Subgroups — C₃²×#5717 #13179095

Order 1728.

3000 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×C₃²×D₃×D₄	C₂	1	4	yes	maximal, complemented
2	C₃³×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#938	C₂	1	4	yes	maximal, complemented
4	C₃²×#937	C₂	1	4	yes	maximal, complemented
5	C₃²×C₄×D₁₂	C₂	1	2	yes	maximal, complemented
6	C₃²×D₄²	—	3	1	no	maximal
7	C₃×#5717	C₃	1	4	yes	maximal, direct product factor
8	C₂³×C₃²×D₃	C₂²	1	4	yes	complemented
9	C₂×C₃³×D₄	C₂²	1	4	yes	complemented
10	C₂×C₃³×D₄	C₂²	1	2	yes	complemented
11	C₂×C₃²×#67	C₂²	1	8	yes	complemented
12	C₃²×D₃×D₄	—	2	16	no	—
13	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
14	C₂×C₃²×C₄×D₃	C₂²	1	4	yes	complemented
15	C₃³×C₄²	C₂²	1	1	yes	characteristic, complemented
16	C₃²×#217	C₂²	1	4	yes	complemented
17	C₃²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
18	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	complemented
19	C₂²×C₃²×D₄	—	3	4	no	—
20	C₂×C₃×D₃×D₄	C₂×C₃	1	16	yes	complemented
21	C₃²×#127	—	2	4	no	—
22	C₃²×#127	D₃	1	1	yes	characteristic, complemented
23	C₃²×#127	C₂×C₃	1	4	yes	complemented
24	C₃²×#121	—	3	4	no	—
25	C₃²×#120	—	3	4	no	—
26	C₃²×C₄×D₄	—	3	2	no	—
27	C₃×#938	C₂×C₃	1	16	yes	complemented
28	C₃×#937	C₂×C₃	1	16	yes	complemented
29	C₃×C₄×D₁₂	C₂×C₃	1	8	yes	complemented
30	C₂³×C₃³	C₂³	1	4	yes	—
31	C₂²×C₃²×D₃	—	2	16	no	—
32	C₂²×C₃²×D₃	—	4	4	no	—
33	C₂²×C₃²×D₃	C₂³	1	4	yes	—
34	C₃³×D₄	—	2	8	no	—
35	C₃³×D₄	—	4	2	no	—
36	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂³	1	2	yes	—
38	C₃²×#67	—	2	16	no	—
39	C₂×C₃²×Dic₃	C₂³	1	4	yes	—
40	C₃²×D₁₂	D₄	1	8	yes	complemented
41	C₃²×C₄×D₃	—	2	8	no	—
42	C₃×D₄²	—	3	4	no	—
43	#5717	C₃²	1	1	yes	characteristic, direct product factor
44	C₂⁴×C₃²	—	3	4	no	—
45	C₂³×C₃×D₃	C₂²×C₃	1	16	yes	complemented
46	C₂×C₃²×D₄	—	2	16	no	—
47	C₂×C₃²×D₄	—	2	8	no	—
48	C₂×C₃²×D₄	—	3	8	no	—
49	C₂×C₃²×D₄	—	3	2	no	—
50	C₂×C₃²×D₄	—	6	16	no	—
51	C₂×C₃²×D₄	C₂×D₃	1	4	yes	complemented
52	C₂×C₃²×D₄	C₂×D₃	1	2	yes	complemented
53	C₂×C₃²×D₄	C₂²×C₃	1	16	yes	complemented
54	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	complemented
55	C₂²×C₃²×C₄	—	3	4	no	—
56	C₂×C₃×#67	C₂²×C₃	1	32	yes	complemented
57	C₃×D₃×D₄	—	2	64	no	—
58	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
59	C₂×C₃×C₄×D₃	C₂²×C₃	1	16	yes	complemented
60	C₃²×C₄⋊C₄	—	3	2	no	—
61	C₃²×C₂²⋊C₄	—	3	4	no	—
62	C₃²×C₂²⋊C₄	—	3	4	no	—
63	C₃²×C₄²	—	2	4	no	—
64	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic, complemented
