Subgroups — C₂×C₃×#2213 #10652199

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#509	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2213	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#361	C₂	1	2	yes	maximal, complemented
7	C₂×#2213	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₃×D₈	—	2	1	no	—
10	C₂²×C₃×D₈	C₂²	1	2	yes	—
11	C₂×C₃×#131	—	2	1	no	—
12	C₂×C₃×#127	—	2	1	no	—
13	C₃×#509	—	2	4	no	—
14	C₃×#509	C₂²	1	8	yes	complemented
15	C₂×C₃×SD₃₂	—	2	2	no	—
16	C₂×C₃×D₁₆	—	2	2	no	—
17	C₂×C₃×M₃₂	C₂²	1	2	yes	—
18	C₃×#493	—	2	2	no	—
19	C₃×#493	C₂²	1	4	yes	complemented
20	C₃×#443	—	2	2	no	—
21	C₃×#443	C₂²	1	4	yes	complemented
22	C₂×C₃×#108	C₂²	1	1	yes	characteristic
23	C₂×C₃×#104	—	2	1	no	—
24	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₃×#364	C₂²	1	4	yes	complemented
26	C₃×#361	C₂²	1	8	yes	complemented
27	C₂×#509	C₂×C₃	1	2	yes	complemented
28	C₂×#493	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#443	C₂×C₃	1	1	yes	characteristic, complemented
30	#2213	C₂×C₃	1	8	yes	direct product factor
31	C₂×#364	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#361	C₂×C₃	1	2	yes	complemented
33	C₂×C₃×#41	—	2	1	no	—
34	C₂²×C₃×D₄	—	2	2	no	—
35	C₂²×C₃×D₄	—	4	2	no	—
36	C₃×#135	—	2	1	no	—
37	C₃×#135	—	2	1	no	—
38	C₃×#135	D₄	1	2	yes	complemented
39	C₃×#135	C₂³	1	2	yes	—
40	C₂×C₃×Q₁₆	D₄	1	1	yes	characteristic, complemented
41	C₂×C₃×SD₁₆	—	2	1	no	—
42	C₂×C₃×D₈	—	2	2	no	—
43	C₂×C₃×D₈	—	2	2	no	—
44	C₂×C₃×D₈	—	2	4	no	—
45	C₂×C₃×D₈	—	2	2	no	—
46	C₂×C₃×D₈	—	2	2	no	—
47	C₂×C₃×D₈	—	2	2	no	—
48	C₂×C₃×D₈	—	4	1	no	—
49	C₂×C₃×D₈	—	4	1	no	—
50	C₂×C₃×D₈	—	4	2	no	—
51	C₂×C₃×D₈	D₄	1	1	yes	characteristic, complemented
52	C₂×C₃×D₈	C₂³	1	4	yes	—
53	C₃×#131	—	2	4	no	—
54	C₃×#131	—	4	2	no	—
55	C₂×C₃×M₁₆	—	2	1	no	—
56	C₂×C₃×M₁₆	—	4	1	no	—
57	C₂²×C₃×C₈	—	4	1	no	—
58	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
60	C₃×#127	—	2	4	no	—
61	C₃×#127	—	4	2	no	—
62	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
63	C₃×SD₃₂	—	2	8	no	—
64	C₃×D₁₆	—	2	8	no	—
65	C₃×M₃₂	—	2	2	no	—
66	C₃×M₃₂	C₂³	1	4	yes	—
67	C₂×C₃×C₁₆	—	2	2	no	—
68	C₃×#108	—	2	1	no	—
69	C₃×#108	C₂³	1	2	yes	—
70	C₃×#104	—	2	4	no	—
71	C₃×C₄×C₈	D₄	1	2	yes	—
72	C₃×C₄×C₈	C₂³	1	2	yes	—
73	C₂×#135	C₂²×C₃	1	1	yes	characteristic
74	C₂²×D₈	—	2	1	no	—
75	C₂²×D₈	C₂²×C₃	1	2	yes	—
76	C₂×#131	—	2	1	no	—
77	C₂×#127	—	2	1	no	—
78	#509	—	2	4	no	—
79	#509	C₂²×C₃	1	8	yes	complemented
80	C₂×SD₃₂	—	2	2	no	—
81	C₂×D₁₆	—	2	2	no	—
82	C₂×M₃₂	C₂²×C₃	1	2	yes	—
83	#493	—	2	2	no	—
84	#493	C₂²×C₃	1	4	yes	complemented
85	#443	—	2	2	no	—
86	#443	C₂²×C₃	1	4	yes	complemented
87	C₂×#108	C₂²×C₃	1	1	yes	characteristic
