Subgroups — C₂⁴×#1368 #72287465

Order 2048.

30088 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#346	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#334	C₂	1	2	yes	maximal
3	C₂³×#1368	C₂	1	60	yes	maximal, direct product factor
4	C₂⁵×C₁₆	C₄	1	1	yes	characteristic
5	C₂⁵×C₁₆	C₄	1	1	yes	characteristic
6	C₂⁴×C₄⋊C₈	—	2	2	no	—
7	C₂⁴×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂³×#346	C₄	1	30	yes	—
9	C₂³×#346	C₂²	1	30	yes	—
10	C₂³×#334	—	2	30	no	—
11	C₂³×#334	C₂²	1	60	yes	—
12	C₂²×#1368	C₂²	1	560	yes	direct product factor
13	C₂⁵×C₈	—	4	2	no	—
14	C₂⁵×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂⁵×C₈	Q₈	1	1	yes	characteristic
16	C₂⁴×C₄²	D₄	1	1	yes	characteristic
17	C₂⁴×C₁₆	—	2	16	no	—
18	C₂⁴×C₁₆	C₈	1	32	yes	complemented
19	C₂⁴×C₁₆	C₂×C₄	1	30	yes	—
20	C₂⁴×C₁₆	C₂×C₄	1	30	yes	—
21	C₂³×C₄⋊C₈	—	2	60	no	—
22	C₂³×C₄⋊C₈	—	4	30	no	—
23	C₂³×C₄×C₈	C₂×C₄	1	15	yes	—
24	C₂³×C₄×C₈	D₄	1	15	yes	—
25	C₂³×C₄×C₈	Q₈	1	15	yes	—
26	C₂³×C₄×C₈	C₂³	1	15	yes	—
27	C₂²×#346	C₂×C₄	1	420	yes	—
28	C₂²×#346	C₂³	1	140	yes	—
29	C₂²×#334	—	2	420	no	—
30	C₂²×#334	C₂³	1	280	yes	—
31	C₂×#1368	C₂³	1	960	yes	direct product factor
32	C₂⁵×C₄	C₄⋊C₄	1	1	yes	characteristic
33	C₂⁵×C₄	SD₁₆	1	2	yes	—
34	C₂⁴×C₈	—	2	16	no	—
35	C₂⁴×C₈	—	4	60	no	—
36	C₂⁴×C₈	—	8	32	no	—
37	C₂⁴×C₈	C₄⋊C₄	1	15	yes	—
38	C₂⁴×C₈	C₄⋊C₄	1	15	yes	—
39	C₂⁴×C₈	C₂×C₈	1	16	yes	—
40	C₂⁴×C₈	M₁₆	1	16	yes	—
41	C₂⁴×C₈	C₂²×C₄	1	15	yes	—
42	C₂⁴×C₈	C₂×Q₈	1	15	yes	—
43	C₂³×C₄²	C₄⋊C₄	1	15	yes	—
44	C₂³×C₄²	SD₁₆	1	30	yes	—
45	C₂³×C₄²	C₂×D₄	1	15	yes	—
46	C₂³×C₁₆	—	2	240	no	—
47	C₂³×C₁₆	C₂×C₈	1	480	yes	complemented
48	C₂³×C₁₆	C₂²×C₄	1	140	yes	—
49	C₂³×C₁₆	C₂²×C₄	1	140	yes	—
50	C₂²×C₄⋊C₈	—	2	280	no	—
51	C₂²×C₄⋊C₈	—	4	420	no	—
52	C₂²×C₄×C₈	C₄⋊C₄	1	210	yes	—
53	C₂²×C₄×C₈	C₂²×C₄	1	105	yes	—
54	C₂²×C₄×C₈	C₂×D₄	1	105	yes	—
55	C₂²×C₄×C₈	C₂×Q₈	1	105	yes	—
56	C₂²×C₄×C₈	C₂⁴	1	35	yes	—
57	C₂×#346	C₂²×C₄	1	840	yes	—
58	C₂×#346	C₂⁴	1	120	yes	—
59	C₂×#334	—	2	840	no	—
60	C₂×#334	C₂⁴	1	240	yes	—
61	#1368	C₂⁴	1	256	yes	direct product factor
62	C₂⁶	#106	1	1	yes	characteristic
63	C₂⁴×C₄	—	2	32	no	—
64	C₂⁴×C₄	C₄⋊C₈	1	16	yes	—
65	C₂⁴×C₄	#106	1	30	yes	—
66	C₂⁴×C₄	#106	1	15	yes	—
67	C₂⁴×C₄	#108	1	16	yes	—
