Subgroups — C₂×#11723352 #13176854

Order 1728.

1868 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3828	C₂	1	1	yes	maximal, characteristic
2	C₂×#88855	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88787	C₂	1	2	yes	maximal, complemented
5	#11723352	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃⋊D₃×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂×#64110	—	3	1	no	maximal
8	C₂×#63814	—	3	4	no	maximal
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×#638	C₄	1	2	yes	—
11	C₃×#3828	C₄	1	4	yes	—
12	C₃×#3828	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
14	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	2	yes	—
15	C₂×C₃×#634	C₄	1	1	yes	characteristic
16	#88855	C₂²	1	4	yes	complemented
17	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
18	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₃⋊C₈	C₂²	1	4	yes	complemented
21	#88787	C₂²	1	8	yes	complemented
22	C₃⋊D₃×C₃⋊C₈	C₂²	1	8	yes	complemented
23	C₂×#3828	D₃	1	1	yes	characteristic
24	C₂×C₃×#235	—	3	4	no	—
25	C₂²×C₃²×C₈	—	3	1	no	—
26	C₂×#3748	—	3	1	no	—
27	C₂×#3748	—	3	4	no	—
28	C₂×#3748	—	6	4	no	—
29	#64110	—	3	8	no	—
30	C₂×#3743	—	3	2	no	—
31	C₂×C₈×C₃⋊D₃	—	3	2	no	—
32	C₂²×C₃×C₃⋊C₈	D₃	1	4	yes	complemented
33	C₂×#3711	—	3	8	no	—
34	#63814	—	3	32	no	—
35	C₂×D₃×C₃⋊C₈	—	3	8	no	—
36	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	2	yes	—
38	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
40	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
41	C₂×C₃³×C₄	C₂³	1	2	yes	—
42	C₃×#638	C₂×C₄	1	8	yes	—
43	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
44	C₃×C₄⋊(C₃⋊D₃)	C₂×C₄	1	4	yes	—
45	C₃×C₄×C₃⋊D₃	C₂×C₄	1	4	yes	—
46	C₃×C₄×C₃⋊D₃	C₂³	1	4	yes	—
47	C₃×#634	C₂×C₄	1	4	yes	—
48	C₃⋊(C₃⋊(C₃⋊C₈))	C₂³	1	4	yes	—
49	C₃²×C₃⋊C₈	C₂³	1	2	yes	—
50	C₃²×C₃⋊C₈	C₂³	1	2	yes	—
51	C₂×#948	—	9	1	no	—
52	C₂×#901	—	9	4	no	—
53	C₂²×C₃²×C₄	—	2	4	no	—
54	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
55	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂×#638	Dic₃	1	2	yes	—
57	#3828	Dic₃	1	4	yes	—
58	#3828	C₂×D₃	1	4	yes	—
59	C₂×C₄⋊(C₃⋊D₃)	Dic₃	1	1	yes	characteristic
60	C₂×C₄×C₃⋊D₃	C₂×D₃	1	2	yes	—
61	C₂×#634	Dic₃	1	1	yes	characteristic
62	C₂×C₃×#67	—	3	8	no	—
63	C₃×#235	—	3	16	no	—
64	C₃×#235	—	3	16	no	—
65	C₂×C₃×D₁₂	—	3	4	no	—
66	C₂×C₃×C₄×D₃	—	3	8	no	—
67	C₂×C₃×Dic₆	—	3	4	no	—
68	C₂×C₃²×C₈	—	3	1	no	—
69	C₂×C₃²×C₈	—	3	1	no	—
70	C₂×C₃²×C₈	—	3	4	no	—
71	#3748	—	3	4	no	—
72	#3748	—	3	16	no	—
73	#3748	—	6	16	no	—
74	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
75	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
76	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
77	#3743	—	3	8	no	—
78	C₈×C₃⋊D₃	—	3	8	no	—
79	C₂×C₃×C₃⋊C₈	C₂×D₃	1	4	yes	—
80	C₂×C₃×C₃⋊C₈	C₂×D₃	1	4	yes	—
81	C₂×C₃×C₃⋊C₈	C₂×D₃	1	16	yes	complemented
82	#3711	—	3	32	no	—
83	D₃×C₃⋊C₈	—	3	32	no	—
84	C₂²×C₃³	—	2	2	no	—
85	C₂²×C₃³	C₂²×C₄	1	2	yes	—
86	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₃⋊D₃	—	2	4	no	—
88	C₂×C₃×C₃⋊D₃	C₂²×C₄	1	4	yes	—
89	C₃³×C₄	C₂²×C₄	1	2	yes	—
