Subgroups — C₃²×#5088 #13162525

Order 1728.

980 conjugacy classes of subgroups.

The subgroup lattice has 254 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, complemented
2	C₃²×#874	C₂	1	1	yes	maximal, characteristic
3	C₃²×#809	C₂	1	4	yes	maximal, complemented
4	C₃²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#486	—	3	1	no	maximal
6	C₃×#5088	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃³×Q₈	—	2	1	no	—
8	C₂×C₃²×D₁₂	—	2	1	no	—
9	C₂×C₃²×D₁₂	—	2	1	no	—
10	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
11	C₃³×C₄⋊C₄	—	2	1	no	—
12	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃²×C₄⋊Q₈	—	2	4	no	—
16	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
17	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	complemented
18	C₃²×#127	—	3	1	no	—
19	C₃×#874	C₂×C₃	1	4	yes	—
20	C₃²×#102	—	3	4	no	—
21	C₃²×C₄×C₈	—	3	1	no	—
22	C₃×#809	C₂×C₃	1	16	yes	complemented
23	C₃×C₄×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
24	C₂²×C₃²×D₃	—	2	2	no	—
25	C₃³×Q₈	—	4	2	no	—
26	C₂×C₃³×C₄	—	2	2	no	—
27	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₃²×D₁₂	—	2	4	no	—
31	C₃²×D₁₂	—	4	2	no	—
32	C₃²×D₁₂	—	4	2	no	—
33	C₃²×C₃⋊C₈	—	2	2	no	—
34	C₃×#486	—	3	4	no	—
35	#5088	C₃²	1	1	yes	characteristic, direct product factor
36	C₂×C₃²×Q₈	—	2	8	no	—
37	C₂×C₃²×Q₈	—	2	1	no	—
38	C₂×C₃²×Q₈	—	2	4	no	—
39	C₂×C₃²×D₄	—	3	2	no	—
40	C₂×C₃²×D₄	—	6	1	no	—
41	C₂×C₃²×D₄	—	6	1	no	—
42	C₂×C₃×D₁₂	—	2	4	no	—
43	C₂×C₃×D₁₂	—	2	4	no	—
44	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
45	C₂×C₃²×C₈	—	3	2	no	—
46	C₃²×C₄⋊C₄	—	2	8	no	—
47	C₃²×C₄⋊C₄	—	2	8	no	—
48	C₃²×C₄⋊C₄	—	2	1	no	—
49	C₃²×C₄⋊C₄	—	2	4	no	—
50	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
52	C₃²×C₄²	—	2	4	no	—
53	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₃²×C₄²	C₂²×C₃	1	4	yes	—
55	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
56	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃²×D₃	—	4	2	no	—
58	C₂×C₃²×D₃	—	4	2	no	—
59	C₂×C₃²×D₃	—	4	2	no	—
60	C₃³×C₄	—	4	2	no	—
61	C₃³×C₄	D₈	1	2	yes	—
62	C₃³×C₄	SD₁₆	1	2	yes	—
63	C₃³×C₄	#41	1	2	yes	—
64	C₃×C₄⋊Q₈	—	2	4	no	—
65	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
66	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
67	C₃×#127	—	3	4	no	—
68	#874	C₂×C₃²	1	1	yes	characteristic
69	C₃×#102	—	3	16	no	—
70	C₃×C₄×C₈	—	3	4	no	—
71	#809	C₂×C₃²	1	4	yes	complemented
72	C₄×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
73	C₂³×C₃²	—	6	2	no	—
74	C₂²×C₃×D₃	—	2	8	no	—
75	C₃²×Q₈	—	4	16	no	—
76	C₃²×Q₈	—	4	2	no	—
77	C₃²×Q₈	—	4	8	no	—
78	C₃²×D₄	—	6	4	no	—
79	C₃²×D₄	—	12	2	no	—
80	C₃²×D₄	—	12	2	no	—
81	C₂×C₃²×C₄	—	2	16	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	#67	1	1	yes	characteristic
88	C₂×C₃²×C₄	#67	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
90	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