65	C₃²×C₄²	C₂²×C₃	1	4	yes	complemented
66	C₃×#217	C₂²×C₃	1	16	yes	complemented
67	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	16	yes	complemented
68	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	complemented
69	C₂²×C₃³	—	2	8	no	—
70	C₂²×C₃³	—	2	4	no	—
71	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
72	C₂×C₃²×D₃	—	2	4	no	—
73	C₂×C₃²×D₃	—	2	4	no	—
74	C₂×C₃²×D₃	—	4	16	no	—
75	C₂×C₃²×D₃	C₂×D₄	1	8	yes	complemented
76	C₃³×C₄	—	2	1	no	—
77	C₃³×C₄	C₂×D₄	1	4	yes	complemented
78	C₃²×Dic₃	—	2	4	no	—
79	C₂²×C₃×D₄	—	3	16	no	—
80	C₂×D₃×D₄	C₂×C₃²	1	4	yes	complemented
81	C₃×#127	—	2	4	no	—
82	C₃×#127	C₃×D₃	1	4	yes	complemented
83	C₃×#127	C₂×C₃²	1	1	yes	characteristic, complemented
84	C₃×#121	—	3	16	no	—
85	C₃×#120	—	3	16	no	—
86	C₃×C₄×D₄	—	3	8	no	—
87	#938	C₂×C₃²	1	4	yes	complemented
88	#937	C₂×C₃²	1	4	yes	complemented
89	C₄×D₁₂	C₂×C₃²	1	2	yes	complemented
90	C₂³×C₃²	—	2	16	no	—
91	C₂³×C₃²	—	3	4	no	—
92	C₂³×C₃²	—	6	16	no	—
93	C₂³×C₃²	—	12	4	no	—
94	C₂³×C₃²	C₂²×D₃	1	4	yes	—
95	C₂³×C₃²	C₂³×C₃	1	16	yes	—
96	C₂²×C₃×D₃	—	2	64	no	—
97	C₂²×C₃×D₃	—	4	16	no	—
98	C₂²×C₃×D₃	C₂³×C₃	1	16	yes	—
99	C₃²×D₄	—	2	8	no	—
100	C₃²×D₄	—	2	32	no	—
101	C₃²×D₄	—	3	8	no	—
102	C₃²×D₄	—	4	32	no	—
103	C₃²×D₄	—	4	2	no	—
104	C₃²×D₄	—	4	8	no	—
105	C₃²×D₄	—	4	16	no	—
106	C₃²×D₄	—	6	16	no	—
107	C₂×C₃²×C₄	—	2	4	no	—
108	C₂×C₃²×C₄	—	2	8	no	—
109	C₂×C₃²×C₄	—	3	4	no	—
110	C₂×C₃²×C₄	—	6	8	no	—
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
113	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
114	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
115	C₃×#67	—	2	64	no	—
116	C₂×C₃×Dic₃	C₂³×C₃	1	16	yes	—
117	C₃×D₁₂	C₃×D₄	1	32	yes	complemented
118	C₃×C₄×D₃	—	2	32	no	—
119	D₄²	—	3	1	no	Sylow
120	C₂×C₃³	—	4	4	no	—
121	C₂×C₃³	C₂²×D₄	1	2	yes	—
122	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
123	C₃²×D₃	—	2	8	no	—
124	C₂⁴×C₃	—	3	16	no	—
125	C₂³×D₃	C₂²×C₃²	1	4	yes	complemented
126	C₂×C₃×D₄	—	2	16	no	—
127	C₂×C₃×D₄	—	2	8	no	—
128	C₂×C₃×D₄	—	3	32	no	—
129	C₂×C₃×D₄	—	3	8	no	—
130	C₂×C₃×D₄	—	6	64	no	—
131	C₂×C₃×D₄	C₂×C₃×D₃	1	16	yes	complemented
132	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	complemented
133	C₂×C₃×D₄	C₂²×C₃²	1	4	yes	complemented
134	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	complemented
135	C₂²×C₃×C₄	—	3	16	no	—
136	C₂×#67	C₂²×C₃²	1	8	yes	complemented
137	D₃×D₄	—	2	16	no	—
138	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
