Subgroups — C₂³×D₄×Dic₃ #11701752

Order 768.

19998 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₂³×#217	C₂	1	2	yes	maximal, complemented
4	C₂²×D₄×Dic₃	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₄	—	3	1	no	maximal, Sylow
8	C₂⁶×C₃	C₂²	1	2	yes	—
9	C₂³×C₃×D₄	C₄	1	32	yes	complemented
10	C₂³×C₃×D₄	C₂²	1	28	yes	—
11	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₃	—	2	32	no	—
13	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₃	C₂²	1	2	yes	—
15	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
16	C₂⁴×Dic₃	C₂²	1	56	yes	complemented
17	C₂²×#217	C₂²	1	56	yes	complemented
18	C₂×D₄×Dic₃	C₂²	1	448	yes	direct product factor
19	C₂²×C₄⋊Dic₃	C₂²	1	28	yes	complemented
20	C₂²×C₄×Dic₃	C₂²	1	28	yes	complemented
21	C₂⁴×D₄	D₃	1	1	yes	characteristic
22	C₂⁵×C₄	—	3	2	no	—
23	C₂²×C₄×D₄	—	3	56	no	—
24	C₂³×C₄⋊C₄	—	3	1	no	—
25	C₂³×C₂²⋊C₄	—	3	2	no	—
26	C₂³×C₄²	—	3	1	no	—
27	C₂⁵×C₃	—	2	16	no	—
28	C₂⁵×C₃	—	2	16	no	—
29	C₂⁵×C₃	C₂×C₄	1	32	yes	complemented
30	C₂⁵×C₃	C₂³	1	28	yes	—
31	C₂⁵×C₃	C₂³	1	1	yes	characteristic
32	C₂²×C₃×D₄	C₂×C₄	1	448	yes	complemented
33	C₂²×C₃×D₄	C₂³	1	112	yes	—
34	C₂³×C₃×C₄	C₂×C₄	1	16	yes	complemented
35	C₂³×C₃×C₄	C₂³	1	14	yes	—
36	C₂³×Dic₃	—	2	16	no	—
37	C₂³×Dic₃	—	2	8	no	—
38	C₂³×Dic₃	—	2	448	no	—
39	C₂³×Dic₃	D₄	1	16	yes	direct product factor
40	C₂³×Dic₃	C₂³	1	28	yes	—
41	C₂³×Dic₃	C₂³	1	14	yes	—
42	C₂³×Dic₃	C₂³	1	14	yes	—
43	C₂³×Dic₃	C₂³	1	224	yes	complemented
44	C₂×#217	C₂³	1	224	yes	complemented
45	D₄×Dic₃	C₂³	1	512	yes	direct product factor
46	C₂×C₄⋊Dic₃	C₂³	1	112	yes	complemented
47	C₂×C₄×Dic₃	C₂³	1	112	yes	complemented
48	C₂⁶	C₂×D₃	1	2	yes	—
49	C₂³×D₄	Dic₃	1	32	yes	direct product factor
50	C₂³×D₄	C₂×D₃	1	28	yes	—
51	C₂⁴×C₄	—	3	1	no	—
52	C₂⁴×C₄	—	3	56	no	—
53	C₂⁴×C₄	—	3	2	no	—
54	C₂⁴×C₄	—	3	1	no	—
55	C₂⁴×C₄	—	6	32	no	—
56	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄×D₄	—	3	448	no	—
58	C₂²×C₄⋊C₄	—	3	28	no	—
59	C₂²×C₂²⋊C₄	—	3	56	no	—
60	C₂²×C₄²	—	3	28	no	—
61	C₂⁴×C₃	—	2	256	no	—
62	C₂⁴×C₃	—	2	112	no	—
63	C₂⁴×C₃	—	2	112	no	—
64	C₂⁴×C₃	C₂²×C₄	1	8	yes	—
65	C₂⁴×C₃	C₂²×C₄	1	224	yes	complemented
66	C₂⁴×C₃	C₂×D₄	1	8	yes	—
