Subgroups — C₃²×#5115 #13162550

Order 1728.

1108 conjugacy classes of subgroups.

The subgroup lattice has 312 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, complemented
2	C₃²×#930	C₂	1	2	yes	maximal, complemented
3	C₂×C₃²×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#821	C₂	1	2	yes	maximal, complemented
5	C₃²×#818	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#466	—	3	1	no	maximal
7	C₃×#5115	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃²×#67	—	2	2	no	—
10	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
11	C₂×C₃³×C₈	—	2	2	no	—
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₃²×(C₂×D₃)⋊C₄	—	2	2	no	—
15	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₃²×D₂₄	—	2	2	no	—
17	C₂×C₃²×D₈	—	3	1	no	—
18	C₂²×C₃²×C₈	—	2	4	no	—
19	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic, complemented
20	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	complemented
21	C₃²×#121	—	3	2	no	—
22	C₃×#930	C₂×C₃	1	8	yes	complemented
23	C₂×C₃×D₂₄	C₂×C₃	1	4	yes	complemented
24	C₃²×#107	—	3	1	no	—
25	C₃²×#102	—	3	2	no	—
26	C₃×#821	C₂×C₃	1	8	yes	complemented
27	C₃×#818	C₂×C₃	1	4	yes	complemented
28	C₂³×C₃³	D₄	1	1	yes	characteristic
29	C₂²×C₃²×D₃	—	2	2	no	—
30	C₂×C₃³×C₄	—	2	1	no	—
31	C₂×C₃³×C₄	—	2	1	no	—
32	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₃²×#67	—	4	4	no	—
35	C₂×C₃²×Dic₃	—	2	2	no	—
36	C₃²×D₁₂	—	2	2	no	—
37	C₃²×D₁₂	—	2	2	no	—
38	C₃³×C₈	—	2	1	no	—
39	C₃³×C₈	D₄	1	2	yes	complemented
40	C₃×#466	—	3	4	no	—
41	#5115	C₃²	1	1	yes	characteristic, direct product factor
42	C₂×C₃²×D₄	—	3	2	no	—
43	C₂×C₃²×D₄	—	6	2	no	—
44	C₂²×C₃²×C₄	—	2	4	no	—
45	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
47	C₂×C₃×#67	—	2	8	no	—
48	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
49	C₃²×D₈	—	6	2	no	—
50	C₂×C₃²×C₈	—	2	16	no	—
51	C₂×C₃²×C₈	—	2	4	no	—
52	C₂×C₃²×C₈	—	2	4	no	—
53	C₂×C₃²×C₈	—	2	2	no	—
54	C₂×C₃²×C₈	—	2	8	no	—
55	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
58	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
59	C₃²×C₄⋊C₄	—	3	2	no	—
60	C₃²×C₂²⋊C₄	—	6	2	no	—
61	C₃×(C₂×D₃)⋊C₄	—	2	8	no	—
62	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
63	C₃×D₂₄	—	2	8	no	—
64	C₂²×C₃³	—	2	1	no	—
65	C₂²×C₃³	—	2	1	no	—
66	C₂²×C₃³	D₈	1	2	yes	complemented
67	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃²×D₃	—	4	2	no	—
69	C₂×C₃²×D₃	—	4	2	no	—
70	C₂×C₃²×D₃	—	4	2	no	—
71	C₃³×C₄	—	2	1	no	—
72	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
73	C₃³×C₄	#41	1	1	yes	characteristic
74	C₃²×Dic₃	—	4	2	no	—
75	C₂×C₃×D₈	—	3	4	no	—
76	C₂²×C₃×C₈	—	2	4	no	—
77	C₂²×C₃×C₈	C₃×D₃	1	4	yes	complemented
78	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
