Subgroups — C₂²×#89014 #13181595

Order 1728.

5964 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	C₂²×#6308	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#6290	C₂	1	2	yes	maximal, complemented
5	C₂²×D₃×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
6	C₂×#89014	C₂	1	24	yes	maximal, direct product factor
7	C₂²×#3830	—	3	1	no	maximal
8	C₂²×#3795	—	3	4	no	maximal
9	C₂³×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	C₂³×C₃²×D₃	C₂²	1	2	yes	—
11	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
13	C₂×#6308	C₂²	1	12	yes	complemented
14	C₂²×C₃×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	12	yes	complemented
16	C₂×C₃²×D₁₂	C₂²	1	12	yes	complemented
17	C₂×#6290	C₂²	1	24	yes	complemented
18	C₂×D₃×C₃⋊Dic₃	C₂²	1	24	yes	complemented
19	#89014	C₂²	1	64	yes	direct product factor
20	C₂²×C₃²×D₄	—	3	1	no	—
21	C₂²×#638	—	3	2	no	—
22	C₂³×C₃⋊Dic₃	—	3	2	no	—
23	C₂×#3830	—	3	24	no	—
24	C₂²×C₄×C₃⋊D₃	D₃	1	1	yes	characteristic, complemented
25	C₂²×#634	—	3	1	no	—
26	C₂²×#634	—	3	4	no	—
27	C₂²×#634	—	6	4	no	—
28	C₂²×C₃×D₁₂	D₃	1	4	yes	complemented
29	C₂²×C₃×C₄×D₃	—	3	4	no	—
30	C₂²×#625	—	3	8	no	—
31	C₂²×D₃×Dic₃	—	3	8	no	—
32	C₂×#3795	—	3	96	no	—
33	C₂³×C₃³	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₃⋊D₃	—	2	4	no	—
35	C₂²×C₃×C₃⋊D₃	C₂³	1	6	yes	—
36	C₂²×C₃²×D₃	—	2	8	no	—
37	C₂²×C₃²×D₃	C₂³	1	12	yes	—
38	C₂×C₃³×C₄	C₂³	1	6	yes	—
39	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	12	yes	—
40	#6308	C₂³	1	16	yes	complemented
41	C₂×C₃×C₃⋊Dic₃	C₂³	1	6	yes	—
42	C₃×C₄×C₃⋊D₃	C₂³	1	16	yes	complemented
43	C₃²×D₁₂	C₂³	1	16	yes	complemented
44	#6290	C₂³	1	32	yes	complemented
45	D₃×C₃⋊Dic₃	C₂³	1	32	yes	complemented
46	C₂²×#237	—	9	4	no	—
47	C₂²×#235	—	9	1	no	—
48	C₂⁴×C₃²	—	3	2	no	—
49	C₂³×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
50	C₂³×C₃×D₃	—	3	4	no	—
51	C₂³×C₃×D₃	C₂×D₃	1	8	yes	—
52	C₂×C₃²×D₄	—	3	12	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	—	3	24	no	—
57	C₂²×C₃⋊Dic₃	—	3	2	no	—
58	C₂²×C₃⋊Dic₃	—	3	8	no	—
59	C₂²×C₃⋊Dic₃	—	3	24	no	—
60	C₂²×C₃⋊Dic₃	—	6	8	no	—
61	C₂²×C₃⋊Dic₃	C₂×D₃	1	1	yes	characteristic