67	C₂⁴×C₃	#41	1	8	yes	—
68	C₂⁴×C₃	C₂⁴	1	56	yes	—
69	C₂⁴×C₃	C₂⁴	1	7	yes	—
70	C₂×C₃×D₄	C₂²×C₄	1	896	yes	complemented
71	C₂×C₃×D₄	C₂⁴	1	64	yes	—
72	C₂²×C₃×C₄	C₂²×C₄	1	112	yes	complemented
73	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
74	C₂²×Dic₃	—	2	112	no	—
75	C₂²×Dic₃	—	2	56	no	—
76	C₂²×Dic₃	—	2	896	no	—
77	C₂²×Dic₃	C₂×D₄	1	112	yes	direct product factor
78	C₂²×Dic₃	C₂⁴	1	28	yes	—
79	C₂²×Dic₃	C₂⁴	1	56	yes	—
80	C₂²×Dic₃	C₂⁴	1	28	yes	—
81	C₂²×Dic₃	C₂⁴	1	128	yes	complemented
82	#217	C₂⁴	1	128	yes	complemented
83	C₄⋊Dic₃	C₂⁴	1	64	yes	complemented
84	C₄×Dic₃	C₂⁴	1	64	yes	complemented
85	C₂⁵	—	2	16	no	—
86	C₂⁵	—	2	16	no	—
87	C₂⁵	C₂×Dic₃	1	32	yes	complemented
88	C₂⁵	C₂²×D₃	1	28	yes	—
89	C₂⁵	C₂²×D₃	1	1	yes	characteristic
90	C₂²×D₄	C₂×Dic₃	1	448	yes	direct product factor
91	C₂²×D₄	C₂²×D₃	1	112	yes	—
92	C₂³×C₄	—	3	16	no	—
93	C₂³×C₄	—	3	14	no	—
94	C₂³×C₄	—	3	14	no	—
95	C₂³×C₄	—	3	28	no	—
96	C₂³×C₄	—	3	224	no	—
97	C₂³×C₄	—	6	448	no	—
98	C₂³×C₄	—	6	16	no	—
99	C₂³×C₄	—	6	8	no	—
100	C₂³×C₄	C₂×Dic₃	1	16	yes	complemented
101	C₂³×C₄	C₂²×D₃	1	14	yes	—
102	C₄×D₄	—	3	512	no	—
103	C₂×C₄⋊C₄	—	3	112	no	—
104	C₂×C₂²⋊C₄	—	3	224	no	—
105	C₂×C₄²	—	3	112	no	—
106	C₂³×C₃	—	2	896	no	—
107	C₂³×C₃	—	2	112	no	—
108	C₂³×C₃	—	2	112	no	—
109	C₂³×C₃	C₄×D₄	1	64	yes	complemented
110	C₂³×C₃	C₂³×C₄	1	28	yes	—
111	C₂³×C₃	C₂³×C₄	1	224	yes	complemented
112	C₂³×C₃	C₂²×D₄	1	28	yes	—
113	C₂³×C₃	C₂×#41	1	28	yes	—
114	C₂³×C₃	C₂⁵	1	16	yes	—
115	C₂³×C₃	C₂⁵	1	7	yes	—
116	C₃×D₄	C₂³×C₄	1	256	yes	complemented
117	C₂×C₃×C₄	C₂³×C₄	1	112	yes	complemented
118	C₂×C₃×C₄	C₂⁵	1	8	yes	—
119	C₂×Dic₃	—	2	112	no	—
120	C₂×Dic₃	—	2	56	no	—
121	C₂×Dic₃	—	2	256	no	—
122	C₂×Dic₃	C₂²×D₄	1	112	yes	direct product factor
123	C₂×Dic₃	C₂⁵	1	8	yes	—
124	C₂×Dic₃	C₂⁵	1	16	yes	—
125	C₂×Dic₃	C₂⁵	1	8	yes	—
126	C₂⁴	—	2	256	no	—
127	C₂⁴	—	2	112	no	—
128	C₂⁴	—	2	112	no	—
129	C₂⁴	D₃×D₄	1	8	yes	—
130	C₂⁴	#237	1	8	yes	—
131	C₂⁴	C₂²×Dic₃	1	8	yes	—
132	C₂⁴	C₂²×Dic₃	1	224	yes	complemented
133	C₂⁴	C₂³×D₃	1	56	yes	—
134	C₂⁴	C₂³×D₃	1	7	yes	—
135	C₂×D₄	C₂²×Dic₃	1	896	yes	direct product factor
136	C₂×D₄	C₂³×D₃	1	64	yes	—
137	C₂²×C₄	—	3	112	no	—
