Subgroups — C₂²×#88787 #13176867

Order 1728.

2092 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×C₃×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88787	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#3743	—	3	1	no	maximal
6	C₂²×#3711	—	3	4	no	maximal
7	C₂³×C₃×C₃⋊D₃	C₄	1	1	yes	characteristic
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₃⋊Dic₃	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×C₃⋊D₃	C₄	1	6	yes	—
11	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	6	yes	—
12	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	6	yes	complemented
13	C₂×C₃²×C₃⋊C₈	C₂²	1	6	yes	complemented
14	#88787	C₂²	1	16	yes	direct product factor
15	C₂²×C₄×C₃⋊D₃	D₃	1	1	yes	characteristic
16	C₂²×C₃×C₄×D₃	—	3	4	no	—
17	C₂²×C₃²×C₈	—	3	1	no	—
18	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
19	C₂²×C₃⋊(C₃⋊C₈)	—	3	4	no	—
20	C₂²×C₃⋊(C₃⋊C₈)	—	6	4	no	—
21	C₂×#3743	—	3	12	no	—
22	C₂²×C₃×C₃⋊C₈	D₃	1	4	yes	complemented
23	C₂×#3711	—	3	48	no	—
24	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₃⋊D₃	—	2	4	no	—
26	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	6	yes	—
27	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
28	C₂×C₃³×C₄	C₂³	1	3	yes	—
29	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	6	yes	—
30	C₃×C₄×C₃⋊D₃	C₂×C₄	1	12	yes	—
31	C₃×C₄×C₃⋊D₃	C₂³	1	4	yes	—
32	C₃⋊(C₃⋊(C₃⋊C₈))	C₂³	1	4	yes	complemented
33	C₃²×C₃⋊C₈	C₂³	1	4	yes	complemented
34	C₂²×#208	—	9	1	no	—
35	C₂²×ΓL(1,𝔽₂₅)	—	9	4	no	—
36	C₂³×C₃⋊D₃	Dic₃	1	1	yes	characteristic
37	C₂³×C₃×D₃	—	3	4	no	—
38	C₂²×C₃²×C₄	—	2	4	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃⋊Dic₃	Dic₃	1	1	yes	characteristic
42	C₂×C₄×C₃⋊D₃	Dic₃	1	6	yes	—
43	C₂×C₄×C₃⋊D₃	C₂×D₃	1	6	yes	—
44	C₂²×C₃×Dic₃	—	3	4	no	—
45	C₂×C₃×C₄×D₃	—	3	24	no	—
46	C₂×C₃×C₄×D₃	—	3	24	no	—
47	C₂×C₃²×C₈	—	3	6	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	3	6	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	3	24	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	6	24	no	—
51	#3743	—	3	16	no	—
52	C₂×C₃×C₃⋊C₈	C₂×D₃	1	24	yes	complemented
53	#3711	—	3	64	no	—
54	C₂²×C₃³	M₁₆	1	4	yes	complemented
55	C₂²×C₃³	C₂²×C₄	1	3	yes	—
56	C₂×C₃×C₃⋊D₃	—	2	12	no	—
57	C₂×C₃×C₃⋊D₃	C₂²×C₄	1	4	yes	—
58	C₃³×C₄	C₂²×C₄	1	3	yes	—
59	C₃³×C₄	C₂⁴	1	1	yes	characteristic