90	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₃³×C₄	C₂⁴	1	1	yes	characteristic
92	C₃×C₃⋊Dic₃	C₂²×C₄	1	4	yes	—
93	C₂×C₃×#41	—	9	1	no	—
94	C₂×#235	—	3	4	no	—
95	C₂²×C₃×C₈	—	3	4	no	—
96	#948	—	9	8	no	—
97	C₂×#208	—	9	4	no	—
98	C₂×#208	—	9	1	no	—
99	C₂×#208	—	9	2	no	—
100	C₂×#208	—	18	4	no	—
101	C₂²×C₃⋊C₈	—	9	2	no	—
102	C₂²×C₃⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
103	#901	—	9	32	no	—
104	C₂×ΓL(1,𝔽₂₅)	—	9	8	no	—
105	C₂×D₃×C₈	—	9	8	no	—
106	C₂³×C₃²	—	2	4	no	—
107	C₂³×C₃²	C₄×D₃	1	4	yes	—
108	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
109	C₂²×C₃⋊D₃	C₂×Dic₃	1	2	yes	—
110	C₂²×C₃×D₃	—	3	8	no	—
111	C₂×C₃²×C₄	—	2	4	no	—
112	C₂×C₃²×C₄	—	2	8	no	—
113	C₂×C₃²×C₄	—	2	4	no	—
114	C₂×C₃²×C₄	—	2	8	no	—
115	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
116	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
117	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
118	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
119	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
120	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
121	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
122	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
123	#638	C₂×Dic₃	1	8	yes	—
124	C₂×C₃⋊Dic₃	C₂×Dic₃	1	2	yes	—
125	C₄⋊(C₃⋊D₃)	C₂×Dic₃	1	4	yes	—
126	C₄×C₃⋊D₃	C₂×Dic₃	1	4	yes	—
127	C₄×C₃⋊D₃	C₂²×D₃	1	4	yes	—
128	#634	C₂×Dic₃	1	4	yes	—
129	C₃×#67	—	3	32	no	—
130	C₂×C₃×Dic₃	—	3	8	no	—
131	C₃×D₁₂	—	3	16	no	—
132	C₃×C₄×D₃	—	3	16	no	—
133	C₃×C₄×D₃	—	3	16	no	—
134	C₃×Dic₆	—	3	16	no	—
135	C₃²×C₈	—	3	2	no	—
136	C₃²×C₈	—	3	2	no	—
137	C₃⋊(C₃⋊C₈)	—	3	4	no	—
138	C₃⋊(C₃⋊C₈)	—	3	16	no	—
139	C₃⋊(C₃⋊C₈)	—	6	16	no	—
140	C₃×C₃⋊C₈	C₂²×D₃	1	8	yes	—
141	C₃×C₃⋊C₈	C₂²×D₃	1	8	yes	—
142	C₂×#131	—	27	1	no	Sylow
143	C₂×C₃³	—	2	2	no	—
144	C₂×C₃³	#131	1	2	yes	complemented
145	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
146	C₃×C₃⋊D₃	—	2	4	no	—
147	C₃×#41	—	9	4	no	—
148	C₃×#41	—	9	4	no	—
149	C₂×C₃×Q₈	—	9	1	no	—
150	C₂×C₃×D₄	—	9	2	no	—
151	C₂×C₃×D₄	—	9	1	no	—
152	C₂²×C₃×C₄	—	2	4	no	—
153	C₂²×C₃×C₄	—	9	2	no	—
154	C₂²×C₃×C₄	D₃²	1	4	yes	—
155	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
156	C₂×#67	—	3	8	no	—
157	#235	—	3	16	no	—
158	#235	—	3	16	no	—
159	C₂×D₁₂	—	3	4	no	—
160	C₂×C₄×D₃	—	3	8	no	—
161	C₂×Dic₆	—	3	4	no	—
162	C₂×C₃×C₈	—	3	4	no	—
163	C₂×C₃×C₈	—	3	4	no	—
164	C₂×C₃×C₈	—	3	16	no	—
165	#208	—	9	16	no	—
166	#208	—	9	4	no	—
167	#208	—	9	8	no	—
168	#208	—	18	16	no	—
169	C₂×C₃⋊C₈	—	9	8	no	—
170	C₂×C₃⋊C₈	—	9	2	no	—
171	C₂×C₃⋊C₈	—	9	8	no	—
172	C₂×C₃⋊C₈	—	18	8	no	—
173	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
174	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	4	yes	complemented
176	ΓL(1,𝔽₂₅)	—	9	32	no	—
177	D₃×C₈	—	9	32	no	—
178	C₂²×C₃²	—	2	2	no	—
179	C₂²×C₃²	—	2	4	no	—
180	C₂²×C₃²	—	2	8	no	—
181	C₂²×C₃²	—	2	8	no	—
182	C₂²×C₃²	—	4	8	no	—
183	C₂²×C₃²	C₂×C₄×D₃	1	8	yes	—
184	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
185	C₂²×C₃²	C₂²×Dic₃	1	2	yes	—
186	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
