Subgroups — C₃×#29210 #10647817

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×#2059	C₂	1	4	yes	maximal
4	C₃×#1991	C₂	1	2	yes	maximal
5	C₃×#1948	C₂	1	2	yes	maximal
6	C₃×#1947	C₂	1	2	yes	maximal
7	C₃×#1846	C₂	1	2	yes	maximal
8	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#29210	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₃×#125	C₂²	1	2	yes	—
12	C₂×C₃×#107	C₂²	1	4	yes	—
13	C₂×C₃×#106	C₂²	1	4	yes	—
14	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
17	C₃×#389	—	2	2	no	—
18	C₃×#389	—	2	2	no	—
19	C₃×#389	C₂²	1	2	yes	—
20	C₃×#387	—	2	2	no	—
21	C₃×#384	C₂²	1	2	yes	—
22	C₃×#382	C₂²	1	2	yes	—
23	C₃×#382	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₄⋊C₄	—	2	2	no	—
25	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
26	C₃×C₄³	C₂²	1	1	yes	characteristic
27	C₃×#340	C₂²	1	4	yes	—
28	C₃×#340	C₂²	1	4	yes	—
29	C₃×#336	C₂²	1	2	yes	—
30	#2859	C₂×C₃	1	1	yes	characteristic
31	C₄×#125	C₂×C₃	1	1	yes	characteristic
32	#2059	C₂×C₃	1	4	yes	—
33	#1991	C₂×C₃	1	2	yes	—
34	#1948	C₂×C₃	1	2	yes	—
35	#1947	C₂×C₃	1	2	yes	—
36	#1846	C₂×C₃	1	2	yes	—
37	C₄×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
38	C₂²×C₃×C₈	C₂³	1	2	yes	—
39	C₂²×C₃×C₈	C₂³	1	2	yes	—
40	C₃×#125	—	2	2	no	—
41	C₃×#125	—	4	2	no	—
42	C₃×#125	—	4	2	no	—
43	C₃×#125	Q₈	1	4	yes	—
44	C₂×C₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₃×C₄⋊C₄	—	2	4	no	—
49	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
51	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₃×C₄²	—	2	2	no	—
53	C₂×C₃×C₄²	D₄	1	2	yes	—
54	C₂×C₃×C₄²	D₄	1	2	yes	—
55	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
58	C₃×#107	—	4	4	no	—
59	C₃×#106	—	4	4	no	—
60	C₃×C₄⋊C₈	Q₈	1	4	yes	—
61	C₃×C₄⋊C₈	Q₈	1	4	yes	—
62	C₃×C₄×C₈	—	2	4	no	—
63	C₃×#95	—	2	2	no	—
64	C₃×#95	—	2	2	no	—
65	C₃×#95	—	2	2	no	—
66	C₃×#95	—	2	4	no	—
67	C₂×#125	C₂²×C₃	1	1	yes	characteristic
68	C₂×#125	C₂²×C₃	1	2	yes	—
69	C₂×#107	C₂²×C₃	1	4	yes	—
70	C₂×#106	C₂²×C₃	1	4	yes	—
71	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
74	#389	—	2	2	no	—
75	#389	—	2	2	no	—
76	#389	C₂²×C₃	1	2	yes	—
77	#387	—	2	2	no	—
78	#384	C₂²×C₃	1	2	yes	—
79	#382	C₂²×C₃	1	1	yes	characteristic
80	#382	C₂²×C₃	1	2	yes	—
81	C₄×C₄⋊C₄	—	2	2	no	—
82	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₄³	C₂²×C₃	1	1	yes	characteristic
84	#340	C₂²×C₃	1	4	yes	—
85	#340	C₂²×C₃	1	4	yes	—
86	#336	C₂²×C₃	1	2	yes	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	4	no	—
90	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
92	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
100	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
101	C₃×C₄⋊C₄	—	2	8	no	—
102	C₃×C₄⋊C₄	—	2	2	no	—
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₃×C₄⋊C₄	—	2	4	no	—
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₄⋊C₄	—	2	4	no	—
107	C₃×C₄⋊C₄	—	4	4	no	—
108	C₃×C₄⋊C₄	—	4	4	no	—
109	C₃×C₄⋊C₄	—	4	8	no	—
110	C₃×C₄⋊C₄	—	4	4	no	—
111	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
112	C₃×C₄²	—	2	4	no	—
113	C₃×C₄²	—	2	1	no	—
114	C₃×C₄²	—	2	1	no	—
115	C₃×C₄²	—	4	2	no	—
116	C₃×C₄²	—	4	4	no	—
117	C₃×C₄²	C₂×Q₈	1	2	yes	—
118	C₃×C₄²	C₂×Q₈	1	2	yes	—
119	C₃×C₄²	C₂×Q₈	1	2	yes	—
120	C₃×C₄²	#41	1	2	yes	—
121	C₂²×C₈	C₂³×C₃	1	2	yes	—
122	C₂²×C₈	C₂³×C₃	1	2	yes	—
123	#125	—	2	2	no	—
124	#125	—	4	2	no	—
125	#125	—	4	2	no	—
126	#125	C₃×Q₈	1	4	yes	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
133	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₃×D₄	1	2	yes	—
137	C₂×C₄²	C₃×D₄	1	2	yes	—
