Subgroups — #13163879

Order 1728.

2156 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×C₃³×#41	C₂	1	1	yes	maximal, characteristic
2	#11723967	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88855	C₂	1	1	yes	maximal, characteristic, complemented
4	#11722207	C₂	1	4	yes	maximal, complemented
5	#10592935	—	3	13	no	maximal
6	C₃³×#41	—	2	1	no	—
7	C₃³×#41	—	2	1	no	—
8	C₃³×#41	C₄	1	2	yes	complemented
9	C₃³×#41	C₂²	1	2	yes	—
10	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃³×D₄	—	2	1	no	—
12	C₂×C₃³×D₄	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₃³×C₄	—	2	1	no	—
14	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
15	#88864	—	2	1	no	—
16	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	1	no	—
17	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	complemented
18	#88855	—	2	1	no	—
19	#88855	C₂²	1	2	yes	complemented
20	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
21	C₂×C₃²×#41	D₃	1	13	yes	—
22	#64135	—	3	13	no	—
23	C₂×#3748	—	3	13	no	—
24	#63664	—	3	52	no	—
25	C₂³×C₃³	—	2	1	no	—
26	C₂³×C₃³	D₄	1	1	yes	characteristic
27	C₃³×Q₈	—	2	1	no	—
28	C₃³×Q₈	C₂×C₄	1	2	yes	—
29	C₃³×D₄	—	2	2	no	—
30	C₃³×D₄	—	2	2	no	—
31	C₃³×D₄	—	2	1	no	—
32	C₃³×D₄	C₂×C₄	1	2	yes	—
33	C₂×C₃³×C₄	—	2	1	no	—
34	C₂×C₃³×C₄	—	2	2	no	—
35	C₂×C₃³×C₄	—	2	2	no	—
36	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
40	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
41	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
43	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
44	#5236	—	9	13	no	—
45	C₃²×#41	—	2	13	no	—
46	C₃²×#41	—	2	13	no	—
47	C₃²×#41	Dic₃	1	26	yes	complemented
48	C₃²×#41	C₂×D₃	1	26	yes	—
49	C₂×C₃²×Q₈	Dic₃	1	13	yes	complemented
50	C₂×C₃²×D₄	—	2	13	no	—
51	C₂×C₃²×D₄	Dic₃	1	13	yes	complemented
52	C₂²×C₃²×C₄	—	2	13	no	—
53	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
54	#3757	—	6	13	no	—
55	C₄⋊(C₃⋊Dic₃)	—	6	13	no	—
56	C₄×C₃⋊Dic₃	—	3	26	no	—
57	#3748	—	3	26	no	—
58	#3748	—	6	13	no	—
59	C₂×C₃⋊(C₃⋊C₈)	—	6	13	no	—
60	C₂²×C₃³	—	2	2	no	—
61	C₂²×C₃³	—	4	1	no	—
62	C₂²×C₃³	—	4	2	no	—
63	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
66	C₃³×C₄	—	2	2	no	—
67	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
71	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
72	C₂×C₃×#41	C₃⋊D₃	1	13	yes	—
73	#926	—	9	13	no	—
74	C₂×#208	—	9	13	no	—
75	#837	—	9	52	no	—
76	C₂³×C₃²	—	2	13	no	—
77	C₂³×C₃²	#67	1	13	yes	—
78	C₃²×Q₈	—	2	13	no	—
79	C₃²×Q₈	C₂×Dic₃	1	26	yes	—
80	C₃²×D₄	—	2	26	no	—
81	C₃²×D₄	—	2	26	no	—
82	C₃²×D₄	—	2	13	no	—
83	C₃²×D₄	C₂×Dic₃	1	26	yes	—
84	C₂×C₃²×C₄	—	2	13	no	—
85	C₂×C₃²×C₄	—	2	26	no	—
86	C₂×C₃²×C₄	—	2	26	no	—
