Subgroups — C₃×#29221 #10647828

Order 768.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 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×#2100	C₂	1	2	yes	maximal
4	C₃×#2099	C₂	1	2	yes	maximal
5	C₃×#1991	C₂	1	2	yes	maximal
6	C₃×#1989	C₂	1	4	yes	maximal
7	C₃×#1835	C₂	1	2	yes	maximal
8	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#29221	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#125	C₂²	1	2	yes	—
11	C₂×C₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
14	C₃×#389	—	2	2	no	—
15	C₃×#389	—	2	2	no	—
16	C₃×#389	C₂²	1	2	yes	—
17	C₃×#387	—	2	2	no	—
18	C₃×#384	C₂²	1	1	yes	characteristic
19	C₃×#384	C₂²	1	1	yes	characteristic
20	C₃×#382	C₂²	1	2	yes	—
21	C₃×#382	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₄⋊C₄	—	2	2	no	—
23	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄³	C₂²	1	1	yes	characteristic
25	C₃×#340	C₂²	1	4	yes	—
26	C₃×#340	C₂²	1	4	yes	—
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	#2100	C₂×C₃	1	2	yes	—
33	#2099	C₂×C₃	1	2	yes	—
34	#1991	C₂×C₃	1	2	yes	—
35	#1989	C₂×C₃	1	4	yes	—
36	#1835	C₂×C₃	1	2	yes	—
37	C₄×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
38	C₂²×C₃×C₈	C₂³	1	4	yes	—
39	C₃×#125	—	2	2	no	—
40	C₃×#125	—	2	8	no	—
41	C₃×#125	Q₈	1	4	yes	—
42	C₂×C₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₃×C₄⋊C₄	—	2	2	no	—
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	C₂³	1	2	yes	—
49	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₃×C₄⋊C₄	C₂³	1	2	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×C₄⋊C₈	—	2	4	no	—
59	C₃×C₄×C₈	—	2	4	no	—
60	C₃×#95	—	2	2	no	—
61	C₃×#95	—	2	2	no	—
62	C₃×#95	—	2	2	no	—
63	C₃×#95	—	2	4	no	—
64	C₂×#125	C₂²×C₃	1	2	yes	—
65	C₂×#125	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
67	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
68	#389	—	2	2	no	—
69	#389	—	2	2	no	—
70	#389	C₂²×C₃	1	2	yes	—
71	#387	—	2	2	no	—
72	#384	C₂²×C₃	1	1	yes	characteristic
73	#384	C₂²×C₃	1	1	yes	characteristic
74	#382	C₂²×C₃	1	1	yes	characteristic
75	#382	C₂²×C₃	1	2	yes	—
76	C₄×C₄⋊C₄	—	2	2	no	—
77	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₄³	C₂²×C₃	1	1	yes	characteristic
79	#340	C₂²×C₃	1	4	yes	—
80	#340	C₂²×C₃	1	4	yes	—
81	#340	C₂²×C₃	1	4	yes	—
82	#340	C₂²×C₃	1	4	yes	—
83	#336	C₂²×C₃	1	2	yes	—
84	C₂²×C₃×C₄	—	2	2	no	—
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
89	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
92	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
93	C₂×C₃×C₈	—	2	4	no	—
94	C₂×C₃×C₈	—	2	4	no	—
95	C₂×C₃×C₈	—	2	4	no	—
96	C₃×C₄⋊C₄	—	2	8	no	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	—	2	4	no	—
99	C₃×C₄⋊C₄	—	2	4	no	—
100	C₃×C₄⋊C₄	—	2	4	no	—
101	C₃×C₄⋊C₄	—	4	4	no	—
102	C₃×C₄⋊C₄	—	4	4	no	—
103	C₃×C₄⋊C₄	—	4	4	no	—
104	C₃×C₄⋊C₄	—	4	4	no	—
105	C₃×C₄⋊C₄	—	4	4	no	—
106	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
107	C₃×C₄²	—	2	4	no	—
108	C₃×C₄²	—	2	2	no	—
109	C₃×C₄²	—	4	2	no	—
110	C₃×C₄²	—	4	4	no	—
111	C₃×C₄²	C₂×Q₈	1	2	yes	—
112	C₃×C₄²	#41	1	2	yes	—
113	C₃×C₄²	#41	1	4	yes	—
114	C₂²×C₈	C₂³×C₃	1	4	yes	—
115	#125	—	2	2	no	—
116	#125	—	2	8	no	—
117	#125	C₃×Q₈	1	4	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
