Subgroups — C₃×#33581 #10652188

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#510	C₂	1	2	yes	maximal, complemented
2	C₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2211	C₂	1	4	yes	maximal, complemented
5	C₃×#2209	C₂	1	2	yes	maximal, complemented
6	C₃×#2207	C₂	1	2	yes	maximal, complemented
7	C₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2139	C₂	1	2	yes	maximal, complemented
9	#33581	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#135	—	2	1	no	—
11	C₂×C₃×#135	C₂²	1	2	yes	—
12	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₃×#532	—	2	1	no	—
14	C₂×C₃×C₄×Q₈	—	2	1	no	—
15	C₃×#510	—	2	8	no	—
16	C₂×C₃×Q₃₂	C₂²	1	4	yes	—
17	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
18	C₂×C₃×M₃₂	C₂²	1	2	yes	—
19	C₃×#495	—	2	1	no	—
20	C₃×#495	C₂²	1	2	yes	—
21	C₃×#494	C₂²	1	1	yes	characteristic
22	C₃×#493	C₂²	1	1	yes	characteristic
23	C₃×#492	—	2	1	no	—
24	C₃×C₄×Q₁₆	—	2	1	no	—
25	C₃×C₄×Q₁₆	—	2	1	no	—
26	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
27	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
28	C₂×C₃×#107	C₂²	1	1	yes	characteristic
29	C₂×C₃×#103	—	2	1	no	—
30	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₃×#358	C₂²	1	4	yes	—
34	C₃×#357	C₂²	1	4	yes	—
35	C₂×#510	C₂×C₃	1	2	yes	complemented
36	#3147	C₂×C₃	1	1	yes	characteristic
37	C₂×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
38	#2211	C₂×C₃	1	4	yes	complemented
39	#2209	C₂×C₃	1	2	yes	complemented
40	#2207	C₂×C₃	1	2	yes	complemented
41	#2150	C₂×C₃	1	1	yes	characteristic, complemented
42	#2139	C₂×C₃	1	2	yes	complemented
43	C₂×C₃×#41	—	2	2	no	—
44	C₂²×C₃×Q₈	—	2	1	no	—
45	C₃×#135	—	2	4	no	—
46	C₃×#135	—	2	4	no	—
47	C₃×#135	—	4	4	no	—
48	C₂×C₃×Q₁₆	—	2	1	no	—
49	C₂×C₃×Q₁₆	—	2	1	no	—
50	C₂×C₃×Q₁₆	—	2	2	no	—
51	C₂×C₃×Q₁₆	D₄	1	4	yes	complemented
52	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
53	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
54	C₂×C₃×SD₁₆	—	2	2	no	—
55	C₂×C₃×SD₁₆	—	2	2	no	—
56	C₂×C₃×D₈	—	2	1	no	—
57	C₂×C₃×D₈	C₂³	1	2	yes	—
58	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
60	C₃×C₄⋊Q₈	—	2	1	no	—
61	C₃×#127	—	2	1	no	—
62	C₃×#124	—	2	2	no	—
63	C₃×#121	—	4	1	no	—
64	C₃×#121	—	4	1	no	—
65	C₃×C₄×Q₈	—	2	2	no	—
66	C₃×C₄×Q₈	—	2	2	no	—
67	C₃×C₄×Q₈	—	4	1	no	—
68	C₃×C₄×Q₈	—	4	1	no	—
69	C₃×C₄×D₄	—	4	1	no	—
70	C₃×C₄×D₄	—	4	1	no	—
71	C₂×C₃×C₄⋊C₄	—	2	1	no	—
72	C₂×C₃×C₄⋊C₄	—	4	1	no	—
73	C₂×C₃×C₄²	—	4	1	no	—
74	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
75	C₃×Q₃₂	—	2	8	no	—
76	C₃×SD₃₂	—	2	8	no	—
77	C₃×M₃₂	—	2	4	no	—
78	C₂×C₃×C₁₆	C₂³	1	4	yes	—
79	C₃×#107	—	2	1	no	—
80	C₃×#107	C₂³	1	2	yes	—
81	C₃×#103	—	2	2	no	—
82	C₃×#103	—	2	2	no	—
83	C₃×C₄×C₈	D₄	1	1	yes	characteristic
84	C₃×C₄×C₈	D₄	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
87	C₂×#135	—	2	1	no	—
88	C₂×#135	C₂²×C₃	1	2	yes	—
89	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
90	#532	—	2	1	no	—
91	C₂×C₄×Q₈	—	2	1	no	—
92	#510	—	2	8	no	—
93	C₂×Q₃₂	C₂²×C₃	1	4	yes	—
94	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
95	C₂×M₃₂	C₂²×C₃	1	2	yes	—
96	#495	—	2	1	no	—
97	#495	C₂²×C₃	1	2	yes	—
98	#494	C₂²×C₃	1	1	yes	characteristic
99	#493	C₂²×C₃	1	1	yes	characteristic
100	#492	—	2	1	no	—
101	C₄×Q₁₆	—	2	1	no	—
102	C₄×Q₁₆	—	2	1	no	—
103	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
104	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
105	C₂×#107	C₂²×C₃	1	1	yes	characteristic
106	C₂×#103	—	2	1	no	—
107	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
108	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
109	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
