Subgroups — C₃×#33779 #10652386

Order 768.

642 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	C₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#495	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×#2210	C₂	1	4	yes	maximal, complemented
6	C₃×#2185	C₂	1	2	yes	maximal, complemented
7	C₃×#2184	C₂	1	2	yes	maximal, complemented
8	C₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	#33779	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×SD₁₆	—	2	1	no	—
12	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
13	C₃×#532	—	2	1	no	—
14	C₂×C₃×#124	—	2	1	no	—
15	C₂×C₃×SD₃₂	C₂²	1	4	yes	complemented
16	C₂×C₃×D₁₆	C₂²	1	4	yes	—
17	C₃×#500	C₂²	1	1	yes	characteristic, complemented
18	C₃×#495	—	2	1	no	—
19	C₃×#495	—	2	1	no	—
20	C₃×#495	C₂²	1	2	yes	—
21	C₃×#495	C₂²	1	2	yes	—
22	C₃×#493	C₂²	1	1	yes	characteristic, complemented
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	complemented
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
30	C₃×#358	C₂²	1	4	yes	complemented
31	C₃×#357	C₂²	1	4	yes	—
32	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₃×#346	C₂²	1	2	yes	complemented
34	#3169	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×#495	C₂×C₃	1	1	yes	characteristic, complemented
36	#2250	C₂×C₃	1	2	yes	complemented
37	#2243	C₂×C₃	1	2	yes	complemented
38	#2210	C₂×C₃	1	4	yes	complemented
39	#2185	C₂×C₃	1	2	yes	complemented
40	#2184	C₂×C₃	1	2	yes	complemented
41	#2163	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×C₃×#41	—	4	1	no	—
43	C₂²×C₃×Q₈	—	2	1	no	—
44	C₂²×C₃×D₄	—	2	1	no	—
45	C₂×C₃×Q₁₆	—	2	1	no	—
46	C₂×C₃×Q₁₆	—	4	2	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×SD₁₆	—	4	4	no	—
51	C₂×C₃×D₈	—	2	1	no	—
52	C₂×C₃×D₈	—	2	1	no	—
53	C₂×C₃×D₈	—	4	2	no	—
54	C₂×C₃×D₈	C₂³	1	2	yes	—
55	C₂×C₃×D₈	C₂³	1	2	yes	—
56	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
58	C₃×C₄⋊Q₈	—	2	1	no	—
59	C₃×#127	—	2	1	no	—
60	C₃×#124	—	2	2	no	—
61	C₃×#124	—	2	2	no	—
62	C₃×#124	—	4	1	no	—
63	C₃×#124	—	4	1	no	—
64	C₃×#124	—	4	1	no	—
65	C₃×#121	—	2	2	no	—
66	C₃×#121	—	4	1	no	—
67	C₃×C₄×D₄	—	2	2	no	—
68	C₃×C₄×D₄	—	4	1	no	—
69	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
70	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
71	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
72	C₃×SD₃₂	—	4	4	no	—
73	C₃×D₁₆	—	4	4	no	—
74	C₂×C₃×C₁₆	C₂³	1	4	yes	—
75	C₃×#107	—	2	1	no	—
76	C₃×#107	C₂³	1	2	yes	—
77	C₃×#102	—	2	2	no	—
78	C₃×#102	—	2	2	no	—
79	C₃×C₄×C₈	D₄	1	1	yes	characteristic
80	C₃×C₄×C₈	D₄	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
82	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
83	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
84	C₂²×SD₁₆	—	2	1	no	—
85	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
86	#532	—	2	1	no	—
87	C₂×#124	—	2	1	no	—
88	C₂×SD₃₂	C₂²×C₃	1	4	yes	complemented
89	C₂×D₁₆	C₂²×C₃	1	4	yes	—
90	#500	C₂²×C₃	1	1	yes	characteristic, complemented
91	#495	—	2	1	no	—
92	#495	—	2	1	no	—
93	#495	C₂²×C₃	1	2	yes	—
94	#495	C₂²×C₃	1	2	yes	—
95	#493	C₂²×C₃	1	1	yes	characteristic, complemented
96	#486	—	2	1	no	—
97	#466	—	2	1	no	—
98	#466	C₂²×C₃	1	2	yes	—
99	C₄×D₈	—	2	1	no	—
100	C₄×D₈	C₂²×C₃	1	2	yes	complemented
101	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
102	C₄⋊C₁₆	C₂²×C₃	1	2	yes	complemented
