Subgroups — C₃×#29233 #10647840

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2102	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#2050	C₂	1	1	yes	maximal, characteristic
3	C₃×#2031	C₂	1	2	yes	maximal, complemented
4	C₃×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1932	C₂	1	2	yes	maximal, complemented
6	C₃×#1906	C₂	1	2	yes	maximal
7	C₃×#1841	C₂	1	2	yes	maximal, complemented
8	C₃×#1827	C₂	1	2	yes	maximal, complemented
9	C₃×#1759	C₂	1	2	yes	maximal, complemented
10	#29233	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×#575	C₂²	1	1	yes	characteristic
12	C₃×#533	C₂²	1	1	yes	characteristic
13	C₃×C₄×#41	C₂²	1	2	yes	—
14	C₃×#430	C₂²	1	1	yes	characteristic
15	C₃×#423	C₂²	1	1	yes	characteristic
16	C₃×#421	C₂²	1	2	yes	—
17	C₂×C₃×#104	C₂²	1	2	yes	—
18	C₂×C₃×#104	C₂²	1	2	yes	—
19	C₃×#413	C₂²	1	2	yes	—
20	C₃×#410	C₂²	1	2	yes	—
21	C₃×C₄×M₁₆	C₂²	1	2	yes	—
22	C₃×#376	C₂²	1	1	yes	characteristic
23	C₃×#344	C₂²	1	2	yes	—
24	C₃×#344	C₂²	1	2	yes	—
25	C₃×#339	C₂²	1	2	yes	—
26	C₃×#339	C₂²	1	2	yes	—
27	C₃×#339	C₂²	1	2	yes	—
28	C₃×#339	C₂²	1	2	yes	—
29	C₃×#338	C₂²	1	1	yes	characteristic
30	C₃×H(3,ℤ/4)	C₂²	1	2	yes	—
31	C₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
32	#2102	C₂×C₃	1	1	yes	characteristic, complemented
33	#2050	C₂×C₃	1	1	yes	characteristic
34	#2031	C₂×C₃	1	2	yes	complemented
35	#1995	C₂×C₃	1	1	yes	characteristic, complemented
36	#1932	C₂×C₃	1	2	yes	complemented
37	#1906	C₂×C₃	1	2	yes	—
38	#1841	C₂×C₃	1	2	yes	complemented
39	#1827	C₂×C₃	1	2	yes	complemented
40	#1759	C₂×C₃	1	2	yes	complemented
41	C₂×C₃×#41	C₂³	1	2	yes	—
42	C₃×#137	—	2	2	no	—
43	C₃×AGL(1,ℤ/8)	—	2	2	no	—
44	C₃×#135	—	4	1	no	—
45	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₃×M₁₆	C₂³	1	2	yes	—
47	C₂×C₃×M₁₆	C₂³	1	2	yes	—
48	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
49	C₃×C₄⋊Q₈	—	2	1	no	—
50	C₃×#125	—	2	1	no	—
51	C₃×C₂²⋊Q₈	—	4	1	no	—
52	C₃×C₄×Q₈	—	2	2	no	—
53	C₃×C₄×Q₈	—	4	1	no	—
54	C₃×C₄×D₄	—	2	2	no	—
55	C₃×C₄×D₄	—	4	2	no	—
56	C₃×#117	—	2	2	no	—
57	C₃×#117	C₂³	1	2	yes	—
58	C₃×#117	C₂³	1	2	yes	—
59	C₂×C₃×C₄²	—	2	2	no	—
60	C₂×C₃×C₄²	C₂³	1	2	yes	—
61	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
62	C₃×#108	—	4	1	no	—
63	C₃×C₄⋊C₈	—	4	1	no	—
64	C₃×#104	—	2	2	no	—
65	C₃×#104	—	2	2	no	—
66	C₃×#104	—	2	2	no	—
67	C₃×#104	—	2	2	no	—
68	C₃×#104	—	4	2	no	—
69	C₃×#101	—	2	2	no	—
70	C₃×#100	—	2	2	no	—
71	C₃×#99	—	4	2	no	—
72	C₃×#97	—	2	2	no	—
73	C₃×C₄×C₈	—	2	2	no	—
74	C₃×#95	—	4	1	no	—
75	#575	C₂²×C₃	1	1	yes	characteristic
76	#533	C₂²×C₃	1	1	yes	characteristic
77	C₄×#41	C₂²×C₃	1	2	yes	—
78	#430	C₂²×C₃	1	1	yes	characteristic
79	#423	C₂²×C₃	1	1	yes	characteristic
80	#421	C₂²×C₃	1	2	yes	—
81	C₂×#104	C₂²×C₃	1	2	yes	—
82	C₂×#104	C₂²×C₃	1	2	yes	—
