Subgroups — #13144320

Order 1728.

930 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
3	C₂×#88534	C₂	1	4	yes	maximal
4	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
5	#10592674	—	3	1	no	maximal
6	#10588655	—	3	3	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
10	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	—	2	2	no	—
12	C₄⋊(C₃⋊Dic₉)	—	2	2	no	—
13	#88534	—	2	8	no	—
14	C₄×C₃⋊Dic₉	C₄	1	4	yes	complemented
15	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
16	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
17	C₂×#3750	—	3	4	no	—
18	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
19	C₂×C₉×C₄⋊C₄	D₃	1	3	yes	—
20	C₂×C₄⋊Dic₉	—	3	3	no	—
21	C₂×Dic₉⋊C₄	—	3	12	no	—
22	C₂×C₄×Dic₉	—	3	3	no	—
23	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	—	2	4	no	—
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
29	C₂×C₃⋊Dic₉	—	2	4	no	—
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
32	C₂×C₃⋊Dic₉	D₄	1	4	yes	—
33	C₂×C₃⋊Dic₉	Q₈	1	4	yes	—
34	#4975	—	9	3	no	—
35	#4975	—	9	1	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	—	3	2	no	—
39	C₂²×C₃⋊Dic₃	—	3	2	no	—
40	C₃²×C₄⋊C₄	—	2	2	no	—
41	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
42	#3750	—	6	8	no	—
43	C₄×C₃⋊Dic₃	—	3	4	no	—
44	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
45	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
46	C₂²×Dic₉	—	3	6	no	—
47	C₂²×Dic₉	—	3	6	no	—
48	C₉×C₄⋊C₄	—	2	6	no	—
49	C₄⋊Dic₉	—	6	6	no	—
50	Dic₉⋊C₄	—	6	24	no	—
51	C₄×Dic₉	—	3	12	no	—
52	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
54	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
57	C₃×C₄×C₉	—	4	2	no	—
58	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	complemented
59	C₃⋊Dic₉	—	4	2	no	—
60	C₃⋊Dic₉	C₄⋊C₄	1	8	yes	complemented
61	C₂×C₃×C₄⋊C₄	D₉	1	3	yes	—
62	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
63	C₂×C₄⋊Dic₃	—	9	3	no	—
64	C₂×C₄⋊Dic₃	—	9	1	no	—
65	C₂×Dic₃⋊C₄	—	9	12	no	—
66	C₂×Dic₃⋊C₄	—	9	4	no	—
67	C₂×C₄×Dic₃	—	9	3	no	—
68	C₂×C₄×Dic₃	—	9	1	no	—
69	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	—	2	4	no	—
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	Dic₆	1	2	yes	—
73	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
74	C₂×C₃²×C₄	#67	1	2	yes	—
75	C₂×C₃⋊Dic₃	—	3	4	no	—
76	C₂×C₃⋊Dic₃	—	3	4	no	—
77	C₂×C₃⋊Dic₃	—	3	4	no	—
78	C₂×C₃⋊Dic₃	—	6	4	no	—
79	C₂×C₃⋊Dic₃	—	6	2	no	—
80	C₂³×C₉	C₂²×D₃	1	3	yes	—
81	C₂×C₄×C₉	—	2	12	no	—
82	C₂×C₄×C₉	—	2	6	no	—
83	C₂×C₄×C₉	Dic₆	1	6	yes	—
84	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
85	C₂×C₄×C₉	#67	1	6	yes	—
86	C₂×Dic₉	—	3	12	no	—
87	C₂×Dic₉	—	3	12	no	—
88	C₂×Dic₉	—	3	12	no	—
89	C₂×Dic₉	—	6	12	no	—
90	C₂×Dic₉	—	6	6	no	—
91	#384	—	27	1	no	Sylow
92	C₂×C₃×C₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₉	#127	1	1	yes	characteristic
95	C₂×C₃×C₉	C₄⋊Q₈	1	2	yes	—
96	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
98	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
99	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
100	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	9	6	no	—
102	C₂²×Dic₃	—	9	2	no	—
103	C₂²×Dic₃	—	9	6	no	—
104	C₂²×Dic₃	—	9	2	no	—
105	C₃×C₄⋊C₄	—	2	6	no	—
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₄⋊Dic₃	—	18	6	no	—
108	C₄⋊Dic₃	—	18	2	no	—
109	Dic₃⋊C₄	—	18	8	no	—
110	Dic₃⋊C₄	—	18	24	no	—
111	C₄×Dic₃	—	9	12	no	—
112	C₄×Dic₃	—	9	4	no	—
113	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
114	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
115	C₂²×C₃²	D₃×D₄	1	2	yes	—
116	C₂²×C₃²	D₃×Q₈	1	2	yes	—
117	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
118	C₃²×C₄	—	4	2	no	—
119	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
120	C₃⋊Dic₃	—	3	8	no	—
121	C₃⋊Dic₃	—	12	2	no	—
122	C₂²×C₉	C₂×Dic₆	1	3	yes	—
123	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
124	C₂²×C₉	D₃×D₄	1	6	yes	—
