Subgroups — C₃×#33757 #10652364

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3050	C₂	1	2	yes	maximal, complemented
2	C₃×#2237	C₂	1	2	yes	maximal, complemented
3	C₃×#2234	C₂	1	2	yes	maximal, complemented
4	C₃×#2217	C₂	1	2	yes	maximal, complemented
5	C₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2189	C₂	1	2	yes	maximal, complemented
8	C₃×#2186	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#33757	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#135	C₂²	1	2	yes	—
12	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₃×#525	—	2	2	no	—
14	C₂×C₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₃×SD₃₂	C₂²	1	2	yes	complemented
16	C₂×C₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₃×#467	C₂²	1	2	yes	—
20	C₃×#467	C₂²	1	2	yes	—
21	C₃×#466	C₂²	1	2	yes	—
22	C₃×#466	C₂²	1	2	yes	—
23	C₃×#465	—	2	2	no	—
24	C₃×#465	—	2	2	no	—
25	C₃×#427	C₂²	1	2	yes	—
26	C₃×#416	—	2	2	no	—
27	C₃×#367	C₂²	1	2	yes	complemented
28	C₃×#366	C₂²	1	1	yes	characteristic, complemented
29	C₃×#366	C₂²	1	1	yes	characteristic, complemented
30	C₃×#358	C₂²	1	2	yes	complemented
31	C₃×#358	C₂²	1	2	yes	complemented
32	C₃×#357	C₂²	1	2	yes	complemented
33	C₃×#357	C₂²	1	2	yes	complemented
34	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
36	#3050	C₂×C₃	1	2	yes	complemented
37	#2237	C₂×C₃	1	2	yes	complemented
38	#2234	C₂×C₃	1	2	yes	complemented
39	#2217	C₂×C₃	1	2	yes	complemented
40	#2216	C₂×C₃	1	1	yes	characteristic, complemented
41	#2215	C₂×C₃	1	1	yes	characteristic, complemented
42	#2189	C₂×C₃	1	2	yes	complemented
43	#2186	C₂×C₃	1	2	yes	complemented
44	C₂×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×#41	—	2	2	no	—
46	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
47	C₃×#135	—	4	2	no	—
48	C₃×#135	—	4	2	no	—
49	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
50	C₂×C₃×SD₁₆	—	2	2	no	—
51	C₂×C₃×D₈	C₂³	1	2	yes	—
52	C₂²×C₃×C₈	—	4	1	no	—
53	C₂²×C₃×C₈	—	4	1	no	—
54	C₂²×C₃×C₈	D₄	1	2	yes	—
55	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂³	1	2	yes	—
58	C₃×#123	—	4	2	no	—
59	C₃×C₂²⋊Q₈	—	2	2	no	—
60	C₃×C₂²⋊Q₈	—	2	2	no	—
61	C₃×#121	—	2	2	no	—
62	C₃×#121	—	2	2	no	—
63	C₃×#120	—	4	2	no	—
64	C₃×C₄×D₄	—	4	2	no	—
65	C₃×C₄×D₄	—	4	2	no	—
66	C₃×#117	—	2	2	no	—
67	C₃×Q₃₂	—	4	1	no	—
68	C₃×SD₃₂	—	4	2	no	—
69	C₃×D₁₆	—	4	1	no	—
70	C₂×C₃×C₁₆	—	4	1	no	—
71	C₂×C₃×C₁₆	—	4	1	no	—
72	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₁₆	C₂³	1	2	yes	—
74	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
75	C₃×#107	C₂³	1	2	yes	—
76	C₃×#107	C₂³	1	2	yes	—
77	C₃×#106	—	2	2	no	—
78	C₃×#103	—	2	2	no	—
79	C₃×#103	—	2	2	no	—
80	C₃×#102	—	2	2	no	—
81	C₃×#102	—	2	2	no	—
82	C₂×#135	C₂²×C₃	1	2	yes	—
83	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
84	#525	—	2	2	no	—
85	C₂×Q₃₂	C₂²×C₃	1	1	yes	characteristic, complemented
86	C₂×SD₃₂	C₂²×C₃	1	2	yes	complemented
87	C₂×D₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
88	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
90	#467	C₂²×C₃	1	2	yes	—
91	#467	C₂²×C₃	1	2	yes	—
92	#466	C₂²×C₃	1	2	yes	—
93	#466	C₂²×C₃	1	2	yes	—
94	#465	—	2	2	no	—
95	#465	—	2	2	no	—
96	#427	C₂²×C₃	1	2	yes	—
97	#416	—	2	2	no	—
98	#367	C₂²×C₃	1	2	yes	complemented
