Subgroups — C₃²×#5570 #13178947

Order 1728.

1480 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×Q₈×Dic₃	C₂	1	2	yes	maximal
3	C₃²×#890	C₂	1	6	yes	maximal
4	C₃²×#869	C₂	1	3	yes	maximal
5	C₃²×C₄×Dic₆	C₂	1	3	yes	maximal
6	C₃²×#557	—	3	1	no	maximal
7	C₃×#5570	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×Dic₆	C₂²	1	3	yes	—
10	C₃³×C₄⋊C₄	C₂²	1	3	yes	—
11	C₃³×C₄²	C₂²	1	3	yes	—
12	C₃²×C₄⋊Dic₃	C₂²	1	6	yes	—
13	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₃²×C₄⋊Dic₃	C₂²	1	6	yes	—
15	C₃²×Dic₃⋊C₄	C₂²	1	6	yes	—
16	C₃²×C₄×Dic₃	C₂²	1	6	yes	—
17	C₃²×C₄⋊Q₈	—	3	3	no	—
18	C₃²×#125	—	3	6	no	—
19	C₃²×C₄×Q₈	—	2	4	no	—
20	C₃²×C₄×Q₈	—	3	2	no	—
21	C₃²×C₄×Q₈	—	3	3	no	—
22	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
23	C₃²×C₄×Q₈	C₂×C₃	1	4	yes	—
24	C₃×Q₈×Dic₃	C₂×C₃	1	8	yes	—
25	C₃×#890	C₂×C₃	1	24	yes	—
26	C₃×#869	C₂×C₃	1	12	yes	—
27	C₃×C₄×Dic₆	C₂×C₃	1	12	yes	—
28	C₃³×Q₈	Q₈	1	4	yes	complemented
29	C₂×C₃³×C₄	C₂³	1	3	yes	—
30	C₂×C₃³×C₄	C₂³	1	3	yes	—
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃²×Dic₃	C₂³	1	6	yes	—
33	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
34	C₃²×Dic₆	—	2	6	no	—
35	C₃×#557	—	3	4	no	—
36	#5570	C₃²	1	1	yes	characteristic, direct product factor
37	C₂×C₃²×Q₈	—	2	4	no	—
38	C₂×C₃²×Q₈	—	3	3	no	—
39	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
41	C₂×C₃×Dic₆	C₂²×C₃	1	12	yes	—
42	C₃²×C₄⋊C₄	—	2	12	no	—
43	C₃²×C₄⋊C₄	—	3	1	no	—
44	C₃²×C₄⋊C₄	—	3	6	no	—
45	C₃²×C₄⋊C₄	—	3	6	no	—
46	C₃²×C₄⋊C₄	—	3	6	no	—
47	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
48	C₃²×C₄⋊C₄	C₂²×C₃	1	12	yes	—
49	C₃²×C₄²	—	2	12	no	—
50	C₃²×C₄²	—	3	6	no	—
51	C₃²×C₄²	C₂×D₃	1	3	yes	—
52	C₃²×C₄²	C₂²×C₃	1	12	yes	—
53	C₃×C₄⋊Dic₃	C₂²×C₃	1	24	yes	—
54	C₃×C₄⋊Dic₃	C₂²×C₃	1	24	yes	—
55	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
56	C₃×Dic₃⋊C₄	C₂²×C₃	1	24	yes	—
57	C₃×C₄×Dic₃	C₂²×C₃	1	24	yes	—
58	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
59	C₃³×C₄	—	2	3	no	—
60	C₃³×C₄	C₂×Q₈	1	6	yes	—
61	C₃³×C₄	#41	1	2	yes	—
62	C₃²×Dic₃	—	2	6	no	—
63	C₃²×Dic₃	—	2	2	no	—
64	C₃×C₄⋊Q₈	—	3	12	no	—
65	C₃×#125	—	3	24	no	—
66	C₃×C₄×Q₈	—	2	4	no	—
67	C₃×C₄×Q₈	—	3	8	no	—
68	C₃×C₄×Q₈	—	3	12	no	—
69	C₃×C₄×Q₈	C₃×D₃	1	4	yes	—
70	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
71	Q₈×Dic₃	C₂×C₃²	1	2	yes	—
72	#890	C₂×C₃²	1	6	yes	—
73	#869	C₂×C₃²	1	3	yes	—
74	C₄×Dic₆	C₂×C₃²	1	3	yes	—
75	C₃²×Q₈	—	2	16	no	—
76	C₃²×Q₈	—	6	6	no	—
77	C₃²×Q₈	Dic₆	1	4	yes	complemented
78	C₃²×Q₈	C₃×Q₈	1	16	yes	complemented
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	2	12	no	—
81	C₂×C₃²×C₄	—	2	12	no	—
82	C₂×C₃²×C₄	—	3	2	no	—
83	C₂×C₃²×C₄	—	3	6	no	—
84	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
