Subgroups — C₂²×#88803 #13178027

Order 1728.

4988 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	C₂⁴×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88803	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#3752	—	3	1	no	maximal
6	C₂²×(C₂×D₃)⋊Dic₃	—	3	4	no	maximal
7	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₃⋊D₃	—	2	4	no	—
9	C₂³×C₃×C₃⋊D₃	—	2	4	no	—
10	C₂³×C₃×C₃⋊D₃	C₄	1	8	yes	complemented
11	C₂³×C₃×C₃⋊D₃	C₂²	1	6	yes	—
12	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
13	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	6	yes	complemented
14	C₂²×C₃²×Dic₃	—	2	4	no	—
15	C₂²×C₃²×Dic₃	C₂²	1	6	yes	complemented
16	#88803	C₂²	1	16	yes	direct product factor
17	C₂⁴×C₃⋊D₃	D₃	1	1	yes	characteristic
18	C₂⁴×C₃×D₃	—	3	4	no	—
19	C₂³×C₃²×C₄	—	3	1	no	—
20	C₂³×C₃⋊Dic₃	—	3	1	no	—
21	C₂³×C₃⋊Dic₃	—	3	4	no	—
22	C₂³×C₃⋊Dic₃	—	6	4	no	—
23	C₂³×C₃×Dic₃	D₃	1	4	yes	complemented
24	C₂×#3752	—	3	12	no	—
25	C₂×(C₂×D₃)⋊Dic₃	—	3	48	no	—
26	C₂³×C₃³	C₂×C₄	1	4	yes	—
27	C₂³×C₃³	D₄	1	4	yes	—
28	C₂³×C₃³	D₄	1	4	yes	—
29	C₂³×C₃³	C₂³	1	3	yes	—
30	C₂²×C₃×C₃⋊D₃	—	2	32	no	—
31	C₂²×C₃×C₃⋊D₃	—	2	12	no	—
32	C₂²×C₃×C₃⋊D₃	—	2	12	no	—
33	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	24	yes	complemented
34	C₂²×C₃×C₃⋊D₃	C₂³	1	4	yes	—
35	C₂×C₃⋊(C₃⋊Dic₃)	—	2	12	no	—
36	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	complemented
37	C₂×C₃²×Dic₃	—	2	12	no	—
38	C₂×C₃²×Dic₃	C₂³	1	4	yes	complemented
39	C₂²×#217	—	9	1	no	—
40	C₂²×(C₂×D₃)⋊C₄	—	9	4	no	—
41	C₂⁴×C₃²	—	2	4	no	—
42	C₂⁴×C₃²	C₂×D₃	1	4	yes	—
43	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
44	C₂³×C₃⋊D₃	—	2	4	no	—
45	C₂³×C₃⋊D₃	—	2	4	no	—
46	C₂³×C₃⋊D₃	Dic₃	1	8	yes	complemented
47	C₂³×C₃⋊D₃	C₂×D₃	1	6	yes	—
48	C₂³×C₃×D₃	—	3	32	no	—
49	C₂³×C₃×D₃	—	3	24	no	—
50	C₂³×C₃×D₃	—	6	16	no	—
51	C₂³×C₃×D₃	—	6	16	no	—
52	C₂²×C₃²×C₄	—	3	6	no	—
53	C₂²×C₃²×C₄	—	6	4	no	—
54	C₂²×C₃⋊Dic₃	—	3	6	no	—
55	C₂²×C₃⋊Dic₃	—	3	24	no	—
56	C₂²×C₃⋊Dic₃	—	6	24	no	—
57	C₂²×C₃⋊Dic₃	—	6	32	no	—
58	C₂²×C₃⋊Dic₃	—	6	16	no	—
59	C₂²×C₃⋊Dic₃	—	6	4	no	—
60	C₂²×C₃×Dic₃	—	2	16	no	—
61	C₂²×C₃×Dic₃	C₂×D₃	1	24	yes	complemented
62	#3752	—	3	16	no	—
63	(C₂×D₃)⋊Dic₃	—	3	64	no	—
64	C₂²×C₃³	C₂²⋊C₄	1	16	yes	complemented
