Subgroups — C₃×#29225 #10647832

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×#2100	C₂	1	4	yes	maximal
4	C₃×#1992	C₂	1	2	yes	maximal
5	C₃×#1988	C₂	1	4	yes	maximal
6	C₃×#1836	C₂	1	2	yes	maximal
7	C₃×#1767	C₂	1	1	yes	maximal, characteristic
8	#29225	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×C₄⋊Q₈	—	2	2	no	—
10	C₂×C₃×C₄⋊Q₈	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×#97	C₂²	1	2	yes	—
14	C₃×#389	C₂²	1	2	yes	—
15	C₃×#387	—	2	2	no	—
16	C₃×#384	—	2	2	no	—
17	C₃×#384	C₂²	1	2	yes	—
18	C₃×#384	C₂²	1	2	yes	—
19	C₃×#382	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₄⋊C₄	—	2	2	no	—
21	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄³	C₂²	1	1	yes	characteristic
23	C₃×#340	C₂²	1	8	yes	—
24	C₃×#340	C₂²	1	8	yes	—
25	C₃×#336	C₂²	1	2	yes	—
26	#2868	C₂×C₃	1	1	yes	characteristic
27	C₄×#125	C₂×C₃	1	1	yes	characteristic
28	#2100	C₂×C₃	1	4	yes	—
29	#1992	C₂×C₃	1	2	yes	—
30	#1988	C₂×C₃	1	4	yes	—
31	#1836	C₂×C₃	1	2	yes	—
32	#1767	C₂×C₃	1	1	yes	characteristic
33	C₂²×C₃×Q₈	—	2	2	no	—
34	C₂²×C₃×C₈	C₂³	1	4	yes	—
35	C₃×C₄⋊Q₈	—	2	8	no	—
36	C₃×C₄⋊Q₈	—	4	8	no	—
37	C₃×#125	—	2	2	no	—
38	C₃×#125	Q₈	1	4	yes	complemented
39	C₂×C₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₃×C₄⋊C₄	—	2	4	no	—
43	C₂×C₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₃×C₄²	—	2	2	no	—
48	C₂×C₃×C₄²	D₄	1	2	yes	—
49	C₂×C₃×C₄²	D₄	1	2	yes	—
50	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
53	C₃×C₄⋊C₈	—	2	4	no	—
54	C₃×#97	—	2	4	no	—
55	C₃×#95	—	2	2	no	—
56	C₃×#95	—	2	2	no	—
57	C₃×#95	—	2	2	no	—
58	C₂×C₄⋊Q₈	—	2	2	no	—
59	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
60	C₂×#125	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
62	C₂×#97	C₂²×C₃	1	2	yes	—
63	#389	C₂²×C₃	1	2	yes	—
64	#387	—	2	2	no	—
65	#384	—	2	2	no	—
66	#384	C₂²×C₃	1	2	yes	—
67	#384	C₂²×C₃	1	2	yes	—
68	#382	C₂²×C₃	1	1	yes	characteristic
69	C₄×C₄⋊C₄	—	2	2	no	—
70	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₄³	C₂²×C₃	1	1	yes	characteristic
72	#340	C₂²×C₃	1	8	yes	—
73	#340	C₂²×C₃	1	8	yes	—
74	#336	C₂²×C₃	1	2	yes	—
75	C₂×C₃×Q₈	—	4	4	no	—
76	C₂×C₃×Q₈	—	4	8	no	—
77	C₂²×C₃×C₄	—	2	2	no	—
78	C₂²×C₃×C₄	—	2	2	no	—
79	C₂²×C₃×C₄	—	2	4	no	—
80	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
81	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
85	C₂×C₃×C₈	—	2	4	no	—
86	C₂×C₃×C₈	—	2	4	no	—
87	C₂×C₃×C₈	—	2	4	no	—
88	C₃×C₄⋊C₄	—	2	8	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	—	2	8	no	—
91	C₃×C₄⋊C₄	—	2	4	no	—
92	C₃×C₄⋊C₄	—	4	8	no	—
93	C₃×C₄⋊C₄	—	4	8	no	—
94	C₃×C₄⋊C₄	—	4	4	no	—
95	C₃×C₄⋊C₄	—	4	4	no	—
96	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
97	C₃×C₄²	—	2	4	no	—
98	C₃×C₄²	—	2	4	no	—
99	C₃×C₄²	—	2	2	no	—
100	C₃×C₄²	—	4	4	no	—
101	C₃×C₄²	C₂×Q₈	1	2	yes	—
102	C₃×C₄²	#41	1	2	yes	—
103	C₃×C₄²	#41	1	4	yes	—
104	C₂²×Q₈	—	2	2	no	—
105	C₂²×C₈	C₂³×C₃	1	4	yes	—
106	C₄⋊Q₈	—	2	8	no	—
107	C₄⋊Q₈	—	4	8	no	—
108	#125	—	2	2	no	—
109	#125	C₃×Q₈	1	4	yes	complemented
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	4	no	—
112	C₂×C₄⋊C₄	—	2	4	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
116	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
117	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
118	C₂×C₄²	—	2	2	no	—
119	C₂×C₄²	C₃×D₄	1	2	yes	—
120	C₂×C₄²	C₃×D₄	1	2	yes	—