138	C₂²×C₄	—	3	28	no	—
139	C₂²×C₄	—	3	28	no	—
140	C₂²×C₄	—	3	56	no	—
141	C₂²×C₄	—	3	128	no	—
142	C₂²×C₄	—	6	896	no	—
143	C₂²×C₄	—	6	112	no	—
144	C₂²×C₄	—	6	56	no	—
145	C₂²×C₄	C₂²×Dic₃	1	112	yes	complemented
146	C₂²×C₄	C₂³×D₃	1	28	yes	—
147	C₄⋊C₄	—	3	64	no	—
148	C₂²⋊C₄	—	3	128	no	—
149	C₄²	—	3	64	no	—
150	C₂²×C₃	—	2	16	no	—
151	C₂²×C₃	—	2	16	no	—
152	C₂²×C₃	—	2	448	no	—
153	C₂²×C₃	C₂×C₄×D₄	1	112	yes	complemented
154	C₂²×C₃	C₂⁴×C₄	1	14	yes	—
155	C₂²×C₃	C₂⁴×C₄	1	32	yes	complemented
156	C₂²×C₃	C₂³×D₄	1	14	yes	—
157	C₂²×C₃	C₂²×#41	1	14	yes	—
158	C₂²×C₃	C₂⁶	1	1	yes	characteristic
159	C₃×C₄	C₂⁴×C₄	1	16	yes	complemented
160	Dic₃	—	2	16	no	—
161	Dic₃	—	2	8	no	—
162	Dic₃	C₂³×D₄	1	16	yes	direct product factor
163	C₂³	—	2	112	no	—
164	C₂³	—	2	112	no	—
165	C₂³	—	2	896	no	—
166	C₂³	D₄×Dic₃	1	64	yes	direct product factor
167	C₂³	C₂×D₃×D₄	1	28	yes	—
168	C₂³	C₂×#237	1	28	yes	—
169	C₂³	C₂³×Dic₃	1	28	yes	—
170	C₂³	C₂³×Dic₃	1	224	yes	complemented
171	C₂³	C₂⁴×D₃	1	16	yes	—
172	C₂³	C₂⁴×D₃	1	7	yes	—
173	D₄	C₂³×Dic₃	1	256	yes	direct product factor
174	C₂×C₄	—	3	112	no	—
175	C₂×C₄	—	3	8	no	—
176	C₂×C₄	—	3	8	no	—
177	C₂×C₄	—	3	16	no	—
178	C₂×C₄	—	6	256	no	—
179	C₂×C₄	—	6	112	no	—
180	C₂×C₄	—	6	56	no	—
181	C₂×C₄	C₂³×Dic₃	1	112	yes	complemented
182	C₂×C₄	C₂⁴×D₃	1	8	yes	—
183	C₂×C₃	—	2	32	no	—
184	C₂×C₃	C₂²×C₄×D₄	1	28	yes	complemented
185	C₂×C₃	C₂⁵×C₄	1	1	yes	characteristic
186	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
187	C₂×C₃	C₂³×#41	1	1	yes	characteristic
188	C₂²	—	2	16	no	—
189	C₂²	—	2	16	no	—
190	C₂²	—	2	448	no	—
191	C₂²	C₂×D₄×Dic₃	1	112	yes	direct product factor
192	C₂²	C₂²×D₃×D₄	1	14	yes	—
193	C₂²	C₂²×#237	1	14	yes	—
194	C₂²	C₂⁴×Dic₃	1	14	yes	—
195	C₂²	C₂⁴×Dic₃	1	32	yes	complemented
196	C₂²	C₂⁵×D₃	1	1	yes	characteristic
197	C₄	—	3	16	no	—
198	C₄	—	6	8	no	—
199	C₄	—	6	16	no	—
200	C₄	C₂⁴×Dic₃	1	16	yes	complemented
201	C₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
202	C₂	—	2	32	no	—
203	C₂	C₂²×D₄×Dic₃	1	28	yes	direct product factor
204	C₂	C₂³×#237	1	1	yes	characteristic
205	C₂	C₂³×D₃×D₄	1	1	yes	characteristic
206	C₂	C₂⁵×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₃