65	C₂²×C₃³	C₂²×C₄	1	6	yes	—
66	C₂²×C₃³	C₂×D₄	1	6	yes	—
67	C₂²×C₃³	C₂×D₄	1	6	yes	—
68	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
69	C₂×C₃×C₃⋊D₃	—	2	48	no	—
70	C₂×C₃×C₃⋊D₃	—	2	4	no	—
71	C₂×C₃×C₃⋊D₃	—	2	4	no	—
72	C₂×C₃×C₃⋊D₃	C₂²×C₄	1	8	yes	complemented
73	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
74	C₃²×Dic₃	—	2	4	no	—
75	C₂⁵×C₃	—	9	1	no	—
76	C₂⁴×D₃	—	3	4	no	—
77	C₂³×C₃×C₄	—	3	4	no	—
78	C₂³×Dic₃	—	9	4	no	—
79	C₂³×Dic₃	—	9	1	no	—
80	C₂³×Dic₃	—	18	4	no	—
81	C₂³×Dic₃	C₃⋊D₃	1	1	yes	characteristic, complemented
82	C₂×#217	—	9	12	no	—
83	C₂×(C₂×D₃)⋊C₄	—	9	48	no	—
84	C₂³×C₃²	—	2	16	no	—
85	C₂³×C₃²	—	2	16	no	—
86	C₂³×C₃²	—	2	16	no	—
87	C₂³×C₃²	—	2	12	no	—
88	C₂³×C₃²	C₄×D₃	1	16	yes	—
89	C₂³×C₃²	D₁₂	1	16	yes	—
90	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
91	C₂³×C₃²	#67	1	16	yes	—
92	C₂³×C₃²	#67	1	4	yes	—
93	C₂³×C₃²	#67	1	4	yes	—
94	C₂³×C₃²	C₂²×D₃	1	12	yes	—
95	C₂³×C₃²	C₂²×D₃	1	3	yes	—
96	C₂²×C₃⋊D₃	—	2	32	no	—
97	C₂²×C₃⋊D₃	—	2	12	no	—
98	C₂²×C₃⋊D₃	—	2	12	no	—
99	C₂²×C₃⋊D₃	C₂×Dic₃	1	24	yes	complemented
100	C₂²×C₃⋊D₃	C₂²×D₃	1	4	yes	—
101	C₂²×C₃×D₃	—	3	96	no	—
102	C₂²×C₃×D₃	—	3	16	no	—
103	C₂²×C₃×D₃	—	6	128	no	—
104	C₂²×C₃×D₃	—	6	48	no	—
105	C₂²×C₃×D₃	—	6	48	no	—
106	C₂×C₃²×C₄	—	3	4	no	—
107	C₂×C₃²×C₄	—	6	12	no	—
108	C₂×C₃⋊Dic₃	—	3	4	no	—
109	C₂×C₃⋊Dic₃	—	3	16	no	—
110	C₂×C₃⋊Dic₃	—	6	16	no	—
111	C₂×C₃⋊Dic₃	—	6	96	no	—
112	C₂×C₃⋊Dic₃	—	6	48	no	—
113	C₂×C₃⋊Dic₃	—	6	12	no	—
114	C₂×C₃×Dic₃	—	2	48	no	—
115	C₂×C₃×Dic₃	C₂²×D₃	1	16	yes	complemented
116	C₂²×C₂²⋊C₄	—	27	1	no	Sylow
117	C₂×C₃³	C₂×C₂²⋊C₄	1	12	yes	complemented
118	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
121	C₃×C₃⋊D₃	—	2	8	no	—
122	C₂⁴×C₃	—	2	4	no	—
123	C₂⁴×C₃	—	9	8	no	—
124	C₂⁴×C₃	—	9	6	no	—
125	C₂⁴×C₃	—	18	4	no	—
126	C₂⁴×C₃	—	18	4	no	—
127	C₂⁴×C₃	D₃²	1	4	yes	—
128	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
129	C₂³×D₃	—	3	32	no	—
130	C₂³×D₃	—	3	24	no	—
131	C₂³×D₃	—	6	16	no	—
132	C₂³×D₃	—	6	16	no	—
133	C₂²×C₃×C₄	—	3	24	no	—
134	C₂²×C₃×C₄	—	6	16	no	—
135	C₂²×Dic₃	—	2	4	no	—
136	C₂²×Dic₃	—	9	24	no	—
137	C₂²×Dic₃	—	9	6	no	—
138	C₂²×Dic₃	—	18	24	no	—
139	C₂²×Dic₃	—	18	32	no	—
140	C₂²×Dic₃	—	18	16	no	—
141	C₂²×Dic₃	—	18	4	no	—
