Subgroups — C₂²×#89015 #13181590

Order 1728.

6284 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×C₄×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#6291	C₂	1	2	yes	maximal, complemented
5	C₂²×D₃×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
6	C₂×#89015	C₂	1	24	yes	maximal, direct product factor
7	C₂²×#3830	—	3	1	no	maximal
8	C₂²×#3796	—	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	1	yes	characteristic
13	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	12	yes	complemented
14	C₂²×C₃×C₃⋊Dic₃	C₂²	1	2	yes	—
15	C₂×C₃×#634	C₂²	1	12	yes	complemented
16	C₂×C₃²×D₁₂	C₂²	1	12	yes	complemented
17	C₂×#6291	C₂²	1	24	yes	complemented
18	C₂×D₃×C₃⋊Dic₃	C₂²	1	24	yes	complemented
19	#89015	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	—	3	1	no	—
25	C₂²×C₄×C₃⋊D₃	—	3	4	no	—
26	C₂²×C₄×C₃⋊D₃	—	6	4	no	—
27	C₂²×#634	D₃	1	1	yes	characteristic, complemented
28	C₂²×C₃×D₁₂	D₃	1	4	yes	complemented
29	C₂²×C₃×Dic₆	—	3	4	no	—
30	C₂²×#626	—	3	8	no	—
31	C₂²×D₃×Dic₃	—	3	8	no	—
32	C₂×#3796	—	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	6	yes	—
40	C₄×C₃⋊(C₃⋊D₃)	C₂³	1	16	yes	complemented
41	C₂×C₃×C₃⋊Dic₃	C₂³	1	12	yes	—
42	C₃×#634	C₂³	1	16	yes	complemented
43	C₃²×D₁₂	C₂³	1	16	yes	complemented
44	#6291	C₂³	1	32	yes	complemented
45	D₃×C₃⋊Dic₃	C₂³	1	32	yes	complemented
46	C₂²×#239	—	9	1	no	—
47	C₂²×#237	—	9	4	no	—
48	C₂⁴×C₃²	—	3	2	no	—
49	C₂³×C₃⋊D₃	—	3	1	no	—
50	C₂³×C₃⋊D₃	—	3	4	no	—
51	C₂³×C₃⋊D₃	—	6	4	no	—
52	C₂³×C₃×D₃	C₂×D₃	1	8	yes	—
53	C₂×C₃²×D₄	—	3	12	no	—
54	C₂²×C₃²×C₄	—	2	4	no	—
55	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
56	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×#638	—	3	24	no	—
58	C₂²×C₃⋊Dic₃	—	3	1	no	—
59	C₂²×C₃⋊Dic₃	—	3	4	no	—
60	C₂²×C₃⋊Dic₃	—	3	24	no	—
61	C₂²×C₃⋊Dic₃	—	6	4	no	—
62	C₂²×C₃⋊Dic₃	C₂×D₃	1	2	yes	—
63	#3830	—	3	64	no	—
64	C₂×C₄×C₃⋊D₃	—	3	12	no	—
65	C₂×C₄×C₃⋊D₃	—	3	48	no	—
66	C₂×C₄×C₃⋊D₃	—	6	48	no	—
67	C₂×#634	C₂×D₃	1	12	yes	complemented
68	C₂²×C₃×Dic₃	—	3	8	no	—
69	C₂×C₃×D₁₂	C₂×D₃	1	48	yes	complemented
70	C₂×C₃×Dic₆	—	3	48	no	—