139	C₂×C₄×D₃	C₂²×C₃²	1	4	yes	complemented
140	C₃×C₄⋊C₄	—	3	8	no	—
141	C₃×C₂²⋊C₄	—	3	16	no	—
142	C₃×C₂²⋊C₄	—	3	16	no	—
143	C₃×C₄²	—	2	4	no	—
144	C₃×C₄²	C₂×C₃×D₃	1	4	yes	complemented
145	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic, complemented
146	#217	C₂²×C₃²	1	4	yes	complemented
147	(C₂×D₃)⋊C₄	C₂²×C₃²	1	4	yes	complemented
148	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	complemented
149	C₂²×C₃²	—	2	8	no	—
150	C₂²×C₃²	—	2	4	no	—
151	C₂²×C₃²	—	2	4	no	—
152	C₂²×C₃²	—	2	32	no	—
153	C₂²×C₃²	—	2	16	no	—
154	C₂²×C₃²	—	3	8	no	—
155	C₂²×C₃²	—	4	32	no	—
156	C₂²×C₃²	—	4	16	no	—
157	C₂²×C₃²	—	6	4	no	—
158	C₂²×C₃²	—	6	4	no	—
159	C₂²×C₃²	—	12	16	no	—
160	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
161	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
162	C₂×C₃×D₃	—	2	16	no	—
163	C₂×C₃×D₃	—	2	16	no	—
164	C₂×C₃×D₃	—	4	64	no	—
165	C₂×C₃×D₃	C₂×C₃×D₄	1	32	yes	complemented
166	C₃²×C₄	—	2	8	no	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	—	2	4	no	—
169	C₃²×C₄	—	2	16	no	—
170	C₃²×C₄	—	6	4	no	—
171	C₃²×C₄	D₃×D₄	1	4	yes	complemented
172	C₃²×C₄	C₂×C₃×D₄	1	16	yes	complemented
173	C₃×Dic₃	—	2	16	no	—
174	C₂²×D₄	—	3	4	no	—
175	#127	C₃²×D₃	1	1	yes	characteristic, complemented
176	#121	—	3	4	no	—
177	#120	—	3	4	no	—
178	C₄×D₄	—	3	2	no	—
179	C₃³	D₄²	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	2	16	no	—
181	C₂³×C₃	—	3	16	no	—
182	C₂³×C₃	—	6	64	no	—
183	C₂³×C₃	—	12	16	no	—
184	C₂³×C₃	C₂²×C₃×D₃	1	16	yes	—
185	C₂³×C₃	C₂³×C₃²	1	4	yes	—
186	C₂²×D₃	—	2	16	no	—
187	C₂²×D₃	—	4	4	no	—
188	C₂²×D₃	C₂³×C₃²	1	4	yes	—
189	C₃×D₄	—	2	32	no	—
190	C₃×D₄	—	2	8	no	—
191	C₃×D₄	—	3	32	no	—
192	C₃×D₄	—	4	32	no	—
193	C₃×D₄	—	4	8	no	—
194	C₃×D₄	—	4	2	no	—
195	C₃×D₄	—	4	16	no	—
196	C₃×D₄	—	6	64	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	3	16	no	—
200	C₂×C₃×C₄	—	6	32	no	—
201	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
202	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
203	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
205	#67	—	2	16	no	—
206	C₂×Dic₃	C₂³×C₃²	1	4	yes	—
207	D₁₂	C₃²×D₄	1	8	yes	complemented
208	C₄×D₃	—	2	8	no	—
209	C₂×C₃²	—	2	8	no	—
210	C₂×C₃²	—	2	4	no	—
211	C₂×C₃²	—	4	4	no	—
212	C₂×C₃²	—	4	16	no	—
213	C₂×C₃²	—	6	8	no	—
214	C₂×C₃²	—	8	16	no	—
215	C₂×C₃²	C₂×D₃×D₄	1	2	yes	—
216	C₂×C₃²	#1004	1	1	yes	characteristic
217	C₂×C₃²	C₂²×C₃×D₄	1	8	yes	—
218	C₂×C₃²	C₃×H(5,𝔽₂)	1	4	yes	—
219	C₃×D₃	—	2	32	no	—
