Subgroups — C₂²×#88420 #13150041

Order 1728.

970 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88420	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#3711	—	3	1	no	maximal
6	C₂²×#3667	—	3	1	no	maximal
7	C₂³×D₃×C₉	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₉×Dic₃	C₄	1	1	yes	characteristic
10	C₂×C₄×D₃×C₉	C₄	1	6	yes	—
11	C₂×C₄×D₃×C₉	C₂²	1	6	yes	—
12	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	6	yes	complemented
13	C₂×C₃×C₉⋊C₈	C₂²	1	6	yes	complemented
14	#88420	C₂²	1	16	yes	direct product factor
15	C₂²×C₃×C₄×D₃	D₃	1	1	yes	characteristic
16	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
17	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
18	C₂×#3711	—	3	12	no	—
19	C₂³×C₄×C₉	—	3	1	no	—
20	C₂×#3667	—	3	12	no	—
21	C₂²×C₉⋊C₈	—	3	1	no	—
22	C₂²×C₉⋊C₈	—	6	1	no	—
23	C₂²×C₉⋊C₈	D₃	1	1	yes	characteristic, complemented
24	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₃×C₉	—	2	4	no	—
26	C₂²×D₃×C₉	C₂×C₄	1	6	yes	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	3	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
29	C₂×C₉×Dic₃	C₂×C₄	1	6	yes	—
30	C₄×D₃×C₉	C₂×C₄	1	12	yes	—
31	C₄×D₃×C₉	C₂³	1	4	yes	—
32	C₃⋊(C₉⋊C₈)	C₂³	1	4	yes	complemented
33	C₃×C₉⋊C₈	C₂³	1	4	yes	complemented
34	C₂²×#208	—	9	1	no	—
35	C₂²×ΓL(1,𝔽₂₅)	—	9	1	no	—
36	C₂³×C₃×D₃	Dic₃	1	1	yes	characteristic
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃×Dic₃	Dic₃	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	Dic₃	1	6	yes	—
40	C₂×C₃×C₄×D₃	C₂×D₃	1	6	yes	—
41	C₂⁴×C₉	—	3	1	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	6	no	—
43	C₂×C₃×C₃⋊C₈	—	3	6	no	—
44	#3711	—	3	16	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	—	3	1	no	—
47	C₂²×C₄×C₉	—	3	6	no	—
48	C₂²×C₄×C₉	—	3	6	no	—
49	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
50	#3667	—	3	16	no	—
51	C₂×C₉⋊C₈	—	3	6	no	—
52	C₂×C₉⋊C₈	—	6	6	no	—
53	C₂×C₉⋊C₈	C₂×D₃	1	6	yes	complemented
54	C₂²×C₃×C₉	M₁₆	1	4	yes	complemented
55	C₂²×C₃×C₉	C₂²×C₄	1	3	yes	—
56	C₂×D₃×C₉	—	2	12	no	—
57	C₂×D₃×C₉	C₂²×C₄	1	4	yes	—
58	C₃×C₄×C₉	C₂²×C₄	1	3	yes	—
59	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
60	C₉×Dic₃	C₂²×C₄	1	4	yes	—
61	C₂³×C₃×C₄	—	3	1	no	—
62	C₂²×C₄×D₃	D₉	1	1	yes	characteristic
63	C₂²×C₃×C₈	—	9	1	no	—
64	C₂×#208	—	9	12	no	—
65	C₂²×C₃⋊C₈	—	3	1	no	—
66	C₂²×C₃⋊C₈	—	9	1	no	—
67	C₂²×C₃⋊C₈	—	9	1	no	—
68	C₂²×C₃⋊C₈	—	18	1	no	—
69	C₂×ΓL(1,𝔽₂₅)	—	9	12	no	—
70	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
71	C₂²×C₃×D₃	—	2	4	no	—
72	C₂²×C₃×D₃	C₂×Dic₃	1	6	yes	—
73	C₂×C₃²×C₄	C₂×Dic₃	1	3	yes	—
74	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
75	C₂×C₃×Dic₃	C₂×Dic₃	1	6	yes	—
76	C₃×C₄×D₃	C₂×Dic₃	1	12	yes	—
77	C₃×C₄×D₃	C₂²×D₃	1	4	yes	—
78	C₂³×C₉	—	2	1	no	—
79	C₂³×C₉	—	3	6	no	—
80	C₂³×C₉	—	6	4	no	—
81	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
82	C₃⋊(C₃⋊C₈)	—	3	4	no	—
83	C₃×C₃⋊C₈	—	3	4	no	—
