Subgroups — C₂×#11723336 #13176669

Order 1728.

1836 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89021	C₂	1	2	yes	maximal, complemented
2	C₂×#88853	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3745	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88796	C₂	1	2	yes	maximal, complemented
6	#11723336	C₂	1	8	yes	maximal, direct product factor
7	C₂×#64113	—	3	1	no	maximal
8	C₂×#63807	—	3	4	no	maximal
9	C₂×#6308	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
11	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
12	C₂×#6290	—	2	2	no	—
13	C₂×Dic₃×C₃⋊D₃	—	2	2	no	—
14	#89021	—	2	4	no	—
15	#89021	C₂²	1	8	yes	complemented
16	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
17	#88853	C₂²	1	4	yes	complemented
18	C₃×#3745	C₂²	1	4	yes	complemented
19	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃²×D₃×C₈	C₂²	1	4	yes	complemented
21	#88796	C₂²	1	8	yes	complemented
22	C₂×#3828	—	3	2	no	—
23	C₂×#3795	—	3	8	no	—
24	C₂²×C₃²×C₈	—	3	1	no	—
25	C₂×#3746	—	3	1	no	—
26	C₂×#3746	—	3	4	no	—
27	C₂×#3746	—	6	4	no	—
28	#64113	—	3	8	no	—
29	C₂×#3745	D₃	1	1	yes	characteristic, complemented
30	C₂×#3744	—	3	2	no	—
31	C₂×C₃×D₂₄	—	3	4	no	—
32	C₂×C₃×D₃×C₈	D₃	1	4	yes	complemented
33	C₂×#3716	—	3	8	no	—
34	#63807	—	3	32	no	—
35	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
36	C₂²×C₃²×D₃	D₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
39	#6308	—	2	2	no	—
40	#6308	C₂³	1	4	yes	—
41	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
42	C₃×C₄⋊(C₃⋊D₃)	C₂³	1	4	yes	—
43	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
44	C₃²×C₄×D₃	D₄	1	2	yes	—
45	C₃²×C₄×D₃	C₂³	1	2	yes	—
46	#6290	—	2	8	no	—
47	Dic₃×C₃⋊D₃	—	2	8	no	—
48	C₃³×C₈	C₂³	1	2	yes	—
49	C₃²×C₃⋊C₈	C₂³	1	2	yes	—
50	C₂×#912	—	9	1	no	—
51	C₂×#904	—	9	4	no	—
52	C₂²×C₃²×C₄	—	3	1	no	—
53	C₂×#638	—	6	2	no	—
54	#3828	—	3	8	no	—
55	#3828	—	6	4	no	—
56	C₂×C₄⋊(C₃⋊D₃)	C₂×D₃	1	2	yes	—
57	C₂×C₄×C₃⋊D₃	—	6	2	no	—
58	C₂×#634	—	3	2	no	—
59	C₂×#634	—	3	8	no	—
60	C₂×#634	—	6	8	no	—
61	C₂×C₃×D₁₂	—	3	8	no	—
62	C₂×C₃×C₄×D₃	C₂×D₃	1	4	yes	—
63	C₂×#625	—	6	8	no	—
64	C₂×D₃×Dic₃	—	6	8	no	—
65	#3795	—	3	32	no	—
66	#3795	—	6	16	no	—
67	C₂×C₃²×C₈	—	2	4	no	—
68	C₂×C₃²×C₈	—	3	1	no	—
69	C₂×C₃²×C₈	—	3	4	no	—
70	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
71	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
72	#3746	—	3	4	no	—
73	#3746	—	3	16	no	—
74	#3746	—	6	16	no	—
75	#3745	C₂×D₃	1	4	yes	complemented
76	#3744	—	3	8	no	—
77	C₂×C₃×C₃⋊C₈	C₂×D₃	1	4	yes	—
78	C₃×D₂₄	—	3	16	no	—
79	C₃×D₃×C₈	C₂×D₃	1	16	yes	complemented
80	#3716	—	3	32	no	—
81	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₃⋊D₃	—	2	4	no	—
83	C₂×C₃×C₃⋊D₃	—	4	4	no	—
84	C₂×C₃²×D₃	—	2	2	no	—
85	C₂×C₃²×D₃	C₂×D₄	1	2	yes	—