88	C₂×#104	—	2	1	no	—
89	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
90	#364	C₂²×C₃	1	4	yes	complemented
91	#361	C₂²×C₃	1	8	yes	complemented
92	C₂⁴×C₃	—	4	2	no	—
93	C₃×#41	—	2	2	no	—
94	C₃×#41	—	2	2	no	—
95	C₃×#41	—	4	1	no	—
96	C₃×#41	—	4	1	no	—
97	C₂×C₃×Q₈	—	2	1	no	—
98	C₂×C₃×D₄	—	2	1	no	—
99	C₂×C₃×D₄	—	2	4	no	—
100	C₂×C₃×D₄	—	4	8	no	—
101	C₂×C₃×D₄	—	4	1	no	—
102	C₂×C₃×D₄	—	4	2	no	—
103	C₂×C₃×D₄	—	4	2	no	—
104	C₂×C₃×D₄	—	4	2	no	—
105	C₂×C₃×D₄	—	4	2	no	—
106	C₂×C₃×D₄	—	4	2	no	—
107	C₂×C₃×D₄	—	8	8	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	4	1	no	—
110	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
111	C₃×Q₁₆	—	2	1	no	—
112	C₃×Q₁₆	C₂×D₄	1	2	yes	complemented
113	C₃×SD₁₆	—	2	2	no	—
114	C₃×SD₁₆	—	2	2	no	—
115	C₃×D₈	—	2	1	no	—
116	C₃×D₈	—	2	4	no	—
117	C₃×D₈	—	2	4	no	—
118	C₃×D₈	—	4	2	no	—
119	C₃×D₈	—	4	2	no	—
120	C₃×D₈	—	4	4	no	—
121	C₃×D₈	—	4	4	no	—
122	C₃×D₈	C₂×D₄	1	2	yes	complemented
123	C₃×M₁₆	—	2	2	no	—
124	C₃×M₁₆	—	4	1	no	—
125	C₃×M₁₆	—	4	4	no	—
126	C₂×C₃×C₈	—	4	1	no	—
127	C₂×C₃×C₈	—	4	4	no	—
128	C₂×C₃×C₈	D₈	1	1	yes	characteristic
129	C₂×C₃×C₈	D₈	1	1	yes	characteristic
130	C₂×C₃×C₈	D₈	1	2	yes	—
131	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₈	#41	1	1	yes	characteristic
136	C₂×C₃×C₈	#41	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
138	C₃×C₄²	—	2	1	no	—
139	C₃×C₄²	C₂×D₄	1	2	yes	—
140	C₃×C₁₆	—	2	4	no	—
141	C₂×#41	—	2	1	no	—
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	—	4	2	no	—
144	#135	—	2	1	no	—
145	#135	—	2	1	no	—
146	#135	C₃×D₄	1	2	yes	complemented
147	#135	C₂³×C₃	1	2	yes	—
148	C₂×Q₁₆	C₃×D₄	1	1	yes	characteristic, complemented
149	C₂×SD₁₆	—	2	1	no	—
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	—	2	4	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	—	4	1	no	—
157	C₂×D₈	—	4	1	no	—
158	C₂×D₈	—	4	2	no	—
159	C₂×D₈	C₃×D₄	1	1	yes	characteristic, complemented
160	C₂×D₈	C₂³×C₃	1	4	yes	—
161	#131	—	2	4	no	—
162	#131	—	4	2	no	—
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	4	1	no	—
165	C₂²×C₈	—	4	1	no	—
166	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
168	#127	—	2	4	no	—
169	#127	—	4	2	no	—
170	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
171	SD₃₂	—	2	8	no	—
172	D₁₆	—	2	8	no	—
173	M₃₂	—	2	2	no	—
174	M₃₂	C₂³×C₃	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	#108	—	2	1	no	—
177	#108	C₂³×C₃	1	2	yes	—
178	#104	—	2	4	no	—
179	C₄×C₈	C₃×D₄	1	2	yes	—
180	C₄×C₈	C₂³×C₃	1	2	yes	—
181	C₂³×C₃	—	4	1	no	—
182	C₂³×C₃	—	4	4	no	—
183	C₂³×C₃	—	8	8	no	—
184	C₂³×C₃	—	8	2	no	—
185	C₂³×C₃	—	8	2	no	—
186	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₃×Q₈	—	2	2	no	—
188	C₃×Q₈	—	4	1	no	—
