Subgroups — C₂×Q₈×Dic₁₂ #11675637

Order 768.

1830 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	C₂×Q₈×Dic₆	C₂	1	2	yes	maximal
2	C₂×C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₁₂	C₂	1	16	yes	maximal, direct product factor
4	C₂×#4850	C₂	1	6	yes	maximal
5	C₂×C₄×Dic₁₂	C₂	1	3	yes	maximal
6	C₂×#4682	C₂	1	3	yes	maximal
7	C₂×Q₈×Q₁₆	—	3	1	no	maximal, Sylow
8	C₂×C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₃	—	2	2	no	—
10	C₂²×Dic₁₂	C₂²	1	1	yes	characteristic
11	Q₈×Dic₆	—	2	8	no	—
12	Q₈×Dic₆	C₂²	1	16	yes	—
13	C₂×#888	—	2	6	no	—
14	C₂×#869	C₂²	1	6	yes	—
15	C₂×C₄×Dic₆	C₂²	1	6	yes	—
16	C₃×C₈×Q₈	C₂²	1	8	yes	—
17	C₂×C₃×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
19	C₂×#818	C₂²	1	3	yes	—
20	C₂×#818	C₂²	1	6	yes	—
21	C₂×#816	C₂²	1	6	yes	—
22	#4850	C₂²	1	48	yes	—
23	C₄×Dic₁₂	C₂²	1	24	yes	—
24	#4682	C₂²	1	24	yes	—
25	C₂×Q₈²	—	3	2	no	—
26	Q₈×Q₁₆	—	3	16	no	—
27	C₂×#500	—	3	3	no	—
28	C₂×#477	—	3	6	no	—
29	C₂×C₈×Q₈	D₃	1	1	yes	characteristic
30	C₂×C₄×Q₁₆	—	3	3	no	—
31	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
32	C₂²×Dic₆	—	2	6	no	—
33	C₂²×Dic₆	C₂³	1	2	yes	—
34	C₂²×C₃×C₈	C₂³	1	3	yes	—
35	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
36	C₃×C₄×Q₈	D₄	1	4	yes	—
37	C₃×C₄×Q₈	C₂³	1	4	yes	—
38	C₂×C₃×C₄⋊C₄	D₄	1	3	yes	—
39	C₂×C₃×C₄²	C₂³	1	3	yes	—
40	Q₈×Dic₃	—	2	16	no	—
41	C₂×C₄⋊Dic₃	—	2	6	no	—
42	C₂×C₄⋊Dic₃	C₂³	1	6	yes	—
43	C₂×Dic₃⋊C₄	—	2	6	no	—
44	C₂×C₄×Dic₃	—	2	6	no	—
45	C₂×Dic₁₂	Q₈	1	8	yes	direct product factor
46	C₂×Dic₁₂	C₂³	1	4	yes	—
47	#888	—	2	48	no	—
48	#869	—	2	12	no	—
49	#869	C₂³	1	24	yes	—
50	C₄×Dic₆	—	2	12	no	—
51	C₄×Dic₆	C₂³	1	24	yes	—
52	C₃×C₄⋊C₈	C₂³	1	12	yes	—
53	C₃×C₄×C₈	C₂³	1	12	yes	—
54	#818	C₂³	1	12	yes	—
55	#818	C₂³	1	24	yes	—
56	#816	C₂³	1	24	yes	—
57	C₂²×Q₁₆	—	3	1	no	—
58	Q₈²	—	3	16	no	—
59	Q₈²	—	6	8	no	—
60	C₂×C₄⋊Q₈	—	3	6	no	—
61	C₂×C₄⋊Q₈	—	6	6	no	—
62	C₂×C₄×Q₈	—	3	6	no	—
63	C₂×C₄×Q₈	—	6	2	no	—
64	C₂×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
65	#500	—	3	24	no	—
66	#477	—	3	48	no	—
67	C₈×Q₈	C₂×D₃	1	8	yes	—
68	C₄×Q₁₆	—	3	24	no	—
69	C₂×#107	—	3	6	no	—
70	C₂×#107	—	3	3	no	—
71	C₂×C₄⋊C₈	C₂×D₃	1	3	yes	—
72	C₂×#103	—	3	6	no	—
73	C₂×C₄×C₈	C₂×D₃	1	3	yes	—
74	C₂×C₃×Q₈	Q₁₆	1	8	yes	complemented
75	C₂×C₃×Q₈	C₂×D₄	1	4	yes	—
76	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
77	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
78	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₃	—	2	2	no	—
80	C₂²×Dic₃	—	2	6	no	—
81	C₂×Dic₆	—	2	2	no	—
