Subgroups — #13144837

Order 1728.

930 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	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	#11720572	C₂	1	4	yes	maximal
5	#10592928	—	3	1	no	maximal
6	#10588909	—	3	3	no	maximal
7	C₂×C₃×Q₈×C₉	—	2	2	no	—
8	C₂×C₃×Q₈×C₉	—	2	2	no	—
9	C₂×C₃×Q₈×C₉	C₄	1	4	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
13	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
14	C₄⋊(C₃⋊Dic₉)	—	2	2	no	—
15	C₄×C₃⋊Dic₉	—	2	2	no	—
16	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
17	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
18	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
19	C₂²×Q₈×C₉	D₃	1	3	yes	—
20	#63658	—	3	4	no	—
21	C₂×C₄⋊Dic₉	—	3	3	no	—
22	C₂×C₄×Dic₉	—	3	3	no	—
23	C₉⋊#95	—	3	12	no	—
24	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
25	C₃×Q₈×C₉	—	2	8	no	—
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
32	C₂×C₃⋊Dic₉	—	2	2	no	—
33	C₂×C₃⋊Dic₉	—	2	2	no	—
34	C₂×C₃⋊Dic₉	—	2	2	no	—
35	C₂×C₃⋊Dic₉	—	2	4	no	—
36	C₂×C₃⋊Dic₉	Q₈	1	4	yes	complemented
37	#5229	—	9	3	no	—
38	#5229	—	9	1	no	—
39	C₂×C₃²×Q₈	—	2	2	no	—
40	C₂×C₃²×Q₈	—	2	2	no	—
41	C₂×C₃²×Q₈	Dic₃	1	4	yes	—
42	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃⋊Dic₃	—	3	2	no	—
45	C₂²×C₃⋊Dic₃	—	3	2	no	—
46	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
47	C₄×C₃⋊Dic₃	—	6	2	no	—
48	C₂×Q₈×C₉	—	2	6	no	—
49	C₂×Q₈×C₉	—	2	6	no	—
50	C₂×Q₈×C₉	Dic₃	1	12	yes	complemented
51	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
52	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
53	C₂²×Dic₉	—	3	6	no	—
54	C₂²×Dic₉	—	3	6	no	—
55	C₄⋊Dic₉	—	6	6	no	—
56	C₄×Dic₉	—	6	6	no	—
57	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
61	C₂²×C₃×C₉	#41	1	2	yes	—
62	C₃×C₄×C₉	—	2	4	no	—
63	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	—
64	C₃⋊Dic₉	—	2	4	no	—
65	C₃⋊Dic₉	—	4	2	no	—
66	C₂²×C₃×Q₈	D₉	1	3	yes	—
67	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
68	C₂×C₄⋊Dic₃	—	9	3	no	—
69	C₂×C₄⋊Dic₃	—	9	1	no	—
70	C₂×C₄×Dic₃	—	9	3	no	—
71	C₂×C₄×Dic₃	—	9	1	no	—
72	C₃⋊#95	—	9	12	no	—
73	C₃⋊#95	—	9	4	no	—
74	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
75	C₃²×Q₈	—	2	8	no	—
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
79	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
80	C₂×C₃²×C₄	#67	1	2	yes	—
81	C₂×C₃²×C₄	#67	1	2	yes	—
82	C₂×C₃⋊Dic₃	—	3	4	no	—
83	C₂×C₃⋊Dic₃	—	6	2	no	—
84	C₂×C₃⋊Dic₃	—	6	2	no	—
85	C₂×C₃⋊Dic₃	—	6	4	no	—
86	C₂×C₃⋊Dic₃	—	6	2	no	—
87	C₂³×C₉	C₂²×D₃	1	3	yes	—
88	Q₈×C₉	—	2	24	no	—
89	C₂×C₄×C₉	—	2	6	no	—
90	C₂×C₄×C₉	—	2	6	no	—
91	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
92	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
93	C₂×C₄×C₉	#67	1	6	yes	—
94	C₂×C₄×C₉	#67	1	6	yes	—
