Subgroups — C₂×C₃×#2255 #10652334

Order 768.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#499	C₂	1	1	yes	maximal, characteristic
3	C₃×#2255	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#367	C₂	1	2	yes	maximal
5	C₂×C₃×#366	C₂	1	2	yes	maximal
6	C₂×C₃×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#2255	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₃×#125	—	2	1	no	—
10	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
11	C₃×#500	—	2	2	no	—
12	C₃×#500	C₂²	1	4	yes	—
13	C₃×#499	—	2	2	no	—
14	C₃×#499	C₂²	1	4	yes	—
15	C₂×C₃×#107	—	2	1	no	—
16	C₂×C₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₃×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃×#107	C₂²	1	2	yes	—
19	C₂×C₃×#106	—	2	1	no	—
20	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₃×#367	C₂²	1	8	yes	—
22	C₃×#366	C₂²	1	8	yes	—
23	C₃×#346	C₂²	1	4	yes	—
24	C₂×#500	C₂×C₃	1	1	yes	characteristic
25	C₂×#499	C₂×C₃	1	1	yes	characteristic
26	#2255	C₂×C₃	1	8	yes	direct product factor
27	C₂×#367	C₂×C₃	1	2	yes	—
28	C₂×#366	C₂×C₃	1	2	yes	—
29	C₂×#346	C₂×C₃	1	1	yes	characteristic
30	C₂²×C₃×Q₈	—	4	1	no	—
31	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₄⋊Q₈	—	2	4	no	—
34	C₃×C₄⋊Q₈	—	4	2	no	—
35	C₃×#125	—	2	4	no	—
36	C₃×#125	—	4	2	no	—
37	C₂×C₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₃×C₄⋊C₄	—	4	1	no	—
41	C₂×C₃×C₄⋊C₄	—	4	1	no	—
42	C₂×C₃×C₄⋊C₄	—	4	1	no	—
43	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
44	C₂×C₃×C₁₆	Q₈	1	8	yes	complemented
45	C₂×C₃×C₁₆	C₂³	1	4	yes	—
46	C₃×#107	—	2	4	no	—
47	C₃×#107	—	2	1	no	—
48	C₃×#107	—	2	1	no	—
49	C₃×#107	—	2	2	no	—
50	C₃×#107	C₂³	1	2	yes	—
51	C₃×#107	C₂³	1	4	yes	—
52	C₃×#107	C₂³	1	2	yes	—
53	C₃×#106	—	2	4	no	—
54	C₃×C₄×C₈	D₄	1	2	yes	—
55	C₃×C₄×C₈	C₂³	1	2	yes	—
56	C₂×C₄⋊Q₈	—	2	1	no	—
57	C₂×#125	—	2	1	no	—
58	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
59	#500	—	2	2	no	—
60	#500	C₂²×C₃	1	4	yes	—
61	#499	—	2	2	no	—
62	#499	C₂²×C₃	1	4	yes	—
63	C₂×#107	—	2	1	no	—
64	C₂×#107	C₂²×C₃	1	1	yes	characteristic
65	C₂×#107	C₂²×C₃	1	1	yes	characteristic
66	C₂×#107	C₂²×C₃	1	2	yes	—
67	C₂×#106	—	2	1	no	—
68	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
69	#367	C₂²×C₃	1	8	yes	—
70	#366	C₂²×C₃	1	8	yes	—
71	#346	C₂²×C₃	1	4	yes	—
72	C₂×C₃×Q₈	—	4	4	no	—
73	C₂×C₃×Q₈	—	8	4	no	—
74	C₂²×C₃×C₄	—	4	1	no	—
75	C₂²×C₃×C₄	—	4	1	no	—
76	C₂²×C₃×C₄	—	4	2	no	—
77	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
78	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₈	Q₁₆	1	4	yes	—
81	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
84	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	—	2	2	no	—
87	C₃×C₄⋊C₄	—	2	4	no	—
88	C₃×C₄⋊C₄	—	4	1	no	—
89	C₃×C₄⋊C₄	—	4	1	no	—
90	C₃×C₄⋊C₄	—	4	2	no	—
91	C₃×C₄⋊C₄	—	4	4	no	—
92	C₃×C₄⋊C₄	—	4	4	no	—
93	C₃×C₄⋊C₄	—	4	4	no	—
94	C₃×C₄²	D₈	1	2	yes	—
95	C₃×C₄²	C₂×D₄	1	2	yes	—
96	C₃×C₁₆	C₂×Q₈	1	8	yes	complemented
97	C₂²×Q₈	—	4	1	no	—
98	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
100	C₄⋊Q₈	—	2	4	no	—
101	C₄⋊Q₈	—	4	2	no	—
102	#125	—	2	4	no	—
103	#125	—	4	2	no	—
104	C₂×C₄⋊C₄	—	2	1	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	1	no	—
108	C₂×C₄⋊C₄	—	4	1	no	—
109	C₂×C₄⋊C₄	—	4	1	no	—
110	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
111	C₂×C₁₆	C₃×Q₈	1	8	yes	complemented
112	C₂×C₁₆	C₂³×C₃	1	4	yes	—
113	#107	—	2	4	no	—