220	C₂⁴	—	3	4	no	—
221	C₂×D₄	—	3	8	no	—
222	C₂×D₄	—	3	2	no	—
223	C₂×D₄	—	6	16	no	—
224	C₂×D₄	C₂×C₃²×D₃	1	4	yes	complemented
225	C₂×D₄	C₂×C₃²×D₃	1	2	yes	complemented
226	C₂²×C₄	—	3	4	no	—
227	C₄⋊C₄	—	3	2	no	—
228	C₂²⋊C₄	—	3	4	no	—
229	C₂²⋊C₄	—	3	4	no	—
230	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
231	C₂²×C₃	—	2	32	no	—
232	C₂²×C₃	—	2	16	no	—
233	C₂²×C₃	—	2	8	no	—
234	C₂²×C₃	—	2	4	no	—
235	C₂²×C₃	—	2	4	no	—
236	C₂²×C₃	—	3	32	no	—
237	C₂²×C₃	—	4	32	no	—
238	C₂²×C₃	—	4	16	no	—
239	C₂²×C₃	—	6	16	no	—
240	C₂²×C₃	—	6	16	no	—
241	C₂²×C₃	—	12	64	no	—
242	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
243	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
244	C₂×D₃	—	2	4	no	—
245	C₂×D₃	—	2	4	no	—
246	C₂×D₃	—	4	16	no	—
247	C₂×D₃	C₂×C₃²×D₄	1	8	yes	complemented
248	C₃×C₄	—	2	8	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	16	no	—
252	C₃×C₄	—	6	16	no	—
253	C₃×C₄	C₃×D₃×D₄	1	16	yes	complemented
254	C₃×C₄	C₂×C₃²×D₄	1	4	yes	complemented
255	Dic₃	—	2	4	no	—
256	C₃²	—	2	4	no	—
257	C₃²	#5717	1	1	yes	characteristic, direct product factor
258	C₃²	C₃×D₄²	1	4	yes	complemented
259	C₂³	—	3	4	no	—
260	C₂³	—	6	16	no	—
261	C₂³	—	12	4	no	—
262	C₂³	C₂²×C₃²×D₃	1	4	yes	—
263	D₄	—	2	8	no	—
264	D₄	—	3	8	no	—
265	D₄	—	4	2	no	—
266	D₄	—	6	16	no	—
267	C₂×C₄	—	3	4	no	—
268	C₂×C₄	—	6	8	no	—
269	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
271	C₂×C₃	—	2	8	no	—
272	C₂×C₃	—	2	4	no	—
273	C₂×C₃	—	4	16	no	—
274	C₂×C₃	—	4	4	no	—
275	C₂×C₃	—	6	32	no	—
276	C₂×C₃	—	8	16	no	—
277	C₂×C₃	C₂×C₃×D₃×D₄	1	8	yes	—
278	C₂×C₃	C₃×#1004	1	4	yes	—
279	C₂×C₃	C₂²×C₃²×D₄	1	2	yes	—
280	C₂×C₃	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
281	D₃	—	2	8	no	—
282	C₂²	—	2	8	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	3	8	no	—
285	C₂²	—	6	4	no	—
286	C₂²	—	6	4	no	—
287	C₂²	—	12	16	no	—
288	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	6	4	no	—
291	C₄	C₃²×D₃×D₄	1	4	yes	complemented
292	C₃	—	2	4	no	—
293	C₃	C₃×#5717	1	4	yes	direct product factor
294	C₃	C₃²×D₄²	1	1	yes	characteristic, complemented
295	C₂	—	4	4	no	—
296	C₂	—	6	8	no	—
297	C₂	C₂×C₃²×D₃×D₄	1	2	yes	—
298	C₂	C₃²×#1004	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×#127	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃²×#127	D₃
hypocenter	C₃	C₃²×D₄²
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₃×D₄	C₂	semidirect
2. C₃³×#127	C₂	semidirect
3. C₃²×#938	C₂	semidirect
4. C₃²×#937	C₂	semidirect