60	C₃×C₃⋊Dic₃	C₂²×C₄	1	4	yes	—
61	C₂³×C₃×C₄	—	9	1	no	—
62	C₂²×C₄×D₃	—	3	4	no	—
63	C₂²×C₃×C₈	—	3	4	no	—
64	C₂×#208	—	9	12	no	—
65	C₂²×C₃⋊C₈	—	9	4	no	—
66	C₂²×C₃⋊C₈	—	9	1	no	—
67	C₂²×C₃⋊C₈	—	18	4	no	—
68	C₂²×C₃⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
69	C₂×ΓL(1,𝔽₂₅)	—	9	48	no	—
70	C₂³×C₃²	—	2	4	no	—
71	C₂³×C₃²	C₄×D₃	1	4	yes	—
72	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
73	C₂²×C₃⋊D₃	—	2	4	no	—
74	C₂²×C₃⋊D₃	C₂×Dic₃	1	6	yes	—
75	C₂²×C₃×D₃	—	3	24	no	—
76	C₂²×C₃×D₃	—	6	16	no	—
77	C₂×C₃²×C₄	—	2	12	no	—
78	C₂×C₃²×C₄	—	2	12	no	—
79	C₂×C₃²×C₄	C₄×D₃	1	12	yes	—
80	C₂×C₃²×C₄	C₂×Dic₃	1	3	yes	—
81	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
82	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
83	C₂×C₃⋊Dic₃	C₂×Dic₃	1	6	yes	—
84	C₄×C₃⋊D₃	C₂×Dic₃	1	12	yes	—
85	C₄×C₃⋊D₃	C₂²×D₃	1	4	yes	—
86	C₂×C₃×Dic₃	—	3	24	no	—
87	C₃×C₄×D₃	—	3	48	no	—
88	C₃×C₄×D₃	—	3	16	no	—
89	C₃²×C₈	—	3	4	no	—
90	C₃⋊(C₃⋊C₈)	—	3	4	no	—
91	C₃⋊(C₃⋊C₈)	—	3	16	no	—
92	C₃⋊(C₃⋊C₈)	—	6	16	no	—
93	C₃×C₃⋊C₈	C₂²×D₃	1	16	yes	complemented
94	C₂²×M₁₆	—	27	1	no	Sylow
95	C₂×C₃³	C₂×M₁₆	1	6	yes	complemented
96	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
97	C₃×C₃⋊D₃	—	2	4	no	—
98	C₂⁴×C₃	—	9	1	no	—
99	C₂³×D₃	—	3	4	no	—
100	C₂²×C₃×C₄	—	2	4	no	—
101	C₂²×C₃×C₄	—	9	1	no	—
102	C₂²×C₃×C₄	—	9	6	no	—
103	C₂²×C₃×C₄	—	9	6	no	—
104	C₂²×C₃×C₄	D₃²	1	4	yes	—
105	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
106	C₂²×Dic₃	—	3	4	no	—
107	C₂×C₄×D₃	—	3	24	no	—
108	C₂×C₄×D₃	—	3	24	no	—
109	C₂×C₃×C₈	—	3	24	no	—
110	#208	—	9	16	no	—
111	C₂×C₃⋊C₈	—	9	24	no	—
112	C₂×C₃⋊C₈	—	9	6	no	—
113	C₂×C₃⋊C₈	—	18	24	no	—
114	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	6	yes	complemented
115	ΓL(1,𝔽₂₅)	—	9	64	no	—
116	C₂²×C₃²	—	2	16	no	—
117	C₂²×C₃²	—	2	12	no	—
118	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	16	yes	complemented
119	C₂²×C₃²	#208	1	4	yes	complemented
120	C₂²×C₃²	C₂×C₄×D₃	1	12	yes	—
121	C₂²×C₃²	C₂²×Dic₃	1	3	yes	—
122	C₂×C₃⋊D₃	—	2	12	no	—
123	C₂×C₃⋊D₃	C₂²×Dic₃	1	4	yes	—
124	C₂×C₃×D₃	—	3	16	no	—
125	C₂×C₃×D₃	—	6	48	no	—
126	C₃²×C₄	—	2	4	no	—
127	C₃²×C₄	—	2	12	no	—
128	C₃²×C₄	C₂×C₄×D₃	1	12	yes	—
129	C₃²×C₄	C₂²×Dic₃	1	3	yes	—
130	C₃²×C₄	C₂³×D₃	1	4	yes	—
131	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
132	C₃⋊Dic₃	C₂²×Dic₃	1	4	yes	—