110	#358	C₂²×C₃	1	4	yes	—
111	#357	C₂²×C₃	1	4	yes	—
112	C₃×#41	—	4	4	no	—
113	C₃×#41	—	4	4	no	—
114	C₂×C₃×Q₈	—	2	4	no	—
115	C₂×C₃×Q₈	—	2	2	no	—
116	C₂×C₃×Q₈	—	2	2	no	—
117	C₂×C₃×Q₈	—	4	1	no	—
118	C₂×C₃×Q₈	—	4	2	no	—
119	C₂×C₃×D₄	—	2	2	no	—
120	C₂×C₃×D₄	—	4	2	no	—
121	C₂×C₃×D₄	—	4	1	no	—
122	C₂×C₃×D₄	—	4	1	no	—
123	C₂²×C₃×C₄	—	4	1	no	—
124	C₂²×C₃×C₄	—	4	1	no	—
125	C₂²×C₃×C₄	—	4	2	no	—
126	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
127	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
129	C₃×Q₁₆	—	2	4	no	—
130	C₃×Q₁₆	—	2	2	no	—
131	C₃×Q₁₆	—	2	2	no	—
132	C₃×Q₁₆	—	2	2	no	—
133	C₃×Q₁₆	—	4	2	no	—
134	C₃×Q₁₆	C₂×D₄	1	4	yes	—
135	C₃×SD₁₆	—	4	4	no	—
136	C₃×SD₁₆	—	4	4	no	—
137	C₃×D₈	—	2	2	no	—
138	C₃×D₈	—	2	2	no	—
139	C₃×D₈	—	4	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
145	C₂×C₃×C₈	#41	1	2	yes	—
146	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	—	4	1	no	—
149	C₃×C₄⋊C₄	—	4	2	no	—
150	C₃×C₄⋊C₄	—	4	2	no	—
151	C₃×C₄⋊C₄	—	4	1	no	—
152	C₃×C₄⋊C₄	—	4	1	no	—
153	C₃×C₂²⋊C₄	—	4	2	no	—
154	C₃×C₂²⋊C₄	—	4	2	no	—
155	C₃×C₄²	—	4	2	no	—
156	C₃×C₄²	—	4	2	no	—
157	C₃×C₄²	D₈	1	1	yes	characteristic
158	C₃×C₄²	D₈	1	1	yes	characteristic
159	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
160	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
161	C₃×C₁₆	—	2	4	no	—
162	C₂×#41	—	2	2	no	—
163	C₂²×Q₈	—	2	1	no	—
164	#135	—	2	4	no	—
165	#135	—	2	4	no	—
166	#135	—	4	4	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	—	2	1	no	—
169	C₂×Q₁₆	—	2	2	no	—
170	C₂×Q₁₆	C₃×D₄	1	4	yes	complemented
171	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
172	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
173	C₂×SD₁₆	—	2	2	no	—
174	C₂×SD₁₆	—	2	2	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	C₂³×C₃	1	2	yes	—
177	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	1	no	—
180	#127	—	2	1	no	—
181	#124	—	2	2	no	—
182	#121	—	4	1	no	—
183	#121	—	4	1	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	—	4	1	no	—
187	C₄×Q₈	—	4	1	no	—
188	C₄×D₄	—	4	1	no	—
189	C₄×D₄	—	4	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	4	1	no	—
192	C₂×C₄²	—	4	1	no	—
193	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
194	Q₃₂	—	2	8	no	—
195	SD₃₂	—	2	8	no	—
196	M₃₂	—	2	4	no	—
197	C₂×C₁₆	C₂³×C₃	1	4	yes	—
198	#107	—	2	1	no	—
199	#107	C₂³×C₃	1	2	yes	—
200	#103	—	2	2	no	—
201	#103	—	2	2	no	—
202	C₄×C₈	C₃×D₄	1	1	yes	characteristic
203	C₄×C₈	C₃×D₄	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
206	C₂³×C₃	—	4	2	no	—
207	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
208	C₃×Q₈	—	2	4	no	—
209	C₃×Q₈	—	4	2	no	—
210	C₃×Q₈	—	4	4	no	—
211	C₃×Q₈	—	4	2	no	—
212	C₃×Q₈	—	4	2	no	—
213	C₃×D₄	—	4	2	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	8	2	no	—
216	C₃×D₄	—	8	4	no	—
217	C₃×D₄	—	8	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	—	4	2	no	—
221	C₂×C₃×C₄	—	4	4	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	—	4	2	no	—
224	C₂×C₃×C₄	—	8	2	no	—
225	C₂×C₃×C₄	—	8	4	no	—
226	C₂×C₃×C₄	#121	1	2	yes	—
227	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
232	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
233	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
234	C₃×C₈	—	2	1	no	—
235	C₃×C₈	—	2	1	no	—
236	C₃×C₈	#121	1	2	yes	—
237	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
238	C₃×C₈	C₂×#41	1	1	yes	characteristic
239	#41	—	4	4	no	—
240	#41	—	4	4	no	—
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
254	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