84	C₂×C₄×C₉	—	2	3	no	—
85	C₂×C₄×C₉	—	2	3	no	—
86	C₂×C₄×C₉	—	3	6	no	—
87	C₂×C₄×C₉	—	3	12	no	—
88	C₂×C₄×C₉	—	3	4	no	—
89	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
90	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
91	C₉⋊C₈	—	3	4	no	—
92	C₉⋊C₈	—	6	4	no	—
93	C₉⋊C₈	C₂²×D₃	1	4	yes	complemented
94	C₂²×M₁₆	—	27	1	no	Sylow
95	C₂×C₃×C₉	C₂×M₁₆	1	6	yes	complemented
96	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
97	D₃×C₉	—	2	4	no	—
98	C₂⁴×C₃	—	3	1	no	—
99	C₂³×D₃	Dic₉	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	—	3	1	no	—
102	C₂²×C₃×C₄	—	3	6	no	—
103	C₂²×C₃×C₄	—	3	6	no	—
104	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
105	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
106	C₂²×Dic₃	Dic₉	1	1	yes	characteristic
107	C₂×C₄×D₃	Dic₉	1	6	yes	—
108	C₂×C₄×D₃	C₂×D₉	1	6	yes	—
109	C₂×C₃×C₈	—	9	6	no	—
110	#208	—	9	16	no	—
111	C₂×C₃⋊C₈	—	3	6	no	—
112	C₂×C₃⋊C₈	—	9	6	no	—
113	C₂×C₃⋊C₈	—	9	6	no	—
114	C₂×C₃⋊C₈	—	18	6	no	—
115	ΓL(1,𝔽₂₅)	—	9	16	no	—
116	C₂²×C₃²	#208	1	4	yes	—
117	C₂²×C₃²	C₂²×Dic₃	1	3	yes	—
118	C₂×C₃×D₃	—	2	12	no	—
119	C₂×C₃×D₃	C₂²×Dic₃	1	4	yes	—
120	C₃²×C₄	C₂²×Dic₃	1	3	yes	—
121	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
122	C₃×Dic₃	C₂²×Dic₃	1	4	yes	—
123	C₂²×C₉	—	2	4	no	—
124	C₂²×C₉	—	2	3	no	—
125	C₂²×C₉	—	3	4	no	—
126	C₂²×C₉	—	6	12	no	—
127	C₂²×C₉	ΓL(1,𝔽₂₅)	1	4	yes	complemented
128	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
129	C₄×C₉	—	2	1	no	—
130	C₄×C₉	—	2	3	no	—
131	C₄×C₉	—	3	4	no	—
132	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
133	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
134	C₂³×C₄	—	3	1	no	—
135	C₂×M₁₆	—	27	12	no	—
136	C₂²×C₈	—	9	1	no	—
137	C₂²×C₈	—	27	1	no	—
138	C₃×C₉	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	—	3	6	no	—
141	C₂³×C₃	—	6	4	no	—
142	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
143	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
144	C₂²×D₃	—	2	4	no	—
145	C₂²×D₃	C₂×Dic₉	1	6	yes	—
146	C₂×C₃×C₄	—	2	3	no	—
147	C₂×C₃×C₄	—	2	3	no	—
148	C₂×C₃×C₄	—	3	6	no	—
149	C₂×C₃×C₄	—	3	4	no	—
150	C₂×C₃×C₄	—	3	12	no	—
151	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
152	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
153	C₂×C₃×C₄	D₃×Dic₃	1	3	yes	—
154	C₂×C₃×C₄	C₂×D₃²	1	3	yes	—
155	C₂×Dic₃	C₂×Dic₉	1	6	yes	—
156	C₄×D₃	C₂×Dic₉	1	12	yes	—
157	C₄×D₃	C₂²×D₉	1	4	yes	—
158	C₃×C₈	—	9	4	no	—
159	C₃⋊C₈	—	3	4	no	—
160	C₃⋊C₈	—	9	4	no	—
161	C₃⋊C₈	—	9	4	no	—
162	C₃⋊C₈	—	18	4	no	—
163	C₂×C₃²	C₂×#208	1	6	yes	—
164	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
165	C₃×D₃	—	2	4	no	—
166	C₂×C₉	—	2	1	no	—
167	C₂×C₉	—	2	6	no	—
168	C₂×C₉	—	6	4	no	—
169	C₂×C₉	C₂×ΓL(1,𝔽₂₅)	1	6	yes	complemented
170	C₂×C₉	C₂²×C₄×D₃	1	1	yes	characteristic
171	C₂⁴	—	3	1	no	—
172	C₂²×C₄	—	3	1	no	—
173	C₂²×C₄	—	3	6	no	—
174	C₂²×C₄	—	3	6	no	—