71	C₂×#626	—	3	96	no	—
72	C₂×D₃×Dic₃	—	3	96	no	—
73	#3796	—	3	256	no	—
74	C₂²×C₃³	#41	1	4	yes	complemented
75	C₂²×C₃³	C₂⁴	1	3	yes	—
76	C₂×C₃⋊(C₃⋊D₃)	—	2	12	no	—
77	C₂×C₃⋊(C₃⋊D₃)	C₂⁴	1	4	yes	—
78	C₂×C₃²×D₃	—	2	24	no	—
79	C₂×C₃²×D₃	C₂⁴	1	8	yes	—
80	C₃³×C₄	C₂⁴	1	4	yes	—
81	C₃⋊(C₃⋊Dic₃)	C₂⁴	1	4	yes	—
82	C₃×C₃⋊Dic₃	C₂⁴	1	8	yes	—
83	C₂²×C₃×Q₈	—	9	1	no	—
84	C₂²×C₃×D₄	—	3	4	no	—
85	C₂²×#67	—	9	8	no	—
86	C₂³×Dic₃	—	9	8	no	—
87	C₂×#239	—	9	24	no	—
88	C₂×#237	—	9	96	no	—
89	C₂²×D₁₂	—	9	2	no	—
90	C₂²×D₁₂	C₃⋊D₃	1	1	yes	characteristic, complemented
91	C₂²×C₄×D₃	—	9	2	no	—
92	C₂²×C₄×D₃	—	9	4	no	—
93	C₂²×C₄×D₃	—	9	1	no	—
94	C₂²×C₄×D₃	—	18	4	no	—
95	C₂²×Dic₆	—	3	4	no	—
96	C₂³×C₃²	—	2	4	no	—
97	C₂³×C₃²	—	3	12	no	—
98	C₂³×C₃²	—	6	8	no	—
99	C₂³×C₃²	C₂²×D₃	1	4	yes	—
100	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
101	C₂²×C₃⋊D₃	—	3	6	no	—
102	C₂²×C₃⋊D₃	—	3	24	no	—
103	C₂²×C₃⋊D₃	—	6	24	no	—
104	C₂²×C₃⋊D₃	—	6	32	no	—
105	C₂²×C₃⋊D₃	—	6	16	no	—
106	C₂²×C₃⋊D₃	—	6	4	no	—
107	C₂²×C₃×D₃	—	2	32	no	—
108	C₂²×C₃×D₃	C₂²×D₃	1	48	yes	—
109	C₃²×D₄	—	3	16	no	—
110	C₂×C₃²×C₄	—	2	24	no	—
111	C₂×C₃²×C₄	C₂²×D₃	1	24	yes	—
112	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
113	#638	—	3	32	no	—
114	C₂×C₃⋊Dic₃	—	3	6	no	—
115	C₂×C₃⋊Dic₃	—	3	32	no	—
116	C₂×C₃⋊Dic₃	—	3	24	no	—
117	C₂×C₃⋊Dic₃	—	6	24	no	—
118	C₂×C₃⋊Dic₃	C₂²×D₃	1	12	yes	—
119	C₄×C₃⋊D₃	—	3	16	no	—
120	C₄×C₃⋊D₃	—	3	64	no	—
121	C₄×C₃⋊D₃	—	6	64	no	—
122	#634	C₂²×D₃	1	16	yes	complemented
123	C₂×C₃×Dic₃	—	3	48	no	—
124	C₃×D₁₂	C₂²×D₃	1	64	yes	complemented
125	C₃×Dic₆	—	3	64	no	—
126	#626	—	3	128	no	—
127	D₃×Dic₃	—	3	128	no	—
128	C₂²×#41	—	27	1	no	Sylow
129	C₂×C₃³	C₂×#41	1	6	yes	complemented
130	C₂×C₃³	C₂⁵	1	1	yes	characteristic
131	C₃⋊(C₃⋊D₃)	—	2	4	no	—
132	C₃²×D₃	—	2	8	no	—
133	C₂⁴×C₃	—	3	8	no	—
134	C₂³×D₃	—	9	4	no	—
135	C₂³×D₃	—	9	1	no	—
136	C₂³×D₃	—	18	4	no	—
137	C₂³×D₃	C₂×C₃⋊D₃	1	2	yes	—
138	C₂×C₃×Q₈	—	9	12	no	—
139	C₂×C₃×D₄	—	3	48	no	—
140	C₂²×C₃×C₄	—	2	4	no	—
141	C₂²×C₃×C₄	—	9	2	no	—
142	C₂²×C₃×C₄	D₃²	1	4	yes	—
143	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