103	#358	C₂²×C₃	1	4	yes	complemented
104	#357	C₂²×C₃	1	4	yes	—
105	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
106	#346	C₂²×C₃	1	2	yes	complemented
107	C₂⁴×C₃	—	4	1	no	—
108	C₃×#41	—	8	4	no	—
109	C₂×C₃×Q₈	—	2	2	no	—
110	C₂×C₃×Q₈	—	4	2	no	—
111	C₂×C₃×Q₈	—	4	2	no	—
112	C₂×C₃×Q₈	—	4	1	no	—
113	C₂×C₃×Q₈	—	4	1	no	—
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	—	4	2	no	—
117	C₂×C₃×D₄	—	4	2	no	—
118	C₂×C₃×D₄	—	4	2	no	—
119	C₂×C₃×D₄	—	4	1	no	—
120	C₂×C₃×D₄	—	4	1	no	—
121	C₂×C₃×D₄	—	4	1	no	—
122	C₂²×C₃×C₄	—	4	1	no	—
123	C₂²×C₃×C₄	—	4	2	no	—
124	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
125	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
127	C₃×Q₁₆	—	2	4	no	—
128	C₃×Q₁₆	—	4	2	no	—
129	C₃×SD₁₆	—	4	4	no	—
130	C₃×SD₁₆	—	4	4	no	—
131	C₃×D₈	—	2	4	no	—
132	C₃×D₈	—	2	4	no	—
133	C₃×D₈	—	4	2	no	—
134	C₃×D₈	—	4	2	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
140	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
141	C₃×C₄⋊C₄	—	2	2	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×C₂²⋊C₄	—	4	2	no	—
148	C₃×C₂²⋊C₄	—	4	2	no	—
149	C₃×C₄²	D₈	1	1	yes	characteristic
150	C₃×C₄²	D₈	1	1	yes	characteristic
151	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
153	C₃×C₁₆	—	2	4	no	—
154	C₂×#41	—	4	1	no	—
155	C₂²×Q₈	—	2	1	no	—
156	C₂²×D₄	—	2	1	no	—
157	C₂×Q₁₆	—	2	1	no	—
158	C₂×Q₁₆	—	4	2	no	—
159	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	4	4	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	4	2	no	—
166	C₂×D₈	C₂³×C₃	1	2	yes	—
167	C₂×D₈	C₂³×C₃	1	2	yes	—
168	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	1	no	—
171	#127	—	2	1	no	—
172	#124	—	2	2	no	—
173	#124	—	2	2	no	—
174	#124	—	4	1	no	—
175	#124	—	4	1	no	—
176	#124	—	4	1	no	—
177	#121	—	2	2	no	—
178	#121	—	4	1	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	—	4	1	no	—
181	C₂×C₂²⋊C₄	—	4	1	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
184	SD₃₂	—	4	4	no	—
185	D₁₆	—	4	4	no	—
186	C₂×C₁₆	C₂³×C₃	1	4	yes	—
187	#107	—	2	1	no	—
188	#107	C₂³×C₃	1	2	yes	—
189	#102	—	2	2	no	—
190	#102	—	2	2	no	—
191	C₄×C₈	C₃×D₄	1	1	yes	characteristic
192	C₄×C₈	C₃×D₄	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
195	C₂³×C₃	—	4	2	no	—
196	C₂³×C₃	—	4	2	no	—
197	C₂³×C₃	—	8	2	no	—
198	C₂³×C₃	—	8	2	no	—
199	C₂³×C₃	—	8	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×Q₈	—	8	2	no	—
205	C₃×D₄	—	4	2	no	—
206	C₃×D₄	—	4	2	no	—
207	C₃×D₄	—	4	2	no	—
208	C₃×D₄	—	4	2	no	—
209	C₃×D₄	—	8	2	no	—
210	C₃×D₄	—	8	2	no	—
211	C₃×D₄	—	8	2	no	—
212	C₃×D₄	—	8	4	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	4	1	no	—
217	C₂×C₃×C₄	—	4	2	no	—
218	C₂×C₃×C₄	—	4	2	no	—
219	C₂×C₃×C₄	—	8	4	no	—
220	C₂×C₃×C₄	—	8	2	no	—
221	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
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	C₂²×D₄	1	1	yes	characteristic
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
230	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
231	C₂⁴	—	4	1	no	—
232	#41	—	8	4	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
249	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	SD₁₆	—	4	4	no	—
254	SD₁₆	—	4	4	no	—
255	D₈	—	2	4	no	—
256	D₈	—	2	4	no	—
257	D₈	—	4	2	no	—
258	D₈	—	4	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
262	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