79	C₃×#121	—	3	8	no	—
80	#930	C₂×C₃²	1	2	yes	complemented
81	C₂×D₂₄	C₂×C₃²	1	1	yes	characteristic, complemented
82	C₃×#107	—	3	4	no	—
83	C₃×#102	—	3	8	no	—
84	#821	C₂×C₃²	1	2	yes	complemented
85	#818	C₂×C₃²	1	1	yes	characteristic, complemented
86	C₂³×C₃²	—	2	4	no	—
87	C₂³×C₃²	—	6	2	no	—
88	C₂³×C₃²	D₁₂	1	1	yes	characteristic
89	C₂³×C₃²	C₃×D₄	1	4	yes	—
90	C₂²×C₃×D₃	—	2	8	no	—
91	C₃²×D₄	—	6	2	no	—
92	C₃²×D₄	—	6	2	no	—
93	C₃²×D₄	—	12	4	no	—
94	C₂×C₃²×C₄	—	2	8	no	—
95	C₂×C₃²×C₄	—	2	8	no	—
96	C₂×C₃²×C₄	—	2	1	no	—
97	C₂×C₃²×C₄	—	2	1	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	—	2	4	no	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	—	2	4	no	—
102	C₂×C₃²×C₄	—	6	2	no	—
103	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
107	C₃×#67	—	4	16	no	—
108	C₂×C₃×Dic₃	—	2	8	no	—
109	C₃×D₁₂	—	2	8	no	—
110	C₃×D₁₂	—	2	8	no	—
111	C₃²×C₈	—	2	8	no	—
112	C₃²×C₈	—	2	1	no	—
113	C₃²×C₈	—	2	4	no	—
114	C₃²×C₈	—	2	8	no	—
115	C₃²×C₈	#67	1	2	yes	complemented
116	C₃²×C₈	C₃×D₄	1	8	yes	complemented
117	#466	—	3	1	no	Sylow
118	C₂×C₃³	—	2	2	no	—
119	C₂×C₃³	#121	1	1	yes	characteristic
120	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
121	C₂×C₃³	#135	1	1	yes	characteristic
122	C₃²×D₃	—	8	2	no	—
123	C₂×C₃×D₄	—	3	8	no	—
124	C₂×C₃×D₄	—	6	8	no	—
125	C₂²×C₃×C₄	—	2	4	no	—
126	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
127	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
128	C₂×#67	—	2	2	no	—
129	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
130	C₃×D₈	—	6	8	no	—
131	C₂×C₃×C₈	—	2	16	no	—
132	C₂×C₃×C₈	—	2	8	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	4	no	—
135	C₂×C₃×C₈	—	2	4	no	—
136	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
137	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
138	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
140	C₃×C₄⋊C₄	—	3	8	no	—
141	C₃×C₂²⋊C₄	—	6	8	no	—
142	(C₂×D₃)⋊C₄	—	2	2	no	—
143	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
144	D₂₄	—	2	2	no	—
145	C₂²×C₃²	—	2	8	no	—
146	C₂²×C₃²	—	2	8	no	—
147	C₂²×C₃²	—	2	1	no	—
148	C₂²×C₃²	—	2	1	no	—
149	C₂²×C₃²	—	2	4	no	—
150	C₂²×C₃²	—	2	4	no	—
151	C₂²×C₃²	—	2	8	no	—
152	C₂²×C₃²	—	2	4	no	—
153	C₂²×C₃²	—	12	2	no	—
154	C₂²×C₃²	—	12	2	no	—
155	C₂²×C₃²	—	12	2	no	—
156	C₂²×C₃²	D₂₄	1	2	yes	complemented
157	C₂²×C₃²	C₃×D₈	1	8	yes	complemented
158	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
159	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
160	C₂×C₃×D₃	—	4	8	no	—
161	C₂×C₃×D₃	—	4	8	no	—
162	C₂×C₃×D₃	—	4	8	no	—
163	C₃²×C₄	—	2	8	no	—
164	C₃²×C₄	—	2	1	no	—
165	C₃²×C₄	—	2	4	no	—
166	C₃²×C₄	—	2	4	no	—
167	C₃²×C₄	—	2	4	no	—
168	C₃²×C₄	—	12	2	no	—