144	C₂×#67	—	9	96	no	—
145	C₂²×Dic₃	—	3	8	no	—
146	C₂²×Dic₃	—	9	96	no	—
147	C₂²×Dic₃	—	9	4	no	—
148	C₂²×Dic₃	—	9	1	no	—
149	C₂²×Dic₃	—	18	4	no	—
150	#239	—	9	64	no	—
151	#237	—	9	256	no	—
152	C₂×D₁₂	—	9	24	no	—
153	C₂×D₁₂	C₂×C₃⋊D₃	1	12	yes	complemented
154	C₂×C₄×D₃	—	9	24	no	—
155	C₂×C₄×D₃	—	9	48	no	—
156	C₂×C₄×D₃	—	9	12	no	—
157	C₂×C₄×D₃	—	18	48	no	—
158	C₂×Dic₆	—	3	48	no	—
159	C₂²×C₃²	—	2	12	no	—
160	C₂²×C₃²	—	2	16	no	—
161	C₂²×C₃²	—	3	8	no	—
162	C₂²×C₃²	—	6	24	no	—
163	C₂²×C₃²	#237	1	16	yes	complemented
164	C₂²×C₃²	#239	1	4	yes	complemented
165	C₂²×C₃²	C₂³×D₃	1	12	yes	—
166	C₂²×C₃²	C₂³×D₃	1	3	yes	—
167	C₂×C₃⋊D₃	—	3	4	no	—
168	C₂×C₃⋊D₃	—	3	16	no	—
169	C₂×C₃⋊D₃	—	6	16	no	—
170	C₂×C₃⋊D₃	—	6	96	no	—
171	C₂×C₃⋊D₃	—	6	48	no	—
172	C₂×C₃⋊D₃	—	6	12	no	—
173	C₂×C₃×D₃	—	2	96	no	—
174	C₂×C₃×D₃	C₂³×D₃	1	32	yes	—
175	C₃²×C₄	—	2	16	no	—
176	C₃²×C₄	C₂³×D₃	1	16	yes	—
177	C₃²×C₄	C₂³×D₃	1	4	yes	—
178	C₃⋊Dic₃	—	3	4	no	—
179	C₃⋊Dic₃	—	3	16	no	—
180	C₃⋊Dic₃	—	6	16	no	—
181	C₃⋊Dic₃	C₂³×D₃	1	8	yes	—
182	C₃×Dic₃	—	3	32	no	—
183	C₂×#41	—	27	24	no	—
184	C₂²×Q₈	—	9	1	no	—
185	C₂²×D₄	—	3	1	no	—
186	C₂²×D₄	—	27	2	no	—
187	C₂³×C₄	—	27	2	no	—
188	C₂³×C₄	—	27	1	no	—
189	C₃³	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₃	—	2	4	no	—
191	C₂³×C₃	—	3	48	no	—
192	C₂³×C₃	—	6	32	no	—
193	C₂³×C₃	C₂×D₃²	1	4	yes	—
194	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
195	C₂²×D₃	—	2	8	no	—
196	C₂²×D₃	—	9	24	no	—
197	C₂²×D₃	—	9	6	no	—
198	C₂²×D₃	—	18	24	no	—
199	C₂²×D₃	—	18	32	no	—
200	C₂²×D₃	—	18	16	no	—
201	C₂²×D₃	—	18	4	no	—
202	C₂²×D₃	C₂²×C₃⋊D₃	1	12	yes	—
203	C₃×Q₈	—	9	16	no	—
204	C₃×D₄	—	3	64	no	—
205	C₂×C₃×C₄	—	2	24	no	—
206	C₂×C₃×C₄	—	9	12	no	—
207	C₂×C₃×C₄	C₂×D₃²	1	24	yes	—
208	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	6	yes	—
209	#67	—	9	128	no	—
210	C₂×Dic₃	—	3	48	no	—
211	C₂×Dic₃	—	9	128	no	—
212	C₂×Dic₃	—	9	24	no	—
213	C₂×Dic₃	—	9	6	no	—
214	C₂×Dic₃	—	18	24	no	—
215	D₁₂	—	9	32	no	—
216	D₁₂	C₂²×C₃⋊D₃	1	16	yes	complemented
217	C₄×D₃	—	9	32	no	—
218	C₄×D₃	—	9	64	no	—
