Subgroups — C₃×#29294 #10647901

Order 768.

630 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2330	C₂	1	2	yes	maximal
2	C₃×#2087	C₂	1	2	yes	maximal
3	C₃×#2086	C₂	1	2	yes	maximal
4	C₃×#1989	C₂	1	4	yes	maximal
5	C₃×#1947	C₂	1	4	yes	maximal
6	C₃×#1843	C₂	1	1	yes	maximal, characteristic
7	#29294	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×#125	C₂²	1	2	yes	—
9	C₂×C₃×#107	C₂²	1	2	yes	—
10	C₂×C₃×#107	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×#389	—	2	4	no	—
15	C₃×#389	C₂²	1	4	yes	—
16	C₃×#387	—	2	4	no	—
17	C₃×#384	C₂²	1	2	yes	—
18	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
19	C₃×#376	C₂²	1	1	yes	characteristic
20	C₃×#340	C₂²	1	2	yes	—
21	C₃×#340	C₂²	1	2	yes	—
22	C₃×#340	C₂²	1	4	yes	—
23	C₃×#340	C₂²	1	4	yes	—
24	#2330	C₂×C₃	1	2	yes	—
25	#2087	C₂×C₃	1	2	yes	—
26	#2086	C₂×C₃	1	2	yes	—
27	#1989	C₂×C₃	1	4	yes	—
28	#1947	C₂×C₃	1	4	yes	—
29	#1843	C₂×C₃	1	1	yes	characteristic
30	C₂²×C₃×C₈	C₂³	1	2	yes	—
31	C₂²×C₃×C₈	C₂³	1	2	yes	—
32	C₃×#125	—	2	4	no	—
33	C₃×#125	Q₈	1	8	yes	—
34	C₂×C₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₃×C₄⋊C₄	—	2	4	no	—
36	C₂×C₃×C₄⋊C₄	—	2	4	no	—
37	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₃×C₄²	—	2	4	no	—
40	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄²	C₂³	1	2	yes	—
42	C₃×#107	—	4	2	no	—
43	C₃×#107	—	4	2	no	—
44	C₃×C₄⋊C₈	—	2	8	no	—
45	C₃×C₄⋊C₈	—	4	1	no	—
46	C₃×C₄⋊C₈	—	4	1	no	—
47	C₃×#95	—	2	4	no	—
48	C₃×#95	—	2	4	no	—
49	C₃×#95	D₄	1	2	yes	—
50	C₃×#95	D₄	1	2	yes	—
51	C₂×#125	C₂²×C₃	1	2	yes	—
52	C₂×#107	C₂²×C₃	1	2	yes	—
53	C₂×#107	C₂²×C₃	1	2	yes	—
54	C₂×C₄⋊C₈	C₂²×C₃	1	4	yes	—
55	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
57	#389	—	2	4	no	—
58	#389	C₂²×C₃	1	4	yes	—
59	#387	—	2	4	no	—
60	#384	C₂²×C₃	1	2	yes	—
61	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
62	#376	C₂²×C₃	1	1	yes	characteristic
63	#340	C₂²×C₃	1	4	yes	—
64	#340	C₂²×C₃	1	2	yes	—
65	#340	C₂²×C₃	1	4	yes	—
66	#340	C₂²×C₃	1	2	yes	—
67	C₂²×C₃×C₄	—	2	4	no	—
68	C₂²×C₃×C₄	—	2	4	no	—
69	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
72	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
73	C₂×C₃×C₈	—	2	4	no	—
74	C₂×C₃×C₈	—	2	4	no	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	—	2	2	no	—
77	C₃×C₄⋊C₄	—	2	16	no	—
78	C₃×C₄⋊C₄	—	2	4	no	—
79	C₃×C₄⋊C₄	—	2	4	no	—
80	C₃×C₄⋊C₄	—	2	4	no	—
81	C₃×C₄⋊C₄	—	8	4	no	—
82	C₃×C₄⋊C₄	—	8	4	no	—
83	C₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	—	8	4	no	—
87	C₃×C₄²	C₂×Q₈	1	4	yes	—
88	C₃×C₄²	#41	1	4	yes	—
89	C₂²×C₈	C₂³×C₃	1	2	yes	—
90	C₂²×C₈	C₂³×C₃	1	2	yes	—
91	#125	—	2	4	no	—
92	#125	C₃×Q₈	1	8	yes	—
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	4	no	—
95	C₂×C₄⋊C₄	—	2	4	no	—
96	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
97	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
98	C₂×C₄²	—	2	4	no	—