169	C₃²×C₄	#235	1	1	yes	characteristic
170	C₃²×C₄	C₂×#67	1	1	yes	characteristic
171	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
172	C₃²×C₄	C₃×#41	1	4	yes	—
173	C₃×Dic₃	—	4	8	no	—
174	C₂×D₈	—	3	1	no	—
175	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic, complemented
176	#121	—	3	2	no	—
177	#107	—	3	1	no	—
178	#102	—	3	2	no	—
179	C₃³	#466	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	2	4	no	—
181	C₂³×C₃	—	6	8	no	—
182	C₂³×C₃	C₃×D₁₂	1	4	yes	—
183	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
184	C₂²×D₃	—	2	2	no	—
185	C₃×D₄	—	6	8	no	—
186	C₃×D₄	—	6	8	no	—
187	C₃×D₄	—	12	16	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	8	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	6	8	no	—
197	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
198	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
200	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
201	#67	—	4	4	no	—
202	C₂×Dic₃	—	2	2	no	—
203	D₁₂	—	2	2	no	—
204	D₁₂	—	2	2	no	—
205	C₃×C₈	—	2	8	no	—
206	C₃×C₈	—	2	4	no	—
207	C₃×C₈	—	2	1	no	—
208	C₃×C₈	—	2	8	no	—
209	C₃×C₈	C₃×#67	1	8	yes	complemented
210	C₃×C₈	C₃²×D₄	1	2	yes	complemented
211	C₂×C₃²	—	2	16	no	—
212	C₂×C₃²	—	2	8	no	—
213	C₂×C₃²	—	2	2	no	—
214	C₂×C₃²	—	2	4	no	—
215	C₂×C₃²	—	2	4	no	—
216	C₂×C₃²	—	2	4	no	—
217	C₂×C₃²	—	24	2	no	—
218	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
219	C₂×C₃²	#904	1	1	yes	characteristic
220	C₂×C₃²	#930	1	1	yes	characteristic
221	C₂×C₃²	C₃×#121	1	4	yes	—
222	C₂×C₃²	C₂×C₃×D₈	1	4	yes	—
223	C₂×C₃²	C₃×#135	1	4	yes	—
224	C₃×D₃	—	8	8	no	—
225	C₂×D₄	—	3	2	no	—
226	C₂×D₄	—	6	2	no	—
227	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
228	D₈	—	6	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
231	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
232	C₄⋊C₄	—	3	2	no	—
233	C₂²⋊C₄	—	6	2	no	—
234	C₂²×C₃	—	2	8	no	—
235	C₂²×C₃	—	2	8	no	—
236	C₂²×C₃	—	2	4	no	—
237	C₂²×C₃	—	2	4	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	2	8	no	—
242	C₂²×C₃	—	12	8	no	—
243	C₂²×C₃	—	12	8	no	—
244	C₂²×C₃	—	12	8	no	—
245	C₂²×C₃	C₃×D₂₄	1	8	yes	complemented
246	C₂²×C₃	C₃²×D₈	1	2	yes	complemented
247	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
248	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
249	C₂×D₃	—	4	2	no	—
250	C₂×D₃	—	4	2	no	—
251	C₂×D₃	—	4	2	no	—
252	C₃×C₄	—	2	8	no	—
253	C₃×C₄	—	2	4	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	12	8	no	—
258	C₃×C₄	C₃×#235	1	4	yes	—
259	C₃×C₄	C₂×C₃×#67	1	4	yes	—
260	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
261	C₃×C₄	C₃²×#41	1	1	yes	characteristic
262	Dic₃	—	4	2	no	—
263	C₃²	—	2	4	no	—
264	C₃²	#5115	1	1	yes	characteristic, direct product factor