219	C₄×D₃	—	9	16	no	—
220	C₄×D₃	—	18	64	no	—
221	Dic₆	—	3	64	no	—
222	C₂×C₃²	—	2	24	no	—
223	C₂×C₃²	—	2	4	no	—
224	C₂×C₃²	—	6	8	no	—
225	C₂×C₃²	C₂×#237	1	24	yes	complemented
226	C₂×C₃²	C₂×#239	1	6	yes	complemented
227	C₂×C₃²	C₂⁴×D₃	1	4	yes	—
228	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
229	C₃⋊D₃	—	6	32	no	—
230	C₃⋊D₃	—	6	16	no	—
231	C₃⋊D₃	—	6	4	no	—
232	C₃×D₃	—	2	32	no	—
233	C₂⁴	—	3	2	no	—
234	C₂⁴	—	27	1	no	—
235	#41	—	27	64	no	—
236	C₂×Q₈	—	9	12	no	—
237	C₂×D₄	—	3	12	no	—
238	C₂×D₄	—	27	24	no	—
239	C₂²×C₄	—	9	2	no	—
240	C₂²×C₄	—	27	24	no	—
241	C₂²×C₄	—	27	1	no	—
242	C₂²×C₄	—	27	12	no	—
243	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
244	C₂²×C₃	—	2	12	no	—
245	C₂²×C₃	—	2	16	no	—
246	C₂²×C₃	—	3	32	no	—
247	C₂²×C₃	—	6	96	no	—
248	C₂²×C₃	#3796	1	16	yes	complemented
249	C₂²×C₃	#3830	1	4	yes	complemented
250	C₂²×C₃	C₂²×D₃²	1	12	yes	—
251	C₂²×C₃	C₂³×C₃⋊D₃	1	3	yes	—
252	C₂×D₃	—	2	24	no	—
253	C₂×D₃	—	9	16	no	—
254	C₂×D₃	—	9	4	no	—
255	C₂×D₃	—	18	16	no	—
256	C₂×D₃	—	18	96	no	—
257	C₂×D₃	—	18	48	no	—
258	C₂×D₃	—	18	12	no	—
259	C₂×D₃	C₂³×C₃⋊D₃	1	8	yes	—
260	C₃×C₄	—	2	16	no	—
261	C₃×C₄	—	9	8	no	—
262	C₃×C₄	C₂²×D₃²	1	16	yes	—
263	C₃×C₄	C₂³×C₃⋊D₃	1	4	yes	—
264	Dic₃	—	3	32	no	—
265	Dic₃	—	9	16	no	—
266	Dic₃	—	9	4	no	—
267	Dic₃	—	18	16	no	—
268	C₃²	—	2	4	no	—
269	C₃²	C₂²×#237	1	4	yes	complemented
270	C₃²	C₂²×#239	1	1	yes	characteristic, complemented
271	C₂³	—	3	12	no	—
272	C₂³	—	6	8	no	—
273	C₂³	—	27	6	no	—
274	C₂³	—	54	4	no	—
275	C₂³	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
276	Q₈	—	9	16	no	—
277	D₄	—	3	16	no	—
278	D₄	—	27	32	no	—
279	C₂×C₄	—	9	12	no	—
280	C₂×C₄	—	27	32	no	—
281	C₂×C₄	—	27	6	no	—
282	C₂×C₄	—	27	16	no	—
283	C₂×C₄	C₂×D₃×C₃⋊D₃	1	6	yes	—
284	C₂×C₃	—	2	24	no	—
285	C₂×C₃	—	2	4	no	—
286	C₂×C₃	—	6	32	no	—
287	C₂×C₃	C₂×#3796	1	24	yes	complemented
288	C₂×C₃	C₂×#3830	1	6	yes	complemented
289	C₂×C₃	C₂³×D₃²	1	4	yes	—
290	C₂×C₃	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
291	D₃	—	2	8	no	—
292	D₃	—	18	32	no	—
293	D₃	—	18	16	no	—
294	D₃	—	18	4	no	—
295	C₂²	—	3	8	no	—