86	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
87	C₃³×C₄	C₂⁴	1	1	yes	characteristic
88	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
89	C₃²×Dic₃	C₂×D₄	1	2	yes	—
90	C₂×#237	—	9	2	no	—
91	C₂×#235	—	9	8	no	—
92	C₂×C₃×D₈	—	9	1	no	—
93	C₂²×C₃×C₈	—	3	4	no	—
94	C₂×#214	—	9	2	no	—
95	#912	—	9	8	no	—
96	C₂×Dic₁₂	—	9	4	no	—
97	C₂×Dic₁₂	—	9	1	no	—
98	C₂×Dic₁₂	—	18	4	no	—
99	#904	—	9	32	no	—
100	C₂×D₂₄	—	3	4	no	—
101	C₂×#204	—	9	8	no	—
102	C₂×D₃×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
103	C₂³×C₃²	—	3	1	no	—
104	C₂²×C₃⋊D₃	—	2	2	no	—
105	C₂²×C₃×D₃	—	6	8	no	—
106	C₂²×C₃×D₃	D₁₂	1	4	yes	—
107	C₂×C₃²×C₄	—	2	4	no	—
108	C₂×C₃²×C₄	—	3	1	no	—
109	C₂×C₃²×C₄	—	3	2	no	—
110	C₂×C₃²×C₄	—	3	2	no	—
111	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
112	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
113	#638	—	6	8	no	—
114	C₂×C₃⋊Dic₃	—	6	2	no	—
115	C₂×C₃⋊Dic₃	—	6	8	no	—
116	C₂×C₃⋊Dic₃	—	12	8	no	—
117	C₄⋊(C₃⋊D₃)	—	2	2	no	—
118	C₄⋊(C₃⋊D₃)	C₂²×D₃	1	4	yes	—
119	C₄×C₃⋊D₃	—	6	8	no	—
120	#634	—	3	4	no	—
121	#634	—	3	16	no	—
122	#634	—	6	16	no	—
123	#634	—	6	2	no	—
124	#634	—	6	8	no	—
125	#634	—	12	8	no	—
126	C₂×C₃×Dic₃	D₁₂	1	4	yes	—
127	C₃×D₁₂	—	3	16	no	—
128	C₃×D₁₂	—	6	8	no	—
129	C₃×C₄×D₃	D₁₂	1	8	yes	—
130	C₃×C₄×D₃	C₂²×D₃	1	8	yes	—
131	#625	—	6	32	no	—
132	D₃×Dic₃	—	6	32	no	—
133	C₃²×C₈	—	2	8	no	—
134	C₃²×C₈	—	3	2	no	—
135	C₃²×C₈	C₂²×D₃	1	8	yes	—
136	C₃²×C₈	C₂²×D₃	1	2	yes	—
137	C₃×C₃⋊C₈	C₂²×D₃	1	8	yes	—
138	C₂×#135	—	27	1	no	Sylow
139	C₂×C₃³	#135	1	2	yes	complemented
140	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
141	C₃×C₃⋊D₃	—	4	4	no	—
142	C₃²×D₃	—	2	2	no	—
143	C₂×C₃×D₄	—	9	2	no	—
144	C₂²×C₃×C₄	—	3	4	no	—
145	C₂×#67	—	18	8	no	—
146	C₂×#67	—	18	2	no	—
147	C₂²×Dic₃	—	18	2	no	—
148	#237	—	9	8	no	—
149	#237	—	18	4	no	—
150	#235	—	9	32	no	—
151	#235	—	18	16	no	—
152	C₂×D₁₂	—	3	8	no	—
153	C₂×C₄×D₃	—	18	8	no	—
154	C₂×C₄×D₃	C₂×C₃⋊D₃	1	1	yes	characteristic
155	C₂×Dic₆	—	9	8	no	—
156	C₂×Dic₆	—	9	2	no	—
157	C₂×Dic₆	—	18	8	no	—
158	C₃×D₈	—	9	4	no	—
159	C₂×C₃×C₈	—	2	4	no	—
160	C₂×C₃×C₈	—	3	4	no	—
161	C₂×C₃×C₈	—	3	16	no	—
162	C₂×C₃×C₈	D₃²	1	4	yes	—
163	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
164	#214	—	9	8	no	—
165	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
166	Dic₁₂	—	9	16	no	—
167	Dic₁₂	—	9	4	no	—
168	Dic₁₂	—	18	16	no	—
169	D₂₄	—	3	16	no	—
170	#204	—	9	32	no	—
171	D₃×C₈	C₂×C₃⋊D₃	1	4	yes	complemented
172	C₂²×C₃²	—	2	4	no	—
173	C₂²×C₃²	—	3	2	no	—
174	C₂²×C₃²	—	6	2	no	—
175	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
176	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
177	C₂×C₃⋊D₃	—	2	4	no	—
178	C₂×C₃⋊D₃	—	4	4	no	—
179	C₂×C₃×D₃	—	2	8	no	—
180	C₂×C₃×D₃	—	6	16	no	—
