Subgroups — C₃×#33584 #10652191

Order 768.

722 conjugacy classes of subgroups.

The subgroup lattice has 370 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	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#509	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×#135	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2211	C₂	1	4	yes	maximal, complemented
6	C₃×#2210	C₂	1	4	yes	maximal, complemented
7	C₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	#33584	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#135	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₃×SD₁₆	—	2	1	no	—
14	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×#124	—	2	1	no	—
16	C₃×C₄×#41	—	2	1	no	—
17	C₃×#510	—	2	4	no	—
18	C₃×#509	—	2	4	no	—
19	C₂×C₃×Q₃₂	C₂²	1	2	yes	complemented
20	C₂×C₃×SD₃₂	C₂²	1	2	yes	—
21	C₂×C₃×SD₃₂	C₂²	1	2	yes	complemented
22	C₂×C₃×D₁₆	C₂²	1	2	yes	—
23	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
24	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
25	C₃×#495	—	2	2	no	—
26	C₃×#495	C₂²	1	4	yes	—
27	C₃×C₄×Q₁₆	C₂²	1	2	yes	complemented
28	C₃×C₄×SD₁₆	—	2	2	no	—
29	C₃×C₄×D₈	C₂²	1	2	yes	complemented
30	C₃×#427	C₂²	1	1	yes	characteristic
31	C₃×#416	—	2	1	no	—
32	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
34	C₃×#358	C₂²	1	4	yes	complemented
35	C₃×#357	C₂²	1	4	yes	—
36	C₂×#510	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×#509	C₂×C₃	1	1	yes	characteristic
38	C₂×#495	C₂×C₃	1	1	yes	characteristic, complemented
39	C₄×#135	C₂×C₃	1	1	yes	characteristic, complemented
40	#2211	C₂×C₃	1	4	yes	complemented
41	#2210	C₂×C₃	1	4	yes	complemented
42	#2150	C₂×C₃	1	1	yes	characteristic, complemented
43	#2141	C₂×C₃	1	1	yes	characteristic, complemented
44	#2140	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×#41	—	2	1	no	—
46	C₂²×C₃×Q₈	—	2	1	no	—
47	C₂²×C₃×D₄	—	2	1	no	—
48	C₃×#135	—	2	2	no	—
49	C₃×#135	D₄	1	4	yes	—
50	C₂×C₃×Q₁₆	—	2	2	no	—
51	C₂×C₃×Q₁₆	—	2	1	no	—
52	C₂×C₃×Q₁₆	—	2	2	no	—
53	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
54	C₂×C₃×Q₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₃×SD₁₆	—	2	1	no	—
56	C₂×C₃×SD₁₆	—	2	4	no	—
57	C₂×C₃×SD₁₆	—	4	4	no	—
58	C₂×C₃×D₈	—	2	2	no	—
59	C₂×C₃×D₈	—	2	1	no	—
60	C₂×C₃×D₈	—	2	2	no	—
61	C₂×C₃×D₈	C₂³	1	2	yes	—
62	C₂×C₃×D₈	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₃×#124	—	2	4	no	—
66	C₃×#124	—	4	1	no	—
67	C₃×#124	—	4	1	no	—
68	C₃×C₄×Q₈	—	2	2	no	—
69	C₃×C₄×D₄	—	2	2	no	—
70	C₃×C₄×D₄	—	4	2	no	—
71	C₃×#117	—	2	1	no	—
72	C₃×#117	—	4	1	no	—
73	C₂×C₃×C₂²⋊C₄	—	4	2	no	—
74	C₂×C₃×C₄²	—	4	1	no	—
75	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
76	C₃×Q₃₂	—	2	4	no	—
77	C₃×SD₃₂	—	2	4	no	—
78	C₃×SD₃₂	—	2	4	no	—
79	C₃×D₁₆	—	2	4	no	—
80	C₃×M₃₂	—	2	2	no	—
81	C₃×M₃₂	—	2	2	no	—
82	C₂×C₃×C₁₆	C₂³	1	2	yes	—
83	C₂×C₃×C₁₆	C₂³	1	2	yes	—
