Subgroups — C₃×#33770 #10652377

Order 768.

576 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3061	C₂	1	2	yes	maximal, complemented
2	C₃×#2235	C₂	1	2	yes	maximal, complemented
3	C₃×#2233	C₂	1	2	yes	maximal, complemented
4	C₃×#2219	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2218	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2217	C₂	1	2	yes	maximal, complemented
7	C₃×#2189	C₂	1	2	yes	maximal, complemented
8	C₃×#2186	C₂	1	2	yes	maximal, complemented
9	C₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	#33770	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#135	C₂²	1	2	yes	—
12	C₂×C₃×#131	C₂²	1	1	yes	characteristic
13	C₃×#537	—	2	2	no	—
14	C₂×C₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₃×D₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₃×#468	—	2	2	no	—
20	C₃×#468	—	2	2	no	—
21	C₃×#467	C₂²	1	2	yes	—
22	C₃×#467	C₂²	1	2	yes	—
23	C₃×#466	C₂²	1	2	yes	—
24	C₃×#466	C₂²	1	2	yes	—
25	C₂×C₃×#107	C₂²	1	2	yes	—
26	C₃×#417	—	2	2	no	—
27	C₃×#367	C₂²	1	2	yes	—
28	C₃×#367	C₂²	1	1	yes	characteristic
29	C₃×#367	C₂²	1	1	yes	characteristic
30	C₃×#358	C₂²	1	2	yes	—
31	C₃×#358	C₂²	1	2	yes	—
32	C₃×#357	C₂²	1	2	yes	—
33	C₃×#357	C₂²	1	2	yes	—
34	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
36	#3061	C₂×C₃	1	2	yes	complemented
37	#2235	C₂×C₃	1	2	yes	complemented
38	#2233	C₂×C₃	1	2	yes	complemented
39	#2219	C₂×C₃	1	1	yes	characteristic, complemented
40	#2218	C₂×C₃	1	1	yes	characteristic, complemented
41	#2217	C₂×C₃	1	2	yes	complemented
42	#2189	C₂×C₃	1	2	yes	complemented
43	#2186	C₂×C₃	1	2	yes	complemented
44	#2113	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×#41	—	2	2	no	—
46	C₂×C₃×#41	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	C₃×#131	—	4	1	no	—
53	C₃×#131	—	4	1	no	—
54	C₂×C₃×M₁₆	D₄	1	2	yes	—
55	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂³	1	2	yes	—
58	C₃×C₂²⋊Q₈	—	2	2	no	—
59	C₃×C₂²⋊Q₈	—	2	2	no	—
60	C₃×C₂²⋊Q₈	—	4	2	no	—
61	C₃×#121	—	2	2	no	—
62	C₃×#121	—	2	2	no	—
63	C₃×#121	—	4	2	no	—
64	C₃×#121	—	4	2	no	—
65	C₃×#121	—	4	2	no	—
66	C₂×C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×Q₃₂	—	4	1	no	—
68	C₃×SD₃₂	—	4	2	no	—
69	C₃×D₁₆	—	4	1	no	—
70	C₃×M₃₂	—	4	1	no	—
71	C₂×C₃×C₁₆	—	4	1	no	—
72	C₂×C₃×C₁₆	C₂³	1	2	yes	—
73	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
75	C₃×#107	—	2	2	no	—
76	C₃×#107	C₂³	1	2	yes	—
77	C₃×#107	C₂³	1	2	yes	—
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	C₂×#131	C₂²×C₃	1	1	yes	characteristic
84	#537	—	2	2	no	—
85	C₂×Q₃₂	C₂²×C₃	1	1	yes	characteristic
86	C₂×SD₃₂	C₂²×C₃	1	2	yes	—
87	C₂×D₁₆	C₂²×C₃	1	1	yes	characteristic
88	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
90	#468	—	2	2	no	—
91	#468	—	2	2	no	—
92	#467	C₂²×C₃	1	2	yes	—
93	#467	C₂²×C₃	1	2	yes	—
94	#466	C₂²×C₃	1	2	yes	—
95	#466	C₂²×C₃	1	2	yes	—
96	C₂×#107	C₂²×C₃	1	2	yes	—
97	#417	—	2	2	no	—
98	#367	C₂²×C₃	1	2	yes	—
99	#367	C₂²×C₃	1	1	yes	characteristic
100	#367	C₂²×C₃	1	1	yes	characteristic
101	#358	C₂²×C₃	1	2	yes	—
102	#358	C₂²×C₃	1	2	yes	—
103	#357	C₂²×C₃	1	2	yes	—
104	#357	C₂²×C₃	1	2	yes	—
105	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
106	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
107	C₃×#41	—	4	1	no	—
108	C₃×#41	—	4	1	no	—
109	C₃×#41	—	4	4	no	—
110	C₃×#41	—	4	4	no	—
111	C₂×C₃×Q₈	—	2	2	no	—
112	C₂×C₃×Q₈	D₈	1	1	yes	characteristic
113	C₂×C₃×D₄	—	2	2	no	—
114	C₂×C₃×D₄	—	4	2	no	—
115	C₂×C₃×D₄	—	4	2	no	—
116	C₂×C₃×D₄	—	4	2	no	—
117	C₂×C₃×D₄	D₈	1	1	yes	characteristic
118	C₂×C₃×D₄	D₈	1	2	yes	—
119	C₂²×C₃×C₄	—	4	2	no	—
120	C₂²×C₃×C₄	—	4	2	no	—
121	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
122	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
123	C₃×Q₁₆	—	2	2	no	—
124	C₃×Q₁₆	—	2	2	no	—
125	C₃×SD₁₆	—	4	4	no	—
126	C₃×D₈	—	2	2	no	—
127	C₃×D₈	—	2	2	no	—
128	C₃×M₁₆	—	4	2	no	—
129	C₃×M₁₆	—	4	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
136	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	4	2	no	—
140	C₃×C₄⋊C₄	—	4	2	no	—
141	C₃×C₄⋊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×C₂²⋊C₄	—	4	2	no	—
146	C₃×C₁₆	—	2	2	no	—
147	C₃×C₁₆	—	2	1	no	—
148	C₃×C₁₆	—	2	1	no	—
149	C₂×#41	—	2	2	no	—