187	C₂×C₃⋊D₃	—	2	4	no	—
188	C₂×C₃⋊D₃	C₂²×Dic₃	1	4	yes	—
189	C₂×C₃×D₃	—	3	16	no	—
190	C₂×C₃×D₃	—	6	16	no	—
191	C₃²×C₄	—	2	4	no	—
192	C₃²×C₄	—	2	8	no	—
193	C₃²×C₄	—	2	4	no	—
194	C₃²×C₄	C₂×C₄×D₃	1	8	yes	—
195	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
196	C₃²×C₄	C₂²×Dic₃	1	2	yes	—
197	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
198	C₃²×C₄	C₂³×D₃	1	4	yes	—
199	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
200	C₃⋊Dic₃	C₂²×Dic₃	1	4	yes	—
201	C₃×Dic₃	—	3	16	no	—
202	C₂×#41	—	9	1	no	—
203	#131	—	27	8	no	—
204	C₂×M₁₆	—	27	2	no	—
205	C₂×M₁₆	—	27	1	no	—
206	C₂²×C₈	—	3	1	no	—
207	C₂²×C₈	—	27	2	no	—
208	C₃³	C₂×#131	1	1	yes	characteristic, complemented, Sylow
209	C₂³×C₃	—	2	4	no	—
210	C₂³×C₃	—	9	2	no	—
211	C₂³×C₃	D₃×Dic₃	1	4	yes	—
212	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
213	C₂²×D₃	—	3	8	no	—
214	C₃×Q₈	—	9	4	no	—
215	C₃×D₄	—	9	8	no	—
216	C₃×D₄	—	9	4	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	2	8	no	—
219	C₂×C₃×C₄	—	2	8	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	9	2	no	—
222	C₂×C₃×C₄	—	9	4	no	—
223	C₂×C₃×C₄	—	9	4	no	—
224	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
225	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
226	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
227	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
228	C₂×C₃×C₄	C₂×D₃²	1	8	yes	—
229	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
230	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
231	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
232	#67	—	3	32	no	—
233	C₂×Dic₃	—	3	8	no	—
234	D₁₂	—	3	16	no	—
235	C₄×D₃	—	3	16	no	—
236	C₄×D₃	—	3	16	no	—
237	Dic₆	—	3	16	no	—
238	C₃×C₈	—	3	8	no	—
239	C₃×C₈	—	3	8	no	—
240	C₃⋊C₈	—	9	16	no	—
241	C₃⋊C₈	—	9	4	no	—
242	C₃⋊C₈	—	18	16	no	—
243	C₃⋊C₈	C₂²×C₃⋊D₃	1	2	yes	—
244	C₃⋊C₈	C₂²×C₃⋊D₃	1	2	yes	—
245	C₂×C₃²	—	2	2	no	—
246	C₂×C₃²	—	2	4	no	—
247	C₂×C₃²	—	2	8	no	—
248	C₂×C₃²	—	2	8	no	—
249	C₂×C₃²	—	4	8	no	—
250	C₂×C₃²	#901	1	8	yes	complemented
251	C₂×C₃²	#948	1	2	yes	complemented
252	C₂×C₃²	C₂²×C₄×D₃	1	4	yes	—
253	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
254	C₃⋊D₃	—	2	4	no	—
255	C₃×D₃	—	6	16	no	—
256	#41	—	9	4	no	—
257	#41	—	9	4	no	—
258	C₂×Q₈	—	9	1	no	—
259	C₂×D₄	—	9	2	no	—
260	C₂×D₄	—	9	1	no	—
261	C₂²×C₄	—	9	2	no	—
262	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
263	M₁₆	—	27	8	no	—
264	M₁₆	—	27	4	no	—
265	C₂×C₈	—	3	4	no	—
266	C₂×C₈	—	3	1	no	—
267	C₂×C₈	—	3	1	no	—
268	C₂×C₈	—	27	8	no	—
269	C₂×C₈	—	27	2	no	—
270	C₂²×C₃	—	2	8	no	—
271	C₂²×C₃	—	2	2	no	—
272	C₂²×C₃	—	2	4	no	—
273	C₂²×C₃	—	2	8	no	—
274	C₂²×C₃	—	4	8	no	—
275	C₂²×C₃	—	9	4	no	—
276	C₂²×C₃	—	18	4	no	—
277	C₂²×C₃	C₂×D₃×Dic₃	1	8	yes	—
278	C₂²×C₃	C₂×D₃×Dic₃	1	4	yes	—
279	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	2	yes	—
280	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
281	C₂×D₃	—	3	16	no	—
282	C₂×D₃	—	6	16	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	8	no	—
286	C₃×C₄	—	9	4	no	—
287	C₃×C₄	C₂×D₃×Dic₃	1	8	yes	—