133	C₃×Dic₃	—	3	16	no	—
134	C₂³×C₄	—	9	1	no	—
135	C₂×M₁₆	—	27	12	no	—
136	C₂²×C₈	—	3	1	no	—
137	C₂²×C₈	—	27	1	no	—
138	C₃³	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	—	9	6	no	—
141	C₂³×C₃	—	18	4	no	—
142	C₂³×C₃	D₃×Dic₃	1	4	yes	—
143	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
144	C₂²×D₃	—	3	24	no	—
145	C₂²×D₃	—	6	16	no	—
146	C₂×C₃×C₄	—	2	12	no	—
147	C₂×C₃×C₄	—	2	12	no	—
148	C₂×C₃×C₄	—	9	6	no	—
149	C₂×C₃×C₄	—	9	4	no	—
150	C₂×C₃×C₄	—	9	12	no	—
151	C₂×C₃×C₄	D₃×Dic₃	1	12	yes	—
152	C₂×C₃×C₄	C₄×C₃⋊D₃	1	3	yes	—
153	C₂×C₃×C₄	C₂×D₃²	1	12	yes	—
154	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
155	C₂×Dic₃	—	3	24	no	—
156	C₄×D₃	—	3	16	no	—
157	C₄×D₃	—	3	48	no	—
158	C₃×C₈	—	3	16	no	—
159	C₃⋊C₈	—	9	16	no	—
160	C₃⋊C₈	—	9	4	no	—
161	C₃⋊C₈	—	18	16	no	—
162	C₃⋊C₈	C₂²×C₃⋊D₃	1	4	yes	complemented
163	C₂×C₃²	—	2	4	no	—
164	C₂×C₃²	—	2	24	no	—
165	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	24	yes	complemented
166	C₂×C₃²	C₂×#208	1	6	yes	complemented
167	C₂×C₃²	C₂²×C₄×D₃	1	4	yes	—
168	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
169	C₃⋊D₃	—	2	4	no	—
170	C₃×D₃	—	6	16	no	—
171	C₂⁴	—	9	1	no	—
172	C₂²×C₄	—	9	1	no	—
173	C₂²×C₄	—	9	6	no	—
174	C₂²×C₄	—	9	6	no	—
175	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
176	M₁₆	—	27	16	no	—
177	C₂×C₈	—	3	6	no	—
178	C₂×C₈	—	27	6	no	—
179	C₂²×C₃	—	2	12	no	—
180	C₂²×C₃	—	2	16	no	—
181	C₂²×C₃	—	9	4	no	—
182	C₂²×C₃	—	18	12	no	—
183	C₂²×C₃	#3711	1	16	yes	complemented
184	C₂²×C₃	#3743	1	4	yes	complemented
185	C₂²×C₃	C₂×D₃×Dic₃	1	12	yes	—
186	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	3	yes	—
187	C₂×D₃	—	3	16	no	—
188	C₂×D₃	—	6	48	no	—
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	—	2	12	no	—
191	C₃×C₄	—	9	4	no	—
192	C₃×C₄	C₂×D₃×Dic₃	1	12	yes	—
193	C₃×C₄	C₂×C₄×C₃⋊D₃	1	3	yes	—
194	C₃×C₄	C₂²×D₃²	1	4	yes	—
195	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
196	Dic₃	—	3	16	no	—
197	C₃²	—	2	4	no	—
198	C₃²	C₂²×ΓL(1,𝔽₂₅)	1	4	yes	complemented
199	C₃²	C₂²×#208	1	1	yes	characteristic, complemented
200	C₂³	—	9	6	no	—
201	C₂³	—	18	4	no	—
202	C₂³	Dic₃×C₃⋊D₃	1	1	yes	characteristic
203	C₂×C₄	—	9	6	no	—
204	C₂×C₄	—	9	4	no	—
205	C₂×C₄	—	9	12	no	—
206	C₂×C₄	Dic₃×C₃⋊D₃	1	3	yes	—
207	C₂×C₄	C₂×D₃×C₃⋊D₃	1	3	yes	—
208	C₈	—	3	4	no	—
209	C₈	—	27	4	no	—
