Subgroups — C₃×#33780 #10652387

Order 768.

632 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2250	C₂	1	2	yes	maximal, complemented
4	C₃×#2243	C₂	1	2	yes	maximal, complemented
5	C₃×#2208	C₂	1	2	yes	maximal, complemented
6	C₃×#2206	C₂	1	2	yes	maximal, complemented
7	C₃×#2186	C₂	1	4	yes	maximal, complemented
8	C₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	#33780	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	C₃×#532	—	2	1	no	—
13	C₃×#532	—	2	1	no	—
14	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
15	C₂×C₃×D₁₆	C₂²	1	4	yes	—
16	C₃×#500	C₂²	1	1	yes	characteristic
17	C₃×#495	—	2	1	no	—
18	C₃×#495	C₂²	1	2	yes	—
19	C₃×#494	C₂²	1	1	yes	characteristic
20	C₃×#493	C₂²	1	1	yes	characteristic
21	C₃×#493	C₂²	1	1	yes	characteristic
22	C₃×#492	—	2	1	no	—
23	C₃×#486	—	2	1	no	—
24	C₃×#466	—	2	1	no	—
25	C₃×#466	C₂²	1	2	yes	—
26	C₃×C₄×D₈	—	2	1	no	—
27	C₃×C₄×D₈	C₂²	1	2	yes	—
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₃×#358	C₂²	1	4	yes	—
31	C₃×#357	C₂²	1	4	yes	—
32	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₃×#346	C₂²	1	2	yes	—
34	#3169	C₂×C₃	1	1	yes	characteristic, complemented
35	#3147	C₂×C₃	1	1	yes	characteristic, complemented
36	#2250	C₂×C₃	1	2	yes	complemented
37	#2243	C₂×C₃	1	2	yes	complemented
38	#2208	C₂×C₃	1	2	yes	complemented
39	#2206	C₂×C₃	1	2	yes	complemented
40	#2186	C₂×C₃	1	4	yes	complemented
41	#2163	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×C₃×#41	—	2	2	no	—
43	C₂×C₃×#41	—	4	1	no	—
44	C₃×#135	—	4	4	no	—
45	C₃×#135	—	4	4	no	—
46	C₂×C₃×Q₁₆	—	2	1	no	—
47	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
48	C₂×C₃×SD₁₆	—	2	2	no	—
49	C₂×C₃×SD₁₆	—	2	2	no	—
50	C₂×C₃×D₈	—	2	1	no	—
51	C₂×C₃×D₈	—	2	1	no	—
52	C₂×C₃×D₈	C₂³	1	2	yes	—
53	C₂×C₃×D₈	C₂³	1	2	yes	—
54	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₃×C₄⋊Q₈	—	2	1	no	—
57	C₃×C₄⋊Q₈	—	2	1	no	—
58	C₃×#127	—	2	1	no	—
59	C₃×#127	—	2	1	no	—
60	C₃×#124	—	2	2	no	—
61	C₃×#124	—	4	1	no	—
62	C₃×#121	—	2	2	no	—
63	C₃×#121	—	4	1	no	—
64	C₃×#121	—	4	1	no	—
65	C₃×#121	—	4	1	no	—
66	C₃×C₄×D₄	—	2	2	no	—
67	C₃×C₄×D₄	—	4	1	no	—
68	C₃×C₄×D₄	—	4	1	no	—
69	C₃×C₄×D₄	—	4	1	no	—
70	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
71	C₃×SD₃₂	—	4	4	no	—
72	C₃×D₁₆	—	4	4	no	—
73	C₂×C₃×C₁₆	C₂³	1	4	yes	—
74	C₃×#107	—	2	1	no	—
75	C₃×#107	C₂³	1	2	yes	—
76	C₃×#102	—	2	2	no	—
77	C₃×#102	—	2	2	no	—
78	C₃×C₄×C₈	D₄	1	1	yes	characteristic
79	C₃×C₄×C₈	D₄	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×#135	—	2	1	no	—
83	C₂×#135	C₂²×C₃	1	2	yes	—
84	#532	—	2	1	no	—
85	#532	—	2	1	no	—
86	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
87	C₂×D₁₆	C₂²×C₃	1	4	yes	—
88	#500	C₂²×C₃	1	1	yes	characteristic
89	#495	—	2	1	no	—
90	#495	C₂²×C₃	1	2	yes	—
91	#494	C₂²×C₃	1	1	yes	characteristic
92	#493	C₂²×C₃	1	1	yes	characteristic
93	#493	C₂²×C₃	1	1	yes	characteristic
94	#492	—	2	1	no	—