87	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
88	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
89	C₂×C₃²×C₄	#67	1	13	yes	—
90	C₂×C₃²×C₄	#67	1	13	yes	—
91	C₂×C₃²×C₄	#67	1	13	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
93	C₂×C₃⋊Dic₃	—	6	26	no	—
94	C₃⋊(C₃⋊C₈)	—	6	26	no	—
95	#421	—	27	1	no	Sylow
96	C₂×C₃³	—	2	1	no	—
97	C₂×C₃³	—	2	1	no	—
98	C₂×C₃³	—	2	1	no	—
99	C₂×C₃³	—	4	2	no	—
100	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₃×#41	—	2	13	no	—
102	C₃×#41	—	2	13	no	—
103	C₃×#41	C₃⋊Dic₃	1	26	yes	complemented
104	C₃×#41	C₂×C₃⋊D₃	1	26	yes	—
105	C₂×C₃×Q₈	C₃⋊Dic₃	1	13	yes	complemented
106	C₂×C₃×D₄	—	2	13	no	—
107	C₂×C₃×D₄	C₃⋊Dic₃	1	13	yes	complemented
108	C₂²×C₃×C₄	—	2	13	no	—
109	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
110	#217	—	18	13	no	—
111	C₄⋊Dic₃	—	18	13	no	—
112	C₄×Dic₃	—	9	26	no	—
113	#208	—	9	26	no	—
114	#208	—	18	13	no	—
115	C₂×C₃⋊C₈	—	18	13	no	—
116	C₂²×C₃²	—	2	26	no	—
117	C₂²×C₃²	—	4	13	no	—
118	C₂²×C₃²	—	4	26	no	—
119	C₂²×C₃²	#217	1	13	yes	—
120	C₂²×C₃²	#217	1	13	yes	—
121	C₂²×C₃²	C₂×#67	1	13	yes	—
122	C₃²×C₄	—	2	26	no	—
123	C₃²×C₄	#217	1	13	yes	—
124	C₃²×C₄	#217	1	13	yes	—
125	C₃²×C₄	C₂²×Dic₃	1	13	yes	—
126	C₃²×C₄	C₂×#67	1	13	yes	—
127	C₃⋊Dic₃	—	12	26	no	—
128	C₂×#41	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
129	C₂×M₁₆	—	27	1	no	—
130	#117	—	27	1	no	—
131	#104	—	27	4	no	—
132	C₃³	#421	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₃	—	2	13	no	—
134	C₂³×C₃	#638	1	13	yes	—
135	C₃×Q₈	—	2	13	no	—
136	C₃×Q₈	C₂×C₃⋊Dic₃	1	26	yes	—
137	C₃×D₄	—	2	26	no	—
138	C₃×D₄	—	2	26	no	—
139	C₃×D₄	—	2	13	no	—
140	C₃×D₄	C₂×C₃⋊Dic₃	1	26	yes	—
141	C₂×C₃×C₄	—	2	13	no	—
142	C₂×C₃×C₄	—	2	26	no	—
143	C₂×C₃×C₄	—	2	26	no	—
144	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
145	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
146	C₂×C₃×C₄	#638	1	13	yes	—
147	C₂×C₃×C₄	#638	1	13	yes	—
148	C₂×C₃×C₄	#638	1	13	yes	—
149	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
150	C₂×Dic₃	—	18	26	no	—
151	C₃⋊C₈	—	18	26	no	—
152	C₂×C₃²	—	2	13	no	—
153	C₂×C₃²	—	2	13	no	—
154	C₂×C₃²	—	2	13	no	—
155	C₂×C₃²	—	4	26	no	—
156	C₂×C₃²	C₂×#217	1	13	yes	—
157	#41	—	2	1	no	—
158	#41	—	2	1	no	—
159	#41	C₃⋊(C₃⋊Dic₃)	1	2	yes	complemented
160	#41	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
161	C₂×Q₈	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, complemented
162	C₂×D₄	—	2	1	no	—
163	C₂×D₄	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, complemented
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
166	M₁₆	—	27	2	no	—
167	M₁₆	—	54	1	no	—
168	C₂×C₈	—	54	1	no	—
169	C₄⋊C₄	—	54	1	no	—