5. C₃²×C₄×D₁₂	C₂	semidirect
7. C₃×#5717	C₃	direct (N × H)
8. C₂³×C₃²×D₃	C₂²	semidirect
9. C₂×C₃³×D₄	C₂²	semidirect
10. C₂×C₃³×D₄	C₂²	semidirect
11. C₂×C₃²×#67	C₂²	semidirect
14. C₂×C₃²×C₄×D₃	C₂²	semidirect
15. C₃³×C₄²	C₂²	semidirect
16. C₃²×#217	C₂²	semidirect
17. C₃²×(C₂×D₃)⋊C₄	C₂²	semidirect
18. C₃²×C₄⋊Dic₃	C₂²	semidirect
20. C₂×C₃×D₃×D₄	C₂×C₃	semidirect
22. C₃²×#127	D₃	semidirect
23. C₃²×#127	C₂×C₃	semidirect
27. C₃×#938	C₂×C₃	semidirect
28. C₃×#937	C₂×C₃	semidirect
29. C₃×C₄×D₁₂	C₂×C₃	semidirect
40. C₃²×D₁₂	D₄	semidirect
43. #5717	C₃²	direct (N × H)
45. C₂³×C₃×D₃	C₂²×C₃	semidirect
51. C₂×C₃²×D₄	C₂×D₃	semidirect
52. C₂×C₃²×D₄	C₂×D₃	semidirect
53. C₂×C₃²×D₄	C₂²×C₃	semidirect
54. C₂×C₃²×D₄	C₂²×C₃	semidirect
56. C₂×C₃×#67	C₂²×C₃	semidirect
59. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
64. C₃²×C₄²	C₂×D₃	semidirect
65. C₃²×C₄²	C₂²×C₃	semidirect
66. C₃×#217	C₂²×C₃	semidirect
67. C₃×(C₂×D₃)⋊C₄	C₂²×C₃	semidirect
68. C₃×C₄⋊Dic₃	C₂²×C₃	semidirect
75. C₂×C₃²×D₃	C₂×D₄	semidirect
77. C₃³×C₄	C₂×D₄	semidirect
80. C₂×D₃×D₄	C₂×C₃²	semidirect
82. C₃×#127	C₃×D₃	semidirect
83. C₃×#127	C₂×C₃²	semidirect
87. #938	C₂×C₃²	semidirect
88. #937	C₂×C₃²	semidirect
89. C₄×D₁₂	C₂×C₃²	semidirect
117. C₃×D₁₂	C₃×D₄	semidirect
125. C₂³×D₃	C₂²×C₃²	semidirect
131. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
132. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
133. C₂×C₃×D₄	C₂²×C₃²	semidirect
134. C₂×C₃×D₄	C₂²×C₃²	semidirect
136. C₂×#67	C₂²×C₃²	semidirect
139. C₂×C₄×D₃	C₂²×C₃²	semidirect
144. C₃×C₄²	C₂×C₃×D₃	semidirect
145. C₃×C₄²	C₂²×C₃²	semidirect
146. #217	C₂²×C₃²	semidirect
147. (C₂×D₃)⋊C₄	C₂²×C₃²	semidirect
148. C₄⋊Dic₃	C₂²×C₃²	semidirect
165. C₂×C₃×D₃	C₂×C₃×D₄	semidirect
171. C₃²×C₄	D₃×D₄	semidirect
172. C₃²×C₄	C₂×C₃×D₄	semidirect
175. #127	C₃²×D₃	semidirect
179. C₃³	D₄²	semidirect
207. D₁₂	C₃²×D₄	semidirect
224. C₂×D₄	C₂×C₃²×D₃	semidirect
225. C₂×D₄	C₂×C₃²×D₃	semidirect
230. C₄²	C₂×C₃²×D₃	semidirect
247. C₂×D₃	C₂×C₃²×D₄	semidirect
253. C₃×C₄	C₃×D₃×D₄	semidirect
254. C₃×C₄	C₂×C₃²×D₄	semidirect
257. C₃²	#5717	direct (N × H)
258. C₃²	C₃×D₄²	semidirect
291. C₄	C₃²×D₃×D₄	semidirect
293. C₃	C₃×#5717	direct (N × H)
294. C₃	C₃²×D₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₃×D₄	2
C₃³×#127	2
C₃²×#938	2
C₃²×#937	2
C₃²×C₄×D₁₂	2
C₃²×D₄²	3
C₃×#5717	3

Subgroups — C32×#5717 #13179095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5717 #13179095