95	#486	—	2	1	no	—
96	#466	—	2	1	no	—
97	#466	C₂²×C₃	1	2	yes	—
98	C₄×D₈	—	2	1	no	—
99	C₄×D₈	C₂²×C₃	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
101	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
102	#358	C₂²×C₃	1	4	yes	—
103	#357	C₂²×C₃	1	4	yes	—
104	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
105	#346	C₂²×C₃	1	2	yes	—
106	C₃×#41	—	4	4	no	—
107	C₃×#41	—	4	4	no	—
108	C₃×#41	—	8	4	no	—
109	C₂×C₃×Q₈	—	2	2	no	—
110	C₂×C₃×Q₈	—	4	1	no	—
111	C₂×C₃×D₄	—	2	2	no	—
112	C₂×C₃×D₄	—	2	2	no	—
113	C₂×C₃×D₄	—	4	2	no	—
114	C₂×C₃×D₄	—	4	2	no	—
115	C₂×C₃×D₄	—	4	1	no	—
116	C₂×C₃×D₄	—	4	1	no	—
117	C₂×C₃×D₄	—	4	1	no	—
118	C₂×C₃×D₄	—	4	1	no	—
119	C₂²×C₃×C₄	—	4	2	no	—
120	C₂²×C₃×C₄	—	4	2	no	—
121	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
122	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₃×Q₁₆	—	2	4	no	—
125	C₃×Q₁₆	—	4	2	no	—
126	C₃×SD₁₆	—	4	4	no	—
127	C₃×SD₁₆	—	4	4	no	—
128	C₃×D₈	—	2	2	no	—
129	C₃×D₈	—	2	4	no	—
130	C₃×D₈	—	2	2	no	—
131	C₃×D₈	—	4	2	no	—
132	C₃×D₈	—	4	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	4	1	no	—
141	C₃×C₄⋊C₄	—	4	1	no	—
142	C₃×C₄⋊C₄	—	4	1	no	—
143	C₃×C₂²⋊C₄	—	4	2	no	—
144	C₃×C₂²⋊C₄	—	4	2	no	—
145	C₃×C₂²⋊C₄	—	4	2	no	—
146	C₃×C₂²⋊C₄	—	4	2	no	—
147	C₃×C₄²	D₈	1	1	yes	characteristic
148	C₃×C₄²	D₈	1	1	yes	characteristic
149	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
150	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
151	C₃×C₁₆	—	2	4	no	—
152	C₂×#41	—	2	2	no	—
153	C₂×#41	—	4	1	no	—
154	#135	—	4	4	no	—
155	#135	—	4	4	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	C₂³×C₃	1	2	yes	—
163	C₂×D₈	C₂³×C₃	1	2	yes	—
164	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	1	no	—
167	C₄⋊Q₈	—	2	1	no	—
168	#127	—	2	1	no	—
169	#127	—	2	1	no	—
170	#124	—	2	2	no	—
171	#124	—	4	1	no	—
172	#121	—	2	2	no	—
173	#121	—	4	1	no	—
174	#121	—	4	1	no	—
175	#121	—	4	1	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	4	1	no	—
178	C₄×D₄	—	4	1	no	—
179	C₄×D₄	—	4	1	no	—
180	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
181	SD₃₂	—	4	4	no	—
182	D₁₆	—	4	4	no	—
183	C₂×C₁₆	C₂³×C₃	1	4	yes	—
184	#107	—	2	1	no	—
185	#107	C₂³×C₃	1	2	yes	—
186	#102	—	2	2	no	—
187	#102	—	2	2	no	—
188	C₄×C₈	C₃×D₄	1	1	yes	characteristic
189	C₄×C₈	C₃×D₄	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
192	C₂³×C₃	—	4	2	no	—
193	C₂³×C₃	—	4	2	no	—
194	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
195	C₃×Q₈	—	4	2	no	—
196	C₃×Q₈	—	4	2	no	—
197	C₃×Q₈	—	8	2	no	—
198	C₃×D₄	—	4	2	no	—
199	C₃×D₄	—	4	2	no	—
200	C₃×D₄	—	4	2	no	—
201	C₃×D₄	—	4	2	no	—
202	C₃×D₄	—	8	2	no	—
203	C₃×D₄	—	8	2	no	—
204	C₃×D₄	—	8	2	no	—
205	C₃×D₄	—	8	4	no	—
206	C₃×D₄	—	8	4	no	—
207	C₃×D₄	—	8	2	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	8	4	no	—
215	C₂×C₃×C₄	—	8	4	no	—
216	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
222	C₃×C₈	—	2	1	no	—
