Subgroups — C₃×#29227 #10647834

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#2101	C₂	1	4	yes	maximal
4	C₃×#1992	C₂	1	1	yes	maximal, characteristic
5	C₃×#1992	C₂	1	1	yes	maximal, characteristic
6	C₃×#1989	C₂	1	4	yes	maximal
7	C₃×#1836	C₂	1	1	yes	maximal, characteristic
8	C₃×#1836	C₂	1	1	yes	maximal, characteristic
9	C₃×#1767	C₂	1	1	yes	maximal, characteristic
10	#29227	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#125	C₂²	1	2	yes	—
12	C₂×C₃×#125	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃×#97	C₂²	1	1	yes	characteristic
15	C₂×C₃×#97	C₂²	1	1	yes	characteristic
16	C₃×#389	—	2	2	no	—
17	C₃×#389	—	2	2	no	—
18	C₃×#389	C₂²	1	2	yes	—
19	C₃×#387	—	2	2	no	—
20	C₃×#384	C₂²	1	1	yes	characteristic
21	C₃×#384	C₂²	1	1	yes	characteristic
22	C₃×#382	C₂²	1	1	yes	characteristic
23	C₃×#382	C₂²	1	2	yes	—
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×#340	C₂²	1	4	yes	—
30	C₃×#340	C₂²	1	4	yes	—
31	C₃×#336	C₂²	1	2	yes	—
32	#2859	C₂×C₃	1	1	yes	characteristic
33	C₄×#125	C₂×C₃	1	1	yes	characteristic
34	#2101	C₂×C₃	1	4	yes	—
35	#1992	C₂×C₃	1	1	yes	characteristic
36	#1992	C₂×C₃	1	1	yes	characteristic
37	#1989	C₂×C₃	1	4	yes	—
38	#1836	C₂×C₃	1	1	yes	characteristic
39	#1836	C₂×C₃	1	1	yes	characteristic
40	#1767	C₂×C₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂³	1	2	yes	—
42	C₂²×C₃×C₈	C₂³	1	2	yes	—
43	C₃×#125	—	2	2	no	—
44	C₃×#125	—	2	8	no	—
45	C₃×#125	Q₈	1	4	yes	—
46	C₂×C₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₃×C₄²	—	2	2	no	—
57	C₂×C₃×C₄²	D₄	1	2	yes	—
58	C₂×C₃×C₄²	D₄	1	2	yes	—
59	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₈	—	2	4	no	—
63	C₃×#97	—	2	2	no	—
64	C₃×#97	—	2	2	no	—
65	C₃×#95	—	2	2	no	—
66	C₃×#95	—	2	2	no	—
67	C₃×#95	—	2	2	no	—
68	C₃×#95	—	2	4	no	—
69	C₂×#125	C₂²×C₃	1	2	yes	—
70	C₂×#125	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
72	C₂×#97	C₂²×C₃	1	1	yes	characteristic
73	C₂×#97	C₂²×C₃	1	1	yes	characteristic
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	1	yes	characteristic
79	#384	C₂²×C₃	1	1	yes	characteristic
80	#382	C₂²×C₃	1	1	yes	characteristic
81	#382	C₂²×C₃	1	2	yes	—
82	C₄×C₄⋊C₄	—	2	2	no	—
83	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄³	C₂²×C₃	1	1	yes	characteristic
85	#340	C₂²×C₃	1	4	yes	—
86	#340	C₂²×C₃	1	4	yes	—
87	#340	C₂²×C₃	1	4	yes	—
88	#340	C₂²×C₃	1	4	yes	—
89	#336	C₂²×C₃	1	2	yes	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	2	no	—
92	C₂²×C₃×C₄	—	2	2	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
95	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
96	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₃×C₄⋊C₄	—	2	8	no	—
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	—	2	4	no	—
108	C₃×C₄⋊C₄	—	2	4	no	—
109	C₃×C₄⋊C₄	—	2	4	no	—
110	C₃×C₄⋊C₄	—	4	4	no	—
111	C₃×C₄⋊C₄	—	4	4	no	—
112	C₃×C₄⋊C₄	—	4	4	no	—
113	C₃×C₄⋊C₄	—	4	4	no	—
114	C₃×C₄⋊C₄	—	4	4	no	—
115	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
116	C₃×C₄²	—	2	2	no	—
117	C₃×C₄²	—	4	2	no	—
118	C₃×C₄²	—	4	2	no	—
119	C₃×C₄²	—	4	4	no	—
120	C₃×C₄²	C₂×Q₈	1	2	yes	—
121	C₃×C₄²	#41	1	2	yes	—
122	C₃×C₄²	#41	1	4	yes	—
123	C₂²×C₈	C₂³×C₃	1	2	yes	—
124	C₂²×C₈	C₂³×C₃	1	2	yes	—
125	#125	—	2	2	no	—
126	#125	—	2	8	no	—