142	C₂²×Dic₃	C₂×C₃⋊D₃	1	6	yes	complemented
143	#217	—	9	16	no	—
144	(C₂×D₃)⋊C₄	—	9	64	no	—
145	C₂²×C₃²	—	2	4	no	—
146	C₂²×C₃²	—	2	64	no	—
147	C₂²×C₃²	—	2	24	no	—
148	C₂²×C₃²	—	2	24	no	—
149	C₂²×C₃²	—	2	24	no	—
150	C₂²×C₃²	(C₂×D₃)⋊C₄	1	64	yes	complemented
151	C₂²×C₃²	#217	1	16	yes	complemented
152	C₂²×C₃²	C₂×C₄×D₃	1	24	yes	—
153	C₂²×C₃²	C₂×D₁₂	1	24	yes	—
154	C₂²×C₃²	C₂²×Dic₃	1	6	yes	—
155	C₂²×C₃²	C₂×#67	1	24	yes	—
156	C₂²×C₃²	C₂×#67	1	6	yes	—
157	C₂²×C₃²	C₂×#67	1	6	yes	—
158	C₂²×C₃²	C₂³×D₃	1	4	yes	—
159	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
160	C₂×C₃⋊D₃	—	2	48	no	—
161	C₂×C₃⋊D₃	—	2	4	no	—
162	C₂×C₃⋊D₃	—	2	4	no	—
163	C₂×C₃⋊D₃	C₂²×Dic₃	1	8	yes	complemented
164	C₂×C₃×D₃	—	3	32	no	—
165	C₂×C₃×D₃	—	6	192	no	—
166	C₂×C₃×D₃	—	6	16	no	—
167	C₂×C₃×D₃	—	6	16	no	—
168	C₃²×C₄	—	6	4	no	—
169	C₃⋊Dic₃	—	6	32	no	—
170	C₃⋊Dic₃	—	6	16	no	—
171	C₃⋊Dic₃	—	6	4	no	—
172	C₃×Dic₃	—	2	16	no	—
173	C₂⁵	—	9	1	no	—
174	C₂³×C₄	—	3	1	no	—
175	C₂³×C₄	—	27	1	no	—
176	C₂×C₂²⋊C₄	—	27	12	no	—
177	C₃³	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₃	—	2	16	no	—
179	C₂³×C₃	—	2	16	no	—
180	C₂³×C₃	—	2	16	no	—
181	C₂³×C₃	—	2	12	no	—
182	C₂³×C₃	—	9	4	no	—
183	C₂³×C₃	—	9	24	no	—
184	C₂³×C₃	—	18	32	no	—
185	C₂³×C₃	—	18	12	no	—
186	C₂³×C₃	—	18	12	no	—
187	C₂³×C₃	D₃×Dic₃	1	16	yes	—
188	C₂³×C₃	#625	1	16	yes	—
189	C₂³×C₃	#626	1	16	yes	—
190	C₂³×C₃	C₄×C₃⋊D₃	1	4	yes	—
191	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	4	yes	—
192	C₂³×C₃	#638	1	4	yes	—
193	C₂³×C₃	C₂×D₃²	1	12	yes	—
194	C₂³×C₃	C₂²×C₃⋊D₃	1	3	yes	—
195	C₂²×D₃	—	3	16	no	—
196	C₂²×D₃	—	3	96	no	—
197	C₂²×D₃	—	6	128	no	—
198	C₂²×D₃	—	6	48	no	—
199	C₂²×D₃	—	6	48	no	—
200	C₂×C₃×C₄	—	3	16	no	—
201	C₂×C₃×C₄	—	6	48	no	—
202	C₂×Dic₃	—	2	12	no	—
203	C₂×Dic₃	—	9	16	no	—
204	C₂×Dic₃	—	9	4	no	—
205	C₂×Dic₃	—	18	16	no	—
206	C₂×Dic₃	—	18	96	no	—
207	C₂×Dic₃	—	18	48	no	—
208	C₂×Dic₃	—	18	12	no	—
209	C₂×Dic₃	C₂²×C₃⋊D₃	1	4	yes	complemented
210	C₂×C₃²	—	2	4	no	—
211	C₂×C₃²	—	2	4	no	—
212	C₂×C₃²	—	2	4	no	—
213	C₂×C₃²	—	2	48	no	—
214	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	48	yes	complemented
215	C₂×C₃²	C₂×#217	1	12	yes	complemented
216	C₂×C₃²	C₂²×C₄×D₃	1	4	yes	—
217	C₂×C₃²	C₂²×D₁₂	1	4	yes	—