95	C₂×Dic₉	—	3	12	no	—
96	C₂×Dic₉	—	6	6	no	—
97	C₂×Dic₉	—	6	6	no	—
98	C₂×Dic₉	—	6	12	no	—
99	C₂×Dic₉	—	6	6	no	—
100	#391	—	27	1	no	Sylow
101	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₃×C₉	C₄×Q₈	1	2	yes	—
103	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₃×C₉	#124	1	1	yes	characteristic
105	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₃×Q₈	—	2	6	no	—
107	C₂×C₃×Q₈	—	2	6	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	2	no	—
110	C₂×C₃×Q₈	Dic₉	1	12	yes	complemented
111	C₂×C₃×Q₈	C₃⋊Dic₃	1	4	yes	—
112	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
113	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
114	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
116	C₂²×Dic₃	—	9	6	no	—
117	C₂²×Dic₃	—	9	2	no	—
118	C₂²×Dic₃	—	9	6	no	—
119	C₂²×Dic₃	—	9	2	no	—
120	C₄⋊Dic₃	—	18	6	no	—
121	C₄⋊Dic₃	—	18	2	no	—
122	C₄×Dic₃	—	18	6	no	—
123	C₄×Dic₃	—	18	2	no	—
124	C₂²×C₃²	D₃×Q₈	1	2	yes	—
125	C₂²×C₃²	#239	1	2	yes	—
126	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
127	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
128	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
129	C₃²×C₄	—	2	4	no	—
130	C₃²×C₄	#217	1	4	yes	—
131	C₃⋊Dic₃	—	6	4	no	—
132	C₃⋊Dic₃	—	12	2	no	—
133	C₂²×C₉	D₃×Q₈	1	6	yes	—
134	C₂²×C₉	#239	1	6	yes	—
135	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
136	C₂²×C₉	C₂×#67	1	3	yes	—
137	C₂²×C₉	C₂×#67	1	3	yes	—
138	C₄×C₉	—	2	12	no	—
139	C₄×C₉	#217	1	12	yes	—
140	Dic₉	—	6	12	no	—
141	Dic₉	—	12	6	no	—
142	C₂²×Q₈	C₃⋊D₉	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	27	1	no	—
144	C₂×C₄²	—	27	1	no	—
145	#95	—	27	4	no	—
146	C₃×C₉	#391	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₃	C₂²×D₉	1	3	yes	—
148	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
149	C₃×Q₈	—	2	24	no	—
150	C₃×Q₈	—	2	8	no	—
151	C₂×C₃×C₄	—	2	6	no	—
152	C₂×C₃×C₄	—	2	6	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	—
156	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
157	C₂×C₃×C₄	#611	1	6	yes	—
158	C₂×C₃×C₄	#611	1	6	yes	—
159	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
160	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
161	C₂×C₃×C₄	#638	1	2	yes	—
162	C₂×C₃×C₄	#638	1	2	yes	—
163	C₂×Dic₃	—	9	12	no	—
164	C₂×Dic₃	—	9	4	no	—
165	C₂×Dic₃	—	18	6	no	—
166	C₂×Dic₃	—	18	6	no	—
167	C₂×Dic₃	—	18	2	no	—
168	C₂×Dic₃	—	18	2	no	—
169	C₂×Dic₃	—	18	12	no	—
170	C₂×Dic₃	—	18	6	no	—
171	C₂×Dic₃	—	18	4	no	—
172	C₂×Dic₃	—	18	2	no	—
173	C₂×C₃²	#944	1	1	yes	characteristic
174	C₂×C₃²	Q₈×Dic₃	1	2	yes	—
175	C₂×C₃²	#946	1	2	yes	—
176	C₂×C₃²	#947	1	1	yes	characteristic
177	C₂×C₃²	C₂×#217	1	1	yes	characteristic
178	C₂×C₉	#944	1	3	yes	—
179	C₂×C₉	Q₈×Dic₃	1	6	yes	—
180	C₂×C₉	#946	1	6	yes	—
