Subgroups — C₃×#33773 #10652380

Order 768.

636 conjugacy classes of subgroups.

The subgroup lattice has 364 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×#2248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2210	C₂	1	4	yes	maximal, complemented
8	C₃×#2186	C₂	1	4	yes	maximal, complemented
9	C₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	#33773	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#135	—	2	1	no	—
12	C₂×C₃×#135	C₂²	1	2	yes	—
13	C₃×#532	—	2	1	no	—
14	C₃×#532	—	2	1	no	—
15	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
16	C₂×C₃×D₁₆	C₂²	1	4	yes	—
17	C₃×#500	C₂²	1	1	yes	characteristic
18	C₃×#495	—	2	1	no	—
19	C₃×#495	C₂²	1	2	yes	—
20	C₃×#494	C₂²	1	1	yes	characteristic
21	C₃×#493	C₂²	1	1	yes	characteristic
22	C₃×#493	C₂²	1	1	yes	characteristic
23	C₃×#492	—	2	1	no	—
24	C₃×#486	—	2	1	no	—
25	C₃×#466	—	2	1	no	—
26	C₃×#466	C₂²	1	2	yes	—
27	C₃×C₄×D₈	—	2	1	no	—
28	C₃×C₄×D₈	C₂²	1	2	yes	—
29	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₃×#358	C₂²	1	4	yes	—
32	C₃×#357	C₂²	1	4	yes	—
33	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
36	#3169	C₂×C₃	1	1	yes	characteristic, complemented
37	#3147	C₂×C₃	1	1	yes	characteristic, complemented
38	#2248	C₂×C₃	1	1	yes	characteristic, complemented
39	#2246	C₂×C₃	1	1	yes	characteristic, complemented
40	#2241	C₂×C₃	1	1	yes	characteristic, complemented
41	#2240	C₂×C₃	1	1	yes	characteristic, complemented
42	#2210	C₂×C₃	1	4	yes	complemented
43	#2186	C₂×C₃	1	4	yes	complemented
44	#2162	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×#41	—	2	2	no	—
46	C₂×C₃×#41	—	4	1	no	—
47	C₃×#135	—	4	4	no	—
48	C₃×#135	—	4	4	no	—
49	C₂×C₃×Q₁₆	—	2	1	no	—
50	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
51	C₂×C₃×SD₁₆	—	2	2	no	—
52	C₂×C₃×SD₁₆	—	2	2	no	—
53	C₂×C₃×D₈	—	2	1	no	—
54	C₂×C₃×D₈	—	2	1	no	—
55	C₂×C₃×D₈	C₂³	1	2	yes	—
56	C₂×C₃×D₈	C₂³	1	2	yes	—
57	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
59	C₃×C₄⋊Q₈	—	2	1	no	—
60	C₃×C₄⋊Q₈	—	2	1	no	—
61	C₃×#127	—	2	1	no	—
62	C₃×#127	—	2	1	no	—
63	C₃×#124	—	2	2	no	—
64	C₃×#124	—	4	1	no	—
65	C₃×#121	—	2	2	no	—
66	C₃×#121	—	4	1	no	—
67	C₃×#121	—	4	1	no	—
68	C₃×#121	—	4	1	no	—
69	C₃×C₄×D₄	—	2	2	no	—
70	C₃×C₄×D₄	—	4	1	no	—
71	C₃×C₄×D₄	—	4	1	no	—
72	C₃×C₄×D₄	—	4	1	no	—
73	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
74	C₃×SD₃₂	—	4	4	no	—
75	C₃×D₁₆	—	4	4	no	—
76	C₂×C₃×C₁₆	C₂³	1	2	yes	—
77	C₂×C₃×C₁₆	C₂³	1	2	yes	—
78	C₃×#107	—	2	1	no	—
79	C₃×#107	C₂³	1	2	yes	—
80	C₃×#102	—	2	2	no	—
81	C₃×#102	—	2	2	no	—
82	C₃×C₄×C₈	D₄	1	1	yes	characteristic
83	C₃×C₄×C₈	D₄	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₂×#135	—	2	1	no	—
87	C₂×#135	C₂²×C₃	1	2	yes	—
88	#532	—	2	1	no	—
89	#532	—	2	1	no	—
90	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
91	C₂×D₁₆	C₂²×C₃	1	4	yes	—
92	#500	C₂²×C₃	1	1	yes	characteristic
93	#495	—	2	1	no	—
94	#495	C₂²×C₃	1	2	yes	—
95	#494	C₂²×C₃	1	1	yes	characteristic
96	#493	C₂²×C₃	1	1	yes	characteristic
97	#493	C₂²×C₃	1	1	yes	characteristic
98	#492	—	2	1	no	—
99	#486	—	2	1	no	—
100	#466	—	2	1	no	—
101	#466	C₂²×C₃	1	2	yes	—
