Subgroups — C₂×C₃×#63394 #13143580

Order 1728.

594 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3667	C₂	1	2	yes	maximal
3	C₃×#63394	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#836	—	3	1	no	maximal
5	C₂×#63394	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×Q₈×C₉	—	2	2	no	—
7	C₂×C₃×Q₈×C₉	—	2	2	no	—
8	C₂×C₃×Q₈×C₉	C₄	1	2	yes	—
9	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₃×#3667	—	2	2	no	—
13	C₃×#3667	C₂²	1	4	yes	—
14	C₂×C₃×C₉⋊C₈	—	2	2	no	—
15	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
16	C₂×C₃×#208	—	3	2	no	—
17	C₂²×Q₈×C₉	—	2	1	no	—
18	C₂²×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
19	C₃×#836	—	3	4	no	—
20	C₂×#3667	C₂×C₃	1	2	yes	—
21	#63394	C₂×C₃	1	4	yes	direct product factor
22	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
23	C₃×Q₈×C₉	—	2	4	no	—
24	C₃×Q₈×C₉	—	2	4	no	—
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₃×C₉⋊C₈	—	2	4	no	—
33	C₂×C₃×#101	—	9	1	no	—
34	C₂×#836	—	3	1	no	—
35	C₂×C₃²×Q₈	—	2	2	no	—
36	C₂×C₃²×Q₈	—	2	2	no	—
37	C₂×C₃²×Q₈	Dic₃	1	2	yes	—
38	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
39	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₃×#208	—	3	4	no	—
42	C₃×#208	—	6	2	no	—
43	C₂×C₃×C₃⋊C₈	—	6	2	no	—
44	C₂×Q₈×C₉	—	2	4	no	—
45	C₂×Q₈×C₉	—	2	4	no	—
46	C₂×Q₈×C₉	—	2	2	no	—
47	C₂×Q₈×C₉	—	2	2	no	—
48	C₂×Q₈×C₉	—	2	2	no	—
49	C₂×Q₈×C₉	—	2	2	no	—
50	C₂×Q₈×C₉	C₃×C₄	1	2	yes	—
51	C₂×Q₈×C₉	C₂²×C₃	1	2	yes	—
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂²×C₄×C₉	C₃×C₄	1	2	yes	—
55	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
56	#3667	—	2	2	no	—
57	#3667	C₂²×C₃	1	4	yes	—
58	C₂×C₉⋊C₈	—	2	2	no	—
59	C₂²×C₃×C₉	—	2	2	no	—
60	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	—	2	4	no	—
64	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
65	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
66	C₂²×C₃×Q₈	—	2	1	no	—
67	C₂²×C₃×Q₈	D₉	1	1	yes	characteristic
68	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
69	C₂×C₃×M₁₆	—	9	2	no	—
70	C₂×#208	—	3	2	no	—
71	C₃×#101	—	9	4	no	—
72	#836	—	3	4	no	—
73	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
74	C₃²×Q₈	—	2	4	no	—
75	C₃²×Q₈	—	2	4	no	—
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
79	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	#67	1	2	yes	—
81	C₂×C₃²×C₄	#67	1	2	yes	—
82	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₂³×C₉	—	2	1	no	—
84	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
85	C₃×C₃⋊C₈	—	6	4	no	—
86	Q₈×C₉	—	2	8	no	—
87	Q₈×C₉	—	2	8	no	—
88	Q₈×C₉	—	2	4	no	—
89	Q₈×C₉	—	2	4	no	—
90	C₂×C₄×C₉	—	2	4	no	—
91	C₂×C₄×C₉	—	2	4	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
100	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
102	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
103	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
104	C₉⋊C₈	—	2	4	no	—
105	C₂×#101	—	9	1	no	Sylow
106	C₂×C₃×C₉	—	2	1	no	—
107	C₂×C₃×C₉	—	2	1	no	—
108	C₂×C₃×C₉	#101	1	2	yes	complemented
109	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₃×Q₈	—	2	4	no	—
111	C₂×C₃×Q₈	—	2	4	no	—
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×Q₈	—	2	2	no	—
114	C₂×C₃×Q₈	—	2	2	no	—
115	C₂×C₃×Q₈	—	2	2	no	—
116	C₂×C₃×Q₈	—	2	2	no	—
117	C₂×C₃×Q₈	—	2	2	no	—
118	C₂×C₃×Q₈	Dic₉	1	2	yes	—
119	C₂×C₃×Q₈	C₂×D₉	1	2	yes	—
120	C₂×C₃×Q₈	C₃×Dic₃	1	2	yes	—
121	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	—	2	1	no	—
124	C₂²×C₃×C₄	Dic₉	1	2	yes	—
125	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
127	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
128	C₃×M₁₆	—	9	4	no	—
129	C₃×M₁₆	—	18	2	no	—
130	C₂×C₃×C₈	—	18	2	no	—
131	#208	—	3	4	no	—
132	#208	—	6	2	no	—
133	C₂×C₃⋊C₈	—	6	2	no	—
134	C₂²×C₃²	—	2	2	no	—
135	C₂²×C₃²	#217	1	1	yes	characteristic
136	C₂²×C₃²	#217	1	1	yes	characteristic
137	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
138	C₃²×C₄	—	2	4	no	—
139	C₃²×C₄	#217	1	2	yes	—
140	C₃²×C₄	C₂×#67	1	2	yes	—
141	C₂²×C₉	—	2	4	no	—
142	C₂²×C₉	—	2	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	—	2	1	no	—
146	C₂²×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₄×C₉	—	2	8	no	—