181	C₂×C₃×D₃	—	12	16	no	—
182	C₂×C₃×D₃	C₂×D₁₂	1	8	yes	—
183	C₃²×C₄	—	2	4	no	—
184	C₃²×C₄	—	2	4	no	—
185	C₃²×C₄	—	3	2	no	—
186	C₃²×C₄	C₂×D₁₂	1	4	yes	—
187	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
188	C₃²×C₄	C₂³×D₃	1	4	yes	—
189	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
190	C₃⋊Dic₃	—	6	4	no	—
191	C₃⋊Dic₃	—	6	16	no	—
192	C₃⋊Dic₃	—	12	16	no	—
193	C₃×Dic₃	C₂×D₁₂	1	8	yes	—
194	C₂×#41	—	27	2	no	—
195	#135	—	27	8	no	—
196	C₂×Q₁₆	—	27	1	no	—
197	C₂×SD₁₆	—	27	2	no	—
198	C₂×D₈	—	9	1	no	—
199	C₂²×C₈	—	3	1	no	—
200	C₃³	C₂×#135	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₃	—	3	4	no	—
202	C₂³×C₃	—	18	2	no	—
203	C₂²×D₃	—	6	8	no	—
204	C₂²×D₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
205	C₃×D₄	—	9	4	no	—
206	C₃×D₄	—	18	2	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	3	4	no	—
209	C₂×C₃×C₄	—	3	8	no	—
210	C₂×C₃×C₄	—	3	8	no	—
211	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
212	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
213	#67	—	18	32	no	—
214	#67	—	18	8	no	—
215	C₂×Dic₃	—	18	8	no	—
216	C₂×Dic₃	—	18	2	no	—
217	C₂×Dic₃	—	18	8	no	—
218	C₂×Dic₃	—	36	8	no	—
219	C₂×Dic₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
220	D₁₂	—	3	16	no	—
221	D₁₂	—	6	8	no	—
222	C₄×D₃	—	18	32	no	—
223	C₄×D₃	C₄⋊(C₃⋊D₃)	1	2	yes	—
224	C₄×D₃	C₂²×C₃⋊D₃	1	2	yes	—
225	Dic₆	—	9	16	no	—
226	Dic₆	—	9	4	no	—
227	Dic₆	—	18	16	no	—
228	Dic₆	—	18	8	no	—
229	Dic₆	—	18	2	no	—
230	Dic₆	—	36	8	no	—
231	C₃×C₈	—	2	8	no	—
232	C₃×C₈	—	3	8	no	—
233	C₃×C₈	C₂×D₃²	1	8	yes	—
234	C₃×C₈	C₂²×C₃⋊D₃	1	2	yes	—
235	C₃⋊C₈	C₂²×C₃⋊D₃	1	2	yes	—
236	C₂×C₃²	—	2	4	no	—
237	C₂×C₃²	—	2	8	no	—
238	C₂×C₃²	—	6	2	no	—
239	C₂×C₃²	#904	1	8	yes	complemented
240	C₂×C₃²	#912	1	2	yes	complemented
241	C₂×C₃²	C₂²×D₁₂	1	4	yes	—
242	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
243	C₃⋊D₃	—	4	4	no	—
244	C₃×D₃	—	2	8	no	—
245	C₃×D₃	—	12	16	no	—
246	#41	—	27	8	no	—
247	#41	—	54	4	no	—
248	C₂×Q₈	—	27	2	no	—
249	C₂×D₄	—	9	2	no	—
250	C₂×D₄	—	54	2	no	—
251	C₂²×C₄	—	3	1	no	—
252	C₂²×C₄	—	54	2	no	—
253	Q₁₆	—	27	4	no	—
254	SD₁₆	—	27	8	no	—
255	D₈	—	9	4	no	—
256	C₂×C₈	—	3	1	no	—
257	C₂×C₈	—	3	4	no	—
258	C₂×C₈	D₃×C₃⋊D₃	1	1	yes	characteristic
259	C₂²×C₃	—	2	4	no	—
260	C₂²×C₃	—	3	8	no	—
261	C₂²×C₃	—	6	8	no	—
262	C₂²×C₃	—	18	4	no	—
263	C₂²×C₃	—	36	4	no	—
264	C₂²×C₃	D₃×D₁₂	1	4	yes	—
265	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
266	C₂×D₃	—	2	2	no	—
267	C₂×D₃	—	6	16	no	—
268	C₂×D₃	—	12	16	no	—
269	C₂×D₃	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
270	C₃×C₄	—	2	4	no	—
271	C₃×C₄	—	2	4	no	—
272	C₃×C₄	—	3	8	no	—
273	C₃×C₄	D₃×D₁₂	1	4	yes	—
274	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
275	C₃×C₄	C₂²×D₃²	1	4	yes	—
276	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
