yes	characteristic
70	C₃³×C₄	—	2	2	no	—
71	C₃³×C₄	—	2	2	no	—
72	C₃³×C₄	—	4	1	no	—
73	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
74	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
75	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
76	C₃⋊(C₃⋊Dic₃)	C₄⋊C₄	1	4	yes	complemented
77	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	13	yes	—
78	C₂×Dic₃⋊C₄	—	9	13	no	—
79	C₂×Dic₃⋊C₄	—	9	13	no	—
80	C₂×Dic₃⋊C₄	—	9	13	no	—
81	C₂×C₄×Dic₃	—	9	13	no	—
82	C₃⋊#95	—	9	13	no	—
83	C₃⋊#95	—	9	13	no	—
84	C₂³×C₃²	C₂²×D₃	1	13	yes	—
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	—	2	13	no	—
91	C₂×C₃²×C₄	—	2	13	no	—
92	C₂×C₃²×C₄	C₄×D₃	1	26	yes	—
93	C₂×C₃²×C₄	#67	1	26	yes	—
94	C₂×C₃⋊Dic₃	—	3	26	no	—
95	C₂×C₃⋊Dic₃	—	3	26	no	—
96	C₂×C₃⋊Dic₃	—	3	26	no	—
97	C₂×C₃⋊Dic₃	—	3	26	no	—
98	C₂×C₃⋊Dic₃	—	3	26	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	C₂×C₃⋊Dic₃	—	6	13	no	—
105	C₂×C₃⋊Dic₃	—	6	13	no	—
106	#389	—	27	1	no	Sylow
107	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
109	C₂×C₃³	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₃³	#121	1	1	yes	characteristic
111	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₃³	#125	1	1	yes	characteristic
113	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
115	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
116	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
117	C₂²×Dic₃	—	9	13	no	—
118	C₂²×Dic₃	—	9	13	no	—
119	C₂²×Dic₃	—	9	13	no	—
120	C₂²×Dic₃	—	9	13	no	—
121	C₃×C₄⋊C₄	—	2	26	no	—
122	Dic₃⋊C₄	—	9	52	no	—
123	Dic₃⋊C₄	—	18	26	no	—
124	Dic₃⋊C₄	—	18	26	no	—
125	C₄×Dic₃	—	18	26	no	—
126	C₂²×C₃²	C₂×C₄×D₃	1	13	yes	—
127	C₂²×C₃²	#235	1	13	yes	—
128	C₂²×C₃²	D₃×D₄	1	13	yes	—
129	C₂²×C₃²	#237	1	13	yes	—
130	C₂²×C₃²	D₃×Q₈	1	13	yes	—
131	C₂²×C₃²	D₃×Q₈	1	13	yes	—
132	C₂²×C₃²	C₂×#67	1	13	yes	—
133	C₃²×C₄	—	2	26	no	—
134	C₃²×C₄	—	2	26	no	—
135	C₃²×C₄	—	4	13	no	—
136	C₃⋊Dic₃	—	3	52	no	—
137	C₃⋊Dic₃	—	6	26	no	—
138	C₃⋊Dic₃	—	6	26	no	—
139	C₃⋊Dic₃	—	12	13	no	—
140	C₂×C₄⋊C₄	—	27	1	no	—
141	C₂×C₄⋊C₄	—	27	1	no	—
142	C₂×C₄⋊C₄	—	27	1	no	—
143	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
144	C₂×C₄²	—	27	1	no	—
145	#95	—	27	1	no	—
146	#95	—	27	1	no	—
147	C₃³	#389	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
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	—	2	13	no	—
154	C₂×C₃×C₄	—	2	13	no	—
155	C₂×C₃×C₄	—	2	13	no	—
156	C₂×C₃×C₄	C₄×C₃⋊D₃	1	26	yes	—
157	C₂×C₃×C₄	#638	1	26	yes	—
158	C₂×Dic₃	—	9	26	no	—
159	C₂×Dic₃	—	9	26	no	—
160	C₂×Dic₃	—	9	26	no	—
161	C₂×Dic₃	—	9	26	no	—
162	C₂×Dic₃	—	9	26	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×Dic₃	—	18	13	no	—