121	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
124	C₄⋊C₈	—	2	4	no	—
125	#97	—	2	4	no	—
126	#95	—	2	2	no	—
127	#95	—	2	2	no	—
128	#95	—	2	2	no	—
129	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
130	C₃×Q₈	—	8	8	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	4	4	no	—
140	C₂×C₃×C₄	—	4	2	no	—
141	C₂×C₃×C₄	—	4	2	no	—
142	C₂×C₃×C₄	—	4	8	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
145	C₂×C₃×C₄	#124	1	2	yes	—
146	C₂×C₃×C₄	#124	1	2	yes	—
147	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
148	C₂×C₃×C₄	#137	1	4	yes	—
149	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×#41	1	2	yes	—
152	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
153	C₃×C₈	—	4	4	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂×Q₈	—	4	8	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
160	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
162	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₄⋊C₄	—	2	8	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	8	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	8	no	—
172	C₄⋊C₄	—	4	8	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	4	4	no	—
180	C₄²	C₂×C₃×Q₈	1	2	yes	—
181	C₄²	C₃×#41	1	2	yes	—
182	C₄²	C₃×#41	1	4	yes	—
183	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₂²×C₃	C₂×#124	1	1	yes	characteristic
185	C₂²×C₃	#536	1	1	yes	characteristic
186	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
187	C₂²×C₃	C₂×#137	1	2	yes	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	4	4	no	—
192	C₃×C₄	—	8	4	no	—
193	C₃×C₄	—	8	2	no	—
194	C₃×C₄	#490	1	4	yes	—
195	C₃×C₄	#556	1	2	yes	—
196	C₃×C₄	#557	1	2	yes	—
197	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
198	Q₈	—	8	8	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
213	C₂×C₄	C₃×#124	1	2	yes	—
214	C₂×C₄	C₃×#124	1	2	yes	—
215	C₂×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
216	C₂×C₄	C₃×#137	1	4	yes	—
217	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
219	C₂×C₄	C₂×C₃×#41	1	2	yes	—
220	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
221	C₈	—	4	4	no	—
222	C₂×C₃	#2568	1	1	yes	characteristic
223	C₂×C₃	#3079	1	2	yes	—
224	C₂×C₃	C₂×#490	1	1	yes	characteristic
225	C₂×C₃	#3139	1	1	yes	characteristic
226	C₂×C₃	#3239	1	2	yes	—
227	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
228	C₂²	C₂×C₃×#124	1	1	yes	characteristic
229	C₂²	C₃×#536	1	1	yes	characteristic
230	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
231	C₂²	C₂×C₃×#137	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	4	no	—
237	C₄	—	8	2	no	—
238	C₄	C₃×#490	1	4	yes	—
239	C₄	C₃×#556	1	2	yes	—
240	C₄	C₃×#557	1	2	yes	—
241	C₃	#29225	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	C₃×#2568	1	1	yes	characteristic
243	C₂	C₃×#3079	1	2	yes	—
244	C₂	C₂×C₃×#490	1	1	yes	characteristic
245	C₂	C₃×#3139	1	1	yes	characteristic
246	C₂	C₃×#3239	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
8. #29225	C₃	direct (N × H)
38. C₃×#125	Q₈	semidirect
109. #125	C₃×Q₈	semidirect
241. C₃	#29225	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×#2868	2
C₃×C₄×#125	2
C₃×#2100	2
C₃×#1992	2
C₃×#1988	2
C₃×#1836	2
C₃×#1767	2
#29225	3

Subgroups — C3×#29225 #10647832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29225 #10647832