175	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
176	M₁₆	—	27	16	no	—
177	C₂×C₈	—	9	6	no	—
178	C₂×C₈	—	27	6	no	—
179	C₂²×C₃	—	2	3	no	—
180	C₂²×C₃	—	2	4	no	—
181	C₂²×C₃	—	3	4	no	—
182	C₂²×C₃	—	6	12	no	—
183	C₂²×C₃	#3667	1	4	yes	complemented
184	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
185	C₂²×C₃	#3711	1	4	yes	—
186	C₂²×C₃	C₂×D₃×Dic₃	1	3	yes	—
187	C₂×D₃	—	2	12	no	—
188	C₂×D₃	C₂²×Dic₉	1	4	yes	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	3	no	—
191	C₃×C₄	—	3	4	no	—
192	C₃×C₄	C₂²×Dic₉	1	3	yes	—
193	C₃×C₄	C₂³×D₉	1	1	yes	characteristic
194	C₃×C₄	C₂×D₃×Dic₃	1	3	yes	—
195	C₃×C₄	C₂²×D₃²	1	1	yes	characteristic
196	Dic₃	C₂²×Dic₉	1	4	yes	—
197	C₃²	C₂²×#208	1	1	yes	characteristic
198	C₉	—	2	1	no	—
199	C₉	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
200	C₂³	—	3	6	no	—
201	C₂³	—	6	4	no	—
202	C₂³	D₃×Dic₉	1	1	yes	characteristic
203	C₂×C₄	—	3	6	no	—
204	C₂×C₄	—	3	4	no	—
205	C₂×C₄	—	3	12	no	—
206	C₂×C₄	D₃×Dic₉	1	3	yes	—
207	C₂×C₄	C₂×D₃×D₉	1	3	yes	—
208	C₈	—	9	4	no	—
209	C₈	—	27	4	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	2	6	no	—
212	C₂×C₃	—	6	4	no	—
213	C₂×C₃	C₂×#3667	1	6	yes	complemented
214	C₂×C₃	C₂³×Dic₉	1	1	yes	characteristic
215	C₂×C₃	C₂×#3711	1	6	yes	—
216	C₂×C₃	C₂²×D₃×Dic₃	1	1	yes	characteristic
217	D₃	—	2	4	no	—
218	C₂²	—	3	4	no	—
219	C₂²	—	6	12	no	—
220	C₂²	#88420	1	4	yes	direct product factor
221	C₂²	C₂×D₃×Dic₉	1	3	yes	—
222	C₄	—	3	4	no	—
223	C₄	C₂×D₃×Dic₉	1	3	yes	—
224	C₄	C₂²×D₃×D₉	1	1	yes	characteristic
225	C₃	—	2	1	no	—
226	C₃	C₂²×#3667	1	1	yes	characteristic, complemented
227	C₃	C₂²×#3711	1	1	yes	characteristic
228	C₂	—	6	4	no	—
229	C₂	C₂×#88420	1	6	yes	direct product factor
230	C₂	C₂²×D₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×D₉
commutator	C₂×C₃×C₉	C₂³×C₄
Frattini	C₃×C₄	C₂²×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂³×C₃²	C₂×Dic₃
prime radical	C₂³×C₃×D₃	Dic₃
cosocle	C₃×C₄×C₉	C₂⁴
hypocenter	C₃×C₉	C₂²×M₁₆
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
3. C₂²×C₃×C₉⋊C₈	C₂	semidirect
4. C₂×#88420	C₂	direct (N × H)
12. C₂×C₃⋊(C₉⋊C₈)	C₂²	semidirect
13. C₂×C₃×C₉⋊C₈	C₂²	semidirect
14. #88420	C₂²	direct (N × H)
23. C₂²×C₉⋊C₈	D₃	semidirect
32. C₃⋊(C₉⋊C₈)	C₂³	semidirect
33. C₃×C₉⋊C₈	C₂³	semidirect
53. C₂×C₉⋊C₈	C₂×D₃	semidirect
54. C₂²×C₃×C₉	M₁₆	semidirect
93. C₉⋊C₈	C₂²×D₃	semidirect
95. C₂×C₃×C₉	C₂×M₁₆	semidirect
127. C₂²×C₉	ΓL(1,𝔽₂₅)	semidirect
138. C₃×C₉	C₂²×M₁₆	semidirect
169. C₂×C₉	C₂×ΓL(1,𝔽₂₅)	semidirect
183. C₂²×C₃	#3667	semidirect
199. C₉	C₂²×ΓL(1,𝔽₂₅)	semidirect
213. C₂×C₃	C₂×#3667	semidirect
220. C₂²	#88420	direct (N × H)
226. C₃	C₂²×#3667	semidirect
229. C₂	C₂×#88420	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₃×C₉	2
C₂²×C₃⋊(C₉⋊C₈)	2
C₂²×C₃×C₉⋊C₈	2
C₂×#88420	2
C₂²×#3711	3
C₂²×#3667	3

Subgroups — C22×#88420 #13150041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88420 #13150041