150	C₂×#41	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	#131	—	4	1	no	—
157	#131	—	4	1	no	—
158	C₂×M₁₆	C₃×D₄	1	2	yes	—
159	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₃	1	2	yes	—
161	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
162	C₂²⋊Q₈	—	2	2	no	—
163	C₂²⋊Q₈	—	2	2	no	—
164	C₂²⋊Q₈	—	4	2	no	—
165	#121	—	2	2	no	—
166	#121	—	2	2	no	—
167	#121	—	4	2	no	—
168	#121	—	4	2	no	—
169	#121	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	Q₃₂	—	4	1	no	—
172	SD₃₂	—	4	2	no	—
173	D₁₆	—	4	1	no	—
174	M₃₂	—	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	—	2	2	no	—
180	#107	C₂³×C₃	1	2	yes	—
181	#107	C₂³×C₃	1	2	yes	—
182	#103	—	2	2	no	—
183	#103	—	2	2	no	—
184	#102	—	2	2	no	—
185	#102	—	2	2	no	—
186	C₂³×C₃	—	4	2	no	—
187	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
188	C₂³×C₃	C₂×D₈	1	2	yes	—
189	C₃×Q₈	—	4	1	no	—
190	C₃×Q₈	—	4	2	no	—
191	C₃×Q₈	—	4	2	no	—
192	C₃×D₄	—	4	2	no	—
193	C₃×D₄	—	4	1	no	—
194	C₃×D₄	—	4	2	no	—
195	C₃×D₄	—	4	2	no	—
196	C₃×D₄	—	8	4	no	—
197	C₃×D₄	—	16	2	no	—
198	C₃×D₄	—	16	2	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	2	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	—	8	4	no	—
207	C₂×C₃×C₄	—	16	2	no	—
208	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
210	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	—	2	2	no	—
213	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
214	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
215	#41	—	4	1	no	—
216	#41	—	4	1	no	—
217	#41	—	4	4	no	—
218	#41	—	4	4	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	C₃×D₈	1	1	yes	characteristic
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	C₃×D₈	1	1	yes	characteristic
226	C₂×D₄	C₃×D₈	1	2	yes	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
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	M₁₆	—	4	1	no	—
237	M₁₆	—	4	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	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	C₄⋊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	C₁₆	—	2	2	no	—
255	C₁₆	—	2	1	no	—
256	C₁₆	—	2	1	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	8	2	no	—
263	C₂²×C₃	—	8	2	no	—
264	C₂²×C₃	—	8	2	no	—
265	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	8	2	no	—
269	C₃×C₄	—	8	2	no	—
270	C₃×C₄	—	8	2	no	—
271	C₃×C₄	#534	1	1	yes	characteristic
272	C₃×C₄	#575	1	1	yes	characteristic
273	C₂³	—	4	2	no	—
274	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
275	C₂³	C₂×C₃×D₈	1	2	yes	—
276	Q₈	—	4	1	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	4	no	—
284	D₄	—	16	2	no	—
285	D₄	—	16	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
296	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
297	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	C₃×H(5,𝔽₂)	1	1	yes	characteristic
301	C₈	C₃×H(5,𝔽₂)	1	1	yes	characteristic
302	C₂×C₃	—	4	2	no	—
303	C₂×C₃	—	4	1	no	—
304	C₂×C₃	—	16	2	no	—
305	C₂×C₃	#3186	1	1	yes	characteristic
306	C₂×C₃	#3416	1	2	yes	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₃×#534	1	1	yes	characteristic
322	C₄	C₃×#575	1	1	yes	characteristic
323	C₃	#33770	1	1	yes	characteristic, direct product factor, Sylow
324	C₂	—	4	2	no	—
325	C₂	—	4	1	no	—
326	C₂	—	16	2	no	—
327	C₂	C₃×#3186	1	1	yes	characteristic
328	C₂	C₃×#3416	1	2	yes	—

	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×#41 ◁ C₂×C₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3061	C₂	semidirect
2. C₃×#2235	C₂	semidirect
3. C₃×#2233	C₂	semidirect
4. C₃×#2219	C₂	semidirect
5. C₃×#2218	C₂	semidirect
6. C₃×#2217	C₂	semidirect
7. C₃×#2189	C₂	semidirect
8. C₃×#2186	C₂	semidirect
9. C₃×#2113	C₂	semidirect
10. #33770	C₃	direct (N × H)
36. #3061	C₂×C₃	semidirect
37. #2235	C₂×C₃	semidirect
38. #2233	C₂×C₃	semidirect
39. #2219	C₂×C₃	semidirect
40. #2218	C₂×C₃	semidirect
41. #2217	C₂×C₃	semidirect
42. #2189	C₂×C₃	semidirect
43. #2186	C₂×C₃	semidirect
44. #2113	C₂×C₃	semidirect
323. C₃	#33770	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×#3061	2
C₃×#2235	2
C₃×#2233	2
C₃×#2219	2
C₃×#2218	2
C₃×#2217	2
C₃×#2189	2
C₃×#2186	2
C₃×#2113	2
#33770	3

Subgroups — C3×#33770 #10652377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33770 #10652377