263	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
264	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	2	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	C₂²⋊C₄	—	4	2	no	—
273	C₄²	C₃×D₈	1	1	yes	characteristic
274	C₄²	C₃×D₈	1	1	yes	characteristic
275	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
276	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
277	C₁₆	—	2	4	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	2	1	no	—
280	C₂²×C₃	—	8	2	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	—	16	4	no	—
287	C₂²×C₃	#509	1	2	yes	—
288	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	8	2	no	—
295	C₃×C₄	—	8	2	no	—
296	C₃×C₄	#535	1	1	yes	characteristic
297	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	8	2	no	—
301	C₂³	—	8	2	no	—
302	C₂³	—	8	2	no	—
303	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	8	2	no	—
307	Q₈	—	8	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	2	no	—
313	D₄	—	8	2	no	—
314	D₄	—	8	2	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	4	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
325	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
326	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
327	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
328	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
333	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
334	C₂×C₃	—	2	2	no	—
335	C₂×C₃	—	16	2	no	—
336	C₂×C₃	—	16	2	no	—
337	C₂×C₃	#3205	1	1	yes	characteristic
338	C₂×C₃	C₂×#509	1	1	yes	characteristic
339	C₂×C₃	#3414	1	1	yes	characteristic
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	16	4	no	—
349	C₂²	C₃×#509	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	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	C₃×#535	1	1	yes	characteristic
359	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
360	C₃	#33779	1	1	yes	characteristic, direct product factor, Sylow
361	C₂	—	2	2	no	—
362	C₂	—	16	2	no	—
363	C₂	—	16	2	no	—
364	C₂	C₃×#3205	1	1	yes	characteristic
365	C₂	C₂×C₃×#509	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
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. C₂×C₃×#495	C₂	semidirect
3. C₃×#2250	C₂	semidirect
4. C₃×#2243	C₂	semidirect
5. C₃×#2210	C₂	semidirect
6. C₃×#2185	C₂	semidirect
7. C₃×#2184	C₂	semidirect
8. C₃×#2163	C₂	semidirect
9. #33779	C₃	direct (N × H)
15. C₂×C₃×SD₃₂	C₂²	semidirect
17. C₃×#500	C₂²	semidirect
22. C₃×#493	C₂²	semidirect
27. C₃×C₄×D₈	C₂²	semidirect
29. C₃×C₄⋊C₁₆	C₂²	semidirect
30. C₃×#358	C₂²	semidirect
33. C₃×#346	C₂²	semidirect
34. #3169	C₂×C₃	semidirect
35. C₂×#495	C₂×C₃	semidirect
36. #2250	C₂×C₃	semidirect
37. #2243	C₂×C₃	semidirect
38. #2210	C₂×C₃	semidirect
39. #2185	C₂×C₃	semidirect
40. #2184	C₂×C₃	semidirect
41. #2163	C₂×C₃	semidirect
88. C₂×SD₃₂	C₂²×C₃	semidirect
90. #500	C₂²×C₃	semidirect
95. #493	C₂²×C₃	semidirect
100. C₄×D₈	C₂²×C₃	semidirect
102. C₄⋊C₁₆	C₂²×C₃	semidirect
103. #358	C₂²×C₃	semidirect
106. #346	C₂²×C₃	semidirect
360. C₃	#33779	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
C₂×C₃×#495	2
C₃×#2250	2
C₃×#2243	2
C₃×#2210	2
C₃×#2185	2
C₃×#2184	2
C₃×#2163	2
#33779	3

Subgroups — C3×#33779 #10652386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33779 #10652386