99	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
100	C₂×C₄²	C₂³×C₃	1	2	yes	—
101	#107	—	4	2	no	—
102	#107	—	4	2	no	—
103	C₄⋊C₈	—	2	8	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄⋊C₈	—	4	1	no	—
106	#95	—	2	4	no	—
107	#95	—	2	4	no	—
108	#95	C₃×D₄	1	2	yes	—
109	#95	C₃×D₄	1	2	yes	—
110	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	—	2	4	no	—
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	2	8	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	4	4	no	—
119	C₂×C₃×C₄	—	4	4	no	—
120	C₂×C₃×C₄	—	4	4	no	—
121	C₂×C₃×C₄	—	4	4	no	—
122	C₂×C₃×C₄	—	4	4	no	—
123	C₂×C₃×C₄	C₂²⋊Q₈	1	8	yes	—
124	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
125	C₂×C₃×C₄	C₂×#41	1	2	yes	—
126	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
128	C₃×C₈	—	4	2	no	—
129	C₃×C₈	—	4	2	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
135	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₄⋊C₄	—	2	16	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	8	4	no	—
145	C₄⋊C₄	—	8	4	no	—
146	C₄⋊C₄	C₂×C₃×Q₈	1	8	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	8	4	no	—
150	C₄²	C₂×C₃×Q₈	1	4	yes	—
151	C₄²	C₃×#41	1	4	yes	—
152	C₂²×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
153	C₂²×C₃	#537	1	1	yes	characteristic
154	C₂²×C₃	#576	1	2	yes	—
155	C₂²×C₃	#578	1	2	yes	—
156	C₃×C₄	—	2	8	no	—
157	C₃×C₄	—	4	8	no	—
158	C₃×C₄	—	4	1	no	—
159	C₃×C₄	—	4	1	no	—
160	C₃×C₄	—	8	4	no	—
161	C₃×C₄	#557	1	2	yes	—
162	C₃×C₄	Q₈²	1	1	yes	characteristic
163	C₃×C₄	#559	1	1	yes	characteristic
164	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₃×C₂²⋊Q₈	1	8	yes	—
178	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
179	C₂×C₄	C₂×C₃×#41	1	2	yes	—
180	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
182	C₈	—	4	2	no	—
183	C₈	—	4	2	no	—
184	C₂×C₃	#2582	1	1	yes	characteristic
185	C₂×C₃	#3083	1	4	yes	—
186	C₂×C₃	#3259	1	1	yes	characteristic
187	C₂×C₃	#3261	1	1	yes	characteristic
188	C₂²	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
189	C₂²	C₃×#537	1	1	yes	characteristic
190	C₂²	C₃×#576	1	2	yes	—
191	C₂²	C₃×#578	1	2	yes	—
192	C₄	—	2	8	no	—
193	C₄	—	4	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	8	no	—
196	C₄	—	8	4	no	—
197	C₄	C₃×#557	1	2	yes	—
198	C₄	C₃×Q₈²	1	1	yes	characteristic
199	C₄	C₃×#559	1	1	yes	characteristic
200	C₃	#29294	1	1	yes	characteristic, direct product factor, Sylow
201	C₂	C₃×#2582	1	1	yes	characteristic
202	C₂	C₃×#3083	1	4	yes	—
203	C₂	C₃×#3259	1	1	yes	characteristic
204	C₂	C₃×#3261	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
7. #29294	C₃	direct (N × H)
200. C₃	#29294	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×#2330	2
C₃×#2087	2
C₃×#2086	2
C₃×#1989	2
C₃×#1947	2
C₃×#1843	2
#29294	3

Subgroups — C3×#29294 #10647901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29294 #10647901