84	C₃×#107	C₂³	1	2	yes	—
85	C₃×#106	—	2	1	no	—
86	C₃×#103	—	2	2	no	—
87	C₃×#102	—	2	2	no	—
88	C₃×C₄×C₈	D₄	1	2	yes	—
89	C₃×C₄×C₈	C₂³	1	2	yes	—
90	C₂×#135	C₂²×C₃	1	1	yes	characteristic
91	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
92	C₂²×SD₁₆	—	2	1	no	—
93	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
94	C₂×#124	—	2	1	no	—
95	C₄×#41	—	2	1	no	—
96	#510	—	2	4	no	—
97	#509	—	2	4	no	—
98	C₂×Q₃₂	C₂²×C₃	1	2	yes	complemented
99	C₂×SD₃₂	C₂²×C₃	1	2	yes	—
100	C₂×SD₃₂	C₂²×C₃	1	2	yes	complemented
101	C₂×D₁₆	C₂²×C₃	1	2	yes	—
102	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
103	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
104	#495	—	2	2	no	—
105	#495	C₂²×C₃	1	4	yes	—
106	C₄×Q₁₆	C₂²×C₃	1	2	yes	complemented
107	C₄×SD₁₆	—	2	2	no	—
108	C₄×D₈	C₂²×C₃	1	2	yes	complemented
109	#427	C₂²×C₃	1	1	yes	characteristic
110	#416	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
112	C₄⋊C₁₆	C₂²×C₃	1	4	yes	complemented
113	#358	C₂²×C₃	1	4	yes	complemented
114	#357	C₂²×C₃	1	4	yes	—
115	C₂⁴×C₃	—	4	1	no	—
116	C₃×#41	—	2	4	no	—
117	C₃×#41	—	4	2	no	—
118	C₂×C₃×Q₈	—	2	1	no	—
119	C₂×C₃×Q₈	—	2	2	no	—
120	C₂×C₃×Q₈	—	4	1	no	—
121	C₂×C₃×Q₈	—	4	2	no	—
122	C₂×C₃×Q₈	—	4	2	no	—
123	C₂×C₃×D₄	—	2	1	no	—
124	C₂×C₃×D₄	—	2	2	no	—
125	C₂×C₃×D₄	—	4	1	no	—
126	C₂×C₃×D₄	—	4	2	no	—
127	C₂×C₃×D₄	—	4	1	no	—
128	C₂×C₃×D₄	—	4	2	no	—
129	C₂²×C₃×C₄	—	4	1	no	—
130	C₂²×C₃×C₄	—	4	1	no	—
131	C₂²×C₃×C₄	—	4	2	no	—
132	C₂²×C₃×C₄	D₈	1	2	yes	—
133	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
134	C₃×Q₁₆	—	2	2	no	—
135	C₃×Q₁₆	—	2	1	no	—
136	C₃×Q₁₆	—	2	2	no	—
137	C₃×Q₁₆	—	4	2	no	—
138	C₃×Q₁₆	C₂×D₄	1	2	yes	—
139	C₃×SD₁₆	—	2	4	no	—
140	C₃×SD₁₆	—	4	8	no	—
141	C₃×D₈	—	2	2	no	—
142	C₃×D₈	—	2	1	no	—
143	C₃×D₈	—	2	2	no	—
144	C₃×D₈	—	4	2	no	—
145	C₃×D₈	C₂×D₄	1	2	yes	—
146	C₂×C₃×C₈	—	2	2	no	—
147	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
148	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
150	C₂×C₃×C₈	#41	1	2	yes	—
151	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	—	4	2	no	—
154	C₃×C₂²⋊C₄	—	4	1	no	—
155	C₃×C₂²⋊C₄	—	4	4	no	—
156	C₃×C₂²⋊C₄	—	4	4	no	—
157	C₃×C₂²⋊C₄	—	4	2	no	—
158	C₃×C₄²	—	4	1	no	—
159	C₃×C₄²	—	4	2	no	—
160	C₃×C₄²	D₈	1	1	yes	characteristic
161	C₃×C₄²	D₈	1	1	yes	characteristic
162	C₃×C₄²	C₂×D₄	1	2	yes	—
163	C₃×C₁₆	—	2	2	no	—
164	C₃×C₁₆	—	2	2	no	—
165	C₂×#41	—	2	1	no	—
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×D₄	—	2	1	no	—
168	#135	—	2	2	no	—
169	#135	C₃×D₄	1	4	yes	—
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×Q₁₆	—	2	1	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
174	C₂×Q₁₆	C₂³×C₃	1	1	yes	characteristic
175	C₂×SD₁₆	—	2	1	no	—
176	C₂×SD₁₆	—	2	4	no	—
177	C₂×SD₁₆	—	4	4	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	C₂³×C₃	1	2	yes	—
182	C₂×D₈	C₂³×C₃	1	1	yes	characteristic
