Subgroups — C₂×C₃×#4562 #11796589

Order 1152.

713 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4562	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×#357	—	3	1	no	maximal
6	C₂×#4562	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×D₄	—	2	1	no	—
8	C₂×C₃²×D₈	—	2	1	no	—
9	C₂×C₃²×D₈	—	2	2	no	—
10	C₂×C₃²×D₈	C₄	1	4	yes	complemented
11	C₂×C₃²×D₈	C₂²	1	2	yes	—
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
14	C₃×#818	—	2	1	no	—
15	C₃×#818	C₂²	1	2	yes	complemented
16	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	complemented
18	C₂²×C₃×D₈	—	2	1	no	—
19	C₂²×C₃×D₈	D₃	1	1	yes	characteristic
20	C₂²×C₃×D₈	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₃×C₁₆	—	3	1	no	—
22	C₂×C₃×#107	—	3	1	no	—
23	C₂×#818	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×#357	—	3	4	no	—
26	#4562	C₂×C₃	1	4	yes	direct product factor
27	C₂⁴×C₃²	—	4	1	no	—
28	C₂×C₃²×D₄	—	2	4	no	—
29	C₂×C₃²×D₄	—	2	2	no	—
30	C₂×C₃²×D₄	—	4	1	no	—
31	C₂×C₃²×D₄	—	4	2	no	—
32	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	—	4	1	no	—
34	C₃²×D₈	—	2	4	no	—
35	C₃²×D₈	—	2	2	no	—
36	C₃²×D₈	C₂×C₄	1	4	yes	complemented
37	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	D₄	1	2	yes	—
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	4	1	no	—
43	C₃×C₃⋊C₁₆	—	2	2	no	—
44	C₂×#357	—	3	1	no	Sylow
45	C₂²×C₃×D₄	—	2	2	no	—
46	C₂²×C₃×D₄	—	2	1	no	—
47	C₂²×C₃×D₄	—	2	1	no	—
48	C₂×C₃×D₈	—	2	2	no	—
49	C₂×C₃×D₈	—	2	4	no	—
50	C₂×C₃×D₈	—	2	1	no	—
51	C₂×C₃×D₈	—	2	2	no	—
52	C₂×C₃×D₈	—	2	1	no	—
53	C₂×C₃×D₈	—	2	2	no	—
54	C₂×C₃×D₈	—	2	4	no	—
55	C₂×C₃×D₈	—	2	2	no	—
56	C₂×C₃×D₈	Dic₃	1	4	yes	complemented
57	C₂×C₃×D₈	C₃×C₄	1	4	yes	complemented
58	C₂×C₃×D₈	C₂×D₃	1	2	yes	—
59	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
60	C₂²×C₃×C₈	—	2	1	no	—
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×C₄⋊C₄	—	6	1	no	—
64	C₂×C₄⋊Dic₃	—	2	1	no	—
65	C₂×C₃×C₁₆	—	3	2	no	—
66	C₂×C₃×C₁₆	—	6	2	no	—
67	C₃×#107	—	3	2	no	—
68	C₃×#107	—	6	1	no	—
69	#818	—	2	1	no	—
70	#818	C₂²×C₃	1	2	yes	complemented
71	C₂×C₃⋊C₁₆	—	2	2	no	—
72	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	complemented
73	C₂³×C₃²	—	4	4	no	—
74	C₂³×C₃²	—	4	2	no	—
75	C₂³×C₃²	—	8	2	no	—
76	C₂³×C₃²	—	8	2	no	—
77	C₂³×C₃²	D₈	1	1	yes	characteristic
78	C₃²×D₄	—	2	4	no	—
79	C₃²×D₄	—	4	4	no	—
80	C₃²×D₄	—	4	2	no	—
81	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
82	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×Dic₃	—	4	2	no	—
86	C₂×C₃×Dic₃	—	8	2	no	—
87	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
88	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₈	C₃×D₃	1	1	yes	characteristic
91	C₂²×C₁₆	—	3	1	no	—
92	C₂×#107	—	3	1	no	—
