Subgroups — C₂×#11720525 #13144308

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 228 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	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11720525	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63634	—	3	1	no	maximal
6	C₂×#63366	—	3	3	no	maximal
7	C₂²×C₃×C₄×C₉	—	2	1	no	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	—	2	1	no	—
10	C₃×C₉×C₄⋊C₄	—	2	1	no	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
13	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
14	C₂×C₃⋊(C₉⋊C₈)	C₄	1	4	yes	complemented
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
17	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
18	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
19	#63634	—	3	4	no	—
20	C₂×C₉×C₄⋊C₄	D₃	1	3	yes	—
21	C₂×C₄⋊Dic₉	—	3	3	no	—
22	C₂²×C₉⋊C₈	—	3	3	no	—
23	#63366	—	3	12	no	—
24	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	—	4	2	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₃⋊Dic₉	—	2	2	no	—
32	C₂×C₃⋊Dic₉	—	4	2	no	—
33	C₃⋊(C₉⋊C₈)	C₂×C₄	1	4	yes	complemented
34	C₂×#807	—	9	3	no	—
35	C₂×#807	—	9	1	no	—
36	C₂²×C₃²×C₄	—	2	1	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	—	6	1	no	—
39	C₃²×C₄⋊C₄	—	2	1	no	—
40	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
41	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
42	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
44	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
45	C₂²×C₄×C₉	—	2	3	no	—
46	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
47	C₂²×Dic₉	—	6	3	no	—
48	C₉×C₄⋊C₄	—	2	3	no	—
49	C₉×C₄⋊C₄	C₂×D₃	1	6	yes	—
50	C₄⋊Dic₉	—	3	6	no	—
51	C₄⋊Dic₉	—	6	3	no	—
52	C₂×C₉⋊C₈	—	3	12	no	—
53	C₂×C₉⋊C₈	—	3	6	no	—
54	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₃×C₉	D₈	1	2	yes	—
56	C₂²×C₃×C₉	Q₁₆	1	2	yes	—
57	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
58	C₃×C₄×C₉	—	4	2	no	—
59	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
60	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
62	C₃⋊Dic₉	—	4	2	no	—
63	C₂×C₃×C₄⋊C₄	D₉	1	3	yes	—
64	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
65	C₂×C₄⋊Dic₃	—	9	3	no	—
66	C₂×C₄⋊Dic₃	—	9	1	no	—
67	C₂²×C₃⋊C₈	—	9	3	no	—
68	C₂²×C₃⋊C₈	—	9	1	no	—
69	#807	—	9	12	no	—
70	#807	—	9	4	no	—
71	C₂³×C₃²	#67	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	—	4	2	no	—
74	C₂×C₃²×C₄	Dic₆	1	2	yes	—
75	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
76	C₂×C₃²×C₄	#67	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	—	6	2	no	—
79	C₂×C₃⋊Dic₃	—	12	2	no	—
80	C₂³×C₉	#67	1	3	yes	—
81	C₃⋊(C₃⋊C₈)	—	3	4	no	—
82	C₂×C₄×C₉	—	2	6	no	—
83	C₂×C₄×C₉	—	4	6	no	—
84	C₂×C₄×C₉	Dic₆	1	6	yes	—
85	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
86	C₂×C₄×C₉	#67	1	3	yes	—
87	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
88	C₂×Dic₉	—	6	6	no	—
89	C₂×Dic₉	—	12	6	no	—
90	C₉⋊C₈	—	3	12	no	—
91	C₂×#107	—	27	1	no	Sylow
92	C₂×C₃×C₉	#107	1	4	yes	complemented
93	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
96	C₂²×C₃×C₄	—	2	3	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
99	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
100	C₂²×Dic₃	—	18	3	no	—
101	C₂²×Dic₃	—	18	1	no	—
102	C₃×C₄⋊C₄	—	2	3	no	—
103	C₃×C₄⋊C₄	—	2	1	no	—
104	C₃×C₄⋊C₄	C₂×D₉	1	6	yes	—
105	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
106	C₄⋊Dic₃	—	9	6	no	—
107	C₄⋊Dic₃	—	9	2	no	—
108	C₄⋊Dic₃	—	18	3	no	—
109	C₄⋊Dic₃	—	18	1	no	—
110	C₂×C₃⋊C₈	—	9	12	no	—
111	C₂×C₃⋊C₈	—	9	6	no	—
112	C₂×C₃⋊C₈	—	9	4	no	—
113	C₂×C₃⋊C₈	—	9	2	no	—
114	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
115	C₂²×C₃²	#213	1	2	yes	—
116	C₂²×C₃²	#216	1	2	yes	—
117	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
118	C₃²×C₄	—	4	2	no	—
119	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
120	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
121	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
122	C₃⋊Dic₃	—	12	2	no	—
123	C₂²×C₉	Dic₃⋊C₄	1	6	yes	—
124	C₂²×C₉	#213	1	6	yes	—
125	C₂²×C₉	#216	1	6	yes	—
126	C₂²×C₉	C₂×#67	1	3	yes	—
127	C₄×C₉	—	4	6	no	—
128	C₄×C₉	Dic₃⋊C₄	1	6	yes	—
129	C₄×C₉	C₂×Dic₆	1	3	yes	—