210	C₂×C₃	—	2	4	no	—
211	C₂×C₃	—	2	24	no	—
212	C₂×C₃	—	18	4	no	—
213	C₂×C₃	C₂×#3711	1	24	yes	complemented
214	C₂×C₃	C₂×#3743	1	6	yes	complemented
215	C₂×C₃	C₂²×D₃×Dic₃	1	4	yes	—
216	C₂×C₃	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
217	D₃	—	6	16	no	—
218	C₂²	—	9	4	no	—
219	C₂²	—	18	12	no	—
220	C₂²	#88787	1	4	yes	direct product factor
221	C₂²	C₂×Dic₃×C₃⋊D₃	1	3	yes	—
222	C₄	—	9	4	no	—
223	C₄	C₂×Dic₃×C₃⋊D₃	1	3	yes	—
224	C₄	C₂²×D₃×C₃⋊D₃	1	1	yes	characteristic
225	C₃	—	2	4	no	—
226	C₃	C₂²×#3711	1	4	yes	complemented
227	C₃	C₂²×#3743	1	1	yes	characteristic, complemented
228	C₂	—	18	4	no	—
229	C₂	C₂×#88787	1	6	yes	direct product factor
230	C₂	C₂²×Dic₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×C₃⋊D₃
commutator	C₂×C₃³	C₂³×C₄
Frattini	C₄	C₂²×D₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₄	C₂²
socle	C₂³×C₃³	C₂×C₄
prime radical	C₂³×C₃×C₃⋊D₃	C₄
cosocle	C₃³×C₄	C₂⁴
hypocenter	C₃³	C₂²×M₁₆
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
3. C₂²×C₃²×C₃⋊C₈	C₂	semidirect
4. C₂×#88787	C₂	direct (N × H)
12. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	semidirect
13. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
14. #88787	C₂²	direct (N × H)
22. C₂²×C₃×C₃⋊C₈	D₃	semidirect
32. C₃⋊(C₃⋊(C₃⋊C₈))	C₂³	semidirect
33. C₃²×C₃⋊C₈	C₂³	semidirect
52. C₂×C₃×C₃⋊C₈	C₂×D₃	semidirect
54. C₂²×C₃³	M₁₆	semidirect
68. C₂²×C₃⋊C₈	C₃⋊D₃	semidirect
93. C₃×C₃⋊C₈	C₂²×D₃	semidirect
95. C₂×C₃³	C₂×M₁₆	semidirect
114. C₂×C₃⋊C₈	C₂×C₃⋊D₃	semidirect
118. C₂²×C₃²	ΓL(1,𝔽₂₅)	semidirect
119. C₂²×C₃²	#208	semidirect
138. C₃³	C₂²×M₁₆	semidirect
162. C₃⋊C₈	C₂²×C₃⋊D₃	semidirect
165. C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	semidirect
166. C₂×C₃²	C₂×#208	semidirect
183. C₂²×C₃	#3711	semidirect
184. C₂²×C₃	#3743	semidirect
198. C₃²	C₂²×ΓL(1,𝔽₂₅)	semidirect
199. C₃²	C₂²×#208	semidirect
213. C₂×C₃	C₂×#3711	semidirect
214. C₂×C₃	C₂×#3743	semidirect
220. C₂²	#88787	direct (N × H)
226. C₃	C₂²×#3711	semidirect
227. C₃	C₂²×#3743	semidirect
229. C₂	C₂×#88787	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×C₃×C₄×C₃⋊D₃	2
C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	2
C₂²×C₃²×C₃⋊C₈	2
C₂×#88787	2
C₂²×#3743	3
C₂²×#3711	3

Subgroups — C22×#88787 #13176867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88787 #13176867