Subgroups — C₄×#88524 #13143755

Order 1728.

610 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88524	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₃⋊D₉)	C₂	1	1	yes	maximal, characteristic
3	C₄×#6226	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720543	C₂	1	1	yes	maximal, characteristic
6	#11720540	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720539	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#3744	—	3	1	no	maximal
9	C₄×#3664	—	3	3	no	maximal
10	C₂×C₄⋊(C₃⋊D₉)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₃⋊D₉	—	2	1	no	—
12	C₂×#6226	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	#88536	—	2	1	no	—
17	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
19	#88534	—	2	1	no	—
20	C₄×C₃⋊Dic₉	—	2	1	no	—
21	#88524	C₄	1	4	yes	direct product factor
22	C₂×#3744	—	3	1	no	—
23	C₄×C₄⋊(C₃⋊D₃)	—	3	1	no	—
24	C₄×#634	—	3	1	no	—
25	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
26	#63648	—	3	1	no	—
27	#63644	—	3	1	no	—
28	#63643	—	3	1	no	—
29	C₂×#3664	—	3	3	no	—
30	C₄×D₃₆	—	3	3	no	—
31	C₄×Dic₁₈	—	3	3	no	—
32	C₄×C₈×C₉	D₃	1	3	yes	complemented
33	#63380	—	3	3	no	—
34	#63377	—	3	3	no	—
35	#63376	—	3	3	no	—
36	C₂²×C₃⋊D₉	—	2	1	no	—
37	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₃⋊Dic₉	—	2	1	no	—
41	C₂×C₃⋊Dic₉	—	2	1	no	—
42	C₂×C₃⋊Dic₉	—	2	1	no	—
43	C₄⋊(C₃⋊D₉)	—	2	1	no	—
44	C₄⋊(C₃⋊D₉)	C₂×C₄	1	2	yes	—
45	C₄×C₃⋊D₉	—	4	2	no	—
46	#6226	—	2	1	no	—
47	#6226	C₂×C₄	1	2	yes	complemented
48	C₃×C₈×C₉	—	2	1	no	—
49	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
50	C₄×#204	—	9	3	no	—
51	C₄×#204	—	9	1	no	—
52	C₂×C₄⋊(C₃⋊D₃)	—	3	1	no	—
53	C₂×C₄×C₃⋊D₃	—	6	1	no	—
54	C₂×#634	—	3	1	no	—
55	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
57	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
58	#3752	—	6	1	no	—
59	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
60	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
61	#3750	—	6	1	no	—
62	C₄×C₃⋊Dic₃	—	6	1	no	—
63	#3744	—	3	4	no	—
64	C₂×D₃₆	—	3	3	no	—
65	C₂×C₄×D₉	—	6	3	no	—
66	C₂×Dic₁₈	—	3	3	no	—
67	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
68	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
69	C₄²×C₉	C₂×D₃	1	3	yes	—
70	(C₂×D₉)⋊C₄	—	6	3	no	—
71	C₄⋊Dic₉	—	3	3	no	—
72	C₄⋊Dic₉	—	3	3	no	—
73	Dic₉⋊C₄	—	6	3	no	—
74	C₄×Dic₉	—	6	3	no	—
75	#3664	—	3	12	no	—
76	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃⋊D₉	—	2	2	no	—
78	C₂×C₃⋊D₉	—	4	1	no	—
79	C₂×C₃⋊D₉	—	4	1	no	—
80	C₃×C₄×C₉	—	2	1	no	—
81	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
82	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₃×C₄×C₉	#41	1	1	yes	characteristic
84	C₃⋊Dic₉	—	2	2	no	—
85	C₃⋊Dic₉	—	4	1	no	—
86	C₃⋊Dic₉	—	4	1	no	—
87	C₂×#204	—	9	3	no	—
88	C₂×#204	—	9	1	no	—
89	C₄×D₁₂	—	9	3	no	—
90	C₄×D₁₂	—	9	1	no	—
91	C₄×Dic₆	—	9	3	no	—
92	C₄×Dic₆	—	9	1	no	—
93	C₃×C₄×C₈	D₉	1	3	yes	complemented
94	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
95	#821	—	9	3	no	—
96	#821	—	9	1	no	—
97	#817	—	9	3	no	—
98	#817	—	9	1	no	—
99	#816	—	9	3	no	—
100	#816	—	9	1	no	—
101	C₂²×C₃⋊D₃	—	6	1	no	—
102	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
103	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃⋊Dic₃	—	6	1	no	—
106	C₂×C₃⋊Dic₃	—	6	1	no	—
107	C₂×C₃⋊Dic₃	—	6	1	no	—
108	C₄⋊(C₃⋊D₃)	—	3	2	no	—
109	C₄⋊(C₃⋊D₃)	—	6	1	no	—
110	C₄×C₃⋊D₃	—	12	2	no	—