62	#3830	—	3	64	no	—
63	C₂×C₄×C₃⋊D₃	C₂×D₃	1	12	yes	complemented
64	C₂×#634	—	3	12	no	—
65	C₂×#634	—	3	48	no	—
66	C₂×#634	—	6	48	no	—
67	C₂²×C₃×Dic₃	—	3	4	no	—
68	C₂×C₃×D₁₂	C₂×D₃	1	48	yes	complemented
69	C₂×C₃×C₄×D₃	—	3	48	no	—
70	C₂×#625	—	3	96	no	—
71	C₂×D₃×Dic₃	—	3	96	no	—
72	#3795	—	3	256	no	—
73	C₂²×C₃³	#41	1	4	yes	complemented
74	C₂²×C₃³	C₂⁴	1	3	yes	—
75	C₂×C₃×C₃⋊D₃	—	2	12	no	—
76	C₂×C₃×C₃⋊D₃	C₂⁴	1	4	yes	—
77	C₂×C₃²×D₃	—	2	24	no	—
78	C₂×C₃²×D₃	C₂⁴	1	8	yes	—
79	C₃³×C₄	C₂⁴	1	4	yes	—
80	C₃⋊(C₃⋊Dic₃)	C₂⁴	1	8	yes	—
81	C₃×C₃⋊Dic₃	C₂⁴	1	4	yes	—
82	C₂²×C₃×D₄	—	3	4	no	—
83	C₂³×C₃×C₄	—	9	1	no	—
84	C₂²×#67	—	9	8	no	—
85	C₂²×#67	—	9	2	no	—
86	C₂³×Dic₃	—	9	8	no	—
87	C₂×#237	—	9	96	no	—
88	C₂×#235	—	9	24	no	—
89	C₂²×D₁₂	C₃⋊D₃	1	1	yes	characteristic, complemented
90	C₂²×C₄×D₃	—	3	4	no	—
91	C₂²×C₄×D₃	—	9	2	no	—
92	C₂²×Dic₆	—	9	4	no	—
93	C₂²×Dic₆	—	9	1	no	—
94	C₂²×Dic₆	—	18	4	no	—
95	C₂³×C₃²	—	2	4	no	—
96	C₂³×C₃²	—	3	12	no	—
97	C₂³×C₃²	—	6	8	no	—
98	C₂³×C₃²	C₂²×D₃	1	4	yes	—
99	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃⋊D₃	—	2	4	no	—
101	C₂²×C₃⋊D₃	C₂²×D₃	1	6	yes	—
102	C₂²×C₃×D₃	—	2	32	no	—
103	C₂²×C₃×D₃	—	3	24	no	—
104	C₂²×C₃×D₃	—	6	16	no	—
105	C₂²×C₃×D₃	C₂²×D₃	1	48	yes	—
106	C₃²×D₄	—	3	16	no	—
107	C₂×C₃²×C₄	—	2	24	no	—
108	C₂×C₃²×C₄	C₂²×D₃	1	24	yes	—
109	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
110	#638	—	3	32	no	—
111	C₂×C₃⋊Dic₃	—	3	12	no	—
112	C₂×C₃⋊Dic₃	—	3	32	no	—
113	C₂×C₃⋊Dic₃	—	3	48	no	—
114	C₂×C₃⋊Dic₃	—	6	48	no	—
115	C₂×C₃⋊Dic₃	C₂²×D₃	1	6	yes	—
116	C₄×C₃⋊D₃	C₂²×D₃	1	16	yes	complemented
117	#634	—	3	16	no	—
118	#634	—	3	64	no	—
119	#634	—	6	64	no	—
120	C₂×C₃×Dic₃	—	3	24	no	—
121	C₃×D₁₂	C₂²×D₃	1	64	yes	complemented
122	C₃×C₄×D₃	—	3	64	no	—
123	#625	—	3	128	no	—
124	D₃×Dic₃	—	3	128	no	—
125	C₂²×#41	—	27	1	no	Sylow
126	C₂×C₃³	C₂×#41	1	6	yes	complemented
127	C₂×C₃³	C₂⁵	1	1	yes	characteristic
128	C₃×C₃⋊D₃	—	2	4	no	—
129	C₃²×D₃	—	2	8	no	—
130	C₂⁴×C₃	—	3	8	no	—
131	C₂⁴×C₃	—	9	1	no	—
132	C₂³×D₃	—	3	4	no	—
133	C₂³×D₃	C₂×C₃⋊D₃	1	2	yes	—
134	C₂×C₃×D₄	—	3	48	no	—
135	C₂²×C₃×C₄	—	2	4	no	—
136	C₂²×C₃×C₄	—	9	1	no	—
