Subgroups — C₃²×#5106 #13162541

Order 1728.

988 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×#926	C₂	1	2	yes	maximal
3	C₃²×#818	C₂	1	2	yes	maximal, complemented
4	C₃²×#817	C₂	1	2	yes	maximal, complemented
5	C₃²×#427	—	3	1	no	maximal
6	C₃×#5106	C₃	1	4	yes	maximal, direct product factor
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃³×C₈	C₄	1	4	yes	complemented
9	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃²×#217	—	2	2	no	—
12	C₃²×C₄⋊Dic₃	C₂²	1	4	yes	—
13	C₃²×C₄×Dic₃	—	2	2	no	—
14	C₂²×C₃²×C₈	—	2	4	no	—
15	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
16	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	—
17	C₃²×#117	—	3	2	no	—
18	C₃×#926	C₂×C₃	1	8	yes	—
19	C₃²×#107	—	3	2	no	—
20	C₃²×#106	—	3	2	no	—
21	C₃×#818	C₂×C₃	1	8	yes	complemented
22	C₃×#817	C₂×C₃	1	8	yes	complemented
23	C₂³×C₃³	D₄	1	1	yes	characteristic
24	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
25	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	Q₈	1	2	yes	—
27	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₃²×Dic₃	—	2	4	no	—
29	C₃³×C₈	C₂×C₄	1	2	yes	—
30	C₃³×C₈	C₂×C₄	1	2	yes	—
31	C₃×#427	—	3	4	no	—
32	#5106	C₃²	1	1	yes	characteristic, direct product factor
33	C₂²×C₃²×C₄	—	2	4	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
36	C₂×C₃²×C₈	—	2	16	no	—
37	C₂×C₃²×C₈	—	2	4	no	—
38	C₂×C₃²×C₈	—	2	4	no	—
39	C₂×C₃²×C₈	Dic₃	1	4	yes	complemented
40	C₂×C₃²×C₈	C₃×C₄	1	16	yes	complemented
41	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
44	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
45	C₃²×C₄⋊C₄	—	3	4	no	—
46	C₃²×C₂²⋊C₄	—	6	2	no	—
47	C₃²×C₄²	—	6	2	no	—
48	C₃×#217	—	2	8	no	—
49	C₃×C₄⋊Dic₃	C₂²×C₃	1	16	yes	—
50	C₃×C₄×Dic₃	—	2	8	no	—
51	C₂²×C₃³	—	2	2	no	—
52	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
53	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
54	C₃³×C₄	C₄⋊C₄	1	2	yes	—
55	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
57	C₃²×Dic₃	—	4	4	no	—
58	C₂²×C₃×C₈	—	2	4	no	—
59	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
60	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
61	C₃×#117	—	3	8	no	—
62	#926	C₂×C₃²	1	2	yes	—
63	C₃×#107	—	3	8	no	—
64	C₃×#106	—	3	8	no	—
65	#818	C₂×C₃²	1	2	yes	complemented
66	#817	C₂×C₃²	1	2	yes	complemented
67	C₂³×C₃²	—	2	4	no	—
68	C₂³×C₃²	D₁₂	1	1	yes	characteristic
69	C₂³×C₃²	C₃×D₄	1	4	yes	—
70	C₂×C₃²×C₄	—	2	8	no	—
71	C₂×C₃²×C₄	—	2	4	no	—
72	C₂×C₃²×C₄	—	2	8	no	—
73	C₂×C₃²×C₄	—	2	4	no	—
74	C₂×C₃²×C₄	—	6	4	no	—
75	C₂×C₃²×C₄	Dic₆	1	2	yes	—
76	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
78	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
79	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
80	C₂×C₃²×C₄	C₃×Q₈	1	8	yes	—
81	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
83	C₂×C₃×Dic₃	—	2	16	no	—
84	C₃²×C₈	—	2	8	no	—
85	C₃²×C₈	—	2	8	no	—
86	C₃²×C₈	C₂×Dic₃	1	2	yes	—
87	C₃²×C₈	C₂×Dic₃	1	2	yes	—
88	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
89	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
90	#427	—	3	1	no	Sylow
91	C₂×C₃³	—	2	2	no	—
92	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₃³	#135	1	2	yes	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
96	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
97	C₂×C₃×C₈	—	2	16	no	—
98	C₂×C₃×C₈	—	2	4	no	—
99	C₂×C₃×C₈	—	2	4	no	—
100	C₂×C₃×C₈	C₃×Dic₃	1	16	yes	complemented
101	C₂×C₃×C₈	C₃²×C₄	1	4	yes	complemented
102	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
103	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
104	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	3	16	no	—
107	C₃×C₂²⋊C₄	—	6	8	no	—
108	C₃×C₄²	—	6	8	no	—
109	#217	—	2	2	no	—
110	C₄⋊Dic₃	C₂²×C₃²	1	4	yes	—
111	C₄×Dic₃	—	2	2	no	—
112	C₂²×C₃²	—	2	16	no	—
113	C₂²×C₃²	—	2	2	no	—
114	C₂²×C₃²	—	2	8	no	—
115	C₂²×C₃²	—	2	8	no	—
116	C₂²×C₃²	—	2	4	no	—
117	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
118	C₂²×C₃²	C₃×C₄⋊C₄	1	8	yes	—
119	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
120	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
121	C₃²×C₄	—	2	8	no	—
122	C₃²×C₄	—	2	4	no	—
123	C₃²×C₄	—	2	4	no	—
124	C₃²×C₄	—	12	4	no	—
125	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
126	C₃²×C₄	C₃×C₄⋊C₄	1	8	yes	—
127	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
128	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
129	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
130	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
131	C₃×Dic₃	—	4	16	no	—
132	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic
133	#117	—	3	2	no	—
134	#107	—	3	2	no	—
135	#106	—	3	2	no	—
136	C₃³	#427	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₃	—	2	4	no	—
138	C₂³×C₃	C₃×D₁₂	1	4	yes	—
139	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	8	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	8	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	6	16	no	—
145	C₂×C₃×C₄	C₃×Dic₆	1	8	yes	—
146	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
147	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
148	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
149	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
151	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
152	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
153	C₂×Dic₃	—	2	4	no	—
154	C₃×C₈	—	2	8	no	—
155	C₃×C₈	—	2	8	no	—
156	C₃×C₈	C₂×C₃×Dic₃	1	8	yes	—
157	C₃×C₈	C₂×C₃×Dic₃	1	8	yes	—
158	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
159	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
160	C₂×C₃²	—	2	16	no	—
161	C₂×C₃²	—	2	8	no	—
162	C₂×C₃²	—	2	2	no	—
163	C₂×C₃²	—	2	8	no	—
164	C₂×C₃²	—	2	4	no	—
165	C₂×C₃²	#904	1	2	yes	—
166	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
167	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
168	C₂×C₃²	C₃×#135	1	8	yes	—
169	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
170	C₂×C₈	C₃²×Dic₃	1	4	yes	complemented
171	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
172	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
173	C₄⋊C₄	—	3	4	no	—
174	C₂²⋊C₄	—	6	2	no	—
175	C₄²	—	6	2	no	—
176	C₂²×C₃	—	2	16	no	—
177	C₂²×C₃	—	2	8	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	2	8	no	—
181	C₂²×C₃	C₃×C₄⋊Dic₃	1	8	yes	—
182	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
183	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
184	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	2	4	no	—
187	C₃×C₄	—	2	8	no	—
188	C₃×C₄	—	12	16	no	—
189	C₃×C₄	C₃×C₄⋊Dic₃	1	8	yes	—
190	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
191	C₃×C₄	C₂×C₃×Dic₆	1	4	yes	—
192	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
193	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
194	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
195	Dic₃	—	4	4	no	—
196	C₃²	—	2	4	no	—
197	C₃²	#5106	1	1	yes	characteristic, direct product factor
198	C₃²	C₃×#427	1	4	yes	complemented
199	C₂³	C₃²×D₁₂	1	1	yes	characteristic
200	C₂×C₄	—	6	4	no	—
201	C₂×C₄	C₃²×Dic₆	1	2	yes	—
202	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
204	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
205	C₈	C₂×C₃²×Dic₃	1	2	yes	—
206	C₈	C₂×C₃²×Dic₃	1	2	yes	—
207	C₂×C₃	—	2	16	no	—
208	C₂×C₃	—	2	8	no	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	2	4	no	—
211	C₂×C₃	—	2	8	no	—
212	C₂×C₃	C₃×#904	1	8	yes	—
213	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	4	yes	—
214	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₃	C₃²×#135	1	2	yes	—
216	C₂²	—	2	2	no	—
217	C₂²	C₃²×C₄⋊Dic₃	1	2	yes	—
218	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
219	C₄	—	12	4	no	—
220	C₄	C₃²×C₄⋊Dic₃	1	2	yes	—
221	C₄	C₂×C₃²×Dic₆	1	1	yes	characteristic
222	C₄	C₂²×C₃²×Dic₃	1	1	yes	characteristic
223	C₃	—	2	4	no	—
224	C₃	C₃×#5106	1	4	yes	direct product factor
225	C₃	C₃²×#427	1	1	yes	characteristic, complemented
226	C₂	—	2	2	no	—
227	C₂	C₃²×#904	1	2	yes	—
228	C₂	C₂×C₃²×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	D₁₂
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂³×C₃³	D₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₂²×C₃²×C₈	D₃
hypocenter	C₃	C₃²×#427
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈

Complements

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

N	H ≅ G/N	type
3. C₃²×#818	C₂	semidirect
4. C₃²×#817	C₂	semidirect
6. C₃×#5106	C₃	direct (N × H)
8. C₂×C₃³×C₈	C₄	semidirect
21. C₃×#818	C₂×C₃	semidirect
22. C₃×#817	C₂×C₃	semidirect
32. #5106	C₃²	direct (N × H)
39. C₂×C₃²×C₈	Dic₃	semidirect
40. C₂×C₃²×C₈	C₃×C₄	semidirect
65. #818	C₂×C₃²	semidirect
66. #817	C₂×C₃²	semidirect
100. C₂×C₃×C₈	C₃×Dic₃	semidirect
101. C₂×C₃×C₈	C₃²×C₄	semidirect
136. C₃³	#427	semidirect
170. C₂×C₈	C₃²×Dic₃	semidirect
197. C₃²	#5106	direct (N × H)
198. C₃²	C₃×#427	semidirect
224. C₃	C₃×#5106	direct (N × H)
225. C₃	C₃²×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃³×C₈	2
C₃²×#926	2
C₃²×#818	2
C₃²×#817	2
C₃²×#427	3
C₃×#5106	3

Subgroups — C32×#5106 #13162541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5106 #13162541