111	#634	—	3	2	no	—
112	#634	—	6	1	no	—
113	C₂²×D₉	—	6	3	no	—
114	C₃²×C₈	—	2	1	no	—
115	C₃²×C₈	C₄×D₃	1	2	yes	—
116	C₂×C₄×C₉	D₁₂	1	3	yes	—
117	C₂×C₄×C₉	D₁₂	1	3	yes	—
118	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
119	C₂×Dic₉	—	6	3	no	—
120	C₂×Dic₉	—	6	3	no	—
121	C₂×Dic₉	—	6	3	no	—
122	D₃₆	—	3	6	no	—
123	D₃₆	—	6	3	no	—
124	C₄×D₉	—	12	6	no	—
125	Dic₁₈	—	3	6	no	—
126	Dic₁₈	—	6	3	no	—
127	C₈×C₉	—	2	3	no	—
128	C₈×C₉	C₄×D₃	1	6	yes	complemented
129	C₄×SD₁₆	—	27	1	no	Sylow
130	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₃×C₉	#135	1	1	yes	characteristic
133	C₃⋊D₉	—	4	2	no	—
134	C₂×D₁₂	—	9	3	no	—
135	C₂×D₁₂	—	9	1	no	—
136	C₂×C₄×D₃	—	18	3	no	—
137	C₂×C₄×D₃	—	18	1	no	—
138	C₂×Dic₆	—	9	3	no	—
139	C₂×Dic₆	—	9	1	no	—
140	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
141	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
142	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
144	C₃×C₄²	C₂×D₉	1	3	yes	—
145	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
146	(C₂×D₃)⋊C₄	—	18	3	no	—
147	(C₂×D₃)⋊C₄	—	18	1	no	—
148	C₄⋊Dic₃	—	9	3	no	—
149	C₄⋊Dic₃	—	9	3	no	—
150	C₄⋊Dic₃	—	9	1	no	—
151	C₄⋊Dic₃	—	9	1	no	—
152	Dic₃⋊C₄	—	18	3	no	—
153	Dic₃⋊C₄	—	18	1	no	—
154	C₄×Dic₃	—	18	3	no	—
155	C₄×Dic₃	—	18	1	no	—
156	#204	—	9	12	no	—
157	#204	—	9	4	no	—
158	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
159	C₂×C₃⋊D₃	—	6	2	no	—
160	C₂×C₃⋊D₃	—	12	1	no	—
161	C₂×C₃⋊D₃	—	12	1	no	—
162	C₃²×C₄	—	2	1	no	—
163	C₃²×C₄	#204	1	2	yes	—
164	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
165	C₃²×C₄	#235	1	1	yes	characteristic
166	C₃⋊Dic₃	—	6	2	no	—
167	C₃⋊Dic₃	—	12	1	no	—
168	C₃⋊Dic₃	—	12	1	no	—
169	C₂²×C₉	C₂×D₁₂	1	3	yes	—
170	C₂×D₉	—	6	6	no	—
171	C₂×D₉	—	12	3	no	—
172	C₂×D₉	—	12	3	no	—
173	C₄×C₉	—	2	3	no	—
174	C₄×C₉	#204	1	6	yes	complemented
175	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
176	C₄×C₉	#235	1	3	yes	—
177	Dic₉	—	6	6	no	—
178	Dic₉	—	12	3	no	—
179	Dic₉	—	12	3	no	—
180	C₂×SD₁₆	—	27	1	no	—
181	C₄×Q₈	—	27	1	no	—
182	C₄×D₄	—	27	1	no	—
183	#106	—	27	1	no	—
184	#103	—	27	1	no	—
185	#102	—	27	1	no	—
186	C₄×C₈	C₃⋊D₉	1	1	yes	characteristic, complemented
187	C₃×C₉	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
188	C₂²×D₃	—	18	3	no	—
189	C₂²×D₃	—	18	1	no	—
190	C₂×C₃×C₄	D₃₆	1	3	yes	—
191	C₂×C₃×C₄	D₃₆	1	3	yes	—
192	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
193	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
194	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
196	C₂×Dic₃	—	18	3	no	—
197	C₂×Dic₃	—	18	1	no	—
198	C₂×Dic₃	—	18	3	no	—
199	C₂×Dic₃	—	18	3	no	—
200	C₂×Dic₃	—	18	1	no	—
201	C₂×Dic₃	—	18	1	no	—
202	D₁₂	—	9	6	no	—
203	D₁₂	—	9	2	no	—
204	D₁₂	—	18	3	no	—
205	D₁₂	—	18	1	no	—
206	C₄×D₃	—	36	6	no	—
207	C₄×D₃	—	36	2	no	—
208	Dic₆	—	9	6	no	—
209	Dic₆	—	9	2	no	—
210	Dic₆	—	18	3	no	—
211	Dic₆	—	18	1	no	—
212	C₃×C₈	—	2	3	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	C₄×D₉	1	6	yes	complemented
215	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	—
216	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
217	C₂×C₃²	C₂×#204	1	1	yes	characteristic
218	C₂×C₃²	#904	1	1	yes	characteristic
219	C₃⋊D₃	—	12	2	no	—
220	C₂×C₉	C₄×D₁₂	1	3	yes	—
221	C₂×C₉	C₂×#204	1	3	yes	—
222	C₂×C₉	#904	1	3	yes	—
223	D₉	—	12	6	no	—
224	C₂×Q₈	—	27	1	no	—
225	C₂×D₄	—	27	1	no	—
