Subgroups — C₂×3₋¹⁺²×C₄⋊C₈ #13145026

Order 1728.

552 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×C₈×3₋¹⁺²	C₂	1	2	yes	maximal
2	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×C₄⋊C₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄⋊C₈	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×C₄⋊C₈	C₃	1	3	yes	maximal, complemented
6	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
7	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
8	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₂×C₈×3₋¹⁺²	—	2	4	no	—
10	C₂×C₈×3₋¹⁺²	C₂²	1	4	yes	—
11	C₄²×3₋¹⁺²	C₄	1	2	yes	—
12	C₄²×3₋¹⁺²	C₂²	1	2	yes	—
13	C₂²×C₃²×C₈	C₂×C₃	1	2	yes	—
14	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
15	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	—
16	C₂²×C₈×C₉	C₂×C₃	1	6	yes	—
17	C₂×C₄²×C₉	C₂×C₃	1	3	yes	—
18	C₉×C₄⋊C₈	C₂×C₃	1	12	yes	complemented
19	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	—	2	2	no	—
21	C₂×C₄×3₋¹⁺²	C₈	1	4	yes	complemented
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
23	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
25	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
26	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
27	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
28	C₈×3₋¹⁺²	—	2	4	no	—
29	C₂×C₃×C₄⋊C₈	—	3	1	no	—
30	C₂×C₃×C₄⋊C₈	C₃²	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₂×C₃²×C₈	—	2	4	no	—
35	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
36	C₃²×C₄²	C₃×C₄	1	2	yes	—
37	C₃²×C₄²	C₂²×C₃	1	2	yes	—
38	C₂²×C₄×C₉	C₃×C₄	1	3	yes	—
39	C₂²×C₄×C₉	C₃×C₄	1	3	yes	—
40	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
41	C₂×C₈×C₉	—	2	12	no	—
42	C₂×C₈×C₉	C₂²×C₃	1	12	yes	—
43	C₄²×C₉	C₃×C₄	1	6	yes	—
44	C₄²×C₉	C₂²×C₃	1	6	yes	—
45	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
46	C₂²×3₋¹⁺²	C₂×C₈	1	2	yes	—
47	C₂²×3₋¹⁺²	M₁₆	1	2	yes	—
48	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
49	C₄×3₋¹⁺²	—	2	2	no	—
50	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
51	C₄×3₋¹⁺²	C₂×C₈	1	4	yes	complemented
52	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
53	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₃×C₈	—	3	2	no	—
55	C₂²×C₃×C₈	C₂×C₃²	1	2	yes	—
56	C₂×C₃×C₄²	—	3	1	no	—
57	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
58	C₃×C₄⋊C₈	—	3	4	no	—
59	C₃×C₄⋊C₈	C₂×C₃²	1	4	yes	—
60	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	C₃×C₈	1	4	yes	—
63	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
64	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
66	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
67	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
68	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
69	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
70	C₃²×C₈	—	2	4	no	—
71	C₂×C₄×C₉	—	2	6	no	—
72	C₂×C₄×C₉	C₃×C₈	1	12	yes	complemented
73	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
74	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
75	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
76	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
77	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
78	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
79	C₈×C₉	—	2	12	no	—
80	C₂×C₄⋊C₈	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×3₋¹⁺²	C₄⋊C₈	1	4	yes	direct product factor
82	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×3₋¹⁺²	C₂²×C₈	1	1	yes	characteristic
84	C₂×3₋¹⁺²	C₂×M₁₆	1	1	yes	characteristic
85	C₂²×C₃×C₄	—	3	1	no	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	3	4	no	—
93	C₂×C₃×C₈	—	6	4	no	—
94	C₂×C₃×C₈	C₂²×C₃²	1	4	yes	—
95	C₃×C₄²	—	3	2	no	—
96	C₃×C₄²	—	3	2	no	—
97	C₃×C₄²	C₃²×C₄	1	2	yes	—
98	C₃×C₄²	C₂²×C₃²	1	2	yes	—
99	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
100	C₂²×C₃²	C₂×C₃×C₈	1	2	yes	—
101	C₂²×C₃²	C₃×M₁₆	1	2	yes	—
102	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
103	C₃²×C₄	—	2	2	no	—
104	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
105	C₃²×C₄	C₂×C₃×C₈	1	4	yes	—
106	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
107	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
108	C₂²×C₉	C₃×C₄⋊C₄	1	6	yes	—
109	C₂²×C₉	C₂×C₃×C₈	1	6	yes	—
110	C₂²×C₉	C₃×M₁₆	1	6	yes	—
111	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
