Subgroups — C₂×#11720271 #13139349

Order 1728.

594 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720271	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63553	—	3	1	no	maximal
6	C₂×C₈×Dic₉	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₄	1	2	yes	—
9	C₄×C₃⋊Dic₉	C₄	1	2	yes	—
10	C₄×C₃⋊Dic₉	C₂²	1	2	yes	—
11	C₂×C₉×C₃⋊C₈	C₄	1	4	yes	complemented
12	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃×C₉⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
16	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
17	C₂²×C₃×C₃⋊C₈	D₃	1	1	yes	characteristic
18	#63553	—	3	4	no	—
19	C₂²×C₈×C₉	—	3	1	no	—
20	C₂×C₄×Dic₉	—	3	1	no	—
21	C₂×C₄×Dic₉	—	6	1	no	—
22	C₂²×C₉⋊C₈	D₃	1	1	yes	characteristic
23	C₈×Dic₉	—	3	4	no	—
24	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊Dic₉	C₈	1	8	yes	complemented
30	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
31	C₉×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
32	C₃×C₉⋊C₈	C₂×C₄	1	4	yes	complemented
33	C₂×C₈×Dic₃	—	9	1	no	—
34	C₂×C₈×Dic₃	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₃⋊Dic₃	—	3	2	no	—
37	C₄×C₃⋊Dic₃	—	3	2	no	—
38	C₄×C₃⋊Dic₃	—	3	2	no	—
39	C₂×C₃×C₃⋊C₈	—	3	4	no	—
40	C₂×C₃×C₃⋊C₈	—	3	2	no	—
41	C₂×C₃×C₃⋊C₈	Dic₃	1	4	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
43	C₂²×C₄×C₉	—	2	1	no	—
44	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
45	C₂²×Dic₉	—	3	2	no	—
46	C₂²×Dic₉	—	6	2	no	—
47	C₂×C₈×C₉	—	3	4	no	—
48	C₂×C₈×C₉	—	3	2	no	—
49	C₄×Dic₉	—	3	2	no	—
50	C₄×Dic₉	—	3	2	no	—
51	C₄×Dic₉	—	6	2	no	—
52	C₄×Dic₉	—	6	2	no	—
53	C₂×C₉⋊C₈	Dic₃	1	4	yes	complemented
54	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
55	C₂²×C₃×C₉	C₄²	1	2	yes	—
56	C₂²×C₃×C₉	C₂×C₈	1	4	yes	—
57	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₄×C₉	C₄²	1	2	yes	—
59	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₃⋊Dic₉	C₂×C₈	1	8	yes	complemented
62	C₂²×C₃×C₈	—	3	1	no	—
63	C₂²×C₃×C₈	—	9	1	no	—
64	C₂×C₄×Dic₃	—	9	1	no	—
65	C₂×C₄×Dic₃	—	9	1	no	—
66	C₂×C₄×Dic₃	—	18	1	no	—
67	C₂²×C₃⋊C₈	—	3	1	no	—
68	C₂²×C₃⋊C₈	D₉	1	1	yes	characteristic
69	C₈×Dic₃	—	9	4	no	—
70	C₈×Dic₃	—	9	4	no	—
71	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
73	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
75	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₂×C₃⋊Dic₃	—	3	8	no	—
77	C₂×C₃⋊Dic₃	—	3	4	no	—
78	C₂³×C₉	—	2	1	no	—
79	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
80	C₃×C₃⋊C₈	—	3	4	no	—
81	C₃×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
87	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
89	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
90	C₂×Dic₉	—	3	8	no	—
91	C₂×Dic₉	—	3	4	no	—
92	C₂×Dic₉	—	6	4	no	—
93	C₂×Dic₉	—	6	8	no	—
94	C₈×C₉	—	3	4	no	—
95	C₉⋊C₈	C₂×Dic₃	1	4	yes	complemented
96	C₂×C₄×C₈	—	27	1	no	Sylow
97	C₂×C₃×C₉	C₄×C₈	1	4	yes	complemented
98	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₃×C₉	C₂²×C₈	1	2	yes	—
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
102	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
103	C₂²×Dic₃	—	9	2	no	—
104	C₂²×Dic₃	—	9	2	no	—
105	C₂²×Dic₃	—	18	2	no	—
106	C₂×C₃×C₈	—	3	4	no	—
107	C₂×C₃×C₈	—	3	2	no	—
108	C₂×C₃×C₈	—	9	4	no	—
109	C₂×C₃×C₈	—	9	2	no	—
110	C₄×Dic₃	—	9	2	no	—
111	C₄×Dic₃	—	9	2	no	—
112	C₄×Dic₃	—	9	2	no	—
113	C₄×Dic₃	—	9	2	no	—
114	C₄×Dic₃	—	18	2	no	—
115	C₄×Dic₃	—	18	2	no	—
116	C₂×C₃⋊C₈	—	3	4	no	—
117	C₂×C₃⋊C₈	—	3	2	no	—
118	C₂×C₃⋊C₈	Dic₉	1	4	yes	complemented
119	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
120	C₂²×C₃²	D₃×C₈	1	4	yes	—
121	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
122	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
123	C₃²×C₄	C₄×Dic₃	1	2	yes	—
124	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
125	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
126	C₃⋊Dic₃	—	3	8	no	—
127	C₂²×C₉	—	2	4	no	—
128	C₂²×C₉	—	2	2	no	—
129	C₂²×C₉	—	2	1	no	—
130	C₂²×C₉	D₃×C₈	1	4	yes	—
131	C₂²×C₉	C₄×Dic₃	1	2	yes	—
132	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	—	2	1	no	—
135	C₄×C₉	—	2	1	no	—
136	C₄×C₉	C₄×Dic₃	1	2	yes	—
137	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
138	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
