Subgroups — C₂²×C₈×Dic₆ #11659538

Order 768.

2478 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₃⋊C₈	C₂	1	2	yes	maximal
4	C₂²×C₈×Dic₃	C₂	1	2	yes	maximal
5	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×Dic₆	C₂	1	24	yes	maximal, direct product factor
7	C₂²×C₈×Q₈	—	3	1	no	maximal, Sylow
8	C₂³×Dic₆	C₄	1	1	yes	characteristic
9	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊Dic₃	C₄	1	1	yes	characteristic
13	C₂²×Dic₃⋊C₄	C₄	1	2	yes	—
14	C₂²×C₄×Dic₃	C₂²	1	2	yes	—
15	C₂³×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×Dic₆	C₄	1	12	yes	—
17	C₂×C₄×Dic₆	C₂²	1	12	yes	—
18	C₂×C₃×C₄×C₈	C₂²	1	12	yes	—
19	C₂×Dic₃⋊C₈	C₂²	1	24	yes	—
20	C₂×C₈×Dic₃	C₂²	1	24	yes	—
21	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	12	yes	—
22	C₈×Dic₆	C₂²	1	64	yes	direct product factor
23	C₂²×C₄×Q₈	—	3	1	no	—
24	C₂×C₈×Q₈	—	3	24	no	—
25	C₂²×C₄⋊C₈	—	3	1	no	—
26	C₂²×C₄⋊C₈	—	3	2	no	—
27	C₂²×C₄×C₈	—	3	2	no	—
28	C₂²×C₄×C₈	D₃	1	1	yes	characteristic
29	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂³×Dic₃	C₂×C₄	1	2	yes	—
33	C₂³×Dic₃	C₂×C₄	1	2	yes	—
34	C₂²×Dic₆	C₈	1	16	yes	direct product factor
35	C₂²×Dic₆	C₂×C₄	1	12	yes	—
36	C₂²×C₃×C₈	—	2	4	no	—
37	C₂²×C₃×C₈	Q₈	1	8	yes	complemented
38	C₂²×C₃×C₈	C₂³	1	6	yes	—
39	C₂²×C₃×C₈	C₂³	1	6	yes	—
40	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
41	C₂×C₃×C₄²	C₂³	1	6	yes	—
42	C₂×C₄⋊Dic₃	C₂×C₄	1	12	yes	—
43	C₂×Dic₃⋊C₄	C₂×C₄	1	24	yes	—
44	C₂×C₄×Dic₃	C₂×C₄	1	12	yes	—
45	C₂×C₄×Dic₃	C₂³	1	12	yes	—
46	C₂²×C₃⋊C₈	—	2	8	no	—
47	C₂²×C₃⋊C₈	C₂³	1	12	yes	—
48	C₄×Dic₆	C₂×C₄	1	48	yes	—
49	C₄×Dic₆	C₂³	1	16	yes	—
50	C₃×C₄×C₈	C₂³	1	16	yes	—
51	Dic₃⋊C₈	C₂³	1	32	yes	—
52	C₈×Dic₃	C₂³	1	32	yes	—
53	C₄⋊(C₃⋊C₈)	C₂³	1	16	yes	—
54	C₂³×Q₈	—	3	1	no	—
55	C₂³×C₈	—	3	2	no	—
56	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₄×Q₈	—	3	12	no	—
59	C₂×C₄×Q₈	—	3	12	no	—
60	C₂²×C₄⋊C₄	—	3	2	no	—
61	C₂²×C₄⋊C₄	—	3	1	no	—
62	C₂²×C₄²	—	3	2	no	—
63	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
64	C₈×Q₈	—	3	64	no	—
65	C₂×C₄⋊C₈	—	3	24	no	—
66	C₂×C₄⋊C₈	—	3	12	no	—
67	C₂×C₄×C₈	—	3	24	no	—
68	C₂×C₄×C₈	C₂×D₃	1	12	yes	—
69	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	—	2	4	no	—
71	C₂²×C₃×C₄	C₂×C₈	1	8	yes	—
72	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
73	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
74	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
75	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
76	C₂²×C₃×C₄	#41	1	4	yes	—
77	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
78	C₂²×Dic₃	—	2	8	no	—
79	C₂²×Dic₃	C₂×C₈	1	16	yes	—
80	C₂²×Dic₃	C₂²×C₄	1	12	yes	—
81	C₂²×Dic₃	C₂²×C₄	1	12	yes	—
82	C₂×Dic₆	C₂×C₈	1	96	yes	direct product factor