218	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
219	C₂×C₃²	C₂²×#67	1	4	yes	—
220	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
221	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
222	C₃⋊D₃	—	2	8	no	—
223	C₃×D₃	—	6	32	no	—
224	C₂⁴	—	9	8	no	—
225	C₂⁴	—	9	6	no	—
226	C₂⁴	—	18	4	no	—
227	C₂⁴	—	18	4	no	—
228	C₂⁴	D₃×C₃⋊D₃	1	1	yes	characteristic
229	C₂²×C₄	—	3	6	no	—
230	C₂²×C₄	—	6	4	no	—
231	C₂²×C₄	—	27	6	no	—
232	C₂²×C₄	—	54	4	no	—
233	C₂²⋊C₄	—	27	16	no	—
234	C₂²×C₃	—	2	4	no	—
235	C₂²×C₃	—	2	24	no	—
236	C₂²×C₃	—	2	24	no	—
237	C₂²×C₃	—	2	24	no	—
238	C₂²×C₃	—	2	64	no	—
239	C₂²×C₃	—	9	8	no	—
240	C₂²×C₃	—	18	4	no	—
241	C₂²×C₃	—	18	4	no	—
242	C₂²×C₃	—	18	48	no	—
243	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	64	yes	complemented
244	C₂²×C₃	#3752	1	16	yes	complemented
245	C₂²×C₃	C₂×D₃×Dic₃	1	24	yes	—
246	C₂²×C₃	C₂×#625	1	24	yes	—
247	C₂²×C₃	C₂×#626	1	24	yes	—
248	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	6	yes	—
249	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	6	yes	—
250	C₂²×C₃	C₂×#638	1	6	yes	—
251	C₂²×C₃	C₂²×D₃²	1	4	yes	—
252	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
253	C₂×D₃	—	3	32	no	—
254	C₂×D₃	—	6	16	no	—
255	C₂×D₃	—	6	16	no	—
256	C₂×D₃	—	6	192	no	—
257	C₃×C₄	—	6	16	no	—
258	Dic₃	—	2	4	no	—
259	Dic₃	—	18	32	no	—
260	Dic₃	—	18	16	no	—
261	Dic₃	—	18	4	no	—
262	C₃²	—	2	4	no	—
263	C₃²	C₂²×(C₂×D₃)⋊C₄	1	4	yes	complemented
264	C₃²	C₂²×#217	1	1	yes	characteristic, complemented
265	C₂³	—	9	4	no	—
266	C₂³	—	9	24	no	—
267	C₂³	—	18	12	no	—
268	C₂³	—	18	12	no	—
269	C₂³	—	18	32	no	—
270	C₂³	Dic₃×C₃⋊D₃	1	4	yes	—
271	C₂³	#6290	1	4	yes	—
272	C₂³	#6292	1	4	yes	—
273	C₂³	C₂×D₃×C₃⋊D₃	1	3	yes	—
274	C₂×C₄	—	3	4	no	—
275	C₂×C₄	—	6	12	no	—
276	C₂×C₄	—	27	4	no	—
277	C₂×C₄	—	54	12	no	—
278	C₂×C₃	—	2	4	no	—
279	C₂×C₃	—	2	4	no	—
280	C₂×C₃	—	2	4	no	—
281	C₂×C₃	—	2	48	no	—
282	C₂×C₃	—	18	8	no	—
283	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	48	yes	complemented
284	C₂×C₃	C₂×#3752	1	12	yes	complemented
285	C₂×C₃	C₂²×D₃×Dic₃	1	4	yes	—
286	C₂×C₃	C₂²×#625	1	4	yes	—
287	C₂×C₃	C₂²×#626	1	4	yes	—
288	C₂×C₃	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
289	C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