83	#413	C₂²×C₃	1	2	yes	—
84	#410	C₂²×C₃	1	2	yes	—
85	C₄×M₁₆	C₂²×C₃	1	2	yes	—
86	#376	C₂²×C₃	1	1	yes	characteristic
87	#344	C₂²×C₃	1	2	yes	—
88	#344	C₂²×C₃	1	2	yes	—
89	#339	C₂²×C₃	1	2	yes	—
90	#339	C₂²×C₃	1	2	yes	—
91	#339	C₂²×C₃	1	2	yes	—
92	#339	C₂²×C₃	1	2	yes	—
93	#338	C₂²×C₃	1	1	yes	characteristic
94	H(3,ℤ/4)	C₂²×C₃	1	1	yes	characteristic
95	H(3,ℤ/4)	C₂²×C₃	1	2	yes	—
96	C₃×#41	—	2	2	no	—
97	C₃×#41	—	2	2	no	—
98	C₃×#41	—	2	2	no	—
99	C₃×#41	—	2	2	no	—
100	C₂×C₃×Q₈	—	4	1	no	—
101	C₂×C₃×Q₈	#41	1	2	yes	—
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	#41	1	2	yes	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	—	4	2	no	—
108	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
109	C₃×Q₁₆	—	4	1	no	—
110	C₃×SD₁₆	—	4	2	no	—
111	C₃×D₈	—	4	1	no	—
112	C₃×M₁₆	—	2	2	no	—
113	C₃×M₁₆	—	2	2	no	—
114	C₃×M₁₆	—	2	2	no	—
115	C₃×M₁₆	—	2	2	no	—
116	C₃×M₁₆	—	2	2	no	—
117	C₃×M₁₆	—	2	2	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	4	2	no	—
125	C₃×C₄⋊C₄	—	4	2	no	—
126	C₃×C₄⋊C₄	—	4	1	no	—
127	C₃×C₂²⋊C₄	—	2	2	no	—
128	C₃×C₂²⋊C₄	—	2	2	no	—
129	C₃×C₂²⋊C₄	—	4	2	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	2	1	no	—
132	C₃×C₄²	—	2	2	no	—
133	C₃×C₄²	—	2	2	no	—
134	C₃×C₄²	—	4	2	no	—
135	C₃×C₄²	#41	1	2	yes	—
136	C₃×C₄²	#41	1	2	yes	—
137	C₃×C₄²	#41	1	1	yes	characteristic
138	C₃×C₄²	#41	1	1	yes	characteristic
139	C₂×#41	C₂³×C₃	1	2	yes	—
140	#137	—	2	2	no	—
141	AGL(1,ℤ/8)	—	2	2	no	—
142	#135	—	4	1	no	—
143	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
144	C₂×M₁₆	C₂³×C₃	1	2	yes	—
145	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
146	C₂×M₁₆	C₂³×C₃	1	2	yes	—
147	C₄⋊Q₈	—	2	1	no	—
148	#125	—	2	1	no	—
149	C₂²⋊Q₈	—	4	1	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	—	4	1	no	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	4	2	no	—
154	#117	—	2	2	no	—
155	#117	C₂³×C₃	1	2	yes	—
156	#117	C₂³×C₃	1	2	yes	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₂³×C₃	1	2	yes	—
159	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
160	#108	—	4	1	no	—
161	C₄⋊C₈	—	4	1	no	—
162	#104	—	2	2	no	—
163	#104	—	2	2	no	—
164	#104	—	2	2	no	—
165	#104	—	2	2	no	—
166	#104	—	4	2	no	—
167	#101	—	2	2	no	—
168	#100	—	2	2	no	—
169	#99	—	4	2	no	—
170	#97	—	2	2	no	—
171	C₄×C₈	—	2	2	no	—
172	#95	—	4	1	no	—
173	C₂³×C₃	—	2	2	no	—
174	C₂³×C₃	H(5,𝔽₂)	1	1	yes	characteristic
175	C₃×Q₈	—	2	2	no	—
176	C₃×Q₈	—	2	2	no	—
177	C₃×Q₈	—	8	2	no	—
178	C₃×D₄	—	2	2	no	—
179	C₃×D₄	—	2	2	no	—
180	C₃×D₄	—	4	2	no	—
181	C₃×D₄	—	4	2	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	4	2	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	—	8	2	no	—
192	C₂×C₃×C₄	—	8	1	no	—
193	C₂×C₃×C₄	—	8	4	no	—
194	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×#41	1	2	yes	—