91	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
93	C₃×D₁₂	—	2	16	no	—
94	C₃×D₁₂	—	4	8	no	—
95	C₃×D₁₂	—	4	8	no	—
96	C₃²×C₈	—	6	2	no	—
97	C₃×C₃⋊C₈	—	2	8	no	—
98	#486	—	3	1	no	Sylow
99	C₂×C₃³	#124	1	1	yes	characteristic
100	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
101	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
102	C₃²×D₃	—	8	2	no	—
103	C₂×C₃×Q₈	—	2	8	no	—
104	C₂×C₃×Q₈	—	2	4	no	—
105	C₂×C₃×Q₈	—	2	1	no	—
106	C₂×C₃×D₄	—	3	8	no	—
107	C₂×C₃×D₄	—	6	4	no	—
108	C₂×C₃×D₄	—	6	4	no	—
109	C₂×D₁₂	—	2	1	no	—
110	C₂×D₁₂	—	2	1	no	—
111	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
112	C₂×C₃×C₈	—	3	8	no	—
113	C₃×C₄⋊C₄	—	2	8	no	—
114	C₃×C₄⋊C₄	—	2	4	no	—
115	C₃×C₄⋊C₄	—	2	1	no	—
116	C₃×C₄⋊C₄	—	2	8	no	—
117	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
118	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
119	C₃×C₄²	—	2	4	no	—
120	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
121	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
122	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
123	C₂²×C₃²	—	2	4	no	—
124	C₂²×C₃²	—	12	2	no	—
125	C₂²×C₃²	—	12	2	no	—
126	C₂²×C₃²	—	12	2	no	—
127	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
128	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
129	C₂×C₃×D₃	—	4	8	no	—
130	C₂×C₃×D₃	—	4	8	no	—
131	C₂×C₃×D₃	—	4	8	no	—
132	C₃²×C₄	—	2	8	no	—
133	C₃²×C₄	—	2	8	no	—
134	C₃²×C₄	—	2	8	no	—
135	C₃²×C₄	—	4	16	no	—
136	C₃²×C₄	—	4	2	no	—
137	C₃²×C₄	—	4	8	no	—
138	C₃²×C₄	#213	1	2	yes	—
139	C₃²×C₄	#215	1	2	yes	—
140	C₃²×C₄	C₃×D₈	1	8	yes	—
141	C₃²×C₄	C₃×SD₁₆	1	8	yes	—
142	C₃²×C₄	#239	1	2	yes	—
143	C₃²×C₄	C₃×#41	1	8	yes	—
144	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
145	#127	—	3	1	no	—
146	#102	—	3	4	no	—
147	C₄×C₈	—	3	1	no	—
148	C₃³	#486	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	—	6	8	no	—
150	C₂²×D₃	—	2	2	no	—
151	C₃×Q₈	—	4	16	no	—
152	C₃×Q₈	—	4	8	no	—
153	C₃×Q₈	—	4	2	no	—
154	C₃×D₄	—	6	16	no	—
155	C₃×D₄	—	12	8	no	—
156	C₃×D₄	—	12	8	no	—
157	C₂×C₃×C₄	—	2	16	no	—
158	C₂×C₃×C₄	—	2	8	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	C₃×#67	1	4	yes	—
164	C₂×C₃×C₄	C₃×#67	1	4	yes	—
165	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
168	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
169	D₁₂	—	2	4	no	—
170	D₁₂	—	4	2	no	—
171	D₁₂	—	4	2	no	—
172	C₃×C₈	—	6	8	no	—
173	C₃⋊C₈	—	2	2	no	—
174	C₂×C₃²	—	2	4	no	—
175	C₂×C₃²	—	2	4	no	—
176	C₂×C₃²	—	2	4	no	—
177	C₂×C₃²	—	24	2	no	—
178	C₂×C₃²	C₂×#213	1	1	yes	characteristic
179	C₂×C₃²	C₂×#215	1	1	yes	characteristic
180	C₂×C₃²	#947	1	1	yes	characteristic
181	C₂×C₃²	C₃×#124	1	4	yes	—
182	C₂×C₃²	C₂×C₃×D₈	1	4	yes	—
183	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
184	C₃×D₃	—	8	8	no	—
185	C₂×Q₈	—	2	1	no	—
186	C₂×D₄	—	3	2	no	—
187	C₂×D₄	—	6	1	no	—
188	C₂×D₄	—	6	1	no	—