256	Q₁₆	—	2	4	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	2	2	no	—
259	Q₁₆	—	2	2	no	—
260	Q₁₆	—	4	2	no	—
261	Q₁₆	C₂×C₃×D₄	1	4	yes	—
262	SD₁₆	—	4	4	no	—
263	SD₁₆	—	4	4	no	—
264	D₈	—	2	2	no	—
265	D₈	—	2	2	no	—
266	D₈	—	4	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
269	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
270	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
271	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
272	C₂×C₈	C₃×#41	1	2	yes	—
273	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₃×D₈	1	1	yes	characteristic
285	C₄²	C₃×D₈	1	1	yes	characteristic
286	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
287	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
288	C₁₆	—	2	4	no	—
289	C₂²×C₃	—	2	1	no	—
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	—	8	2	no	—
292	C₂²×C₃	—	8	2	no	—
293	C₂²×C₃	—	8	2	no	—
294	C₂²×C₃	#459	1	2	yes	—
295	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
296	C₃×C₄	—	2	1	no	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	4	4	no	—
300	C₃×C₄	—	8	2	no	—
301	C₃×C₄	—	8	2	no	—
302	C₃×C₄	#459	1	2	yes	—
303	C₃×C₄	#510	1	2	yes	complemented
304	C₃×C₄	C₂×#121	1	1	yes	characteristic
305	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂³	—	4	2	no	—
307	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
308	Q₈	—	2	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	2	no	—
316	D₄	—	8	4	no	—
317	D₄	—	8	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	C₃×#121	1	2	yes	—
327	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
328	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
329	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
330	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
331	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
332	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
333	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	C₃×#121	1	2	yes	—
337	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
338	C₈	C₂×C₃×#41	1	1	yes	characteristic
339	C₂×C₃	—	2	2	no	—
340	C₂×C₃	—	16	2	no	—
341	C₂×C₃	C₂×#459	1	1	yes	characteristic
342	C₂×C₃	C₂×#510	1	1	yes	characteristic
343	C₂×C₃	#3414	1	1	yes	characteristic
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	C₃×#459	1	2	yes	—
350	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	4	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	C₃×#459	1	2	yes	—
358	C₄	C₃×#510	1	2	yes	complemented
359	C₄	C₂×C₃×#121	1	1	yes	characteristic
360	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
361	C₃	#33581	1	1	yes	characteristic, direct product factor, Sylow
362	C₂	—	2	2	no	—
363	C₂	—	16	2	no	—
364	C₂	C₂×C₃×#459	1	1	yes	characteristic
365	C₂	C₂×C₃×#510	1	1	yes	characteristic
366	C₂	C₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#510	C₂	semidirect
3. C₂×C₃×C₄×Q₁₆	C₂	semidirect
4. C₃×#2211	C₂	semidirect
5. C₃×#2209	C₂	semidirect
6. C₃×#2207	C₂	semidirect
7. C₃×#2150	C₂	semidirect
8. C₃×#2139	C₂	semidirect
9. #33581	C₃	direct (N × H)
35. C₂×#510	C₂×C₃	semidirect
37. C₂×C₄×Q₁₆	C₂×C₃	semidirect
38. #2211	C₂×C₃	semidirect
39. #2209	C₂×C₃	semidirect
40. #2207	C₂×C₃	semidirect
41. #2150	C₂×C₃	semidirect
42. #2139	C₂×C₃	semidirect
51. C₂×C₃×Q₁₆	D₄	semidirect
170. C₂×Q₁₆	C₃×D₄	semidirect
303. C₃×C₄	#510	semidirect
358. C₄	C₃×#510	semidirect
361. C₃	#33581	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×#510	2
C₃×#3147	2
C₂×C₃×C₄×Q₁₆	2
C₃×#2211	2
C₃×#2209	2
C₃×#2207	2
C₃×#2150	2
C₃×#2139	2
#33581	3

Subgroups — C3×#33581 #10652188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33581 #10652188