Subgroups — C₂×Q₈×3₋¹⁺²⋊C₄ #13153668

Order 1728.

972 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	Q₈×3₋¹⁺²⋊C₄	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(3₋¹⁺²⋊C₄)	C₂	1	3	yes	maximal
4	C₂×C₄×3₋¹⁺²⋊C₄	C₂	1	3	yes	maximal
5	C₂×C₃×Q₈×Dic₃	—	3	1	no	maximal
6	C₂×Q₈×Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×3₋¹⁺²	C₄	1	8	yes	complemented
8	C₂×Q₈×3₋¹⁺²	C₂²	1	4	yes	—
9	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
10	C₂²×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
11	C₂²×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
12	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	12	yes	—
13	C₄×3₋¹⁺²⋊C₄	C₂²	1	12	yes	—
14	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
15	C₃×Q₈×Dic₃	—	3	8	no	—
16	C₂×C₃×C₄⋊Dic₃	—	3	3	no	—
17	C₂×C₃×C₄×Dic₃	—	3	3	no	—
18	C₂²×Q₈×C₉	—	2	1	no	—
19	C₂²×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
20	Q₈×Dic₉	C₂×C₃	1	8	yes	complemented
21	C₂×C₄⋊Dic₉	C₂×C₃	1	3	yes	—
22	C₂×C₄×Dic₉	C₂×C₃	1	3	yes	—
23	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
24	Q₈×3₋¹⁺²	C₂×C₄	1	16	yes	complemented
25	C₂×C₄×3₋¹⁺²	C₂×C₄	1	12	yes	—
26	C₂×C₄×3₋¹⁺²	C₂³	1	6	yes	—
27	C₂×3₋¹⁺²⋊C₄	—	2	6	no	—
28	C₂×3₋¹⁺²⋊C₄	Q₈	1	4	yes	direct product factor
29	C₂×3₋¹⁺²⋊C₄	C₂³	1	6	yes	—
30	C₂×3₋¹⁺²⋊C₄	C₂³	1	2	yes	—
31	C₂×C₃×C₄×Q₈	—	9	1	no	—
32	C₂×Q₈×Dic₃	—	3	1	no	—
33	C₂×C₃²×Q₈	Dic₃	1	8	yes	—
34	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
35	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
36	C₂²×C₃×Dic₃	—	3	1	no	—
37	C₂²×C₃×Dic₃	—	3	3	no	—
38	C₃×C₄⋊Dic₃	—	3	12	no	—
39	C₃×C₄×Dic₃	—	3	12	no	—
40	C₂×Q₈×C₉	—	2	8	no	—
41	C₂×Q₈×C₉	—	2	4	no	—
42	C₂×Q₈×C₉	C₃×C₄	1	8	yes	complemented
43	C₂×Q₈×C₉	C₂²×C₃	1	4	yes	—
44	C₂²×C₄×C₉	—	2	3	no	—
45	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
46	C₂²×Dic₉	C₂²×C₃	1	3	yes	—
47	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
48	C₄⋊Dic₉	C₂²×C₃	1	12	yes	—
49	C₄×Dic₉	C₂²×C₃	1	12	yes	—
50	C₂²×3₋¹⁺²	C₂²×C₄	1	2	yes	—
51	C₂²×3₋¹⁺²	C₂×Q₈	1	2	yes	—
52	C₂²×3₋¹⁺²	#41	1	2	yes	—
53	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
54	C₄×3₋¹⁺²	C₂²×C₄	1	12	yes	—
55	3₋¹⁺²⋊C₄	—	2	6	no	—
56	3₋¹⁺²⋊C₄	C₂×Q₈	1	4	yes	direct product factor
57	C₂²×C₃×Q₈	—	3	1	no	—
58	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
59	C₃×C₄×Q₈	—	9	8	no	—
60	C₂×C₃×C₄⋊C₄	—	9	3	no	—
61	C₂×C₃×C₄²	—	9	3	no	—
62	Q₈×Dic₃	—	3	8	no	—
63	C₂×C₄⋊Dic₃	—	3	3	no	—
64	C₂×C₄×Dic₃	—	3	3	no	—
65	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
66	C₃²×Q₈	C₂×Dic₃	1	16	yes	—
67	C₂×C₃²×C₄	C₂×Dic₃	1	12	yes	—
68	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
69	C₂×C₃×Dic₃	—	3	4	no	—
70	C₂×C₃×Dic₃	—	3	2	no	—
71	C₂×C₃×Dic₃	—	3	6	no	—
72	C₂×C₃×Dic₃	—	6	6	no	—
73	C₂³×C₉	—	2	1	no	—
74	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
75	Q₈×C₉	—	2	16	no	—