114	#107	—	2	1	no	—
115	#107	—	2	1	no	—
116	#107	—	2	2	no	—
117	#107	C₂³×C₃	1	2	yes	—
118	#107	C₂³×C₃	1	4	yes	—
119	#107	C₂³×C₃	1	2	yes	—
120	#106	—	2	4	no	—
121	C₄×C₈	C₃×D₄	1	2	yes	—
122	C₄×C₈	C₂³×C₃	1	2	yes	—
123	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
124	C₃×Q₈	—	8	8	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	—	4	2	no	—
129	C₂×C₃×C₄	—	4	4	no	—
130	C₂×C₃×C₄	—	8	2	no	—
131	C₂×C₃×C₄	—	8	4	no	—
132	C₂×C₃×C₄	—	8	2	no	—
133	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
134	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
135	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
136	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
138	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₃×C₈	C₄⋊Q₈	1	2	yes	—
140	C₃×C₈	C₂×Q₁₆	1	4	yes	—
141	C₃×C₈	C₂²×Q₈	1	2	yes	—
142	C₂×Q₈	—	4	4	no	—
143	C₂×Q₈	—	8	4	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
148	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₈	C₃×Q₁₆	1	4	yes	—
151	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
153	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
154	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	1	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄²	C₃×D₈	1	2	yes	—
165	C₄²	C₂×C₃×D₄	1	2	yes	—
166	C₁₆	C₂×C₃×Q₈	1	8	yes	complemented
167	C₂²×C₃	#500	1	2	yes	—
168	C₂²×C₃	#509	1	2	yes	—
169	C₂²×C₃	#510	1	2	yes	—
170	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	8	2	no	—
174	C₃×C₄	—	8	4	no	—
175	C₃×C₄	—	8	2	no	—
176	C₃×C₄	#500	1	2	yes	—
177	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
178	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
179	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
190	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
191	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
192	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
194	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
195	C₈	C₃×C₄⋊Q₈	1	2	yes	—
196	C₈	C₂×C₃×Q₁₆	1	4	yes	—
197	C₈	C₂²×C₃×Q₈	1	2	yes	—
198	C₂×C₃	#2255	1	4	yes	direct product factor
199	C₂×C₃	C₂×#500	1	1	yes	characteristic
200	C₂×C₃	C₂×#509	1	1	yes	characteristic
201	C₂×C₃	C₂×#510	1	1	yes	characteristic
202	C₂²	C₃×#500	1	2	yes	—
203	C₂²	C₃×#509	1	2	yes	—
204	C₂²	C₃×#510	1	2	yes	—
205	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	8	4	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	2	no	—
211	C₄	C₃×#500	1	2	yes	—
212	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
213	C₄	C₂²×C₃×Q₁₆	1	1	yes	characteristic
214	C₃	C₂×#2255	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	C₃×#2255	1	4	yes	direct product factor
216	C₂	C₂×C₃×#500	1	1	yes	characteristic
217	C₂	C₂×C₃×#509	1	1	yes	characteristic
218	C₂	C₂×C₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#2255	C₂	direct (N × H)
7. C₂×#2255	C₃	direct (N × H)
26. #2255	C₂×C₃	direct (N × H)
44. C₂×C₃×C₁₆	Q₈	semidirect
96. C₃×C₁₆	C₂×Q₈	semidirect
111. C₂×C₁₆	C₃×Q₈	semidirect
166. C₁₆	C₂×C₃×Q₈	semidirect
198. C₂×C₃	#2255	direct (N × H)
214. C₃	C₂×#2255	direct (N × H)
215. C₂	C₃×#2255	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#500	2
C₂×C₃×#499	2
C₃×#2255	2
C₂×C₃×#367	2
C₂×C₃×#366	2
C₂×C₃×#346	2
C₂×#2255	3

Subgroups — C2×C3×#2255 #10652334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2255 #10652334