183	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
185	#124	—	2	4	no	—
186	#124	—	4	1	no	—
187	#124	—	4	1	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×D₄	—	2	2	no	—
190	C₄×D₄	—	4	2	no	—
191	#117	—	2	1	no	—
192	#117	—	4	1	no	—
193	C₂×C₂²⋊C₄	—	4	2	no	—
194	C₂×C₄²	—	4	1	no	—
195	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
196	Q₃₂	—	2	4	no	—
197	SD₃₂	—	2	4	no	—
198	SD₃₂	—	2	4	no	—
199	D₁₆	—	2	4	no	—
200	M₃₂	—	2	2	no	—
201	M₃₂	—	2	2	no	—
202	C₂×C₁₆	C₂³×C₃	1	2	yes	—
203	C₂×C₁₆	C₂³×C₃	1	2	yes	—
204	#107	C₂³×C₃	1	2	yes	—
205	#106	—	2	1	no	—
206	#103	—	2	2	no	—
207	#102	—	2	2	no	—
208	C₄×C₈	C₃×D₄	1	2	yes	—
209	C₄×C₈	C₂³×C₃	1	2	yes	—
210	C₂³×C₃	—	4	1	no	—
211	C₂³×C₃	—	4	2	no	—
212	C₂³×C₃	—	8	4	no	—
213	C₂³×C₃	—	8	2	no	—
214	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
215	C₃×Q₈	—	2	2	no	—
216	C₃×Q₈	—	4	1	no	—
217	C₃×Q₈	—	4	2	no	—
218	C₃×Q₈	—	4	2	no	—
219	C₃×Q₈	—	8	2	no	—
220	C₃×D₄	—	2	2	no	—
221	C₃×D₄	—	4	4	no	—
222	C₃×D₄	—	4	2	no	—
223	C₃×D₄	—	4	1	no	—
224	C₃×D₄	—	4	2	no	—
225	C₃×D₄	—	8	2	no	—
226	C₂×C₃×C₄	—	4	2	no	—
227	C₂×C₃×C₄	—	4	1	no	—
228	C₂×C₃×C₄	—	4	4	no	—
229	C₂×C₃×C₄	—	4	2	no	—
230	C₂×C₃×C₄	—	4	2	no	—
231	C₂×C₃×C₄	—	8	4	no	—
232	C₂×C₃×C₄	—	8	2	no	—
233	C₂×C₃×C₄	#121	1	2	yes	—
234	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
235	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
236	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
237	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
238	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
239	C₃×C₈	—	2	2	no	—
240	C₃×C₈	#121	1	2	yes	—
241	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
242	C₃×C₈	C₂×#41	1	1	yes	characteristic
243	C₂⁴	—	4	1	no	—
244	#41	—	2	4	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	C₃×D₈	1	2	yes	—
261	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	4	2	no	—
266	Q₁₆	C₂×C₃×D₄	1	2	yes	—
267	SD₁₆	—	2	4	no	—
268	SD₁₆	—	4	8	no	—
269	D₈	—	2	2	no	—
270	D₈	—	2	1	no	—
271	D₈	—	2	2	no	—
272	D₈	—	4	2	no	—
273	D₈	C₂×C₃×D₄	1	2	yes	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
276	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
277	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
278	C₂×C₈	C₃×#41	1	2	yes	—
279	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₂²⋊C₄	—	4	4	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₄²	—	4	1	no	—
287	C₄²	—	4	2	no	—
288	C₄²	C₃×D₈	1	1	yes	characteristic
289	C₄²	C₃×D₈	1	1	yes	characteristic
290	C₄²	C₂×C₃×D₄	1	2	yes	—
291	C₁₆	—	2	2	no	—
292	C₁₆	—	2	2	no	—
293	C₂²×C₃	—	2	2	no	—
294	C₂²×C₃	—	4	2	no	—
295	C₂²×C₃	—	8	2	no	—
296	C₂²×C₃	—	8	4	no	—
297	C₂²×C₃	—	8	2	no	—
298	C₂²×C₃	—	16	4	no	—
299	C₂²×C₃	#459	1	2	yes	—
300	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
301	C₃×C₄	—	2	2	no	—
302	C₃×C₄	—	2	2	no	—
303	C₃×C₄	—	4	2	no	—
304	C₃×C₄	—	8	2	no	—
305	C₃×C₄	—	8	2	no	—