170	C₂²⋊C₄	—	54	1	no	—
171	C₄²	—	27	2	no	—
172	C₂²×C₃	—	2	26	no	—
173	C₂²×C₃	—	4	13	no	—
174	C₂²×C₃	—	4	26	no	—
175	C₂²×C₃	#3757	1	13	yes	—
176	C₂²×C₃	#3757	1	13	yes	—
177	C₂²×C₃	C₂×#638	1	13	yes	—
178	C₃×C₄	—	2	26	no	—
179	C₃×C₄	#3757	1	13	yes	—
180	C₃×C₄	#3757	1	13	yes	—
181	C₃×C₄	C₂²×C₃⋊Dic₃	1	13	yes	—
182	C₃×C₄	C₂×#638	1	13	yes	—
183	Dic₃	—	36	26	no	—
184	C₃²	#5236	1	13	yes	complemented
185	C₂³	—	2	1	no	—
186	C₂³	#6312	1	1	yes	characteristic
187	Q₈	—	2	1	no	—
188	Q₈	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
189	D₄	—	2	2	no	—
190	D₄	—	2	2	no	—
191	D₄	—	2	1	no	—
192	D₄	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	54	2	no	—
197	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
198	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
199	C₂×C₄	#6312	1	1	yes	characteristic
200	C₂×C₄	#6312	1	1	yes	characteristic
201	C₂×C₄	#6312	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
203	C₈	—	54	2	no	—
204	C₂×C₃	—	2	13	no	—
205	C₂×C₃	—	2	13	no	—
206	C₂×C₃	—	2	13	no	—
207	C₂×C₃	—	4	26	no	—
208	C₂×C₃	C₂×#3757	1	13	yes	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	2	no	—
212	C₂²	#88864	1	1	yes	characteristic
213	C₂²	#88864	1	1	yes	characteristic
214	C₂²	C₂×#6312	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	108	2	no	—
217	C₄	#88864	1	1	yes	characteristic
218	C₄	#88864	1	1	yes	characteristic
219	C₄	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
220	C₄	C₂×#6312	1	1	yes	characteristic
221	C₃	#10592935	1	13	yes	complemented
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂×#88864	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#6312
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting, prime radical	C₂×C₃³×#41	C₂
socle	C₂×C₃³	C₂×C₂²⋊C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂×#41	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#421
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #11723967	C₂	semidirect
3. C₂×#88855	C₂	semidirect
4. #11722207	C₂	semidirect
8. C₃³×#41	C₄	semidirect
10. C₂×C₃³×Q₈	C₄	semidirect
12. C₂×C₃³×D₄	C₄	semidirect
17. C₄×C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
19. #88855	C₂²	semidirect
47. C₃²×#41	Dic₃	semidirect
49. C₂×C₃²×Q₈	Dic₃	semidirect
51. C₂×C₃²×D₄	Dic₃	semidirect
103. C₃×#41	C₃⋊Dic₃	semidirect
105. C₂×C₃×Q₈	C₃⋊Dic₃	semidirect
107. C₂×C₃×D₄	C₃⋊Dic₃	semidirect
132. C₃³	#421	semidirect
159. #41	C₃⋊(C₃⋊Dic₃)	semidirect
161. C₂×Q₈	C₃⋊(C₃⋊Dic₃)	semidirect
163. C₂×D₄	C₃⋊(C₃⋊Dic₃)	semidirect
184. C₃²	#5236	semidirect
221. C₃	#10592935	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×#41	2
#11723967	2
C₂×#88855	2
#11722207	2
#10592935	3