99	#366	C₂²×C₃	1	1	yes	characteristic, complemented
100	#366	C₂²×C₃	1	1	yes	characteristic, complemented
101	#358	C₂²×C₃	1	2	yes	complemented
102	#358	C₂²×C₃	1	2	yes	complemented
103	#357	C₂²×C₃	1	2	yes	complemented
104	#357	C₂²×C₃	1	2	yes	complemented
105	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
106	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
107	C₂⁴×C₃	D₈	1	1	yes	characteristic
108	C₃×#41	—	4	4	no	—
109	C₃×#41	—	4	4	no	—
110	C₂×C₃×Q₈	—	2	2	no	—
111	C₂×C₃×D₄	—	2	2	no	—
112	C₂×C₃×D₄	—	4	2	no	—
113	C₂×C₃×D₄	—	4	2	no	—
114	C₂×C₃×D₄	—	4	2	no	—
115	C₂²×C₃×C₄	—	4	1	no	—
116	C₂²×C₃×C₄	—	4	1	no	—
117	C₂²×C₃×C₄	—	4	2	no	—
118	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	D₈	1	2	yes	—
120	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
121	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₃×Q₁₆	—	2	2	no	—
123	C₃×Q₁₆	—	2	2	no	—
124	C₃×SD₁₆	—	4	4	no	—
125	C₃×D₈	—	2	2	no	—
126	C₃×D₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	4	2	no	—
132	C₂×C₃×C₈	—	4	1	no	—
133	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	4	2	no	—
139	C₃×C₄⋊C₄	—	4	2	no	—
140	C₃×C₂²⋊C₄	—	4	2	no	—
141	C₃×C₂²⋊C₄	—	4	2	no	—
142	C₃×C₂²⋊C₄	—	4	2	no	—
143	C₃×C₂²⋊C₄	—	4	2	no	—
144	C₃×C₂²⋊C₄	—	4	2	no	—
145	C₃×C₄²	—	4	2	no	—
146	C₃×C₁₆	—	2	1	no	—
147	C₃×C₁₆	—	2	2	no	—
148	C₃×C₁₆	—	2	1	no	—
149	C₂×#41	—	2	2	no	—
150	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
151	#135	—	4	2	no	—
152	#135	—	4	2	no	—
153	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
154	C₂×SD₁₆	—	2	2	no	—
155	C₂×D₈	C₂³×C₃	1	2	yes	—
156	C₂²×C₈	—	4	1	no	—
157	C₂²×C₈	—	4	1	no	—
158	C₂²×C₈	C₃×D₄	1	2	yes	—
159	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₃	1	2	yes	—
161	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
162	#123	—	4	2	no	—
163	C₂²⋊Q₈	—	2	2	no	—
164	C₂²⋊Q₈	—	2	2	no	—
165	#121	—	2	2	no	—
166	#121	—	2	2	no	—
167	#120	—	4	2	no	—
168	C₄×D₄	—	4	2	no	—
169	C₄×D₄	—	4	2	no	—
170	#117	—	2	2	no	—
171	Q₃₂	—	4	1	no	—
172	SD₃₂	—	4	2	no	—
173	D₁₆	—	4	1	no	—
174	C₂×C₁₆	—	4	1	no	—
175	C₂×C₁₆	—	4	1	no	—
176	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂³×C₃	1	2	yes	—
178	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
179	#107	C₂³×C₃	1	2	yes	—
180	#107	C₂³×C₃	1	2	yes	—
181	#106	—	2	2	no	—
182	#103	—	2	2	no	—
183	#103	—	2	2	no	—
184	#102	—	2	2	no	—
185	#102	—	2	2	no	—
186	C₂³×C₃	—	2	2	no	—
187	C₂³×C₃	—	4	1	no	—
188	C₂³×C₃	—	4	1	no	—
189	C₂³×C₃	—	4	2	no	—
190	C₂³×C₃	C₂×D₈	1	2	yes	—
191	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
192	C₃×Q₈	—	4	2	no	—
193	C₃×Q₈	—	4	2	no	—
194	C₃×D₄	—	4	2	no	—
195	C₃×D₄	—	4	2	no	—
196	C₃×D₄	—	8	4	no	—
197	C₃×D₄	—	8	4	no	—
198	C₃×D₄	—	8	4	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	4	1	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	—	4	2	no	—
207	C₂×C₃×C₄	—	4	2	no	—
208	C₂×C₃×C₄	—	8	4	no	—
209	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
210	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	2	2	no	—
214	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
215	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂⁴	C₃×D₈	1	1	yes	characteristic
217	#41	—	4	4	no	—
218	#41	—	4	4	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