93	#357	—	3	4	no	—
94	C₂⁴×C₃	—	4	2	no	—
95	C₂⁴×C₃	—	4	1	no	—
96	C₂⁴×C₃	—	4	1	no	—
97	C₂×C₃×D₄	—	2	8	no	—
98	C₂×C₃×D₄	—	2	4	no	—
99	C₂×C₃×D₄	—	2	4	no	—
100	C₂×C₃×D₄	—	2	4	no	—
101	C₂×C₃×D₄	—	2	2	no	—
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	—	4	2	no	—
104	C₂×C₃×D₄	—	4	4	no	—
105	C₂×C₃×D₄	—	4	1	no	—
106	C₂×C₃×D₄	—	4	2	no	—
107	C₂×C₃×D₄	—	4	1	no	—
108	C₂×C₃×D₄	—	4	2	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	12	1	no	—
111	C₂²×C₃×C₄	#67	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
113	C₂²×Dic₃	—	4	1	no	—
114	C₃×D₈	—	2	8	no	—
115	C₃×D₈	—	2	4	no	—
116	C₃×D₈	—	2	4	no	—
117	C₃×D₈	—	2	4	no	—
118	C₃×D₈	—	2	2	no	—
119	C₃×D₈	—	2	2	no	—
120	C₃×D₈	—	2	4	no	—
121	C₃×D₈	C₂×Dic₃	1	4	yes	complemented
122	C₃×D₈	C₂×C₃×C₄	1	4	yes	complemented
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
128	C₂×C₃×C₈	#67	1	1	yes	characteristic
129	C₂×C₃×C₈	#67	1	2	yes	—
130	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
131	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
132	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
133	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
135	C₃×C₄⋊C₄	—	6	2	no	—
136	C₃×C₄⋊C₄	—	12	1	no	—
137	C₄⋊Dic₃	—	2	2	no	—
138	C₄⋊Dic₃	—	4	1	no	—
139	C₃×C₁₆	—	6	2	no	—
140	C₃⋊C₁₆	—	2	2	no	—
141	C₂²×C₃²	—	4	4	no	—
142	C₂²×C₃²	—	8	8	no	—
143	C₂²×C₃²	—	8	2	no	—
144	C₂²×C₃²	—	8	2	no	—
145	C₂²×C₃²	#102	1	2	yes	—
146	C₂²×C₃²	D₁₆	1	2	yes	—
147	C₂²×C₃²	SD₃₂	1	2	yes	—
148	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
149	C₃²×C₄	#102	1	2	yes	—
150	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
152	C₃×Dic₃	—	8	2	no	—
153	C₂²×D₄	—	2	1	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	C₃×Dic₃	1	4	yes	complemented
157	C₂×D₈	C₂×C₃×D₃	1	2	yes	—
158	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
159	C₂×C₄⋊C₄	—	6	1	no	—
160	C₂×C₁₆	—	3	2	no	—
161	C₂×C₁₆	—	6	2	no	—
162	#107	—	3	2	no	—
163	#107	—	6	1	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	—	4	4	no	—
166	C₂³×C₃	—	4	8	no	—
167	C₂³×C₃	—	4	4	no	—
168	C₂³×C₃	—	4	4	no	—
169	C₂³×C₃	—	4	2	no	—
170	C₂³×C₃	—	4	2	no	—
171	C₂³×C₃	—	8	4	no	—
172	C₂³×C₃	—	8	4	no	—
173	C₂³×C₃	—	8	2	no	—
174	C₂³×C₃	—	8	2	no	—
175	C₂³×C₃	—	8	2	no	—
176	C₂³×C₃	—	8	2	no	—
177	C₂³×C₃	#213	1	1	yes	characteristic
178	C₂³×C₃	C₃×D₈	1	1	yes	characteristic
179	C₃×D₄	—	2	8	no	—
180	C₃×D₄	—	2	4	no	—
181	C₃×D₄	—	2	4	no	—
182	C₃×D₄	—	4	4	no	—
183	C₃×D₄	—	4	8	no	—
184	C₃×D₄	—	4	4	no	—
185	C₃×D₄	—	4	4	no	—
186	C₃×D₄	—	4	2	no	—
187	C₃×D₄	—	4	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	12	2	no	—
193	C₂×C₃×C₄	—	24	2	no	—
194	C₂×C₃×C₄	#213	1	1	yes	characteristic