139	Dic₉	—	3	8	no	—
140	Dic₉	—	6	8	no	—
141	C₂²×C₈	—	3	1	no	—
142	C₂²×C₈	—	9	1	no	—
143	C₂×C₄²	—	27	1	no	—
144	C₄×C₈	—	27	4	no	—
145	C₃×C₉	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
148	C₂³×C₃	#624	1	1	yes	characteristic
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
154	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
156	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
157	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
158	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
159	C₂×C₃×C₄	#624	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
161	C₂×Dic₃	—	9	8	no	—
162	C₂×Dic₃	—	9	4	no	—
163	C₂×Dic₃	—	9	8	no	—
164	C₂×Dic₃	—	9	4	no	—
165	C₂×Dic₃	—	18	4	no	—
166	C₂×Dic₃	—	18	8	no	—
167	C₃×C₈	—	3	4	no	—
168	C₃×C₈	—	9	4	no	—
169	C₃⋊C₈	—	3	4	no	—
170	C₃⋊C₈	C₂×Dic₉	1	4	yes	complemented
171	C₂×C₃²	C₈×Dic₃	1	4	yes	—
172	C₂×C₃²	C₂×D₃×C₈	1	2	yes	—
173	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
174	C₂×C₉	—	2	4	no	—
175	C₂×C₉	—	2	2	no	—
176	C₂×C₉	—	2	1	no	—
177	C₂×C₉	C₈×Dic₃	1	4	yes	complemented
178	C₂×C₉	C₂×D₃×C₈	1	2	yes	—
179	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
180	C₂²×C₄	—	27	2	no	—
181	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
182	C₂×C₈	—	3	4	no	—
183	C₂×C₈	—	3	2	no	—
184	C₂×C₈	—	9	4	no	—
185	C₂×C₈	—	9	2	no	—
186	C₄²	—	27	2	no	—
187	C₄²	—	27	2	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	C₈×D₉	1	4	yes	—
192	C₂²×C₃	C₄×Dic₉	1	2	yes	—
193	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
194	C₂²×C₃	#3710	1	4	yes	—
195	C₂²×C₃	Dic₃²	1	2	yes	—
196	C₂²×C₃	C₂×#624	1	1	yes	characteristic
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	C₄×Dic₉	1	2	yes	—
201	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
202	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
203	C₃×C₄	Dic₃²	1	2	yes	—
204	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
205	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
206	Dic₃	—	9	8	no	—
207	Dic₃	—	9	8	no	—
208	Dic₃	—	18	8	no	—
209	C₃²	C₂×C₈×Dic₃	1	1	yes	characteristic
210	C₉	—	2	1	no	—
211	C₉	C₂×C₈×Dic₃	1	1	yes	characteristic, complemented
212	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
213	C₂×C₄	—	27	4	no	—
214	C₂×C₄	—	27	8	no	—
215	C₂×C₄	Dic₃×D₉	1	2	yes	—
216	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
217	C₂×C₄	D₃×Dic₉	1	2	yes	—
218	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
219	C₈	—	3	4	no	—
220	C₈	—	9	4	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	4	no	—
223	C₂×C₃	—	2	2	no	—
224	C₂×C₃	C₈×Dic₉	1	4	yes	complemented
225	C₂×C₃	C₂×C₈×D₉	1	2	yes	—
226	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
227	C₂×C₃	#63553	1	4	yes	—
228	C₂×C₃	C₂×#3710	1	2	yes	—
229	C₂×C₃	C₂×Dic₃²	1	1	yes	characteristic
230	C₂²	(C₃⋊D₉)⋊C₈	1	4	yes	—
231	C₂²	Dic₃×Dic₉	1	2	yes	—
232	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
233	C₄	—	27	8	no	—
234	C₄	Dic₃×Dic₉	1	2	yes	—
235	C₄	C₂×Dic₃×D₉	1	1	yes	characteristic
236	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
237	C₃	—	2	1	no	—
238	C₃	C₂×C₈×Dic₉	1	1	yes	characteristic, complemented
239	C₃	C₂×#63553	1	1	yes	characteristic
240	C₂	#11720271	1	4	yes	direct product factor
241	C₂	C₂×(C₃⋊D₉)⋊C₈	1	2	yes	—
242	C₂	C₂×Dic₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×D₉
commutator, hypocenter	C₃×C₉	C₂×C₄×C₈
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720271	C₂	direct (N × H)
11. C₂×C₉×C₃⋊C₈	C₄	semidirect
13. C₂×C₃×C₉⋊C₈	C₄	semidirect
29. C₂×C₃⋊Dic₉	C₈	semidirect
31. C₉×C₃⋊C₈	C₂×C₄	semidirect
32. C₃×C₉⋊C₈	C₂×C₄	semidirect
53. C₂×C₉⋊C₈	Dic₃	semidirect
61. C₃⋊Dic₉	C₂×C₈	semidirect
95. C₉⋊C₈	C₂×Dic₃	semidirect
97. C₂×C₃×C₉	C₄×C₈	semidirect
118. C₂×C₃⋊C₈	Dic₉	semidirect
145. C₃×C₉	C₂×C₄×C₈	semidirect
170. C₃⋊C₈	C₂×Dic₉	semidirect
177. C₂×C₉	C₈×Dic₃	semidirect
211. C₉	C₂×C₈×Dic₃	semidirect
224. C₂×C₃	C₈×Dic₉	semidirect
238. C₃	C₂×C₈×Dic₉	semidirect
240. C₂	#11720271	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊Dic₉	2
C₂²×C₉×C₃⋊C₈	2
C₂²×C₃×C₉⋊C₈	2
#11720271	2
C₂×#63553	3
C₂×C₈×Dic₉	3

Subgroups — C2×#11720271 #13139349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720271 #13139349