189	C₂×C₈	—	3	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
192	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	12	8	no	—
195	C₂²×C₃	—	12	8	no	—
196	C₂²×C₃	—	12	8	no	—
197	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
198	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
199	C₂×D₃	—	4	2	no	—
200	C₂×D₃	—	4	2	no	—
201	C₂×D₃	—	4	2	no	—
202	C₃×C₄	—	2	8	no	—
203	C₃×C₄	—	2	8	no	—
204	C₃×C₄	—	2	8	no	—
205	C₃×C₄	—	4	16	no	—
206	C₃×C₄	—	4	8	no	—
207	C₃×C₄	—	4	2	no	—
208	C₃×C₄	C₃×#213	1	8	yes	—
209	C₃×C₄	C₃×#215	1	8	yes	—
210	C₃×C₄	C₃²×D₈	1	2	yes	—
211	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
212	C₃×C₄	C₃×#239	1	8	yes	—
213	C₃×C₄	C₃²×#41	1	2	yes	—
214	C₃²	—	2	4	no	—
215	C₃²	#5088	1	1	yes	characteristic, direct product factor
216	C₃²	C₃×#486	1	4	yes	complemented
217	C₂³	—	6	2	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	6	4	no	—
220	D₄	—	12	2	no	—
221	D₄	—	12	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	C₃²×#67	1	1	yes	characteristic
224	C₂×C₄	C₃²×#67	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
226	C₈	—	6	2	no	—
227	C₂×C₃	—	2	4	no	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	24	8	no	—
231	C₂×C₃	C₂×C₃×#213	1	4	yes	—
232	C₂×C₃	C₂×C₃×#215	1	4	yes	—
233	C₂×C₃	C₃×#947	1	4	yes	—
234	C₂×C₃	C₃²×#124	1	1	yes	characteristic
235	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
236	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
237	D₃	—	8	2	no	—
238	C₂²	—	12	2	no	—
239	C₂²	—	12	2	no	—
240	C₂²	—	12	2	no	—
241	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
242	C₄	—	4	2	no	—
243	C₄	C₃²×#213	1	2	yes	—
244	C₄	C₃²×#215	1	2	yes	—
245	C₄	C₃²×#239	1	2	yes	—
246	C₃	—	2	4	no	—
247	C₃	C₃×#5088	1	4	yes	direct product factor
248	C₃	C₃²×#486	1	1	yes	characteristic, complemented
249	C₂	—	24	2	no	—
250	C₂	C₂×C₃²×#213	1	1	yes	characteristic
251	C₂	C₂×C₃²×#215	1	1	yes	characteristic
252	C₂	C₃²×#947	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₃²×#874	C₂
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×#486
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄⋊Q₈	C₂	semidirect
3. C₃²×#809	C₂	semidirect
4. C₃²×C₄×C₃⋊C₈	C₂	semidirect
6. C₃×#5088	C₃	direct (N × H)
16. C₃²×C₄⋊Q₈	D₃	semidirect
17. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
22. C₃×#809	C₂×C₃	semidirect
23. C₃×C₄×C₃⋊C₈	C₂×C₃	semidirect
35. #5088	C₃²	direct (N × H)
65. C₃×C₄⋊Q₈	C₃×D₃	semidirect
66. C₃×C₄⋊Q₈	C₂×C₃²	semidirect
71. #809	C₂×C₃²	semidirect
72. C₄×C₃⋊C₈	C₂×C₃²	semidirect
144. C₄⋊Q₈	C₃²×D₃	semidirect
148. C₃³	#486	semidirect
215. C₃²	#5088	direct (N × H)
216. C₃²	C₃×#486	semidirect
247. C₃	C₃×#5088	direct (N × H)
248. C₃	C₃²×#486	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×#874	2
C₃²×#809	2
C₃²×C₄×C₃⋊C₈	2
C₃²×#486	3
C₃×#5088	3

Subgroups — C32×#5088 #13162525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5088 #13162525