83	C₂×Dic₆	C₂²×C₄	1	16	yes	—
84	C₂×C₃×C₈	—	2	12	no	—
85	C₂×C₃×C₈	C₂×Q₈	1	24	yes	complemented
86	C₂×C₃×C₈	C₂⁴	1	4	yes	—
87	C₂×C₃×C₈	C₂⁴	1	4	yes	—
88	C₃×C₄²	C₂²×C₄	1	12	yes	—
89	C₃×C₄²	C₂⁴	1	4	yes	—
90	C₄⋊Dic₃	C₂²×C₄	1	16	yes	—
91	Dic₃⋊C₄	C₂²×C₄	1	32	yes	—
92	C₄×Dic₃	C₂²×C₄	1	24	yes	—
93	C₄×Dic₃	C₂⁴	1	8	yes	—
94	C₂×C₃⋊C₈	—	2	24	no	—
95	C₂×C₃⋊C₈	C₂⁴	1	8	yes	—
96	C₂²×Q₈	—	3	16	no	—
97	C₂²×Q₈	—	3	12	no	—
98	C₂³×C₄	—	3	2	no	—
99	C₂³×C₄	—	3	2	no	—
100	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
101	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
102	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×C₈	—	2	4	no	—
104	C₂²×C₈	—	3	12	no	—
105	C₂²×C₈	—	6	8	no	—
106	C₂²×C₈	Dic₆	1	8	yes	direct product factor
107	C₂²×C₈	C₂²×D₃	1	6	yes	—
108	C₂²×C₈	C₂²×D₃	1	6	yes	—
109	C₄×Q₈	—	3	48	no	—
110	C₄×Q₈	—	3	16	no	—
111	C₂×C₄⋊C₄	—	3	24	no	—
112	C₂×C₄⋊C₄	—	3	12	no	—
113	C₂×C₄²	—	3	12	no	—
114	C₂×C₄²	—	3	12	no	—
115	C₂×C₄²	C₄×D₃	1	6	yes	—
116	C₂×C₄²	C₂²×D₃	1	6	yes	—
117	C₄⋊C₈	—	3	32	no	—
118	C₄⋊C₈	—	3	16	no	—
119	C₄×C₈	—	3	32	no	—
120	C₄×C₈	C₂²×D₃	1	16	yes	—
121	C₂³×C₃	C₄×Q₈	1	4	yes	—
122	C₂³×C₃	C₂²×C₈	1	4	yes	—
123	C₂³×C₃	#131	1	4	yes	—
124	C₂³×C₃	C₂³×C₄	1	3	yes	—
125	C₂×C₃×C₄	—	2	12	no	—
126	C₂×C₃×C₄	C₄×Q₈	1	12	yes	—
127	C₂×C₃×C₄	C₂²×C₈	1	24	yes	—
128	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
129	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
130	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
131	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
132	C₂×C₃×C₄	C₂×#41	1	6	yes	—
133	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
134	C₂×Dic₃	—	2	24	no	—
135	C₂×Dic₃	C₂²×C₈	1	48	yes	—
136	C₂×Dic₃	C₂³×C₄	1	8	yes	—
137	C₂×Dic₃	C₂³×C₄	1	8	yes	—
138	Dic₆	C₂²×C₈	1	64	yes	direct product factor
139	C₃×C₈	—	2	4	no	—
140	C₃×C₈	C₂²×Q₈	1	8	yes	complemented
141	C₃⋊C₈	—	2	8	no	—
142	C₂⁴	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂×Q₈	—	3	96	no	—
144	C₂×Q₈	—	3	16	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	3	16	no	—
147	C₂²×C₄	—	3	12	no	—
148	C₂²×C₄	—	3	12	no	—
149	C₂²×C₄	—	6	8	no	—
150	C₂²×C₄	D₃×C₈	1	8	yes	—
151	C₂²×C₄	C₂×Dic₆	1	4	yes	—
152	C₂²×C₄	C₂×C₄×D₃	1	6	yes	—
153	C₂²×C₄	C₂×C₄×D₃	1	3	yes	—
154	C₂²×C₄	C₂×C₄×D₃	1	6	yes	—
155	C₂²×C₄	#235	1	4	yes	—
156	C₂²×C₄	C₂³×D₃	1	3	yes	—
157	C₂×C₈	—	2	12	no	—
158	C₂×C₈	—	3	8	no	—
159	C₂×C₈	—	6	24	no	—
160	C₂×C₈	C₂×Dic₆	1	24	yes	direct product factor
161	C₂×C₈	C₂³×D₃	1	4	yes	—
162	C₂×C₈	C₂³×D₃	1	4	yes	—
163	C₄⋊C₄	—	3	32	no	—
164	C₄⋊C₄	—	3	16	no	—
165	C₄²	—	3	24	no	—
166	C₄²	—	3	8	no	—
167	C₄²	C₂×C₄×D₃	1	12	yes	—
168	C₄²	C₂³×D₃	1	4	yes	—
169	C₂²×C₃	C₈×Q₈	1	16	yes	complemented
170	C₂²×C₃	C₂×C₄×Q₈	1	6	yes	—
171	C₂²×C₃	C₂³×C₈	1	6	yes	—