195	C₂×C₃×C₄	#214	1	2	yes	—
196	C₂×C₃×C₄	#217	1	2	yes	—
197	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
198	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
199	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
200	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×Dic₃	—	4	2	no	—
203	C₂×Dic₃	—	8	2	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	#217	1	2	yes	—
208	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
209	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
210	C₃×C₈	C₂×#67	1	1	yes	characteristic
211	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
212	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃²	—	8	4	no	—
214	C₂×C₃²	#357	1	4	yes	complemented
215	C₂×C₃²	C₂×#102	1	1	yes	characteristic
216	C₂×C₃²	C₂×D₁₆	1	1	yes	characteristic
217	C₂×C₃²	C₂×SD₃₂	1	1	yes	characteristic
218	C₂⁴	—	4	1	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂²×C₄	—	12	1	no	—
224	C₂²×C₄	C₃×#67	1	1	yes	characteristic
225	D₈	—	2	4	no	—
226	D₈	—	2	2	no	—
227	D₈	C₂×C₃×Dic₃	1	4	yes	complemented
228	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
229	C₂×C₈	C₃×#67	1	1	yes	characteristic
230	C₂×C₈	C₃×#67	1	2	yes	—
231	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
232	C₄⋊C₄	—	6	2	no	—
233	C₄⋊C₄	—	12	1	no	—
234	C₁₆	—	6	2	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	4	8	no	—
240	C₂²×C₃	—	4	4	no	—
241	C₂²×C₃	—	4	4	no	—
242	C₂²×C₃	—	8	16	no	—
243	C₂²×C₃	—	8	4	no	—
244	C₂²×C₃	—	8	4	no	—
245	C₂²×C₃	—	8	2	no	—
246	C₂²×C₃	—	8	2	no	—
247	C₂²×C₃	—	8	2	no	—
248	C₂²×C₃	—	8	2	no	—
249	C₂²×C₃	—	8	8	no	—
250	C₂²×C₃	—	8	8	no	—
251	C₂²×C₃	#826	1	2	yes	—
252	C₂²×C₃	#827	1	2	yes	—
253	C₂²×C₃	#832	1	2	yes	—
254	C₂²×C₃	C₃×#102	1	2	yes	—
255	C₂²×C₃	C₃×D₁₆	1	2	yes	—
256	C₂²×C₃	C₃×SD₃₂	1	2	yes	—
257	C₂²×C₃	C₂×#213	1	1	yes	characteristic
258	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	24	2	no	—
263	C₃×C₄	#832	1	2	yes	—
264	C₃×C₄	C₃×#102	1	2	yes	—
265	C₃×C₄	C₂×#214	1	1	yes	characteristic
266	C₃×C₄	C₂×#217	1	1	yes	characteristic
267	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
268	C₃×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
269	Dic₃	—	8	2	no	—
270	C₃²	C₂×#357	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	4	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	8	2	no	—
274	C₂³	—	8	2	no	—
275	C₂³	C₃×#213	1	1	yes	characteristic
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	4	no	—
279	C₂×C₄	—	12	2	no	—
280	C₂×C₄	—	24	2	no	—
281	C₂×C₄	C₃×#213	1	1	yes	characteristic
282	C₂×C₄	C₃×#214	1	2	yes	—
283	C₂×C₄	C₃×#217	1	2	yes	—
284	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
285	C₈	C₃×#217	1	2	yes	—