189	C₃×D₄	—	2	2	no	—
190	C₃×D₄	—	4	2	no	—
191	C₃×D₄	—	4	2	no	—
192	C₃×D₄	—	4	4	no	—
193	C₃×D₄	—	4	4	no	—
194	C₃×D₄	—	4	1	no	—
195	C₃×D₄	—	8	16	no	—
196	C₃×D₄	—	8	2	no	—
197	C₃×D₄	—	8	2	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	4	1	no	—
201	C₂×C₃×C₄	—	4	2	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	#121	1	1	yes	characteristic
204	C₂×C₃×C₄	#121	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
207	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
209	C₃×C₈	—	4	2	no	—
210	C₃×C₈	#121	1	1	yes	characteristic
211	C₃×C₈	#121	1	1	yes	characteristic
212	C₃×C₈	C₂×D₈	1	2	yes	—
213	C₃×C₈	C₂×D₈	1	2	yes	—
214	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
215	C₃×C₈	C₂×#41	1	1	yes	characteristic
216	C₂⁴	—	4	2	no	—
217	#41	—	2	2	no	—
218	#41	—	2	2	no	—
219	#41	—	4	1	no	—
220	#41	—	4	1	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	4	8	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	8	8	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	Q₁₆	—	2	1	no	—
236	Q₁₆	C₂×C₃×D₄	1	2	yes	complemented
237	SD₁₆	—	2	2	no	—
238	SD₁₆	—	2	2	no	—
239	D₈	—	2	1	no	—
240	D₈	—	2	4	no	—
241	D₈	—	2	4	no	—
242	D₈	—	4	2	no	—
243	D₈	—	4	2	no	—
244	D₈	—	4	4	no	—
245	D₈	—	4	4	no	—
246	D₈	C₂×C₃×D₄	1	2	yes	complemented
247	M₁₆	—	2	2	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	4	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	4	no	—
252	C₂×C₈	C₃×D₈	1	1	yes	characteristic
253	C₂×C₈	C₃×D₈	1	1	yes	characteristic
254	C₂×C₈	C₃×D₈	1	2	yes	—
255	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
257	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
258	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
259	C₂×C₈	C₃×#41	1	1	yes	characteristic
260	C₂×C₈	C₃×#41	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
262	C₄²	—	2	1	no	—
263	C₄²	C₂×C₃×D₄	1	2	yes	—
264	C₁₆	—	2	4	no	—
265	C₂²×C₃	—	2	2	no	—
266	C₂²×C₃	—	4	2	no	—
267	C₂²×C₃	—	8	8	no	—
268	C₂²×C₃	—	8	2	no	—
269	C₂²×C₃	—	8	4	no	—
270	C₂²×C₃	—	16	4	no	—
271	C₂²×C₃	—	16	4	no	—
272	C₂²×C₃	#459	1	1	yes	characteristic
273	C₂²×C₃	#459	1	1	yes	characteristic
274	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	4	2	no	—
277	C₃×C₄	#459	1	1	yes	characteristic
278	C₃×C₄	#459	1	1	yes	characteristic
279	C₃×C₄	C₂×#121	1	1	yes	characteristic
280	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
281	C₂³	—	4	1	no	—
282	C₂³	—	4	4	no	—
283	C₂³	—	8	8	no	—
284	C₂³	—	8	2	no	—
285	C₂³	—	8	2	no	—
286	C₂³	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	4	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	16	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	C₃×#121	1	1	yes	characteristic
