Subgroups — C₃×#29293 #10647900

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2331	C₂	1	2	yes	maximal
2	C₃×#2088	C₂	1	2	yes	maximal
3	C₃×#2087	C₂	1	1	yes	maximal, characteristic
4	C₃×#2086	C₂	1	1	yes	maximal, characteristic
5	C₃×#1988	C₂	1	2	yes	maximal
6	C₃×#1987	C₂	1	2	yes	maximal
7	C₃×#1949	C₂	1	2	yes	maximal
8	C₃×#1949	C₂	1	2	yes	maximal
9	C₃×#1843	C₂	1	1	yes	maximal, characteristic
10	#29293	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₃×#106	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#391	—	2	2	no	—
20	C₃×#391	—	2	2	no	—
21	C₃×#389	C₂²	1	2	yes	—
22	C₃×#389	C₂²	1	2	yes	—
23	C₃×#387	—	2	2	no	—
24	C₃×#387	—	2	2	no	—
25	C₃×#382	C₂²	1	2	yes	—
26	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
27	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
28	C₃×#376	C₂²	1	1	yes	characteristic
29	C₃×#340	C₂²	1	2	yes	—
30	C₃×#340	C₂²	1	1	yes	characteristic
31	C₃×#340	C₂²	1	2	yes	—
32	C₃×#340	C₂²	1	1	yes	characteristic
33	C₃×#340	C₂²	1	2	yes	—
34	C₃×#340	C₂²	1	2	yes	—
35	C₃×#340	C₂²	1	2	yes	—
36	#2331	C₂×C₃	1	2	yes	—
37	#2088	C₂×C₃	1	2	yes	—
38	#2087	C₂×C₃	1	1	yes	characteristic
39	#2086	C₂×C₃	1	1	yes	characteristic
40	#1988	C₂×C₃	1	2	yes	—
41	#1987	C₂×C₃	1	2	yes	—
42	#1949	C₂×C₃	1	2	yes	—
43	#1949	C₂×C₃	1	2	yes	—
44	#1843	C₂×C₃	1	1	yes	characteristic
45	C₂²×C₃×Q₈	—	2	2	no	—
46	C₂²×C₃×C₈	C₂³	1	2	yes	—
47	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
49	C₃×C₄⋊Q₈	—	2	4	no	—
50	C₃×C₄⋊Q₈	Q₈	1	8	yes	complemented
51	C₂×C₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₃×C₄⋊C₄	—	2	2	no	—
53	C₂×C₃×C₄⋊C₄	—	2	2	no	—
54	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
57	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
58	C₂×C₃×C₄²	—	2	2	no	—
59	C₂×C₃×C₄²	—	2	2	no	—
60	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₄²	C₂³	1	2	yes	—
62	C₃×#107	—	4	1	no	—
63	C₃×#107	—	4	1	no	—
64	C₃×#106	—	4	2	no	—
65	C₃×C₄⋊C₈	—	2	4	no	—
66	C₃×C₄⋊C₈	—	2	4	no	—
67	C₃×C₄⋊C₈	—	4	1	no	—
68	C₃×C₄⋊C₈	—	4	1	no	—
69	C₃×#95	—	2	2	no	—
70	C₃×#95	—	2	2	no	—
71	C₃×#95	—	2	2	no	—
72	C₃×#95	—	2	2	no	—
73	C₃×#95	—	2	2	no	—
74	C₃×#95	—	2	2	no	—
75	C₃×#95	D₄	1	1	yes	characteristic
76	C₃×#95	D₄	1	2	yes	—
77	C₃×#95	D₄	1	1	yes	characteristic
78	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
79	C₂×#107	C₂²×C₃	1	1	yes	characteristic
80	C₂×#107	C₂²×C₃	1	1	yes	characteristic
81	C₂×#106	C₂²×C₃	1	2	yes	—
82	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
83	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
84	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
86	#391	—	2	2	no	—
87	#391	—	2	2	no	—
88	#389	C₂²×C₃	1	2	yes	—
89	#389	C₂²×C₃	1	2	yes	—
90	#387	—	2	2	no	—
91	#387	—	2	2	no	—
92	#382	C₂²×C₃	1	2	yes	—
93	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
94	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
95	#376	C₂²×C₃	1	1	yes	characteristic
96	#340	C₂²×C₃	1	2	yes	—
97	#340	C₂²×C₃	1	1	yes	characteristic
98	#340	C₂²×C₃	1	2	yes	—
99	#340	C₂²×C₃	1	1	yes	characteristic
100	#340	C₂²×C₃	1	2	yes	—
101	#340	C₂²×C₃	1	2	yes	—
102	#340	C₂²×C₃	1	2	yes	—
103	C₂×C₃×Q₈	—	2	8	no	—
104	C₂×C₃×Q₈	—	8	2	no	—
105	C₂×C₃×Q₈	—	8	2	no	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	—	2	2	no	—
110	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
113	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
114	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	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	8	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄⋊C₄	—	2	2	no	—
128	C₃×C₄⋊C₄	—	2	2	no	—
129	C₃×C₄⋊C₄	—	8	2	no	—
130	C₃×C₄⋊C₄	—	8	2	no	—
131	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
132	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
133	C₃×C₄²	—	2	1	no	—
134	C₃×C₄²	—	2	1	no	—
135	C₃×C₄²	—	8	2	no	—
136	C₃×C₄²	—	8	2	no	—
137	C₃×C₄²	C₂×Q₈	1	4	yes	—
138	C₃×C₄²	#41	1	4	yes	—
139	C₂²×Q₈	—	2	2	no	—
140	C₂²×C₈	C₂³×C₃	1	2	yes	—