102	C₄×D₈	—	2	1	no	—
103	C₄×D₈	C₂²×C₃	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
105	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
106	#358	C₂²×C₃	1	4	yes	—
107	#357	C₂²×C₃	1	4	yes	—
108	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
109	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
110	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
111	C₃×#41	—	4	4	no	—
112	C₃×#41	—	4	4	no	—
113	C₃×#41	—	8	4	no	—
114	C₂×C₃×Q₈	—	2	2	no	—
115	C₂×C₃×Q₈	—	4	1	no	—
116	C₂×C₃×D₄	—	2	2	no	—
117	C₂×C₃×D₄	—	2	2	no	—
118	C₂×C₃×D₄	—	4	2	no	—
119	C₂×C₃×D₄	—	4	2	no	—
120	C₂×C₃×D₄	—	4	1	no	—
121	C₂×C₃×D₄	—	4	1	no	—
122	C₂×C₃×D₄	—	4	1	no	—
123	C₂×C₃×D₄	—	4	1	no	—
124	C₂²×C₃×C₄	—	4	2	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	—	4	2	no	—
131	C₃×SD₁₆	—	4	4	no	—
132	C₃×SD₁₆	—	4	4	no	—
133	C₃×D₈	—	2	4	no	—
134	C₃×D₈	—	2	4	no	—
135	C₃×D₈	—	4	2	no	—
136	C₃×D₈	—	4	2	no	—
137	C₂×C₃×C₈	—	2	2	no	—
138	C₂×C₃×C₈	—	2	2	no	—
139	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
140	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	—	4	1	no	—
145	C₃×C₄⋊C₄	—	4	1	no	—
146	C₃×C₄⋊C₄	—	4	1	no	—
147	C₃×C₂²⋊C₄	—	4	2	no	—
148	C₃×C₂²⋊C₄	—	4	2	no	—
149	C₃×C₂²⋊C₄	—	4	2	no	—
150	C₃×C₂²⋊C₄	—	4	2	no	—
151	C₃×C₄²	D₈	1	1	yes	characteristic
152	C₃×C₄²	D₈	1	1	yes	characteristic
153	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
154	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
155	C₃×C₁₆	—	2	2	no	—
156	C₃×C₁₆	—	2	2	no	—
157	C₂×#41	—	2	2	no	—
158	C₂×#41	—	4	1	no	—
159	#135	—	4	4	no	—
160	#135	—	4	4	no	—
161	C₂×Q₁₆	—	2	1	no	—
162	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	C₂³×C₃	1	2	yes	—
168	C₂×D₈	C₂³×C₃	1	2	yes	—
169	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	1	no	—
172	C₄⋊Q₈	—	2	1	no	—
173	#127	—	2	1	no	—
174	#127	—	2	1	no	—
175	#124	—	2	2	no	—
176	#124	—	4	1	no	—
177	#121	—	2	2	no	—
178	#121	—	4	1	no	—
179	#121	—	4	1	no	—
180	#121	—	4	1	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	4	1	no	—
183	C₄×D₄	—	4	1	no	—
184	C₄×D₄	—	4	1	no	—
185	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
186	SD₃₂	—	4	4	no	—
187	D₁₆	—	4	4	no	—
188	C₂×C₁₆	C₂³×C₃	1	2	yes	—
189	C₂×C₁₆	C₂³×C₃	1	2	yes	—
190	#107	—	2	1	no	—
191	#107	C₂³×C₃	1	2	yes	—
192	#102	—	2	2	no	—
193	#102	—	2	2	no	—
194	C₄×C₈	C₃×D₄	1	1	yes	characteristic
195	C₄×C₈	C₃×D₄	1	1	yes	characteristic
196	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
197	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
198	C₂³×C₃	—	4	2	no	—
199	C₂³×C₃	—	4	2	no	—
200	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
201	C₃×Q₈	—	4	2	no	—
202	C₃×Q₈	—	4	2	no	—
203	C₃×Q₈	—	8	2	no	—
204	C₃×D₄	—	4	2	no	—
205	C₃×D₄	—	4	2	no	—
206	C₃×D₄	—	4	2	no	—
207	C₃×D₄	—	4	2	no	—
208	C₃×D₄	—	8	4	no	—
209	C₃×D₄	—	8	4	no	—
210	C₃×D₄	—	8	2	no	—
211	C₃×D₄	—	8	2	no	—
212	C₃×D₄	—	8	2	no	—
213	C₃×D₄	—	8	2	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	4	2	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	—	8	4	no	—
221	C₂×C₃×C₄	—	8	4	no	—
222	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