82	C₂×Dic₆	—	2	4	no	—
83	C₂×Dic₆	—	2	24	no	—
84	C₂×Dic₆	—	8	12	no	—
85	C₂×Dic₆	C₂×Q₈	1	8	yes	—
86	C₂×Dic₆	C₂⁴	1	4	yes	—
87	C₂×C₃×C₈	—	2	6	no	—
88	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
89	C₂×C₃×C₈	C₂⁴	1	6	yes	—
90	C₂×C₃×C₈	C₂⁴	1	2	yes	—
91	C₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
92	C₃×C₄²	C₂×D₄	1	6	yes	—
93	C₃×C₄²	C₂⁴	1	6	yes	—
94	C₄⋊Dic₃	—	2	6	no	—
95	C₄⋊Dic₃	—	2	24	no	—
96	C₄⋊Dic₃	C₂⁴	1	12	yes	—
97	Dic₃⋊C₄	—	2	24	no	—
98	C₄×Dic₃	—	2	24	no	—
99	Dic₁₂	C₂×Q₈	1	16	yes	direct product factor
100	C₂²×Q₈	—	3	2	no	—
101	C₂²×Q₈	—	6	6	no	—
102	C₂²×Q₈	D₁₂	1	1	yes	characteristic
103	C₂×Q₁₆	—	3	8	no	—
104	C₂×Q₁₆	—	3	4	no	—
105	C₂²×C₈	C₂²×D₃	1	3	yes	—
106	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₄⋊Q₈	—	3	24	no	—
108	C₄⋊Q₈	—	6	48	no	—
109	C₄⋊Q₈	—	6	12	no	—
110	C₄×Q₈	—	3	24	no	—
111	C₄×Q₈	—	6	16	no	—
112	C₄×Q₈	—	6	12	no	—
113	C₄×Q₈	D₁₂	1	4	yes	—
114	C₄×Q₈	C₂²×D₃	1	4	yes	—
115	C₂×C₄⋊C₄	—	3	6	no	—
116	C₂×C₄⋊C₄	—	6	6	no	—
117	C₂×C₄⋊C₄	—	6	6	no	—
118	C₂×C₄⋊C₄	D₁₂	1	3	yes	—
119	C₂×C₄²	—	6	6	no	—
120	C₂×C₄²	C₂²×D₃	1	3	yes	—
121	#107	—	3	12	no	—
122	#107	—	3	24	no	—
123	C₄⋊C₈	C₂²×D₃	1	12	yes	—
124	#103	—	3	24	no	—
125	C₄×C₈	C₂²×D₃	1	12	yes	—
126	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
127	C₃×Q₈	C₂×Q₁₆	1	16	yes	complemented
128	C₂×C₃×C₄	—	2	6	no	—
129	C₂×C₃×C₄	C₂×Q₁₆	1	12	yes	—
130	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
131	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
133	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
134	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
135	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
136	C₂×Dic₃	—	2	8	no	—
137	C₂×Dic₃	—	2	4	no	—
138	C₂×Dic₃	—	2	12	no	—
139	C₂×Dic₃	—	4	12	no	—
140	Dic₆	—	2	8	no	—
141	Dic₆	—	2	4	no	—
142	Dic₆	—	8	24	no	—
143	Dic₆	C₂²×Q₈	1	8	yes	—
144	C₃×C₈	—	2	6	no	—
145	C₃×C₈	C₂²×Q₈	1	4	yes	—
146	C₂×Q₈	—	3	8	no	—
147	C₂×Q₈	—	3	4	no	—
148	C₂×Q₈	—	6	2	no	—
149	C₂×Q₈	—	6	24	no	—
150	C₂×Q₈	—	6	4	no	—
151	C₂×Q₈	—	24	12	no	—
152	C₂×Q₈	Dic₁₂	1	8	yes	direct product factor
153	C₂×Q₈	C₂×D₁₂	1	4	yes	—
154	C₂²×C₄	—	6	2	no	—
155	C₂²×C₄	—	6	6	no	—
156	C₂²×C₄	C₂×D₁₂	1	3	yes	—
157	C₂²×C₄	C₂×D₁₂	1	3	yes	—
158	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
159	Q₁₆	—	3	16	no	—
160	C₂×C₈	—	2	6	no	—
161	C₂×C₈	D₃×Q₈	1	4	yes	—
162	C₂×C₈	C₂³×D₃	1	6	yes	—
163	C₂×C₈	C₂³×D₃	1	2	yes	—
164	C₄⋊C₄	—	3	12	no	—
165	C₄⋊C₄	—	6	6	no	—
166	C₄⋊C₄	—	6	24	no	—
167	C₄⋊C₄	—	6	24	no	—