181	C₂×C₉	#947	1	3	yes	—
182	C₂×C₉	C₂×#217	1	3	yes	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	C₃⋊Dic₉	1	4	yes	complemented
186	C₂²×C₄	—	27	2	no	—
187	C₂²×C₄	—	27	2	no	—
188	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
189	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
190	C₄⋊C₄	—	54	2	no	—
191	C₄²	—	54	2	no	—
192	C₂²×C₃	Q₈×D₉	1	6	yes	—
193	C₂²×C₃	#3701	1	6	yes	—
194	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
195	C₂²×C₃	C₂×#611	1	3	yes	—
196	C₂²×C₃	C₂×#611	1	3	yes	—
197	C₂²×C₃	Q₈×C₃⋊D₃	1	2	yes	—
198	C₂²×C₃	#3832	1	2	yes	—
199	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
200	C₂²×C₃	C₂×#638	1	1	yes	characteristic
201	C₂²×C₃	C₂×#638	1	1	yes	characteristic
202	C₃×C₄	—	2	12	no	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	#3676	1	12	yes	—
205	C₃×C₄	#3757	1	4	yes	—
206	Dic₃	—	18	12	no	—
207	Dic₃	—	18	4	no	—
208	Dic₃	—	36	6	no	—
209	Dic₃	—	36	2	no	—
210	C₃²	#5229	1	1	yes	characteristic
211	C₉	#5229	1	3	yes	complemented
212	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
213	Q₈	—	2	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	27	4	no	—
217	C₂×C₄	—	54	2	no	—
218	C₂×C₄	—	54	2	no	—
219	C₂×C₄	—	54	4	no	—
220	C₂×C₄	—	54	2	no	—
221	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	—
222	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
223	C₂×C₄	#6233	1	2	yes	—
224	C₂×C₄	#6233	1	2	yes	—
225	C₂×C₃	#63506	1	3	yes	—
226	C₂×C₃	Q₈×Dic₉	1	6	yes	—
227	C₂×C₃	#63508	1	6	yes	—
228	C₂×C₃	#63509	1	3	yes	—
229	C₂×C₃	C₂×#3676	1	3	yes	—
230	C₂×C₃	#64153	1	1	yes	characteristic
231	C₂×C₃	Q₈×C₃⋊Dic₃	1	2	yes	—
232	C₂×C₃	#64155	1	2	yes	—
233	C₂×C₃	#64156	1	1	yes	characteristic
234	C₂×C₃	C₂×#3757	1	1	yes	characteristic
235	C₂²	Q₈×C₃⋊D₉	1	2	yes	—
236	C₂²	#88746	1	2	yes	—
237	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
238	C₂²	C₂×#6233	1	1	yes	characteristic
239	C₂²	C₂×#6233	1	1	yes	characteristic
240	C₄	—	2	4	no	—
241	C₄	—	54	4	no	—
242	C₄	—	108	2	no	—
243	C₄	#88551	1	4	yes	—
244	C₃	#10588909	1	3	yes	complemented
245	C₃	#10592928	1	1	yes	characteristic
246	C₂	#11721854	1	1	yes	characteristic
247	C₂	Q₈×C₃⋊Dic₉	1	2	yes	—
248	C₂	#11721856	1	2	yes	—
249	C₂	#11721857	1	1	yes	characteristic
250	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂²×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#391
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
9. C₂×C₃×Q₈×C₉	C₄	semidirect
36. C₂×C₃⋊Dic₉	Q₈	semidirect
50. C₂×Q₈×C₉	Dic₃	semidirect
110. C₂×C₃×Q₈	Dic₉	semidirect
146. C₃×C₉	#391	semidirect
185. C₂×Q₈	C₃⋊Dic₉	semidirect
211. C₉	#5229	semidirect
244. C₃	#10588909	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈×C₉	2
C₂×C₄⋊(C₃⋊Dic₉)	2
C₂×C₄×C₃⋊Dic₉	2
#11720572	2
#10592928	3
#10588909	3