130	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
131	Dic₉	—	12	6	no	—
132	C₂²×C₈	—	27	1	no	—
133	C₂×C₄⋊C₄	—	27	1	no	—
134	C₂×C₄⋊C₄	C₃⋊D₉	1	1	yes	characteristic
135	#107	—	27	4	no	—
136	C₃×C₉	C₂×#107	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₃	#611	1	3	yes	—
138	C₂³×C₃	#638	1	1	yes	characteristic
139	C₂×C₃×C₄	—	2	6	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	4	6	no	—
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
144	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
145	C₂×C₃×C₄	#611	1	3	yes	—
146	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
147	C₂×C₃×C₄	#634	1	2	yes	—
148	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
149	C₂×C₃×C₄	#638	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
151	C₂×Dic₃	—	18	6	no	—
152	C₂×Dic₃	—	18	2	no	—
153	C₂×Dic₃	—	36	6	no	—
154	C₂×Dic₃	—	36	2	no	—
155	C₃⋊C₈	—	9	12	no	—
156	C₃⋊C₈	—	9	4	no	—
157	C₂×C₃²	#807	1	4	yes	—
158	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
159	C₂×C₃²	C₂×#213	1	1	yes	characteristic
160	C₂×C₃²	C₂×#216	1	1	yes	characteristic
161	C₂×C₉	#807	1	12	yes	complemented
162	C₂×C₉	C₂×Dic₃⋊C₄	1	3	yes	—
163	C₂×C₉	C₂×#213	1	3	yes	—
164	C₂×C₉	C₂×#216	1	3	yes	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	54	1	no	—
167	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
168	C₂×C₈	—	27	4	no	—
169	C₂×C₈	—	27	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	27	2	no	—
172	C₄⋊C₄	—	54	1	no	—
173	C₄⋊C₄	C₂×C₃⋊D₉	1	2	yes	—
174	C₂²×C₃	Dic₉⋊C₄	1	6	yes	—
175	C₂²×C₃	#3673	1	6	yes	—
176	C₂²×C₃	#3674	1	6	yes	—
177	C₂²×C₃	C₂×#611	1	3	yes	—
178	C₂²×C₃	#3750	1	2	yes	—
179	C₂²×C₃	#3753	1	2	yes	—
180	C₂²×C₃	#3756	1	2	yes	—
181	C₂²×C₃	C₂×#638	1	1	yes	characteristic
182	C₃×C₄	—	4	6	no	—
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	Dic₉⋊C₄	1	6	yes	—
185	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
186	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
187	C₃×C₄	#3750	1	2	yes	—
188	C₃×C₄	C₂×#634	1	1	yes	characteristic
189	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
190	Dic₃	—	36	6	no	—
191	Dic₃	—	36	2	no	—
192	C₃²	C₂×#807	1	1	yes	characteristic
193	C₉	C₂×#807	1	3	yes	complemented
194	C₂³	#6233	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	54	2	no	—
198	C₂×C₄	—	108	2	no	—
199	C₂×C₄	#6226	1	2	yes	—
200	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
201	C₂×C₄	#6233	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
203	C₈	—	27	4	no	—
204	C₂×C₃	#63366	1	12	yes	complemented
205	C₂×C₃	C₂×Dic₉⋊C₄	1	3	yes	—
206	C₂×C₃	C₂×#3673	1	3	yes	—
207	C₂×C₃	C₂×#3674	1	3	yes	—
208	C₂×C₃	#63634	1	4	yes	—
209	C₂×C₃	C₂×#3750	1	1	yes	characteristic
210	C₂×C₃	C₂×#3753	1	1	yes	characteristic
211	C₂×C₃	C₂×#3756	1	1	yes	characteristic
212	C₂²	#88534	1	2	yes	—
213	C₂²	#88544	1	2	yes	—
214	C₂²	#88547	1	2	yes	—
215	C₂²	C₂×#6233	1	1	yes	characteristic
216	C₄	—	4	2	no	—
217	C₄	—	108	2	no	—
218	C₄	#88534	1	2	yes	—
219	C₄	C₂×#6226	1	1	yes	characteristic
220	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
221	C₃	C₂×#63366	1	3	yes	complemented
222	C₃	C₂×#63634	1	1	yes	characteristic
223	C₂	#11720525	1	4	yes	direct product factor
224	C₂	C₂×#88534	1	1	yes	characteristic
225	C₂	C₂×#88544	1	1	yes	characteristic
226	C₂	C₂×#88547	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6233
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₄	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#107
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #11720525	C₂	direct (N × H)
14. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
33. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
92. C₂×C₃×C₉	#107	semidirect
136. C₃×C₉	C₂×#107	semidirect
161. C₂×C₉	#807	semidirect
193. C₉	C₂×#807	semidirect
204. C₂×C₃	#63366	semidirect
221. C₃	C₂×#63366	semidirect
223. C₂	#11720525	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×C₃×C₉×C₄⋊C₄	2
C₂×C₄⋊(C₃⋊Dic₉)	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720525	2
C₂×#63634	3
C₂×#63366	3

Subgroups — C2×#11720525 #13144308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720525 #13144308