85	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
88	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
89	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
90	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
91	C₂×C₃×Dic₃	C₂³×C₃	1	24	yes	—
92	C₃×Dic₆	—	2	24	no	—
93	#557	—	3	1	no	Sylow
94	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₃³	C₂×#41	1	1	yes	characteristic
96	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
97	C₂×C₃×Q₈	—	2	4	no	—
98	C₂×C₃×Q₈	—	3	12	no	—
99	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
100	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
101	C₂×Dic₆	C₂²×C₃²	1	3	yes	—
102	C₃×C₄⋊C₄	—	2	12	no	—
103	C₃×C₄⋊C₄	—	3	24	no	—
104	C₃×C₄⋊C₄	—	3	4	no	—
105	C₃×C₄⋊C₄	—	3	24	no	—
106	C₃×C₄⋊C₄	—	3	24	no	—
107	C₃×C₄⋊C₄	C₂×C₃×D₃	1	12	yes	—
108	C₃×C₄⋊C₄	C₂²×C₃²	1	3	yes	—
109	C₃×C₄²	—	2	12	no	—
110	C₃×C₄²	—	3	24	no	—
111	C₃×C₄²	C₂×C₃×D₃	1	12	yes	—
112	C₃×C₄²	C₂²×C₃²	1	3	yes	—
113	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
114	C₄⋊Dic₃	C₂²×C₃²	1	6	yes	—
115	C₄⋊Dic₃	C₂²×C₃²	1	6	yes	—
116	Dic₃⋊C₄	C₂²×C₃²	1	6	yes	—
117	C₄×Dic₃	C₂²×C₃²	1	6	yes	—
118	C₂²×C₃²	—	2	4	no	—
119	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
120	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
121	C₃²×C₄	—	2	3	no	—
122	C₃²×C₄	—	2	24	no	—
123	C₃²×C₄	—	2	8	no	—
124	C₃²×C₄	—	2	12	no	—
125	C₃²×C₄	—	4	12	no	—
126	C₃²×C₄	—	6	6	no	—
127	C₃²×C₄	—	6	2	no	—
128	C₃²×C₄	C₂×Dic₆	1	6	yes	—
129	C₃²×C₄	#239	1	2	yes	—
130	C₃²×C₄	C₂×C₃×Q₈	1	24	yes	—
131	C₃²×C₄	C₃×#41	1	8	yes	—
132	C₃×Dic₃	—	2	24	no	—
133	C₃×Dic₃	—	2	8	no	—
134	C₄⋊Q₈	—	3	3	no	—
135	#125	—	3	6	no	—
136	C₄×Q₈	—	3	2	no	—
137	C₄×Q₈	—	3	3	no	—
138	C₄×Q₈	C₃²×D₃	1	1	yes	characteristic
139	C₃³	#557	1	1	yes	characteristic, complemented, Sylow
140	C₃×Q₈	—	2	16	no	—
141	C₃×Q₈	—	6	24	no	—
142	C₃×Q₈	C₃×Dic₆	1	16	yes	complemented
143	C₃×Q₈	C₃²×Q₈	1	4	yes	complemented
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	12	no	—
146	C₂×C₃×C₄	—	2	12	no	—
147	C₂×C₃×C₄	—	3	8	no	—
148	C₂×C₃×C₄	—	3	24	no	—
149	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
150	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
151	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
152	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
153	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
154	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
155	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
156	C₂×Dic₃	C₂³×C₃²	1	6	yes	—
157	Dic₆	—	2	6	no	—
158	C₂×C₃²	—	2	4	no	—
159	C₂×C₃²	—	2	4	no	—
160	C₂×C₃²	—	2	4	no	—
161	C₂×C₃²	C₂²×Dic₆	1	1	yes	characteristic
162	C₂×C₃²	C₂×#239	1	1	yes	characteristic