296	C₂²	—	6	24	no	—
297	C₂²	—	27	4	no	—
298	C₂²	—	54	12	no	—
299	C₂²	#89015	1	4	yes	direct product factor
300	C₂²	C₂²×D₃×C₃⋊D₃	1	3	yes	—
301	C₄	—	9	8	no	—
302	C₄	—	27	4	no	—
303	C₄	C₂²×D₃×C₃⋊D₃	1	4	yes	—
304	C₃	—	2	4	no	—
305	C₃	C₂²×#3796	1	4	yes	complemented
306	C₃	C₂²×#3830	1	1	yes	characteristic, complemented
307	C₂	—	6	8	no	—
308	C₂	—	54	4	no	—
309	C₂	C₂×#89015	1	6	yes	direct product factor
310	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×C₄×C₃⋊(C₃⋊D₃)	C₂	semidirect
2. C₂²×C₃×#634	C₂	semidirect
3. C₂²×C₃²×D₁₂	C₂	semidirect
4. C₂²×#6291	C₂	semidirect
5. C₂²×D₃×C₃⋊Dic₃	C₂	semidirect
6. C₂×#89015	C₂	direct (N × H)
13. C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	semidirect
15. C₂×C₃×#634	C₂²	semidirect
16. C₂×C₃²×D₁₂	C₂²	semidirect
17. C₂×#6291	C₂²	semidirect
18. C₂×D₃×C₃⋊Dic₃	C₂²	semidirect
19. #89015	C₂²	direct (N × H)
27. C₂²×#634	D₃	semidirect
28. C₂²×C₃×D₁₂	D₃	semidirect
40. C₄×C₃⋊(C₃⋊D₃)	C₂³	semidirect
42. C₃×#634	C₂³	semidirect
43. C₃²×D₁₂	C₂³	semidirect
44. #6291	C₂³	semidirect
45. D₃×C₃⋊Dic₃	C₂³	semidirect
67. C₂×#634	C₂×D₃	semidirect
69. C₂×C₃×D₁₂	C₂×D₃	semidirect
74. C₂²×C₃³	#41	semidirect
90. C₂²×D₁₂	C₃⋊D₃	semidirect
122. #634	C₂²×D₃	semidirect
124. C₃×D₁₂	C₂²×D₃	semidirect
129. C₂×C₃³	C₂×#41	semidirect
153. C₂×D₁₂	C₂×C₃⋊D₃	semidirect
163. C₂²×C₃²	#237	semidirect
164. C₂²×C₃²	#239	semidirect
189. C₃³	C₂²×#41	semidirect
216. D₁₂	C₂²×C₃⋊D₃	semidirect
225. C₂×C₃²	C₂×#237	semidirect
226. C₂×C₃²	C₂×#239	semidirect
248. C₂²×C₃	#3796	semidirect
249. C₂²×C₃	#3830	semidirect
269. C₃²	C₂²×#237	semidirect
270. C₃²	C₂²×#239	semidirect
287. C₂×C₃	C₂×#3796	semidirect
288. C₂×C₃	C₂×#3830	semidirect
299. C₂²	#89015	direct (N × H)
305. C₃	C₂²×#3796	semidirect
306. C₃	C₂²×#3830	semidirect
309. C₂	C₂×#89015	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×C₄×C₃⋊(C₃⋊D₃)	2
C₂²×C₃×#634	2
C₂²×C₃²×D₁₂	2
C₂²×#6291	2
C₂²×D₃×C₃⋊Dic₃	2
C₂×#89015	2
C₂²×#3830	3
C₂²×#3796	3

Subgroups — C22×#89015 #13181590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89015 #13181590