196	C₂×C₃×C₄	C₂×#41	1	2	yes	—
197	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
198	C₃×C₈	—	4	2	no	—
199	C₃×C₈	—	4	1	no	—
200	C₃×C₈	—	4	1	no	—
201	C₃×C₈	—	4	2	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	—	2	2	no	—
205	#41	—	2	2	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	C₃×#41	1	2	yes	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	C₃×#41	1	2	yes	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
215	Q₁₆	—	4	1	no	—
216	SD₁₆	—	4	2	no	—
217	D₈	—	4	1	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₃×#41	1	2	yes	—
242	C₄²	C₃×#41	1	2	yes	—
243	C₄²	C₃×#41	1	1	yes	characteristic
244	C₄²	C₃×#41	1	1	yes	characteristic
245	C₂²×C₃	—	4	1	no	—
246	C₂²×C₃	—	4	2	no	—
247	C₂²×C₃	—	8	2	no	—
248	C₂²×C₃	#538	1	1	yes	characteristic
249	C₂²×C₃	#551	1	2	yes	—
250	C₃×C₄	—	4	1	no	—
251	C₃×C₄	—	4	2	no	—
252	C₃×C₄	—	4	2	no	—
253	C₃×C₄	—	4	1	no	—
254	C₃×C₄	—	4	2	no	—
255	C₃×C₄	—	8	2	no	—
256	C₃×C₄	—	8	2	no	—
257	C₃×C₄	#529	1	2	yes	—
258	C₃×C₄	#551	1	1	yes	characteristic
259	C₃×C₄	#555	1	1	yes	characteristic
260	C₂³	—	2	2	no	—
261	C₂³	C₃×H(5,𝔽₂)	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	2	2	no	—
264	Q₈	—	8	2	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
282	C₂×C₄	C₂×C₃×#41	1	2	yes	—
283	C₂×C₄	C₂×C₃×#41	1	2	yes	—
284	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
285	C₈	—	4	2	no	—
286	C₈	—	4	1	no	—
287	C₈	—	4	1	no	—
288	C₈	—	4	2	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	8	2	no	—
292	C₂×C₃	#2572	1	1	yes	characteristic
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	C₃×#538	1	1	yes	characteristic
297	C₂²	C₃×#551	1	2	yes	—
298	C₄	—	4	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₃×#529	1	2	yes	—
306	C₄	C₃×#551	1	1	yes	characteristic
307	C₄	C₃×#555	1	1	yes	characteristic
308	C₃	#29233	1	1	yes	characteristic, direct product factor, Sylow
309	C₂	—	2	1	no	—
310	C₂	—	2	2	no	—
311	C₂	—	8	2	no	—
312	C₂	C₃×#2572	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	#2572
prime radical	C₃×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#2102	C₂	semidirect
3. C₃×#2031	C₂	semidirect
4. C₃×#1995	C₂	semidirect
5. C₃×#1932	C₂	semidirect
7. C₃×#1841	C₂	semidirect
8. C₃×#1827	C₂	semidirect
9. C₃×#1759	C₂	semidirect
10. #29233	C₃	direct (N × H)
32. #2102	C₂×C₃	semidirect
34. #2031	C₂×C₃	semidirect
35. #1995	C₂×C₃	semidirect
36. #1932	C₂×C₃	semidirect
38. #1841	C₂×C₃	semidirect
39. #1827	C₂×C₃	semidirect
40. #1759	C₂×C₃	semidirect
308. C₃	#29233	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×#2102	2
C₃×#2050	2
C₃×#2031	2
C₃×#1995	2
C₃×#1932	2
C₃×#1906	2
C₃×#1841	2
C₃×#1827	2
C₃×#1759	2
#29233	3

Subgroups — C3×#29233 #10647840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29233 #10647840