137	C₂²×C₃×C₄	—	9	12	no	—
138	C₂²×C₃×C₄	D₃²	1	4	yes	—
139	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
140	C₂×#67	—	9	96	no	—
141	C₂×#67	—	9	24	no	—
142	C₂²×Dic₃	—	3	4	no	—
143	C₂²×Dic₃	—	9	96	no	—
144	C₂²×Dic₃	—	9	8	no	—
145	C₂²×Dic₃	—	9	2	no	—
146	C₂²×Dic₃	—	18	8	no	—
147	#237	—	9	256	no	—
148	#235	—	9	64	no	—
149	C₂×D₁₂	C₂×C₃⋊D₃	1	12	yes	complemented
150	C₂×C₄×D₃	—	3	48	no	—
151	C₂×C₄×D₃	—	9	24	no	—
152	C₂×Dic₆	—	9	48	no	—
153	C₂×Dic₆	—	9	12	no	—
154	C₂×Dic₆	—	18	48	no	—
155	C₂²×C₃²	—	2	12	no	—
156	C₂²×C₃²	—	2	16	no	—
157	C₂²×C₃²	—	3	8	no	—
158	C₂²×C₃²	—	6	24	no	—
159	C₂²×C₃²	#235	1	4	yes	complemented
160	C₂²×C₃²	#237	1	16	yes	complemented
161	C₂²×C₃²	C₂³×D₃	1	12	yes	—
162	C₂²×C₃²	C₂³×D₃	1	3	yes	—
163	C₂×C₃⋊D₃	—	2	12	no	—
164	C₂×C₃⋊D₃	C₂³×D₃	1	4	yes	—
165	C₂×C₃×D₃	—	2	96	no	—
166	C₂×C₃×D₃	—	3	16	no	—
167	C₂×C₃×D₃	—	6	48	no	—
168	C₂×C₃×D₃	C₂³×D₃	1	32	yes	—
169	C₃²×C₄	—	2	16	no	—
170	C₃²×C₄	C₂³×D₃	1	16	yes	—
171	C₃²×C₄	C₂³×D₃	1	4	yes	—
172	C₃⋊Dic₃	—	3	8	no	—
173	C₃⋊Dic₃	—	3	32	no	—
174	C₃⋊Dic₃	—	6	32	no	—
175	C₃⋊Dic₃	C₂³×D₃	1	4	yes	—
176	C₃×Dic₃	—	3	16	no	—
177	C₂×#41	—	27	24	no	—
178	C₂²×Q₈	—	27	1	no	—
179	C₂²×D₄	—	3	1	no	—
180	C₂²×D₄	—	27	2	no	—
181	C₂³×C₄	—	9	1	no	—
182	C₂³×C₄	—	27	2	no	—
183	C₃³	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₃	—	2	4	no	—
185	C₂³×C₃	—	3	48	no	—
186	C₂³×C₃	—	6	32	no	—
187	C₂³×C₃	—	9	6	no	—
188	C₂³×C₃	—	18	4	no	—
189	C₂³×C₃	C₂×D₃²	1	4	yes	—
190	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
191	C₂²×D₃	—	2	8	no	—
192	C₂²×D₃	—	3	24	no	—
193	C₂²×D₃	—	6	16	no	—
194	C₂²×D₃	C₂²×C₃⋊D₃	1	12	yes	—
195	C₃×D₄	—	3	64	no	—
196	C₂×C₃×C₄	—	2	24	no	—
197	C₂×C₃×C₄	—	9	6	no	—
198	C₂×C₃×C₄	—	9	16	no	—
199	C₂×C₃×C₄	C₂×D₃²	1	24	yes	—
200	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	6	yes	—
201	#67	—	9	128	no	—
202	#67	—	9	32	no	—
203	C₂×Dic₃	—	3	24	no	—
204	C₂×Dic₃	—	9	128	no	—
205	C₂×Dic₃	—	9	48	no	—
206	C₂×Dic₃	—	9	12	no	—
207	C₂×Dic₃	—	18	48	no	—
208	D₁₂	C₂²×C₃⋊D₃	1	16	yes	complemented
209	C₄×D₃	—	3	64	no	—
210	C₄×D₃	—	9	32	no	—
211	Dic₆	—	9	64	no	—
212	Dic₆	—	9	16	no	—
213	Dic₆	—	18	64	no	—