231	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
232	#41	—	4	4	no	—
233	#41	—	4	4	no	—
234	#41	—	8	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
248	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
250	Q₁₆	—	2	4	no	—
251	Q₁₆	—	4	2	no	—
252	SD₁₆	—	4	4	no	—
253	SD₁₆	—	4	4	no	—
254	D₈	—	2	4	no	—
255	D₈	—	2	4	no	—
256	D₈	—	4	2	no	—
257	D₈	—	4	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
261	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
262	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
263	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	C₃×D₈	1	1	yes	characteristic
273	C₄²	C₃×D₈	1	1	yes	characteristic
274	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
275	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
276	C₁₆	—	2	2	no	—
277	C₁₆	—	2	2	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	2	1	no	—
280	C₂²×C₃	—	2	1	no	—
281	C₂²×C₃	—	8	2	no	—
282	C₂²×C₃	—	8	2	no	—
283	C₂²×C₃	—	8	2	no	—
284	C₂²×C₃	—	8	2	no	—
285	C₂²×C₃	—	8	2	no	—
286	C₂²×C₃	—	8	2	no	—
287	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	8	2	no	—
292	C₃×C₄	—	8	2	no	—
293	C₃×C₄	#508	1	2	yes	—
294	C₃×C₄	#509	1	2	yes	—
295	C₃×C₄	#535	1	1	yes	characteristic
296	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	4	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	2	no	—
311	D₄	—	8	2	no	—
312	D₄	—	8	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
323	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
324	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
325	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
330	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
331	C₂×C₃	—	4	1	no	—
332	C₂×C₃	—	16	2	no	—
333	C₂×C₃	—	16	2	no	—
334	C₂×C₃	#3205	1	1	yes	characteristic
335	C₂×C₃	C₂×#508	1	1	yes	characteristic
336	C₂×C₃	C₂×#509	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	8	2	no	—
351	C₄	—	8	2	no	—
352	C₄	C₃×#508	1	2	yes	—
353	C₄	C₃×#509	1	2	yes	—
354	C₄	C₃×#535	1	1	yes	characteristic
355	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
356	C₃	#33773	1	1	yes	characteristic, direct product factor, Sylow
357	C₂	—	4	1	no	—
358	C₂	—	16	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₃×#3205	1	1	yes	characteristic
361	C₂	C₂×C₃×#508	1	1	yes	characteristic
362	C₂	C₂×C₃×#509	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×#2248	C₂	semidirect
4. C₃×#2246	C₂	semidirect
5. C₃×#2241	C₂	semidirect
6. C₃×#2240	C₂	semidirect
7. C₃×#2210	C₂	semidirect
8. C₃×#2186	C₂	semidirect
9. C₃×#2162	C₂	semidirect
10. #33773	C₃	direct (N × H)
36. #3169	C₂×C₃	semidirect
37. #3147	C₂×C₃	semidirect
38. #2248	C₂×C₃	semidirect
39. #2246	C₂×C₃	semidirect
40. #2241	C₂×C₃	semidirect
41. #2240	C₂×C₃	semidirect
42. #2210	C₂×C₃	semidirect
43. #2186	C₂×C₃	semidirect
44. #2162	C₂×C₃	semidirect
356. C₃	#33773	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×#2248	2
C₃×#2246	2
C₃×#2241	2
C₃×#2240	2
C₃×#2210	2
C₃×#2186	2
C₃×#2162	2
#33773	3

Subgroups — C3×#33773 #10652380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33773 #10652380