286	C₈	C₂²×C₃×Dic₃	1	1	yes	characteristic
287	C₈	C₂×C₃×#67	1	1	yes	characteristic
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	4	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	8	8	no	—
293	C₂×C₃	—	8	4	no	—
294	C₂×C₃	—	8	4	no	—
295	C₂×C₃	#4562	1	4	yes	direct product factor
296	C₂×C₃	C₃×#357	1	4	yes	complemented
297	C₂×C₃	C₂×#826	1	1	yes	characteristic
298	C₂×C₃	C₂×#827	1	1	yes	characteristic
299	C₂×C₃	C₂×#832	1	1	yes	characteristic
300	C₂×C₃	C₂×C₃×#102	1	1	yes	characteristic
301	C₂×C₃	C₂×C₃×D₁₆	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×SD₃₂	1	1	yes	characteristic
303	C₂²	—	4	4	no	—
304	C₂²	—	8	8	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	C₃×#826	1	2	yes	—
308	C₂²	C₃×#827	1	2	yes	—
309	C₂²	C₃×#832	1	2	yes	—
310	C₂²	C₂×C₃×#213	1	1	yes	characteristic
311	C₄	—	24	2	no	—
312	C₄	C₃×#832	1	2	yes	—
313	C₄	C₂×C₃×#214	1	1	yes	characteristic
314	C₄	C₂×C₃×#217	1	1	yes	characteristic
315	C₃	—	2	1	no	—
316	C₃	C₂×#4562	1	1	yes	characteristic, direct product factor
317	C₃	C₂×C₃×#357	1	1	yes	characteristic, complemented
318	C₂	—	8	4	no	—
319	C₂	C₃×#4562	1	4	yes	direct product factor
320	C₂	C₂×C₃×#826	1	1	yes	characteristic
321	C₂	C₂×C₃×#827	1	1	yes	characteristic
322	C₂	C₂×C₃×#832	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	#213
commutator	C₃×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting, prime radical	C₂²×C₃²×D₈	C₂
socle	C₂³×C₃²	D₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₂²×C₃×D₈	D₃
hypocenter	C₃	C₂×C₃×#357
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂²×C₃×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#818	C₂	semidirect
3. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
4. C₃×#4562	C₂	direct (N × H)
6. C₂×#4562	C₃	direct (N × H)
10. C₂×C₃²×D₈	C₄	semidirect
15. C₃×#818	C₂²	semidirect
17. C₂×C₃×C₃⋊C₁₆	C₂²	semidirect
23. C₂×#818	C₂×C₃	semidirect
24. C₂²×C₃⋊C₁₆	C₂×C₃	semidirect
26. #4562	C₂×C₃	direct (N × H)
36. C₃²×D₈	C₂×C₄	semidirect
56. C₂×C₃×D₈	Dic₃	semidirect
57. C₂×C₃×D₈	C₃×C₄	semidirect
70. #818	C₂²×C₃	semidirect
72. C₂×C₃⋊C₁₆	C₂²×C₃	semidirect
121. C₃×D₈	C₂×Dic₃	semidirect
122. C₃×D₈	C₂×C₃×C₄	semidirect
156. C₂×D₈	C₃×Dic₃	semidirect
214. C₂×C₃²	#357	semidirect
227. D₈	C₂×C₃×Dic₃	semidirect
270. C₃²	C₂×#357	semidirect
295. C₂×C₃	#4562	direct (N × H)
296. C₂×C₃	C₃×#357	semidirect
316. C₃	C₂×#4562	direct (N × H)
317. C₃	C₂×C₃×#357	semidirect
319. C₂	C₃×#4562	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×C₃²×D₈	2
C₂×C₃×#818	2
C₂²×C₃×C₃⋊C₁₆	2
C₃×#4562	2
C₂×C₃×#357	3
C₂×#4562	3

Subgroups — C2×C3×#4562 #11796589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4562 #11796589