138	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
141	#107	—	4	4	no	—
142	#106	—	4	4	no	—
143	C₄⋊C₈	C₃×Q₈	1	4	yes	—
144	C₄⋊C₈	C₃×Q₈	1	4	yes	—
145	C₄×C₈	—	2	4	no	—
146	#95	—	2	2	no	—
147	#95	—	2	2	no	—
148	#95	—	2	2	no	—
149	#95	—	2	4	no	—
150	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	4	4	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	4	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	4	4	no	—
166	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
167	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
168	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
169	C₂×C₃×C₄	#135	1	4	yes	—
170	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
174	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
176	C₃×C₈	—	2	4	no	—
177	C₃×C₈	—	2	4	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
182	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
191	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	8	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	4	2	no	—
207	C₄²	—	4	4	no	—
208	C₄²	C₂×C₃×Q₈	1	2	yes	—
209	C₄²	C₂×C₃×Q₈	1	2	yes	—
210	C₄²	C₂×C₃×Q₈	1	2	yes	—
211	C₄²	C₃×#41	1	2	yes	—
212	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
213	C₂²×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
214	C₂²×C₃	#535	1	1	yes	characteristic
215	C₂²×C₃	C₂×#135	1	2	yes	—
216	C₂²×C₃	#575	1	2	yes	—
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	4	4	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	8	4	no	—
222	C₃×C₄	—	8	2	no	—
223	C₃×C₄	#499	1	4	yes	—
224	C₃×C₄	#554	1	1	yes	characteristic
225	C₃×C₄	#554	1	1	yes	characteristic
226	C₃×C₄	Q₈²	1	1	yes	characteristic
227	C₃×C₄	Q₈²	1	1	yes	characteristic
228	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
245	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
246	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
247	C₂×C₄	C₃×#135	1	4	yes	—
248	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
252	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
253	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
254	C₈	—	2	4	no	—
255	C₈	—	2	4	no	—
256	C₂×C₃	#2567	1	1	yes	characteristic
257	C₂×C₃	#3076	1	2	yes	—
258	C₂×C₃	C₂×#499	1	1	yes	characteristic
259	C₂×C₃	#3166	1	1	yes	characteristic
260	C₂×C₃	#3212	1	2	yes	—
261	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
262	C₂²	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
263	C₂²	C₃×#535	1	1	yes	characteristic
264	C₂²	C₂×C₃×#135	1	2	yes	—
265	C₂²	C₃×#575	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	4	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	4	no	—
272	C₄	C₃×#499	1	4	yes	—
273	C₄	C₃×#554	1	1	yes	characteristic
274	C₄	C₃×#554	1	1	yes	characteristic
275	C₄	C₃×Q₈²	1	1	yes	characteristic
276	C₄	C₃×Q₈²	1	1	yes	characteristic
277	C₃	#29210	1	1	yes	characteristic, direct product factor, Sylow
278	C₂	C₃×#2567	1	1	yes	characteristic
279	C₂	C₃×#3076	1	2	yes	—
280	C₂	C₂×C₃×#499	1	1	yes	characteristic
281	C₂	C₃×#3166	1	1	yes	characteristic
282	C₂	C₃×#3212	1	2	yes	—

	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×C₄³ ◁ G

Complements

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

N	H ≅ G/N	type
9. #29210	C₃	direct (N × H)
277. C₃	#29210	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×#2859	2
C₃×C₄×#125	2
C₃×#2059	2
C₃×#1991	2
C₃×#1948	2
C₃×#1947	2
C₃×#1846	2
C₃×C₄×C₄⋊C₈	2
#29210	3

Subgroups — C3×#29210 #10647817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29210 #10647817