290	C₂×C₃	C₂²×#638	1	1	yes	characteristic
291	D₃	—	6	32	no	—
292	C₂²	—	9	8	no	—
293	C₂²	—	18	4	no	—
294	C₂²	—	18	4	no	—
295	C₂²	—	18	48	no	—
296	C₂²	#88803	1	16	yes	direct product factor
297	C₂²	C₂×Dic₃×C₃⋊D₃	1	6	yes	—
298	C₂²	C₂×#6290	1	6	yes	—
299	C₂²	C₂×#6292	1	6	yes	—
300	C₂²	C₂²×D₃×C₃⋊D₃	1	1	yes	characteristic
301	C₄	—	6	4	no	—
302	C₄	—	54	4	no	—
303	C₃	—	2	4	no	—
304	C₃	C₂²×(C₂×D₃)⋊Dic₃	1	4	yes	complemented
305	C₃	C₂²×#3752	1	1	yes	characteristic, complemented
306	C₂	—	18	8	no	—
307	C₂	C₂×#88803	1	12	yes	direct product factor
308	C₂	C₂²×Dic₃×C₃⋊D₃	1	1	yes	characteristic
309	C₂	C₂²×#6290	1	1	yes	characteristic
310	C₂	C₂²×#6292	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
3. C₂³×C₃²×Dic₃	C₂	semidirect
4. C₂×#88803	C₂	direct (N × H)
10. C₂³×C₃×C₃⋊D₃	C₄	semidirect
13. C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
15. C₂²×C₃²×Dic₃	C₂²	semidirect
16. #88803	C₂²	direct (N × H)
23. C₂³×C₃×Dic₃	D₃	semidirect
33. C₂²×C₃×C₃⋊D₃	C₂×C₄	semidirect
36. C₂×C₃⋊(C₃⋊Dic₃)	C₂³	semidirect
38. C₂×C₃²×Dic₃	C₂³	semidirect
46. C₂³×C₃⋊D₃	Dic₃	semidirect
61. C₂²×C₃×Dic₃	C₂×D₃	semidirect
64. C₂²×C₃³	C₂²⋊C₄	semidirect
72. C₂×C₃×C₃⋊D₃	C₂²×C₄	semidirect
81. C₂³×Dic₃	C₃⋊D₃	semidirect
99. C₂²×C₃⋊D₃	C₂×Dic₃	semidirect
115. C₂×C₃×Dic₃	C₂²×D₃	semidirect
117. C₂×C₃³	C₂×C₂²⋊C₄	semidirect
142. C₂²×Dic₃	C₂×C₃⋊D₃	semidirect
150. C₂²×C₃²	(C₂×D₃)⋊C₄	semidirect
151. C₂²×C₃²	#217	semidirect
163. C₂×C₃⋊D₃	C₂²×Dic₃	semidirect
177. C₃³	C₂²×C₂²⋊C₄	semidirect
209. C₂×Dic₃	C₂²×C₃⋊D₃	semidirect
214. C₂×C₃²	C₂×(C₂×D₃)⋊C₄	semidirect
215. C₂×C₃²	C₂×#217	semidirect
243. C₂²×C₃	(C₂×D₃)⋊Dic₃	semidirect
244. C₂²×C₃	#3752	semidirect
263. C₃²	C₂²×(C₂×D₃)⋊C₄	semidirect
264. C₃²	C₂²×#217	semidirect
283. C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	semidirect
284. C₂×C₃	C₂×#3752	semidirect
296. C₂²	#88803	direct (N × H)
304. C₃	C₂²×(C₂×D₃)⋊Dic₃	semidirect
305. C₃	C₂²×#3752	semidirect
307. C₂	C₂×#88803	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₃⋊D₃	2
C₂³×C₃⋊(C₃⋊Dic₃)	2
C₂³×C₃²×Dic₃	2
C₂×#88803	2
C₂²×#3752	3
C₂²×(C₂×D₃)⋊Dic₃	3

Subgroups — C22×#88803 #13178027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88803 #13178027