76	Q₈×C₉	C₂×C₃×C₄	1	16	yes	complemented
77	C₂×C₄×C₉	—	2	12	no	—
78	C₂×C₄×C₉	—	2	6	no	—
79	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
80	C₂×C₄×C₉	C₂³×C₃	1	6	yes	—
81	C₂×Dic₉	—	2	6	no	—
82	C₂×Dic₉	C₃×Q₈	1	4	yes	complemented
83	C₂×Dic₉	C₂³×C₃	1	6	yes	—
84	C₂×Dic₉	C₂³×C₃	1	2	yes	—
85	C₂×C₄×Q₈	—	9	1	no	Sylow
86	C₂×3₋¹⁺²	C₄×Q₈	1	4	yes	complemented
87	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
88	C₂×3₋¹⁺²	C₂²×Q₈	1	1	yes	characteristic
89	C₂×3₋¹⁺²	C₂×#41	1	1	yes	characteristic
90	C₂×C₃×Q₈	—	3	8	no	—
91	C₂×C₃×Q₈	—	3	4	no	—
92	C₂×C₃×Q₈	C₃×Dic₃	1	8	yes	—
93	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
94	C₂²×C₃×C₄	—	3	3	no	—
95	C₂²×C₃×C₄	—	9	1	no	—
96	C₂²×C₃×C₄	—	9	3	no	—
97	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
98	C₂²×Dic₃	—	3	1	no	—
99	C₂²×Dic₃	—	3	3	no	—
100	C₃×C₄⋊C₄	—	9	12	no	—
101	C₃×C₄²	—	9	12	no	—
102	C₄⋊Dic₃	—	3	12	no	—
103	C₄×Dic₃	—	3	12	no	—
104	C₂²×C₃²	D₃×Q₈	1	2	yes	—
105	C₂²×C₃²	#239	1	2	yes	—
106	C₂²×C₃²	C₂²×Dic₃	1	2	yes	—
107	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
108	C₃²×C₄	C₂²×Dic₃	1	12	yes	—
109	C₃×Dic₃	—	3	4	no	—
110	C₃×Dic₃	—	6	6	no	—
111	C₂²×C₉	—	2	1	no	—
112	C₂²×C₉	—	2	2	no	—
113	C₂²×C₉	—	2	2	no	—
114	C₂²×C₉	—	2	2	no	—
115	C₂²×C₉	C₂²×C₃×C₄	1	2	yes	—
116	C₂²×C₉	C₂×C₃×Q₈	1	2	yes	—
117	C₂²×C₉	C₃×#41	1	2	yes	—
118	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
119	C₄×C₉	—	2	12	no	—
120	C₄×C₉	C₂²×C₃×C₄	1	12	yes	—
121	Dic₉	—	2	6	no	—
122	Dic₉	C₂×C₃×Q₈	1	4	yes	complemented
123	C₂²×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic
124	C₄×Q₈	—	9	8	no	—
125	C₂×C₄⋊C₄	—	9	3	no	—
126	C₂×C₄²	—	9	3	no	—
127	3₋¹⁺²	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₃	—	3	1	no	—
129	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
130	C₃×Q₈	—	3	16	no	—
131	C₃×Q₈	C₂×C₃×Dic₃	1	16	yes	—
132	C₂×C₃×C₄	—	3	12	no	—
133	C₂×C₃×C₄	—	3	6	no	—
134	C₂×C₃×C₄	—	9	2	no	—
135	C₂×C₃×C₄	—	9	4	no	—
136	C₂×C₃×C₄	—	9	6	no	—
137	C₂×C₃×C₄	—	18	6	no	—
138	C₂×C₃×C₄	C₂×C₃×Dic₃	1	12	yes	—
139	C₂×C₃×C₄	C₂²×C₃×D₃	1	6	yes	—
140	C₂×Dic₃	—	3	2	no	—
141	C₂×Dic₃	—	3	4	no	—
142	C₂×Dic₃	—	3	6	no	—
143	C₂×Dic₃	—	6	6	no	—
144	C₂×C₃²	Q₈×Dic₃	1	4	yes	—
145	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
146	C₂×C₃²	C₂×#239	1	1	yes	characteristic
147	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
148	C₂×C₉	—	2	4	no	—
149	C₂×C₉	—	2	1	no	—
150	C₂×C₉	—	2	1	no	—
151	C₂×C₉	—	2	1	no	—
152	C₂×C₉	C₃×C₄×Q₈	1	4	yes	complemented
153	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
154	C₂×C₉	C₂²×C₃×Q₈	1	1	yes	characteristic
155	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
156	C₂×Q₈	3₋¹⁺²⋊C₄	1	8	yes	direct product factor
157	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	4	yes	—
158	C₂²×C₄	—	9	1	no	—
159	C₂²×C₄	—	9	3	no	—
160	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