288	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
289	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
290	C₃×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
291	C₃×C₄	C₂²×D₃²	1	4	yes	—
292	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
293	Dic₃	—	3	16	no	—
294	C₃²	—	2	4	no	—
295	C₃²	C₂×#901	1	4	yes	complemented
296	C₃²	C₂×#948	1	1	yes	characteristic, complemented
297	C₂³	—	9	2	no	—
298	C₂³	Dic₃×C₃⋊D₃	1	1	yes	characteristic
299	Q₈	—	9	4	no	—
300	D₄	—	9	8	no	—
301	D₄	—	9	4	no	—
302	C₂×C₄	—	9	2	no	—
303	C₂×C₄	—	9	4	no	—
304	C₂×C₄	—	9	4	no	—
305	C₂×C₄	Dic₃×C₃⋊D₃	1	1	yes	characteristic
306	C₂×C₄	Dic₃×C₃⋊D₃	1	2	yes	—
307	C₂×C₄	C₂×D₃×C₃⋊D₃	1	2	yes	—
308	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
309	C₈	—	3	2	no	—
310	C₈	—	3	2	no	—
311	C₈	—	27	4	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	2	8	no	—
314	C₂×C₃	—	2	4	no	—
315	C₂×C₃	—	2	8	no	—
316	C₂×C₃	—	4	8	no	—
317	C₂×C₃	—	18	4	no	—
318	C₂×C₃	#63814	1	8	yes	complemented
319	C₂×C₃	#64110	1	2	yes	complemented
320	C₂×C₃	C₂²×D₃×Dic₃	1	4	yes	—
321	C₂×C₃	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
322	D₃	—	6	16	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	9	4	no	—
325	C₂²	—	18	4	no	—
326	C₂²	C₂×Dic₃×C₃⋊D₃	1	2	yes	—
327	C₂²	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
328	C₄	—	9	4	no	—
329	C₄	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
330	C₄	C₂×Dic₃×C₃⋊D₃	1	2	yes	—
331	C₄	C₂²×D₃×C₃⋊D₃	1	1	yes	characteristic
332	C₃	—	2	4	no	—
333	C₃	C₂×#63814	1	4	yes	complemented
334	C₃	C₂×#64110	1	1	yes	characteristic, complemented
335	C₂	—	2	2	no	—
336	C₂	—	18	4	no	—
337	C₂	#11723352	1	2	yes	direct product factor
338	C₂	C₂²×Dic₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator	C₂×C₃³	C₂³×C₄
Frattini	C₄	C₂²×D₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₄	C₂²
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂×C₃×#3828	C₂
cosocle	C₃³×C₄	C₂⁴
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	C₂×#131
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88855	C₂	semidirect
3. C₂²×C₃²×C₃⋊C₈	C₂	semidirect
4. C₂×#88787	C₂	semidirect
5. #11723352	C₂	direct (N × H)
6. C₂×C₃⋊D₃×C₃⋊C₈	C₂	semidirect
16. #88855	C₂²	semidirect
20. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
21. #88787	C₂²	semidirect
22. C₃⋊D₃×C₃⋊C₈	C₂²	semidirect
32. C₂²×C₃×C₃⋊C₈	D₃	semidirect
81. C₂×C₃×C₃⋊C₈	C₂×D₃	semidirect
102. C₂²×C₃⋊C₈	C₃⋊D₃	semidirect
144. C₂×C₃³	#131	semidirect
175. C₂×C₃⋊C₈	C₂×C₃⋊D₃	semidirect
208. C₃³	C₂×#131	semidirect
250. C₂×C₃²	#901	semidirect
251. C₂×C₃²	#948	semidirect
295. C₃²	C₂×#901	semidirect
296. C₃²	C₂×#948	semidirect
318. C₂×C₃	#63814	semidirect
319. C₂×C₃	#64110	semidirect
333. C₃	C₂×#63814	semidirect
334. C₃	C₂×#64110	semidirect
337. C₂	#11723352	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3828	2
C₂×#88855	2
C₂²×C₃²×C₃⋊C₈	2
C₂×#88787	2
#11723352	2
C₂×C₃⋊D₃×C₃⋊C₈	2
C₂×#64110	3
C₂×#63814	3

Subgroups — C2×#11723352 #13176854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11723352 #13176854