163	C₂×C₃²	#1010	1	1	yes	characteristic
164	C₂×C₃²	C₂²×C₃×Q₈	1	4	yes	—
165	C₂×C₃²	C₂×C₃×#41	1	4	yes	—
166	C₂×C₃²	C₃×2₋¹⁺⁴	1	4	yes	—
167	C₂×Q₈	—	3	3	no	—
168	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
169	C₄⋊C₄	—	3	1	no	—
170	C₄⋊C₄	—	3	6	no	—
171	C₄⋊C₄	—	3	6	no	—
172	C₄⋊C₄	—	3	6	no	—
173	C₄⋊C₄	C₂×C₃²×D₃	1	3	yes	—
174	C₄²	—	3	6	no	—
175	C₄²	C₂×C₃²×D₃	1	3	yes	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
178	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
179	C₃×C₄	—	2	12	no	—
180	C₃×C₄	—	2	3	no	—
181	C₃×C₄	—	2	24	no	—
182	C₃×C₄	—	2	8	no	—
183	C₃×C₄	—	4	12	no	—
184	C₃×C₄	—	6	24	no	—
185	C₃×C₄	—	6	8	no	—
186	C₃×C₄	C₂×C₃×Dic₆	1	24	yes	—
187	C₃×C₄	C₃×#239	1	8	yes	—
188	C₃×C₄	C₂×C₃²×Q₈	1	6	yes	—
189	C₃×C₄	C₃²×#41	1	2	yes	—
190	Dic₃	—	2	6	no	—
191	Dic₃	—	2	2	no	—
192	C₃²	—	2	4	no	—
193	C₃²	#5570	1	1	yes	characteristic, direct product factor
194	C₃²	C₃×#557	1	4	yes	complemented
195	Q₈	—	6	6	no	—
196	Q₈	C₃²×Dic₆	1	4	yes	complemented
197	C₂×C₄	—	3	2	no	—
198	C₂×C₄	—	3	6	no	—
199	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
200	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
201	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
202	C₂×C₃	—	2	4	no	—
203	C₂×C₃	—	2	4	no	—
204	C₂×C₃	—	2	4	no	—
205	C₂×C₃	C₂²×C₃×Dic₆	1	4	yes	—
206	C₂×C₃	C₂×C₃×#239	1	4	yes	—
207	C₂×C₃	C₃×#1010	1	4	yes	—
208	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
209	C₂×C₃	C₂×C₃²×#41	1	1	yes	characteristic
210	C₂×C₃	C₃²×2₋¹⁺⁴	1	1	yes	characteristic
211	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
212	C₄	—	2	3	no	—
213	C₄	—	6	6	no	—
214	C₄	—	6	2	no	—
215	C₄	C₂×C₃²×Dic₆	1	6	yes	—
216	C₄	C₃²×#239	1	2	yes	—
217	C₃	—	2	4	no	—
218	C₃	C₃×#5570	1	4	yes	direct product factor
219	C₃	C₃²×#557	1	1	yes	characteristic, complemented
220	C₂	C₂²×C₃²×Dic₆	1	1	yes	characteristic
221	C₂	C₂×C₃²×#239	1	1	yes	characteristic
222	C₂	C₃²×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂⁴
hypercenter	C₃²×C₄×Q₈	D₃
hypocenter	C₃	C₃²×#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
7. C₃×#5570	C₃	direct (N × H)
28. C₃³×Q₈	Q₈	semidirect
36. #5570	C₃²	direct (N × H)
77. C₃²×Q₈	Dic₆	semidirect
78. C₃²×Q₈	C₃×Q₈	semidirect
139. C₃³	#557	semidirect
142. C₃×Q₈	C₃×Dic₆	semidirect
143. C₃×Q₈	C₃²×Q₈	semidirect
193. C₃²	#5570	direct (N × H)
194. C₃²	C₃×#557	semidirect
196. Q₈	C₃²×Dic₆	semidirect
218. C₃	C₃×#5570	direct (N × H)
219. C₃	C₃²×#557	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
C₃²×Q₈×Dic₃	2
C₃²×#890	2
C₃²×#869	2
C₃²×C₄×Dic₆	2
C₃²×#557	3
C₃×#5570	3

Subgroups — C32×#5570 #13178947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5570 #13178947