150	C₄×C₉	—	2	4	no	—
151	C₄×C₉	—	2	2	no	—
152	C₄×C₉	—	2	2	no	—
153	C₄×C₉	C₃×C₂²⋊C₄	1	2	yes	—
154	C₄×C₉	C₂×C₃×D₄	1	2	yes	—
155	C₂²×Q₈	C₃×D₉	1	1	yes	characteristic
156	C₂×M₁₆	—	9	2	no	—
157	#101	—	9	4	no	—
158	C₃×C₉	C₂×#101	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
161	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
162	C₃×Q₈	—	2	8	no	—
163	C₃×Q₈	—	2	8	no	—
164	C₃×Q₈	—	2	4	no	—
165	C₃×Q₈	—	2	4	no	—
166	C₃×Q₈	—	2	4	no	—
167	C₃×Q₈	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	C₂×Dic₉	1	4	yes	—
180	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
181	C₂×C₃×C₄	#611	1	2	yes	—
182	C₂×C₃×C₄	#611	1	2	yes	—
183	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
185	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₃×#67	1	2	yes	—
187	C₂×C₃×C₄	C₃×#67	1	2	yes	—
188	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
189	C₃×C₈	—	18	4	no	—
190	C₃⋊C₈	—	6	4	no	—
191	C₂×C₃²	—	2	1	no	—
192	C₂×C₃²	—	2	1	no	—
193	C₂×C₃²	#836	1	2	yes	—
194	C₂×C₃²	C₂×#217	1	1	yes	characteristic
195	C₂×C₉	—	2	2	no	—
196	C₂×C₉	—	2	2	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	—	2	2	no	—
201	C₂×C₉	C₃×#101	1	2	yes	complemented
202	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	C₃×Dic₉	1	2	yes	—
206	C₂×Q₈	C₂×C₃×D₉	1	2	yes	—
207	C₂²×C₄	C₃×Dic₉	1	2	yes	—
208	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
209	M₁₆	—	9	4	no	—
210	M₁₆	—	18	2	no	—
211	C₂×C₈	—	18	2	no	—
212	C₂²×C₃	—	2	4	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	#3676	1	1	yes	characteristic
219	C₂²×C₃	#3676	1	1	yes	characteristic
220	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
221	C₂²×C₃	C₃×#217	1	1	yes	characteristic
222	C₂²×C₃	C₃×#217	1	1	yes	characteristic
223	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
224	C₃×C₄	—	2	8	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	2	4	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	#3676	1	2	yes	—
230	C₃×C₄	C₂×#611	1	2	yes	—
231	C₃×C₄	C₃×#217	1	2	yes	—
232	C₃×C₄	C₂×C₃×#67	1	2	yes	—
233	C₃²	C₂×#836	1	1	yes	characteristic
234	C₉	—	2	1	no	—
235	C₉	C₂×C₃×#101	1	1	yes	characteristic, complemented
236	C₂³	C₂×C₃×Dic₉	1	1	yes	characteristic
237	Q₈	—	2	4	no	—
238	Q₈	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	C₂×C₃×Dic₉	1	4	yes	—
242	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
243	C₂×C₄	C₃×#611	1	2	yes	—
244	C₂×C₄	C₃×#611	1	2	yes	—
245	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
246	C₈	—	18	4	no	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	2	2	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	2	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	#63394	1	2	yes	direct product factor
256	C₂×C₃	C₂×#3676	1	1	yes	characteristic
257	C₂×C₃	C₃×#836	1	2	yes	—
258	C₂×C₃	C₂×C₃×#217	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	C₃×#3676	1	1	yes	characteristic
261	C₂²	C₃×#3676	1	1	yes	characteristic
262	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	C₃×#3676	1	2	yes	—
265	C₄	C₂×C₃×#611	1	2	yes	—
266	C₃	—	2	1	no	—
267	C₃	C₂×#63394	1	1	yes	characteristic, direct product factor
268	C₃	C₂×C₃×#836	1	1	yes	characteristic
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	C₃×#63394	1	2	yes	direct product factor
272	C₂	C₂×C₃×#3676	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#3676
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle	C₂²×C₃²	#217
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂²×C₃×Q₈	D₉
hypocenter	C₉	C₂×C₃×#101
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
3. C₃×#63394	C₂	direct (N × H)
5. C₂×#63394	C₃	direct (N × H)
21. #63394	C₂×C₃	direct (N × H)
108. C₂×C₃×C₉	#101	semidirect
158. C₃×C₉	C₂×#101	semidirect
201. C₂×C₉	C₃×#101	semidirect
235. C₉	C₂×C₃×#101	semidirect
255. C₂×C₃	#63394	direct (N × H)
267. C₃	C₂×#63394	direct (N × H)
271. C₂	C₃×#63394	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×Q₈×C₉	2
C₂×C₃×#3667	2
C₃×#63394	2
C₂×C₃×#836	3
C₂×#63394	3

Subgroups — C2×C3×#63394 #13143580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63394 #13143580