277	Dic₃	—	18	16	no	—
278	Dic₃	—	18	4	no	—
279	Dic₃	—	36	16	no	—
280	Dic₃	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
281	C₃²	—	2	4	no	—
282	C₃²	C₂×#904	1	4	yes	complemented
283	C₃²	C₂×#912	1	1	yes	characteristic, complemented
284	C₂³	—	3	1	no	—
285	C₂³	—	18	2	no	—
286	Q₈	—	27	4	no	—
287	Q₈	—	54	2	no	—
288	D₄	—	9	4	no	—
289	D₄	—	18	2	no	—
290	D₄	—	54	8	no	—
291	C₂×C₄	—	3	1	no	—
292	C₂×C₄	—	3	2	no	—
293	C₂×C₄	—	3	2	no	—
294	C₂×C₄	—	54	8	no	—
295	C₂×C₄	—	54	2	no	—
296	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
297	C₈	—	3	2	no	—
298	C₈	C₂×D₃×C₃⋊D₃	1	2	yes	—
299	C₂×C₃	—	2	4	no	—
300	C₂×C₃	—	2	8	no	—
301	C₂×C₃	—	6	8	no	—
302	C₂×C₃	—	36	4	no	—
303	C₂×C₃	#63807	1	8	yes	complemented
304	C₂×C₃	#64113	1	2	yes	complemented
305	C₂×C₃	C₂×D₃×D₁₂	1	4	yes	—
306	C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
307	D₃	—	2	2	no	—
308	D₃	—	12	16	no	—
309	C₂²	—	3	2	no	—
310	C₂²	—	6	2	no	—
311	C₂²	—	18	4	no	—
312	C₂²	—	36	4	no	—
313	C₂²	D₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
314	C₄	—	3	2	no	—
315	C₄	—	54	4	no	—
316	C₄	D₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
317	C₄	C₂²×D₃×C₃⋊D₃	1	1	yes	characteristic
318	C₃	—	2	4	no	—
319	C₃	C₂×#63807	1	4	yes	complemented
320	C₃	C₂×#64113	1	1	yes	characteristic, complemented
321	C₂	—	6	2	no	—
322	C₂	—	36	4	no	—
323	C₂	#11723336	1	2	yes	direct product factor
324	C₂	C₂×D₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃³×C₄	C₂⁴
Frattini	C₄	C₂²×D₃×C₃⋊D₃
Fitting	C₂×C₃³×C₈	C₂²
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₂×C₈	D₃×C₃⋊D₃
hypocenter	C₃³	C₂×#135
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#89021	C₂	semidirect
2. C₂×#88853	C₂	semidirect
3. C₂×C₃×#3745	C₂	semidirect
4. C₂×C₃²×D₃×C₈	C₂	semidirect
5. C₂×#88796	C₂	semidirect
6. #11723336	C₂	direct (N × H)
15. #89021	C₂²	semidirect
17. #88853	C₂²	semidirect
18. C₃×#3745	C₂²	semidirect
20. C₃²×D₃×C₈	C₂²	semidirect
21. #88796	C₂²	semidirect
29. C₂×#3745	D₃	semidirect
32. C₂×C₃×D₃×C₈	D₃	semidirect
75. #3745	C₂×D₃	semidirect
79. C₃×D₃×C₈	C₂×D₃	semidirect
102. C₂×D₃×C₈	C₃⋊D₃	semidirect
139. C₂×C₃³	#135	semidirect
171. D₃×C₈	C₂×C₃⋊D₃	semidirect
200. C₃³	C₂×#135	semidirect
239. C₂×C₃²	#904	semidirect
240. C₂×C₃²	#912	semidirect
282. C₃²	C₂×#904	semidirect
283. C₃²	C₂×#912	semidirect
303. C₂×C₃	#63807	semidirect
304. C₂×C₃	#64113	semidirect
319. C₃	C₂×#63807	semidirect
320. C₃	C₂×#64113	semidirect
323. C₂	#11723336	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×#89021	2
C₂×#88853	2
C₂×C₃×#3745	2
C₂×C₃²×D₃×C₈	2
C₂×#88796	2
#11723336	2
C₂×#64113	3
C₂×#63807	3

Subgroups — C2×#11723336 #13176669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11723336 #13176669