125	C₂²×C₉	D₃×Q₈	1	6	yes	—
126	C₂²×C₉	C₂×#67	1	3	yes	—
127	C₄×C₉	—	4	6	no	—
128	C₄×C₉	Dic₃⋊C₄	1	12	yes	complemented
129	Dic₉	—	3	24	no	—
130	Dic₉	—	12	6	no	—
131	C₂×C₄⋊C₄	—	27	4	no	—
132	C₂×C₄⋊C₄	—	27	1	no	—
133	C₂×C₄⋊C₄	C₃⋊D₉	1	1	yes	characteristic
134	C₂×C₄²	—	27	1	no	—
135	C₃×C₉	#384	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₃	C₂²×D₉	1	3	yes	—
137	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	12	no	—
139	C₂×C₃×C₄	—	2	6	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
143	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
144	C₂×C₃×C₄	#611	1	6	yes	—
145	C₂×C₃×C₄	#634	1	2	yes	—
146	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
147	C₂×C₃×C₄	#638	1	2	yes	—
148	C₂×Dic₃	—	9	12	no	—
149	C₂×Dic₃	—	9	12	no	—
150	C₂×Dic₃	—	9	12	no	—
151	C₂×Dic₃	—	9	4	no	—
152	C₂×Dic₃	—	9	4	no	—
153	C₂×Dic₃	—	9	4	no	—
154	C₂×Dic₃	—	18	12	no	—
155	C₂×Dic₃	—	18	6	no	—
156	C₂×Dic₃	—	18	4	no	—
157	C₂×Dic₃	—	18	2	no	—
158	C₂×C₃²	#888	1	2	yes	—
159	C₂×C₃²	D₃×C₄⋊C₄	1	2	yes	—
160	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
161	C₂×C₃²	#940	1	1	yes	characteristic
162	C₂×C₃²	#944	1	1	yes	characteristic
163	C₂×C₉	#888	1	6	yes	—
164	C₂×C₉	D₃×C₄⋊C₄	1	6	yes	—
165	C₂×C₉	C₂×Dic₃⋊C₄	1	3	yes	—
166	C₂×C₉	#940	1	3	yes	—
167	C₂×C₉	#944	1	3	yes	—
168	C₂²×C₄	—	27	2	no	—
169	C₂²×C₄	—	27	2	no	—
170	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
171	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	54	8	no	—
174	C₄⋊C₄	—	54	2	no	—
175	C₄²	—	27	4	no	—
176	C₂²×C₃	C₂×Dic₁₈	1	3	yes	—
177	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
178	C₂²×C₃	D₄×D₉	1	6	yes	—
179	C₂²×C₃	Q₈×D₉	1	6	yes	—
180	C₂²×C₃	C₂×#611	1	3	yes	—
181	C₂²×C₃	C₂×#634	1	1	yes	characteristic
182	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
183	C₂²×C₃	D₄×C₃⋊D₃	1	2	yes	—
184	C₂²×C₃	Q₈×C₃⋊D₃	1	2	yes	—
185	C₂²×C₃	C₂×#638	1	1	yes	characteristic
186	C₃×C₄	—	4	6	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	Dic₉⋊C₄	1	12	yes	complemented
189	C₃×C₄	#3750	1	4	yes	—
190	Dic₃	—	9	8	no	—
191	Dic₃	—	9	24	no	—
192	Dic₃	—	36	6	no	—
193	Dic₃	—	36	2	no	—
194	C₃²	#4975	1	1	yes	characteristic
195	C₉	#4975	1	3	yes	complemented
196	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	27	4	no	—
200	C₂×C₄	—	27	4	no	—
201	C₂×C₄	—	27	4	no	—
202	C₂×C₄	—	54	4	no	—
203	C₂×C₄	—	54	2	no	—
204	C₂×C₄	#6226	1	2	yes	—
205	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
206	C₂×C₄	#6233	1	2	yes	—
207	C₂×C₃	#63450	1	6	yes	—
208	C₂×C₃	C₄⋊C₄×D₉	1	6	yes	—
209	C₂×C₃	C₂×Dic₉⋊C₄	1	3	yes	—
210	C₂×C₃	#63502	1	3	yes	—
211	C₂×C₃	#63506	1	3	yes	—
212	C₂×C₃	#64097	1	2	yes	—
213	C₂×C₃	C₄⋊C₄×C₃⋊D₃	1	2	yes	—
214	C₂×C₃	C₂×#3750	1	1	yes	characteristic
215	C₂×C₃	#64149	1	1	yes	characteristic
216	C₂×C₃	#64153	1	1	yes	characteristic
217	C₂²	C₂×#6226	1	1	yes	characteristic
218	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
219	C₂²	D₄×C₃⋊D₉	1	2	yes	—
220	C₂²	Q₈×C₃⋊D₉	1	2	yes	—
221	C₂²	C₂×#6233	1	1	yes	characteristic
222	C₄	—	4	2	no	—
223	C₄	—	27	8	no	—
224	C₄	—	108	2	no	—
225	C₄	#88534	1	4	yes	complemented
226	C₃	#10588655	1	3	yes	complemented
227	C₃	#10592674	1	1	yes	characteristic
228	C₂	#11721744	1	2	yes	—
229	C₂	C₄⋊C₄×C₃⋊D₉	1	2	yes	—
230	C₂	C₂×#88534	1	1	yes	characteristic
231	C₂	#11721840	1	1	yes	characteristic
232	C₂	#11721854	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	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂×C₄⋊C₄	C₃⋊D₉
hypocenter	C₃×C₉	#384
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. C₄×C₃⋊Dic₉	C₄	semidirect
58. C₃×C₄×C₉	C₄⋊C₄	semidirect
60. C₃⋊Dic₉	C₄⋊C₄	semidirect
128. C₄×C₉	Dic₃⋊C₄	semidirect
135. C₃×C₉	#384	semidirect
188. C₃×C₄	Dic₉⋊C₄	semidirect
195. C₉	#4975	semidirect
225. C₄	#88534	semidirect
226. C₃	#10588655	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₄⋊(C₃⋊Dic₉)	2
C₂×#88534	2
C₂×C₄×C₃⋊Dic₉	2
#10592674	3
#10588655	3