265	C₃²	C₃×#466	1	4	yes	complemented
266	C₂³	—	6	2	no	—
267	C₂³	C₃²×D₁₂	1	1	yes	characteristic
268	D₄	—	6	2	no	—
269	D₄	—	6	2	no	—
270	D₄	—	12	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	6	2	no	—
274	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	C₃²×#67	1	2	yes	complemented
278	C₂×C₃	—	2	16	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	8	no	—
281	C₂×C₃	—	2	4	no	—
282	C₂×C₃	—	2	4	no	—
283	C₂×C₃	—	2	4	no	—
284	C₂×C₃	—	24	8	no	—
285	C₂×C₃	C₂×C₃×D₂₄	1	4	yes	—
286	C₂×C₃	C₃×#904	1	4	yes	—
287	C₂×C₃	C₃×#930	1	4	yes	—
288	C₂×C₃	C₃²×#121	1	1	yes	characteristic
289	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
290	C₂×C₃	C₃²×#135	1	1	yes	characteristic
291	D₃	—	8	2	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	12	2	no	—
295	C₂²	—	12	2	no	—
296	C₂²	—	12	2	no	—
297	C₂²	C₃²×D₂₄	1	2	yes	complemented
298	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	12	2	no	—
301	C₄	C₃²×#235	1	1	yes	characteristic
302	C₄	C₂×C₃²×#67	1	1	yes	characteristic
303	C₃	—	2	4	no	—
304	C₃	C₃×#5115	1	4	yes	direct product factor
305	C₃	C₃²×#466	1	1	yes	characteristic, complemented
306	C₂	—	2	2	no	—
307	C₂	—	24	2	no	—
308	C₂	C₂×C₃²×D₂₄	1	1	yes	characteristic
309	C₂	C₃²×#904	1	1	yes	characteristic
310	C₂	C₃²×#930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₂²×C₃²×C₈	D₃
hypocenter	C₃	C₃²×#466
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×C₈	C₂	semidirect
2. C₃²×#930	C₂	semidirect
3. C₂×C₃²×D₂₄	C₂	semidirect
4. C₃²×#821	C₂	semidirect
5. C₃²×#818	C₂	semidirect
7. C₃×#5115	C₃	direct (N × H)
19. C₂²×C₃²×C₈	D₃	semidirect
20. C₂²×C₃²×C₈	C₂×C₃	semidirect
22. C₃×#930	C₂×C₃	semidirect
23. C₂×C₃×D₂₄	C₂×C₃	semidirect
26. C₃×#821	C₂×C₃	semidirect
27. C₃×#818	C₂×C₃	semidirect
39. C₃³×C₈	D₄	semidirect
41. #5115	C₃²	direct (N × H)
66. C₂²×C₃³	D₈	semidirect
77. C₂²×C₃×C₈	C₃×D₃	semidirect
78. C₂²×C₃×C₈	C₂×C₃²	semidirect
80. #930	C₂×C₃²	semidirect
81. C₂×D₂₄	C₂×C₃²	semidirect
84. #821	C₂×C₃²	semidirect
85. #818	C₂×C₃²	semidirect
115. C₃²×C₈	#67	semidirect
116. C₃²×C₈	C₃×D₄	semidirect
156. C₂²×C₃²	D₂₄	semidirect
157. C₂²×C₃²	C₃×D₈	semidirect
175. C₂²×C₈	C₃²×D₃	semidirect
179. C₃³	#466	semidirect
209. C₃×C₈	C₃×#67	semidirect
210. C₃×C₈	C₃²×D₄	semidirect
245. C₂²×C₃	C₃×D₂₄	semidirect
246. C₂²×C₃	C₃²×D₈	semidirect
264. C₃²	#5115	direct (N × H)
265. C₃²	C₃×#466	semidirect
277. C₈	C₃²×#67	semidirect
297. C₂²	C₃²×D₂₄	semidirect
304. C₃	C₃×#5115	direct (N × H)
305. C₃	C₃²×#466	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×#930	2
C₂×C₃²×D₂₄	2
C₃²×#821	2
C₃²×#818	2
C₃²×#466	3
C₃×#5115	3

Subgroups — C32×#5115 #13162550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5115 #13162550