226	C₂²×C₄	—	54	1	no	—
227	SD₁₆	—	27	4	no	—
228	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
229	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
230	C₄⋊C₄	—	27	1	no	—
231	C₄⋊C₄	—	27	1	no	—
232	C₄⋊C₄	—	54	1	no	—
233	C₂²⋊C₄	—	54	1	no	—
234	C₄²	—	54	1	no	—
235	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
236	C₂²×C₃	C₂×D₃₆	1	3	yes	—
237	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
238	C₂×D₃	—	18	2	no	—
239	C₂×D₃	—	18	6	no	—
240	C₂×D₃	—	36	1	no	—
241	C₂×D₃	—	36	1	no	—
242	C₂×D₃	—	36	3	no	—
243	C₂×D₃	—	36	3	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	3	no	—
246	C₃×C₄	#3664	1	6	yes	complemented
247	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
248	C₃×C₄	#3697	1	3	yes	—
249	C₃×C₄	#3744	1	2	yes	—
250	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
251	C₃×C₄	#3828	1	1	yes	characteristic
252	Dic₃	—	18	2	no	—
253	Dic₃	—	18	6	no	—
254	Dic₃	—	36	1	no	—
255	Dic₃	—	36	1	no	—
256	Dic₃	—	36	3	no	—
257	Dic₃	—	36	3	no	—
258	C₃²	C₄×#204	1	1	yes	characteristic
259	C₉	C₄×#204	1	3	yes	complemented
260	C₂³	—	54	1	no	—
261	Q₈	—	27	2	no	—
262	Q₈	—	54	1	no	—
263	D₄	—	27	2	no	—
264	D₄	—	54	1	no	—
265	C₂×C₄	—	54	1	no	—
266	C₂×C₄	—	54	1	no	—
267	C₂×C₄	—	54	1	no	—
268	C₂×C₄	—	108	2	no	—
269	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
270	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	C₄×C₃⋊D₉	1	2	yes	complemented
274	C₂×C₃	C₄×D₃₆	1	3	yes	—
275	C₂×C₃	C₂×#3664	1	3	yes	—
276	C₂×C₃	#63467	1	3	yes	—
277	C₂×C₃	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
278	C₂×C₃	C₂×#3744	1	1	yes	characteristic
279	C₂×C₃	#64113	1	1	yes	characteristic
280	D₃	—	36	2	no	—
281	D₃	—	36	6	no	—
282	C₂²	—	54	2	no	—
283	C₂²	—	108	1	no	—
284	C₂²	—	108	1	no	—
285	C₂²	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	54	2	no	—
288	C₄	—	108	1	no	—
289	C₄	—	108	1	no	—
290	C₄	#88524	1	2	yes	direct product factor
291	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
292	C₄	#88736	1	1	yes	characteristic
293	C₃	C₄×#3664	1	3	yes	complemented
294	C₃	C₄×#3744	1	1	yes	characteristic
295	C₂	—	108	2	no	—
296	C₂	C₄×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
297	C₂	C₂×#88524	1	1	yes	characteristic
298	C₂	#11721773	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊(C₃⋊D₉)
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×D₁₂
prime radical	C₂×C₄⋊(C₃⋊D₉)	C₂²
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×C₈	C₃⋊D₉
hypocenter	C₃×C₉	C₄×SD₁₆
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#6226	C₂	semidirect
4. C₃×C₄×C₈×C₉	C₂	semidirect
6. #11720540	C₂	semidirect
7. #11720539	C₂	semidirect
21. #88524	C₄	direct (N × H)
32. C₄×C₈×C₉	D₃	semidirect
47. #6226	C₂×C₄	semidirect
49. C₃×C₈×C₉	C₂×C₄	semidirect
81. C₃×C₄×C₉	SD₁₆	semidirect
93. C₃×C₄×C₈	D₉	semidirect
128. C₈×C₉	C₄×D₃	semidirect
174. C₄×C₉	#204	semidirect
186. C₄×C₈	C₃⋊D₉	semidirect
187. C₃×C₉	C₄×SD₁₆	semidirect
214. C₃×C₈	C₄×D₉	semidirect
246. C₃×C₄	#3664	semidirect
259. C₉	C₄×#204	semidirect
273. C₈	C₄×C₃⋊D₉	semidirect
290. C₄	#88524	direct (N × H)
293. C₃	C₄×#3664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88524	2
C₄×C₄⋊(C₃⋊D₉)	2
C₄×#6226	2
C₃×C₄×C₈×C₉	2
#11720543	2
#11720540	2
#11720539	2
C₄×#3744	3
C₄×#3664	3

Subgroups — C4×#88524 #13143755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88524 #13143755