306	C₃×C₄	#459	1	2	yes	—
307	C₃×C₄	C₂×#121	1	1	yes	characteristic
308	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₂³	—	4	1	no	—
310	C₂³	—	4	2	no	—
311	C₂³	—	8	4	no	—
312	C₂³	—	8	2	no	—
313	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	4	1	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	8	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	4	4	no	—
321	D₄	—	4	1	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	8	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	4	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	C₃×#121	1	2	yes	—
333	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
334	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
335	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
336	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
337	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	C₃×#121	1	2	yes	—
340	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
341	C₈	C₂×C₃×#41	1	1	yes	characteristic
342	C₂×C₃	—	2	2	no	—
343	C₂×C₃	—	8	2	no	—
344	C₂×C₃	—	16	2	no	—
345	C₂×C₃	C₂×#459	1	1	yes	characteristic
346	C₂×C₃	#3414	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	4	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	16	4	no	—
353	C₂²	C₃×#459	1	2	yes	—
354	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	2	no	—
360	C₄	C₃×#459	1	2	yes	—
361	C₄	C₂×C₃×#121	1	1	yes	characteristic
362	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
363	C₃	#33584	1	1	yes	characteristic, direct product factor, Sylow
364	C₂	—	2	2	no	—
365	C₂	—	8	2	no	—
366	C₂	—	16	2	no	—
367	C₂	C₂×C₃×#459	1	1	yes	characteristic
368	C₂	C₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#509	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×#495	C₂	semidirect
4. C₃×C₄×#135	C₂	semidirect
5. C₃×#2211	C₂	semidirect
6. C₃×#2210	C₂	semidirect
7. C₃×#2150	C₂	semidirect
8. C₃×#2141	C₂	semidirect
9. C₃×#2140	C₂	semidirect
10. #33584	C₃	direct (N × H)
19. C₂×C₃×Q₃₂	C₂²	semidirect
21. C₂×C₃×SD₃₂	C₂²	semidirect
27. C₃×C₄×Q₁₆	C₂²	semidirect
29. C₃×C₄×D₈	C₂²	semidirect
33. C₃×C₄⋊C₁₆	C₂²	semidirect
34. C₃×#358	C₂²	semidirect
36. C₂×#510	C₂×C₃	semidirect
38. C₂×#495	C₂×C₃	semidirect
39. C₄×#135	C₂×C₃	semidirect
40. #2211	C₂×C₃	semidirect
41. #2210	C₂×C₃	semidirect
42. #2150	C₂×C₃	semidirect
43. #2141	C₂×C₃	semidirect
44. #2140	C₂×C₃	semidirect
98. C₂×Q₃₂	C₂²×C₃	semidirect
100. C₂×SD₃₂	C₂²×C₃	semidirect
106. C₄×Q₁₆	C₂²×C₃	semidirect
108. C₄×D₈	C₂²×C₃	semidirect
112. C₄⋊C₁₆	C₂²×C₃	semidirect
113. #358	C₂²×C₃	semidirect
363. C₃	#33584	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
C₂×C₃×#509	2
C₂×C₃×#495	2
C₃×C₄×#135	2
C₃×#2211	2
C₃×#2210	2
C₃×#2150	2
C₃×#2141	2
C₃×#2140	2
#33584	3

Subgroups — C3×#33584 #10652191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33584 #10652191