168	C₂×Dic₃	—	18	13	no	—
169	C₂×Dic₃	—	18	13	no	—
170	C₂×C₃²	#887	1	13	yes	—
171	C₂×C₃²	#889	1	13	yes	—
172	C₂×C₃²	D₃×C₄⋊C₄	1	13	yes	—
173	C₂×C₃²	#896	1	13	yes	—
174	C₂×C₃²	C₄×#67	1	13	yes	—
175	C₂×C₃²	#939	1	13	yes	—
176	C₂×C₃²	#944	1	13	yes	—
177	C₂²×C₄	—	27	1	no	—
178	C₂²×C₄	—	27	1	no	—
179	C₂²×C₄	—	27	1	no	—
180	C₂²×C₄	—	27	1	no	—
181	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	27	4	no	—
186	C₄⋊C₄	—	54	2	no	—
187	C₄⋊C₄	—	54	2	no	—
188	C₄²	—	54	2	no	—
189	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	13	yes	—
190	C₂²×C₃	#3828	1	13	yes	—
191	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
192	C₂²×C₃	#3830	1	13	yes	—
193	C₂²×C₃	Q₈×C₃⋊D₃	1	13	yes	—
194	C₂²×C₃	Q₈×C₃⋊D₃	1	13	yes	—
195	C₂²×C₃	C₂×#638	1	13	yes	—
196	C₃×C₄	—	2	26	no	—
197	C₃×C₄	—	2	26	no	—
198	C₃×C₄	—	4	13	no	—
199	Dic₃	—	9	52	no	—
200	Dic₃	—	18	26	no	—
201	Dic₃	—	18	26	no	—
202	Dic₃	—	36	13	no	—
203	C₃²	#4976	1	13	yes	complemented
204	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	27	2	no	—
213	C₂×C₄	—	27	2	no	—
214	C₂×C₄	—	27	2	no	—
215	C₂×C₄	—	27	2	no	—
216	C₂×C₄	—	27	2	no	—
217	C₂×C₄	—	54	1	no	—
218	C₂×C₄	—	54	1	no	—
219	C₂×C₄	—	54	1	no	—
220	C₂×C₄	—	54	1	no	—
221	C₂×C₄	—	54	1	no	—
222	C₂×C₄	—	54	1	no	—
223	C₂×C₄	—	54	1	no	—
224	C₂×C₄	C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
225	C₂×C₄	#6312	1	2	yes	—
226	C₂×C₃	#64096	1	13	yes	—
227	C₂×C₃	#64098	1	13	yes	—
228	C₂×C₃	C₄⋊C₄×C₃⋊D₃	1	13	yes	—
229	C₂×C₃	#64105	1	13	yes	—
230	C₂×C₃	C₄×#638	1	13	yes	—
231	C₂×C₃	#64148	1	13	yes	—
232	C₂×C₃	#64153	1	13	yes	—
233	C₂²	C₂×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
234	C₂²	#89076	1	1	yes	characteristic
235	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
236	C₂²	#89078	1	1	yes	characteristic
237	C₂²	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
238	C₂²	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
239	C₂²	C₂×#6312	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	27	4	no	—
244	C₄	—	54	2	no	—
245	C₄	—	54	2	no	—
246	C₄	—	108	1	no	—
247	C₃	#10592675	1	13	yes	complemented
248	C₂	#11723928	1	1	yes	characteristic
249	C₂	#11723930	1	1	yes	characteristic
250	C₂	C₄⋊C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
251	C₂	#11723937	1	1	yes	characteristic
252	C₂	C₄×#6312	1	1	yes	characteristic
253	C₂	#11723980	1	1	yes	characteristic
254	C₂	#11723985	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
19. #88857	C₄	semidirect
76. C₃⋊(C₃⋊Dic₃)	C₄⋊C₄	semidirect
147. C₃³	#389	semidirect
203. C₃²	#4976	semidirect
247. C₃	#10592675	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×#88857	2
C₂×C₄×C₃⋊(C₃⋊Dic₃)	2
#11722201	2
#10592675	3