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
225	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
226	#41	—	4	4	no	—
227	#41	—	4	4	no	—
228	#41	—	8	4	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
242	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
244	Q₁₆	—	2	4	no	—
245	Q₁₆	—	4	2	no	—
246	SD₁₆	—	4	4	no	—
247	SD₁₆	—	4	4	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	4	no	—
250	D₈	—	2	2	no	—
251	D₈	—	4	2	no	—
252	D₈	—	4	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
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	C₂⁴×C₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	C₃×D₈	1	1	yes	characteristic
268	C₄²	C₃×D₈	1	1	yes	characteristic
269	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
270	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
271	C₁₆	—	2	4	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	8	2	no	—
276	C₂²×C₃	—	8	2	no	—
277	C₂²×C₃	—	8	2	no	—
278	C₂²×C₃	—	8	2	no	—
279	C₂²×C₃	—	8	2	no	—
280	C₂²×C₃	—	8	2	no	—
281	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	8	2	no	—
287	C₃×C₄	—	8	2	no	—
288	C₃×C₄	#509	1	2	yes	—
289	C₃×C₄	#535	1	1	yes	characteristic
290	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	4	no	—
305	D₄	—	8	4	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	4	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
316	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
324	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
325	C₂×C₃	—	4	1	no	—
326	C₂×C₃	—	16	2	no	—
327	C₂×C₃	—	16	2	no	—
328	C₂×C₃	#3205	1	1	yes	characteristic
329	C₂×C₃	C₂×#509	1	1	yes	characteristic
330	C₂×C₃	#3414	1	1	yes	characteristic
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₃×#509	1	2	yes	—
348	C₄	C₃×#535	1	1	yes	characteristic
349	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
350	C₃	#33780	1	1	yes	characteristic, direct product factor, Sylow
351	C₂	—	4	1	no	—
352	C₂	—	16	2	no	—
353	C₂	—	16	2	no	—
354	C₂	C₃×#3205	1	1	yes	characteristic
355	C₂	C₂×C₃×#509	1	1	yes	characteristic
356	C₂	C₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
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. C₃×#3169	C₂	semidirect
2. C₃×#3147	C₂	semidirect
3. C₃×#2250	C₂	semidirect
4. C₃×#2243	C₂	semidirect
5. C₃×#2208	C₂	semidirect
6. C₃×#2206	C₂	semidirect
7. C₃×#2186	C₂	semidirect
8. C₃×#2163	C₂	semidirect
9. #33780	C₃	direct (N × H)
34. #3169	C₂×C₃	semidirect
35. #3147	C₂×C₃	semidirect
36. #2250	C₂×C₃	semidirect
37. #2243	C₂×C₃	semidirect
38. #2208	C₂×C₃	semidirect
39. #2206	C₂×C₃	semidirect
40. #2186	C₂×C₃	semidirect
41. #2163	C₂×C₃	semidirect
350. C₃	#33780	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#3169	2
C₃×#3147	2
C₃×#2250	2
C₃×#2243	2
C₃×#2208	2
C₃×#2206	2
C₃×#2186	2
C₃×#2163	2
#33780	3

Subgroups — C3×#33780 #10652387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33780 #10652387