161	C₄⋊C₄	—	9	12	no	—
162	C₄²	—	9	12	no	—
163	C₂²×C₃	—	3	1	no	—
164	C₂²×C₃	—	3	2	no	—
165	C₂²×C₃	—	3	2	no	—
166	C₂²×C₃	—	3	2	no	—
167	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
168	C₂²×C₃	C₃×#239	1	2	yes	—
169	C₂²×C₃	C₂²×C₃×Dic₃	1	2	yes	—
170	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
171	C₃×C₄	—	3	12	no	—
172	C₃×C₄	—	9	4	no	—
173	C₃×C₄	—	18	6	no	—
174	C₃×C₄	C₂²×C₃×Dic₃	1	12	yes	—
175	Dic₃	—	3	4	no	—
176	Dic₃	—	6	6	no	—
177	C₃²	C₂×Q₈×Dic₃	1	1	yes	characteristic
178	C₉	—	2	1	no	—
179	C₉	C₂×C₃×C₄×Q₈	1	1	yes	characteristic, complemented
180	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
181	Q₈	C₂×3₋¹⁺²⋊C₄	1	16	yes	direct product factor
182	C₂×C₄	—	9	2	no	—
183	C₂×C₄	—	9	4	no	—
184	C₂×C₄	—	9	6	no	—
185	C₂×C₄	—	18	6	no	—
186	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	12	yes	—
187	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	6	yes	—
188	C₂×C₃	—	3	1	no	—
189	C₂×C₃	—	3	1	no	—
190	C₂×C₃	—	3	1	no	—
191	C₂×C₃	—	3	4	no	—
192	C₂×C₃	C₃×Q₈×Dic₃	1	4	yes	—
193	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
194	C₂×C₃	C₂×C₃×#239	1	1	yes	characteristic
195	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
196	C₂²	Q₈×AGL(1,ℤ/9)	1	2	yes	—
197	C₂²	#88724	1	2	yes	—
198	C₂²	C₂²×3₋¹⁺²⋊C₄	1	2	yes	—
199	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
200	C₄	—	9	4	no	—
201	C₄	—	18	6	no	—
202	C₄	C₂²×3₋¹⁺²⋊C₄	1	12	yes	—
203	C₃	—	3	1	no	—
204	C₃	C₂×C₃×Q₈×Dic₃	1	1	yes	characteristic
205	C₂	Q₈×3₋¹⁺²⋊C₄	1	4	yes	direct product factor
206	C₂	C₂×Q₈×AGL(1,ℤ/9)	1	1	yes	characteristic
207	C₂	C₂×#88724	1	1	yes	characteristic
208	C₂	C₂³×3₋¹⁺²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂³×C₃×C₄
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂²×Q₈×3₋¹⁺²	C₂
socle	C₂³×C₃	C₂²×C₃×D₃
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂²×C₉	C₂⁴×C₃
hypercenter	C₂²×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×C₄×Q₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×3₋¹⁺²⋊C₄	C₂	direct (N × H)
6. C₂×Q₈×Dic₉	C₃	semidirect
7. C₂×Q₈×3₋¹⁺²	C₄	semidirect
20. Q₈×Dic₉	C₂×C₃	semidirect
24. Q₈×3₋¹⁺²	C₂×C₄	semidirect
28. C₂×3₋¹⁺²⋊C₄	Q₈	direct (N × H)
42. C₂×Q₈×C₉	C₃×C₄	semidirect
56. 3₋¹⁺²⋊C₄	C₂×Q₈	direct (N × H)
76. Q₈×C₉	C₂×C₃×C₄	semidirect
82. C₂×Dic₉	C₃×Q₈	semidirect
86. C₂×3₋¹⁺²	C₄×Q₈	semidirect
122. Dic₉	C₂×C₃×Q₈	semidirect
127. 3₋¹⁺²	C₂×C₄×Q₈	semidirect
152. C₂×C₉	C₃×C₄×Q₈	semidirect
156. C₂×Q₈	3₋¹⁺²⋊C₄	direct (N × H)
179. C₉	C₂×C₃×C₄×Q₈	semidirect
181. Q₈	C₂×3₋¹⁺²⋊C₄	direct (N × H)
205. C₂	Q₈×3₋¹⁺²⋊C₄	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×Q₈×3₋¹⁺²	2
Q₈×3₋¹⁺²⋊C₄	2
C₂×C₄⋊(3₋¹⁺²⋊C₄)	2
C₂×C₄×3₋¹⁺²⋊C₄	2
C₂×C₃×Q₈×Dic₃	3
C₂×Q₈×Dic₉	3

Subgroups — C2×Q8×3−1+2⋊C4 #13153668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×3₋¹⁺²⋊C₄ #13153668