172	C₂²×C₃	C₂×#131	1	6	yes	—
173	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
174	C₃×C₄	—	2	4	no	—
175	C₃×C₄	C₂×C₄×Q₈	1	6	yes	—
176	C₃×C₄	C₂³×C₈	1	8	yes	—
177	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
178	C₃×C₄	C₂²×#41	1	1	yes	characteristic
179	Dic₃	—	2	8	no	—
180	Dic₃	C₂³×C₈	1	16	yes	—
181	C₂³	C₄×Dic₆	1	4	yes	—
182	C₂³	C₂×D₃×C₈	1	4	yes	—
183	C₂³	#901	1	4	yes	—
184	C₂³	C₂²×C₄×D₃	1	3	yes	—
185	Q₈	—	3	64	no	—
186	C₂×C₄	—	2	12	no	—
187	C₂×C₄	—	3	8	no	—
188	C₂×C₄	—	3	48	no	—
189	C₂×C₄	—	3	8	no	—
190	C₂×C₄	—	6	24	no	—
191	C₂×C₄	C₄×Dic₆	1	12	yes	—
192	C₂×C₄	C₂×D₃×C₈	1	24	yes	—
193	C₂×C₄	C₂²×Dic₆	1	6	yes	—
194	C₂×C₄	C₂²×C₄×D₃	1	4	yes	—
195	C₂×C₄	C₂²×C₄×D₃	1	3	yes	—
196	C₂×C₄	C₂²×C₄×D₃	1	4	yes	—
197	C₂×C₄	C₂×#235	1	6	yes	—
198	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
199	C₈	—	2	4	no	—
200	C₈	—	6	8	no	—
201	C₈	C₂²×Dic₆	1	8	yes	direct product factor
202	C₂×C₃	C₂×C₈×Q₈	1	12	yes	complemented
203	C₂×C₃	C₂²×C₄×Q₈	1	1	yes	characteristic
204	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
205	C₂×C₃	C₂²×#131	1	1	yes	characteristic
206	C₂²	C₈×Dic₆	1	16	yes	direct product factor
207	C₂²	C₂×C₄×Dic₆	1	6	yes	—
208	C₂²	C₂²×D₃×C₈	1	6	yes	—
209	C₂²	C₂×#901	1	6	yes	—
210	C₂²	C₂³×C₄×D₃	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	3	16	no	—
213	C₄	—	6	8	no	—
214	C₄	C₂×C₄×Dic₆	1	6	yes	—
215	C₄	C₂²×D₃×C₈	1	8	yes	—
216	C₄	C₂³×Dic₆	1	1	yes	characteristic
217	C₄	C₂²×#235	1	1	yes	characteristic
218	C₃	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
219	C₂	C₂×C₈×Dic₆	1	12	yes	direct product factor
220	C₂	C₂²×C₄×Dic₆	1	1	yes	characteristic
221	C₂	C₂³×D₃×C₈	1	1	yes	characteristic
222	C₂	C₂²×#901	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₃
commutator	C₂×C₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×C₄×C₈	C₂
socle, prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂²×C₄×C₈	D₃
hypocenter	C₃	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₂×C₈×Dic₆	C₂	direct (N × H)
22. C₈×Dic₆	C₂²	direct (N × H)
34. C₂²×Dic₆	C₈	direct (N × H)
37. C₂²×C₃×C₈	Q₈	semidirect
82. C₂×Dic₆	C₂×C₈	direct (N × H)
85. C₂×C₃×C₈	C₂×Q₈	semidirect
106. C₂²×C₈	Dic₆	direct (N × H)
138. Dic₆	C₂²×C₈	direct (N × H)
140. C₃×C₈	C₂²×Q₈	semidirect
160. C₂×C₈	C₂×Dic₆	direct (N × H)
169. C₂²×C₃	C₈×Q₈	semidirect
201. C₈	C₂²×Dic₆	direct (N × H)
202. C₂×C₃	C₂×C₈×Q₈	semidirect
206. C₂²	C₈×Dic₆	direct (N × H)
218. C₃	C₂²×C₈×Q₈	semidirect
219. C₂	C₂×C₈×Dic₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Dic₆	2
C₂²×C₃×C₄×C₈	2
C₂²×Dic₃⋊C₈	2
C₂²×C₈×Dic₃	2
C₂²×C₄⋊(C₃⋊C₈)	2
C₂×C₈×Dic₆	2
C₂²×C₈×Q₈	3

Subgroups — C22×C8×Dic6 #11659538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×Dic₆ #11659538