214	C₂×C₃²	—	2	24	no	—
215	C₂×C₃²	—	2	4	no	—
216	C₂×C₃²	—	6	8	no	—
217	C₂×C₃²	C₂×#235	1	6	yes	complemented
218	C₂×C₃²	C₂×#237	1	24	yes	complemented
219	C₂×C₃²	C₂⁴×D₃	1	4	yes	—
220	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
221	C₃⋊D₃	—	2	4	no	—
222	C₃×D₃	—	2	32	no	—
223	C₃×D₃	—	6	16	no	—
224	C₂⁴	—	3	2	no	—
225	C₂⁴	—	9	1	no	—
226	#41	—	27	64	no	—
227	C₂×Q₈	—	27	12	no	—
228	C₂×D₄	—	3	12	no	—
229	C₂×D₄	—	27	24	no	—
230	C₂²×C₄	—	9	1	no	—
231	C₂²×C₄	—	9	12	no	—
232	C₂²×C₄	—	27	24	no	—
233	C₂²×C₄	—	27	2	no	—
234	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
235	C₂²×C₃	—	2	12	no	—
236	C₂²×C₃	—	2	16	no	—
237	C₂²×C₃	—	3	32	no	—
238	C₂²×C₃	—	6	96	no	—
239	C₂²×C₃	—	9	4	no	—
240	C₂²×C₃	—	18	12	no	—
241	C₂²×C₃	#3795	1	16	yes	complemented
242	C₂²×C₃	#3830	1	4	yes	complemented
243	C₂²×C₃	C₂²×D₃²	1	12	yes	—
244	C₂²×C₃	C₂³×C₃⋊D₃	1	3	yes	—
245	C₂×D₃	—	2	24	no	—
246	C₂×D₃	—	3	16	no	—
247	C₂×D₃	—	6	48	no	—
248	C₂×D₃	C₂³×C₃⋊D₃	1	8	yes	—
249	C₃×C₄	—	2	16	no	—
250	C₃×C₄	—	9	4	no	—
251	C₃×C₄	C₂²×D₃²	1	16	yes	—
252	C₃×C₄	C₂³×C₃⋊D₃	1	4	yes	—
253	Dic₃	—	3	16	no	—
254	Dic₃	—	9	32	no	—
255	Dic₃	—	9	8	no	—
256	Dic₃	—	18	32	no	—
257	C₃²	—	2	4	no	—
258	C₃²	C₂²×#235	1	1	yes	characteristic, complemented
259	C₃²	C₂²×#237	1	4	yes	complemented
260	C₂³	—	3	12	no	—
261	C₂³	—	6	8	no	—
262	C₂³	—	9	6	no	—
263	C₂³	—	18	4	no	—
264	C₂³	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
265	Q₈	—	27	16	no	—
266	D₄	—	3	16	no	—
267	D₄	—	27	32	no	—
268	C₂×C₄	—	9	6	no	—
269	C₂×C₄	—	9	16	no	—
270	C₂×C₄	—	27	32	no	—
271	C₂×C₄	—	27	12	no	—
272	C₂×C₄	C₂×D₃×C₃⋊D₃	1	6	yes	—
273	C₂×C₃	—	2	24	no	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	6	32	no	—
276	C₂×C₃	—	18	4	no	—
277	C₂×C₃	C₂×#3795	1	24	yes	complemented
278	C₂×C₃	C₂×#3830	1	6	yes	complemented
279	C₂×C₃	C₂³×D₃²	1	4	yes	—
280	C₂×C₃	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
281	D₃	—	2	8	no	—
282	D₃	—	6	16	no	—
283	C₂²	—	3	8	no	—
284	C₂²	—	6	24	no	—
285	C₂²	—	9	4	no	—
286	C₂²	—	18	12	no	—
287	C₂²	#89014	1	4	yes	direct product factor
288	C₂²	C₂²×D₃×C₃⋊D₃	1	3	yes	—
289	C₄	—	9	4	no	—
290	C₄	—	27	8	no	—