228	C₂²×C₄	C₃×D₈	1	2	yes	—
229	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
230	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	SD₁₆	—	4	4	no	—
234	D₈	—	2	2	no	—
235	D₈	—	2	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
243	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
244	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₁₆	—	2	1	no	—
256	C₁₆	—	2	2	no	—
257	C₁₆	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	4	4	no	—
264	C₂²×C₃	—	8	2	no	—
265	C₂²×C₃	—	8	2	no	—
266	C₂²×C₃	—	8	2	no	—
267	C₂²×C₃	#508	1	2	yes	—
268	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	8	2	no	—
272	C₃×C₄	—	8	2	no	—
273	C₃×C₄	—	8	2	no	—
274	C₃×C₄	#534	1	1	yes	characteristic
275	C₃×C₄	#575	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂×C₃×D₈	1	2	yes	—
281	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	8	4	no	—
287	D₄	—	8	4	no	—
288	D₄	—	8	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
300	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	C₃×H(5,𝔽₂)	1	1	yes	characteristic
305	C₈	C₃×H(5,𝔽₂)	1	1	yes	characteristic
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	4	2	no	—
308	C₂×C₃	—	16	2	no	—
309	C₂×C₃	#3186	1	1	yes	characteristic
310	C₂×C₃	C₂×#508	1	1	yes	characteristic
311	C₂×C₃	#3414	1	1	yes	characteristic
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	C₃×#508	1	2	yes	—
322	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	C₃×#534	1	1	yes	characteristic
329	C₄	C₃×#575	1	1	yes	characteristic
330	C₃	#33757	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	4	2	no	—
333	C₂	—	16	2	no	—
334	C₂	C₃×#3186	1	1	yes	characteristic
335	C₂	C₂×C₃×#508	1	1	yes	characteristic
336	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 ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3050	C₂	semidirect
2. C₃×#2237	C₂	semidirect
3. C₃×#2234	C₂	semidirect
4. C₃×#2217	C₂	semidirect
5. C₃×#2216	C₂	semidirect
6. C₃×#2215	C₂	semidirect
7. C₃×#2189	C₂	semidirect
8. C₃×#2186	C₂	semidirect
9. C₂×C₃×C₂²⋊C₁₆	C₂	semidirect
10. #33757	C₃	direct (N × H)
14. C₂×C₃×Q₃₂	C₂²	semidirect
15. C₂×C₃×SD₃₂	C₂²	semidirect
16. C₂×C₃×D₁₆	C₂²	semidirect
27. C₃×#367	C₂²	semidirect
28. C₃×#366	C₂²	semidirect
29. C₃×#366	C₂²	semidirect
30. C₃×#358	C₂²	semidirect
31. C₃×#358	C₂²	semidirect
32. C₃×#357	C₂²	semidirect
33. C₃×#357	C₂²	semidirect
36. #3050	C₂×C₃	semidirect
37. #2237	C₂×C₃	semidirect
38. #2234	C₂×C₃	semidirect
39. #2217	C₂×C₃	semidirect
40. #2216	C₂×C₃	semidirect
41. #2215	C₂×C₃	semidirect
42. #2189	C₂×C₃	semidirect
43. #2186	C₂×C₃	semidirect
44. C₂×C₂²⋊C₁₆	C₂×C₃	semidirect
85. C₂×Q₃₂	C₂²×C₃	semidirect
86. C₂×SD₃₂	C₂²×C₃	semidirect
87. C₂×D₁₆	C₂²×C₃	semidirect
98. #367	C₂²×C₃	semidirect
99. #366	C₂²×C₃	semidirect
100. #366	C₂²×C₃	semidirect
101. #358	C₂²×C₃	semidirect
102. #358	C₂²×C₃	semidirect
103. #357	C₂²×C₃	semidirect
104. #357	C₂²×C₃	semidirect
330. C₃	#33757	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×#3050	2
C₃×#2237	2
C₃×#2234	2
C₃×#2217	2
C₃×#2216	2
C₃×#2215	2
C₃×#2189	2
C₃×#2186	2
C₂×C₃×C₂²⋊C₁₆	2
#33757	3

Subgroups — C3×#33757 #10652364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33757 #10652364