125	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
126	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
127	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	C₃×D₄	1	2	yes	—
130	C₂×C₄²	C₃×D₄	1	2	yes	—
131	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
134	C₄⋊C₈	—	2	4	no	—
135	C₄×C₈	—	2	4	no	—
136	#95	—	2	2	no	—
137	#95	—	2	2	no	—
138	#95	—	2	2	no	—
139	#95	—	2	4	no	—
140	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	—	4	4	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	4	no	—
157	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
158	C₂×C₃×C₄	#124	1	2	yes	—
159	C₂×C₃×C₄	#124	1	2	yes	—
160	C₂×C₃×C₄	#135	1	4	yes	—
161	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×#41	1	2	yes	—
164	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
165	C₃×C₈	—	4	4	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
171	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
174	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₄⋊C₄	—	2	8	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
189	C₄²	—	2	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	4	2	no	—
192	C₄²	—	4	4	no	—
193	C₄²	C₂×C₃×Q₈	1	2	yes	—
194	C₄²	C₃×#41	1	2	yes	—
195	C₄²	C₃×#41	1	4	yes	—
196	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
197	C₂²×C₃	C₂×#124	1	1	yes	characteristic
198	C₂²×C₃	#536	1	1	yes	characteristic
199	C₂²×C₃	C₂×#135	1	2	yes	—
200	C₂²×C₃	#575	1	2	yes	—
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	4	4	no	—
206	C₃×C₄	—	8	2	no	—
207	C₃×C₄	—	8	2	no	—
208	C₃×C₄	—	8	2	no	—
209	C₃×C₄	#556	1	2	yes	—
210	C₃×C₄	#557	1	2	yes	—
211	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
229	C₂×C₄	C₃×#124	1	2	yes	—
230	C₂×C₄	C₃×#124	1	2	yes	—
231	C₂×C₄	C₃×#135	1	4	yes	—
232	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
233	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
234	C₂×C₄	C₂×C₃×#41	1	2	yes	—
235	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
236	C₈	—	4	4	no	—
237	C₂×C₃	#2568	1	1	yes	characteristic
238	C₂×C₃	#3076	1	2	yes	—
239	C₂×C₃	#3131	1	1	yes	characteristic
240	C₂×C₃	#3142	1	1	yes	characteristic
241	C₂×C₃	#3232	1	2	yes	—
242	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
243	C₂²	C₂×C₃×#124	1	1	yes	characteristic
244	C₂²	C₃×#536	1	1	yes	characteristic
245	C₂²	C₂×C₃×#135	1	2	yes	—
246	C₂²	C₃×#575	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	4	no	—
250	C₄	—	2	4	no	—
251	C₄	—	4	4	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	2	no	—
255	C₄	C₃×#556	1	2	yes	—
256	C₄	C₃×#557	1	2	yes	—
257	C₃	#29221	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	C₃×#2568	1	1	yes	characteristic
259	C₂	C₃×#3076	1	2	yes	—
260	C₂	C₃×#3131	1	1	yes	characteristic
261	C₂	C₃×#3142	1	1	yes	characteristic
262	C₂	C₃×#3232	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. #29221	C₃	direct (N × H)
257. C₃	#29221	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×#2100	2
C₃×#2099	2
C₃×#1991	2
C₃×#1989	2
C₃×#1835	2
C₃×C₄×C₄⋊C₈	2
#29221	3

Subgroups — C3×#29221 #10647828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29221 #10647828