commutator	C₂×C₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₃
Fitting, prime radical	C₂⁴×C₃×D₄	C₂
socle	C₂⁵×C₃	C₂³
cosocle	C₂²×C₃	C₂⁶
hypercenter	C₂⁴×D₄	D₃
hypocenter	C₃	C₂³×C₄×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
2. C₂⁵×Dic₃	C₂	semidirect
3. C₂³×#217	C₂	semidirect
4. C₂²×D₄×Dic₃	C₂	direct (N × H)
5. C₂³×C₄⋊Dic₃	C₂	semidirect
6. C₂³×C₄×Dic₃	C₂	semidirect
9. C₂³×C₃×D₄	C₄	semidirect
16. C₂⁴×Dic₃	C₂²	semidirect
17. C₂²×#217	C₂²	semidirect
18. C₂×D₄×Dic₃	C₂²	direct (N × H)
19. C₂²×C₄⋊Dic₃	C₂²	semidirect
20. C₂²×C₄×Dic₃	C₂²	semidirect
29. C₂⁵×C₃	C₂×C₄	semidirect
32. C₂²×C₃×D₄	C₂×C₄	semidirect
34. C₂³×C₃×C₄	C₂×C₄	semidirect
39. C₂³×Dic₃	D₄	direct (N × H)
43. C₂³×Dic₃	C₂³	semidirect
44. C₂×#217	C₂³	semidirect
45. D₄×Dic₃	C₂³	direct (N × H)
46. C₂×C₄⋊Dic₃	C₂³	semidirect
47. C₂×C₄×Dic₃	C₂³	semidirect
49. C₂³×D₄	Dic₃	direct (N × H)
65. C₂⁴×C₃	C₂²×C₄	semidirect
70. C₂×C₃×D₄	C₂²×C₄	semidirect
72. C₂²×C₃×C₄	C₂²×C₄	semidirect
77. C₂²×Dic₃	C₂×D₄	direct (N × H)
81. C₂²×Dic₃	C₂⁴	semidirect
82. #217	C₂⁴	semidirect
83. C₄⋊Dic₃	C₂⁴	semidirect
84. C₄×Dic₃	C₂⁴	semidirect
87. C₂⁵	C₂×Dic₃	semidirect
90. C₂²×D₄	C₂×Dic₃	direct (N × H)
100. C₂³×C₄	C₂×Dic₃	semidirect
109. C₂³×C₃	C₄×D₄	semidirect
111. C₂³×C₃	C₂³×C₄	semidirect
116. C₃×D₄	C₂³×C₄	semidirect
117. C₂×C₃×C₄	C₂³×C₄	semidirect
122. C₂×Dic₃	C₂²×D₄	direct (N × H)
132. C₂⁴	C₂²×Dic₃	semidirect
135. C₂×D₄	C₂²×Dic₃	direct (N × H)
145. C₂²×C₄	C₂²×Dic₃	semidirect
153. C₂²×C₃	C₂×C₄×D₄	semidirect
155. C₂²×C₃	C₂⁴×C₄	semidirect
159. C₃×C₄	C₂⁴×C₄	semidirect
162. Dic₃	C₂³×D₄	direct (N × H)
166. C₂³	D₄×Dic₃	direct (N × H)
170. C₂³	C₂³×Dic₃	semidirect
173. D₄	C₂³×Dic₃	direct (N × H)
181. C₂×C₄	C₂³×Dic₃	semidirect
184. C₂×C₃	C₂²×C₄×D₄	semidirect
191. C₂²	C₂×D₄×Dic₃	direct (N × H)
195. C₂²	C₂⁴×Dic₃	semidirect
200. C₄	C₂⁴×Dic₃	semidirect
201. C₃	C₂³×C₄×D₄	semidirect
203. C₂	C₂²×D₄×Dic₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×D₄	2
C₂⁵×Dic₃	2
C₂³×#217	2
C₂²×D₄×Dic₃	2
C₂³×C₄⋊Dic₃	2
C₂³×C₄×Dic₃	2
C₂³×C₄×D₄	3

Subgroups — C23×D4×Dic3 #11701752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₄×Dic₃ #11701752