168	C₄⋊C₄	C₂×D₁₂	1	12	yes	—
169	C₄²	—	6	24	no	—
170	C₄²	C₂×D₁₂	1	6	yes	—
171	C₄²	C₂³×D₃	1	6	yes	—
172	C₂²×C₃	D₄×Q₈	1	2	yes	—
173	C₂²×C₃	C₂²×Q₁₆	1	2	yes	—
174	C₂²×C₃	#576	1	2	yes	—
175	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
176	C₃×C₄	—	2	6	no	—
177	C₃×C₄	D₄×Q₈	1	2	yes	—
178	C₃×C₄	C₂²×Q₁₆	1	12	yes	—
179	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
180	C₃×C₄	C₂×2₋¹⁺⁴	1	1	yes	characteristic
181	Dic₃	—	2	8	no	—
182	Dic₃	—	4	12	no	—
183	C₂³	C₂²×D₁₂	1	1	yes	characteristic
184	Q₈	—	3	8	no	—
185	Q₈	—	6	4	no	—
186	Q₈	—	6	8	no	—
187	Q₈	—	24	24	no	—
188	Q₈	C₂×Dic₁₂	1	16	yes	direct product factor
189	C₂×C₄	—	2	6	no	—
190	C₂×C₄	—	6	4	no	—
191	C₂×C₄	—	6	8	no	—
192	C₂×C₄	—	6	12	no	—
193	C₂×C₄	—	12	12	no	—
194	C₂×C₄	C₂×Dic₁₂	1	12	yes	—
195	C₂×C₄	C₂²×D₁₂	1	6	yes	—
196	C₂×C₄	C₂²×D₁₂	1	6	yes	—
197	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
198	C₂×C₄	C₂×D₃×Q₈	1	2	yes	—
199	C₂×C₄	#1010	1	2	yes	—
200	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
201	C₈	—	2	6	no	—
202	C₈	C₂×D₃×Q₈	1	4	yes	—
203	C₂×C₃	Q₈×Q₁₆	1	4	yes	complemented
204	C₂×C₃	C₂×D₄×Q₈	1	1	yes	characteristic
205	C₂×C₃	C₂³×Q₁₆	1	1	yes	characteristic
206	C₂×C₃	C₂×#576	1	1	yes	characteristic
207	C₂²	Q₈×D₁₂	1	2	yes	—
208	C₂²	C₂²×Dic₁₂	1	2	yes	—
209	C₂²	#5752	1	2	yes	—
210	C₂²	C₂³×D₁₂	1	1	yes	characteristic
211	C₄	—	2	6	no	—
212	C₄	—	6	8	no	—
213	C₄	—	12	12	no	—
214	C₄	Q₈×D₁₂	1	2	yes	—
215	C₄	C₂²×Dic₁₂	1	12	yes	—
216	C₄	C₂²×D₃×Q₈	1	1	yes	characteristic
217	C₄	C₂×#1010	1	1	yes	characteristic
218	C₃	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
219	C₂	Q₈×Dic₁₂	1	4	yes	direct product factor
220	C₂	C₂×Q₈×D₁₂	1	1	yes	characteristic
221	C₂	C₂³×Dic₁₂	1	1	yes	characteristic
222	C₂	C₂×#5752	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂×C₃×C₈×Q₈	C₂
socle, prime radical	C₂³×C₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₃
hypocenter	C₃	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈×Dic₁₂	C₂	direct (N × H)
45. C₂×Dic₁₂	Q₈	direct (N × H)
74. C₂×C₃×Q₈	Q₁₆	semidirect
99. Dic₁₂	C₂×Q₈	direct (N × H)
127. C₃×Q₈	C₂×Q₁₆	semidirect
152. C₂×Q₈	Dic₁₂	direct (N × H)
188. Q₈	C₂×Dic₁₂	direct (N × H)
203. C₂×C₃	Q₈×Q₁₆	semidirect
218. C₃	C₂×Q₈×Q₁₆	semidirect
219. C₂	Q₈×Dic₁₂	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×Q₈×Dic₆	2
C₂×C₃×C₈×Q₈	2
Q₈×Dic₁₂	2
C₂×#4850	2
C₂×C₄×Dic₁₂	2
C₂×#4682	2
C₂×Q₈×Q₁₆	3

Subgroups — C2×Q8×Dic12 #11675637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₁₂ #11675637