112	C₄×C₉	—	2	6	no	—
113	C₄×C₉	C₃×C₄⋊C₄	1	6	yes	—
114	C₄×C₉	C₂×C₃×C₈	1	12	yes	complemented
115	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
116	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
117	C₂²×C₈	C₂×3₋¹⁺²	1	2	yes	—
118	C₂×C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic
119	C₄⋊C₈	C₂×3₋¹⁺²	1	4	yes	direct product factor
120	3₋¹⁺²	C₂×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
121	C₂³×C₃	—	3	1	no	—
122	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	3	4	no	—
125	C₂×C₃×C₄	—	3	1	no	—
126	C₂×C₃×C₄	—	3	2	no	—
127	C₂×C₃×C₄	—	3	2	no	—
128	C₂×C₃×C₄	—	3	2	no	—
129	C₂×C₃×C₄	—	3	2	no	—
130	C₂×C₃×C₄	—	3	1	no	—
131	C₂×C₃×C₄	—	6	2	no	—
132	C₂×C₃×C₄	C₃²×C₈	1	4	yes	—
133	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
136	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
137	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
138	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
139	C₃×C₈	—	2	4	no	—
140	C₃×C₈	—	6	4	no	—
141	C₂×C₃²	C₃×C₄⋊C₈	1	4	yes	—
142	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
144	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
145	C₂×C₉	C₃×C₄⋊C₈	1	12	yes	complemented
146	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
147	C₂×C₉	C₂²×C₃×C₈	1	3	yes	—
148	C₂×C₉	C₂×C₃×M₁₆	1	3	yes	—
149	C₂²×C₄	C₄×3₋¹⁺²	1	1	yes	characteristic
150	C₂²×C₄	C₄×3₋¹⁺²	1	1	yes	characteristic
151	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂²×3₋¹⁺²	1	4	yes	—
154	C₄²	C₄×3₋¹⁺²	1	2	yes	—
155	C₄²	C₂²×3₋¹⁺²	1	2	yes	—
156	C₂²×C₃	—	3	1	no	—
157	C₂²×C₃	—	3	2	no	—
158	C₂²×C₃	—	3	2	no	—
159	C₂²×C₃	—	3	2	no	—
160	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
161	C₂²×C₃	C₂×C₃²×C₈	1	2	yes	—
162	C₂²×C₃	C₃²×M₁₆	1	2	yes	—
163	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	3	1	no	—
166	C₃×C₄	—	3	1	no	—
167	C₃×C₄	—	3	2	no	—
168	C₃×C₄	—	3	4	no	—
169	C₃×C₄	—	6	2	no	—
170	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
171	C₃×C₄	C₂×C₃²×C₈	1	4	yes	—
172	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
173	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
174	C₃²	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
175	C₉	C₂×C₃×C₄⋊C₈	1	3	yes	complemented
176	C₂³	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₈×3₋¹⁺²	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
180	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
182	C₂×C₄	D₄×3₋¹⁺²	1	2	yes	—
183	C₂×C₄	Q₈×3₋¹⁺²	1	2	yes	—
184	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₂×C₃	—	3	1	no	—
187	C₂×C₃	—	3	4	no	—
188	C₂×C₃	—	3	1	no	—
189	C₂×C₃	—	3	1	no	—
190	C₂×C₃	C₃²×C₄⋊C₈	1	4	yes	—
191	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
193	C₂×C₃	C₂×C₃²×M₁₆	1	1	yes	characteristic
194	C₂²	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
195	C₂²	C₂×C₈×3₋¹⁺²	1	2	yes	—
196	C₂²	M₁₆×3₋¹⁺²	1	2	yes	—
197	C₂²	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	C₄⋊C₄×3₋¹⁺²	1	2	yes	—
200	C₄	C₂×C₈×3₋¹⁺²	1	4	yes	complemented
201	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
202	C₄	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
203	C₃	—	3	1	no	—
204	C₃	C₂×C₃²×C₄⋊C₈	1	1	yes	characteristic
205	C₂	3₋¹⁺²×C₄⋊C₈	1	4	yes	direct product factor
206	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
207	C₂	C₂²×C₈×3₋¹⁺²	1	1	yes	characteristic
208	C₂	C₂×M₁₆×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×C₃²
commutator	C₂×C₃	C₂²×C₃²×C₈
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂³×C₃	C₂×C₃²×C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²×C₄⋊C₈	C₂	direct (N × H)
5. C₂×C₉×C₄⋊C₈	C₃	semidirect
18. C₉×C₄⋊C₈	C₂×C₃	semidirect
21. C₂×C₄×3₋¹⁺²	C₈	semidirect
51. C₄×3₋¹⁺²	C₂×C₈	semidirect
72. C₂×C₄×C₉	C₃×C₈	semidirect
80. C₂×C₄⋊C₈	3₋¹⁺²	direct (N × H)
81. C₂×3₋¹⁺²	C₄⋊C₈	direct (N × H)
114. C₄×C₉	C₂×C₃×C₈	semidirect
119. C₄⋊C₈	C₂×3₋¹⁺²	direct (N × H)
120. 3₋¹⁺²	C₂×C₄⋊C₈	direct (N × H)
145. C₂×C₉	C₃×C₄⋊C₈	semidirect
175. C₉	C₂×C₃×C₄⋊C₈	semidirect
178. C₂×C₄	C₈×3₋¹⁺²	semidirect
200. C₄	C₂×C₈×3₋¹⁺²	semidirect
205. C₂	3₋¹⁺²×C₄⋊C₈	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×C₈×3₋¹⁺²	2
C₂×C₄²×3₋¹⁺²	2
3₋¹⁺²×C₄⋊C₈	2
C₂×C₃²×C₄⋊C₈	3
C₂×C₉×C₄⋊C₈	3

Subgroups — C2×3−1+2×C4⋊C8 #13145026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×3₋¹⁺²×C₄⋊C₈ #13145026