127	#125	C₃×Q₈	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	2	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	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
135	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
136	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
137	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₃×D₄	1	2	yes	—
140	C₂×C₄²	C₃×D₄	1	2	yes	—
141	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
143	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
144	C₄⋊C₈	—	2	4	no	—
145	#97	—	2	2	no	—
146	#97	—	2	2	no	—
147	#95	—	2	2	no	—
148	#95	—	2	2	no	—
149	#95	—	2	2	no	—
150	#95	—	2	4	no	—
151	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	—	4	4	no	—
167	C₂×C₃×C₄	—	4	4	no	—
168	C₂×C₃×C₄	—	4	2	no	—
169	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
170	C₂×C₃×C₄	#124	1	2	yes	—
171	C₂×C₃×C₄	#124	1	2	yes	—
172	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×#41	1	2	yes	—
175	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
176	C₃×C₈	—	4	2	no	—
177	C₃×C₈	—	4	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
204	C₄²	—	2	2	no	—
205	C₄²	—	4	2	no	—
206	C₄²	—	4	2	no	—
207	C₄²	—	4	4	no	—
208	C₄²	C₂×C₃×Q₈	1	2	yes	—
209	C₄²	C₃×#41	1	2	yes	—
210	C₄²	C₃×#41	1	4	yes	—
211	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
212	C₂²×C₃	C₂×#124	1	1	yes	characteristic
213	C₂²×C₃	#536	1	1	yes	characteristic
214	C₂²×C₃	#575	1	2	yes	—
215	C₂²×C₃	#575	1	2	yes	—
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	4	4	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	8	2	no	—
222	C₃×C₄	—	8	2	no	—
223	C₃×C₄	—	8	2	no	—
224	C₃×C₄	#556	1	2	yes	—
225	C₃×C₄	#557	1	2	yes	—
226	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
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	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
245	C₂×C₄	C₃×#124	1	2	yes	—
246	C₂×C₄	C₃×#124	1	2	yes	—
247	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
248	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×#41	1	2	yes	—
250	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
251	C₈	—	4	2	no	—
252	C₈	—	4	2	no	—
253	C₂×C₃	#2568	1	1	yes	characteristic
254	C₂×C₃	#3080	1	2	yes	—
255	C₂×C₃	#3135	1	1	yes	characteristic
256	C₂×C₃	#3142	1	1	yes	characteristic
257	C₂×C₃	#3240	1	2	yes	—
258	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
259	C₂²	C₂×C₃×#124	1	1	yes	characteristic
260	C₂²	C₃×#536	1	1	yes	characteristic
261	C₂²	C₃×#575	1	2	yes	—
262	C₂²	C₃×#575	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	4	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	2	no	—
270	C₄	—	8	2	no	—
271	C₄	C₃×#556	1	2	yes	—
272	C₄	C₃×#557	1	2	yes	—
273	C₃	#29227	1	1	yes	characteristic, direct product factor, Sylow
274	C₂	C₃×#2568	1	1	yes	characteristic
275	C₂	C₃×#3080	1	2	yes	—
276	C₂	C₃×#3135	1	1	yes	characteristic
277	C₂	C₃×#3142	1	1	yes	characteristic
278	C₂	C₃×#3240	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
10. #29227	C₃	direct (N × H)
273. C₃	#29227	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×#2101	2
C₃×#1992	2
C₃×#1989	2
C₃×#1836	2
C₃×#1767	2
#29227	3

Subgroups — C3×#29227 #10647834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29227 #10647834