291	C₄	C₂²×D₃×C₃⋊D₃	1	4	yes	—
292	C₃	—	2	4	no	—
293	C₃	C₂²×#3795	1	4	yes	complemented
294	C₃	C₂²×#3830	1	1	yes	characteristic, complemented
295	C₂	—	6	8	no	—
296	C₂	—	18	4	no	—
297	C₂	C₂×#89014	1	6	yes	direct product factor
298	C₂	C₂³×D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃×C₃⋊D₃
commutator, cosocle	C₂×C₃³	C₂⁵
Frattini	C₂	C₂³×D₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₄	C₂²
socle	C₂³×C₃³	C₂³
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	C₂²×#41
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#6308	C₂	semidirect
2. C₂²×C₃×C₄×C₃⋊D₃	C₂	semidirect
3. C₂²×C₃²×D₁₂	C₂	semidirect
4. C₂²×#6290	C₂	semidirect
5. C₂²×D₃×C₃⋊Dic₃	C₂	semidirect
6. C₂×#89014	C₂	direct (N × H)
13. C₂×#6308	C₂²	semidirect
15. C₂×C₃×C₄×C₃⋊D₃	C₂²	semidirect
16. C₂×C₃²×D₁₂	C₂²	semidirect
17. C₂×#6290	C₂²	semidirect
18. C₂×D₃×C₃⋊Dic₃	C₂²	semidirect
19. #89014	C₂²	direct (N × H)
24. C₂²×C₄×C₃⋊D₃	D₃	semidirect
28. C₂²×C₃×D₁₂	D₃	semidirect
40. #6308	C₂³	semidirect
42. C₃×C₄×C₃⋊D₃	C₂³	semidirect
43. C₃²×D₁₂	C₂³	semidirect
44. #6290	C₂³	semidirect
45. D₃×C₃⋊Dic₃	C₂³	semidirect
63. C₂×C₄×C₃⋊D₃	C₂×D₃	semidirect
68. C₂×C₃×D₁₂	C₂×D₃	semidirect
73. C₂²×C₃³	#41	semidirect
89. C₂²×D₁₂	C₃⋊D₃	semidirect
116. C₄×C₃⋊D₃	C₂²×D₃	semidirect
121. C₃×D₁₂	C₂²×D₃	semidirect
126. C₂×C₃³	C₂×#41	semidirect
149. C₂×D₁₂	C₂×C₃⋊D₃	semidirect
159. C₂²×C₃²	#235	semidirect
160. C₂²×C₃²	#237	semidirect
183. C₃³	C₂²×#41	semidirect
208. D₁₂	C₂²×C₃⋊D₃	semidirect
217. C₂×C₃²	C₂×#235	semidirect
218. C₂×C₃²	C₂×#237	semidirect
241. C₂²×C₃	#3795	semidirect
242. C₂²×C₃	#3830	semidirect
258. C₃²	C₂²×#235	semidirect
259. C₃²	C₂²×#237	semidirect
277. C₂×C₃	C₂×#3795	semidirect
278. C₂×C₃	C₂×#3830	semidirect
287. C₂²	#89014	direct (N × H)
293. C₃	C₂²×#3795	semidirect
294. C₃	C₂²×#3830	semidirect
297. C₂	C₂×#89014	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#6308	2
C₂²×C₃×C₄×C₃⋊D₃	2
C₂²×C₃²×D₁₂	2
C₂²×#6290	2
C₂²×D₃×C₃⋊Dic₃	2
C₂×#89014	2
C₂²×#3830	3
C₂²×#3795	3

Subgroups — C22×#89014 #13181595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89014 #13181595