304	C₂×C₄	C₃×#121	1	1	yes	characteristic
305	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
306	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
307	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	C₃×#121	1	1	yes	characteristic
311	C₈	C₃×#121	1	1	yes	characteristic
312	C₈	C₂×C₃×D₈	1	2	yes	—
313	C₈	C₂×C₃×D₈	1	2	yes	—
314	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
315	C₈	C₂×C₃×#41	1	1	yes	characteristic
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	8	2	no	—
319	C₂×C₃	—	16	4	no	—
320	C₂×C₃	#2213	1	2	yes	direct product factor
321	C₂×C₃	C₂×#459	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	8	8	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	4	no	—
327	C₂²	—	16	4	no	—
328	C₂²	—	16	4	no	—
329	C₂²	C₃×#459	1	1	yes	characteristic
330	C₂²	C₃×#459	1	1	yes	characteristic
331	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
332	C₄	—	2	4	no	—
333	C₄	—	4	2	no	—
334	C₄	C₃×#459	1	1	yes	characteristic
335	C₄	C₃×#459	1	1	yes	characteristic
336	C₄	C₂×C₃×#121	1	1	yes	characteristic
337	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
338	C₃	C₂×#2213	1	1	yes	characteristic, direct product factor, Sylow
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	16	4	no	—
343	C₂	C₃×#2213	1	2	yes	direct product factor
344	C₂	C₂×C₃×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#509	C₂	semidirect
2. C₂×C₃×#493	C₂	semidirect
3. C₂×C₃×#443	C₂	semidirect
4. C₃×#2213	C₂	direct (N × H)
5. C₂×C₃×#364	C₂	semidirect
6. C₂×C₃×#361	C₂	semidirect
7. C₂×#2213	C₃	direct (N × H)
14. C₃×#509	C₂²	semidirect
19. C₃×#493	C₂²	semidirect
21. C₃×#443	C₂²	semidirect
25. C₃×#364	C₂²	semidirect
26. C₃×#361	C₂²	semidirect
27. C₂×#509	C₂×C₃	semidirect
28. C₂×#493	C₂×C₃	semidirect
29. C₂×#443	C₂×C₃	semidirect
30. #2213	C₂×C₃	direct (N × H)
31. C₂×#364	C₂×C₃	semidirect
32. C₂×#361	C₂×C₃	semidirect
38. C₃×#135	D₄	semidirect
40. C₂×C₃×Q₁₆	D₄	semidirect
51. C₂×C₃×D₈	D₄	semidirect
79. #509	C₂²×C₃	semidirect
84. #493	C₂²×C₃	semidirect
86. #443	C₂²×C₃	semidirect
90. #364	C₂²×C₃	semidirect
91. #361	C₂²×C₃	semidirect
112. C₃×Q₁₆	C₂×D₄	semidirect
122. C₃×D₈	C₂×D₄	semidirect
146. #135	C₃×D₄	semidirect
148. C₂×Q₁₆	C₃×D₄	semidirect
159. C₂×D₈	C₃×D₄	semidirect
236. Q₁₆	C₂×C₃×D₄	semidirect
246. D₈	C₂×C₃×D₄	semidirect
320. C₂×C₃	#2213	direct (N × H)
338. C₃	C₂×#2213	direct (N × H)
343. C₂	C₃×#2213	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×C₃×#509	2
C₂×C₃×#493	2
C₂×C₃×#443	2
C₃×#2213	2
C₂×C₃×#364	2
C₂×C₃×#361	2
C₂×#2213	3

Subgroups — C2×C3×#2213 #10652199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2213 #10652199