141	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	C₃×Q₈	1	8	yes	complemented
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
149	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
150	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
152	C₂×C₄²	—	2	2	no	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₃	1	2	yes	—
156	#107	—	4	1	no	—
157	#107	—	4	1	no	—
158	#106	—	4	2	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	—	2	4	no	—
161	C₄⋊C₈	—	4	1	no	—
162	C₄⋊C₈	—	4	1	no	—
163	#95	—	2	2	no	—
164	#95	—	2	2	no	—
165	#95	—	2	2	no	—
166	#95	—	2	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	C₃×D₄	1	1	yes	characteristic
170	#95	C₃×D₄	1	2	yes	—
171	#95	C₃×D₄	1	1	yes	characteristic
172	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
173	C₃×Q₈	—	8	8	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	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	—	4	2	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	4	2	no	—
194	C₂×C₃×C₄	—	4	2	no	—
195	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
196	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
197	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
198	C₂×C₃×C₄	C₂×#41	1	2	yes	—
199	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
201	C₃×C₈	—	4	2	no	—
202	C₃×C₈	—	4	1	no	—
203	C₃×C₈	—	4	1	no	—
204	C₂×Q₈	—	2	8	no	—
205	C₂×Q₈	—	8	2	no	—
206	C₂×Q₈	—	8	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
215	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	8	2	no	—
231	C₄⋊C₄	—	8	2	no	—
232	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
233	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	8	2	no	—
237	C₄²	—	8	2	no	—
238	C₄²	C₂×C₃×Q₈	1	4	yes	—
239	C₄²	C₃×#41	1	4	yes	—
240	C₂²×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
241	C₂²×C₃	#537	1	1	yes	characteristic
242	C₂²×C₃	#576	1	1	yes	characteristic
243	C₂²×C₃	#577	1	2	yes	—
244	C₂²×C₃	#578	1	1	yes	characteristic
245	C₃×C₄	—	2	4	no	—
246	C₃×C₄	—	2	4	no	—
247	C₃×C₄	—	4	4	no	—
248	C₃×C₄	—	4	4	no	—
249	C₃×C₄	—	4	1	no	—
250	C₃×C₄	—	4	1	no	—
251	C₃×C₄	—	8	2	no	—
252	C₃×C₄	—	8	2	no	—
253	C₃×C₄	#557	1	2	yes	—
254	C₃×C₄	Q₈²	1	1	yes	characteristic
255	C₃×C₄	#559	1	1	yes	characteristic
256	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
257	Q₈	—	8	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	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	—	4	2	no	—
279	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
280	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
281	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
282	C₂×C₄	C₂×C₃×#41	1	2	yes	—
283	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
284	C₂×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
285	C₈	—	4	2	no	—
286	C₈	—	4	1	no	—
287	C₈	—	4	1	no	—
288	C₂×C₃	#2582	1	1	yes	characteristic
289	C₂×C₃	#3081	1	2	yes	—
290	C₂×C₃	#3082	1	2	yes	—
291	C₂×C₃	#3262	1	1	yes	characteristic
292	C₂×C₃	#3263	1	1	yes	characteristic
293	C₂²	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
294	C₂²	C₃×#537	1	1	yes	characteristic
295	C₂²	C₃×#576	1	1	yes	characteristic
296	C₂²	C₃×#577	1	2	yes	—
297	C₂²	C₃×#578	1	1	yes	characteristic
298	C₄	—	2	4	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₃×#557	1	2	yes	—
307	C₄	C₃×Q₈²	1	1	yes	characteristic
308	C₄	C₃×#559	1	1	yes	characteristic
309	C₃	#29293	1	1	yes	characteristic, direct product factor, Sylow
310	C₂	C₃×#2582	1	1	yes	characteristic
311	C₂	C₃×#3081	1	2	yes	—
312	C₂	C₃×#3082	1	2	yes	—
313	C₂	C₃×#3262	1	1	yes	characteristic
314	C₂	C₃×#3263	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
10. #29293	C₃	direct (N × H)
50. C₃×C₄⋊Q₈	Q₈	semidirect
144. C₄⋊Q₈	C₃×Q₈	semidirect
309. C₃	#29293	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×#2331	2
C₃×#2088	2
C₃×#2087	2
C₃×#2086	2
C₃×#1988	2
C₃×#1987	2
C₃×#1949	2
C₃×#1843	2
#29293	3

Subgroups — C3×#29293 #10647900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29293 #10647900