68	C₂⁴×C₄	C₂×C₄⋊C₄	1	15	yes	—
69	C₂⁴×C₄	C₂×SD₁₆	1	30	yes	—
70	C₂³×C₈	—	2	120	no	—
71	C₂³×C₈	—	2	120	no	—
72	C₂³×C₈	—	4	280	no	—
73	C₂³×C₈	—	8	480	no	—
74	C₂³×C₈	C₄⋊C₈	1	240	yes	—
75	C₂³×C₈	C₂×C₄⋊C₄	1	105	yes	—
76	C₂³×C₈	C₂×C₄⋊C₄	1	105	yes	—
77	C₂³×C₈	C₂²×C₈	1	120	yes	—
78	C₂³×C₈	C₂×M₁₆	1	120	yes	—
79	C₂³×C₈	C₂³×C₄	1	35	yes	—
80	C₂³×C₈	C₂²×Q₈	1	35	yes	—
81	C₂²×C₄²	#106	1	210	yes	—
82	C₂²×C₄²	C₂×C₄⋊C₄	1	105	yes	—
83	C₂²×C₄²	C₂×SD₁₆	1	210	yes	—
84	C₂²×C₄²	C₂²×D₄	1	35	yes	—
85	C₂²×C₁₆	—	2	560	no	—
86	C₂²×C₁₆	C₂²×C₈	1	1120	yes	complemented
87	C₂²×C₁₆	C₂³×C₄	1	120	yes	—
88	C₂²×C₁₆	C₂³×C₄	1	120	yes	—
89	C₂×C₄⋊C₈	—	2	240	no	—
90	C₂×C₄⋊C₈	—	4	840	no	—
91	C₂×C₄×C₈	C₂×C₄⋊C₄	1	630	yes	—
92	C₂×C₄×C₈	C₂³×C₄	1	105	yes	—
93	C₂×C₄×C₈	C₂²×D₄	1	105	yes	—
94	C₂×C₄×C₈	C₂²×Q₈	1	105	yes	—
95	C₂×C₄×C₈	C₂⁵	1	15	yes	—
96	#346	C₂³×C₄	1	240	yes	—
97	#346	C₂⁵	1	16	yes	—
98	#334	—	2	240	no	—
99	#334	C₂⁵	1	32	yes	—
100	C₂⁵	#334	1	16	yes	—
101	C₂⁵	#365	1	32	yes	—
102	C₂⁵	C₂×#106	1	15	yes	—
103	C₂³×C₄	—	2	240	no	—
104	C₂³×C₄	—	2	240	no	—
105	C₂³×C₄	#334	1	240	yes	—
106	C₂³×C₄	C₂×C₄⋊C₈	1	120	yes	—
107	C₂³×C₄	C₂×#106	1	210	yes	—
108	C₂³×C₄	C₂×#106	1	105	yes	—
109	C₂³×C₄	C₂×#108	1	120	yes	—
110	C₂³×C₄	C₂²×C₄⋊C₄	1	35	yes	—
111	C₂³×C₄	C₂²×SD₁₆	1	70	yes	—
112	C₂²×C₈	—	2	420	no	—
113	C₂²×C₈	—	2	140	no	—
114	C₂²×C₈	—	4	240	no	—
115	C₂²×C₈	—	8	1120	no	—
116	C₂²×C₈	C₂×C₄⋊C₈	1	840	yes	—
117	C₂²×C₈	C₂²×C₄⋊C₄	1	105	yes	—
118	C₂²×C₈	C₂²×C₄⋊C₄	1	105	yes	—
119	C₂²×C₈	C₂³×C₈	1	140	yes	—
120	C₂²×C₈	C₂²×M₁₆	1	140	yes	—
121	C₂²×C₈	C₂⁴×C₄	1	15	yes	—
122	C₂²×C₈	C₂³×Q₈	1	15	yes	—
123	C₂×C₄²	C₂×#106	1	630	yes	—
124	C₂×C₄²	C₂²×C₄⋊C₄	1	105	yes	—
125	C₂×C₄²	C₂²×SD₁₆	1	210	yes	—
126	C₂×C₄²	C₂³×D₄	1	15	yes	—
127	C₂×C₁₆	—	2	240	no	—
128	C₂×C₁₆	C₂³×C₈	1	480	yes	complemented
129	C₂×C₁₆	C₂⁴×C₄	1	16	yes	—
130	C₂×C₁₆	C₂⁴×C₄	1	16	yes	—
131	C₄⋊C₈	—	2	32	no	—
132	C₄⋊C₈	—	4	240	no	—
133	C₄×C₈	C₂²×C₄⋊C₄	1	210	yes	—
134	C₄×C₈	C₂⁴×C₄	1	15	yes	—
135	C₄×C₈	C₂³×D₄	1	15	yes	—
136	C₄×C₈	C₂³×Q₈	1	15	yes	—
137	C₄×C₈	C₂⁶	1	1	yes	characteristic
138	C₂⁴	#1368	1	256	yes	direct product factor
139	C₂⁴	C₂×#334	1	120	yes	—
140	C₂⁴	C₂×#365	1	240	yes	—
141	C₂⁴	C₂²×#106	1	35	yes	—
142	C₂²×C₄	—	2	280	no	—
143	C₂²×C₄	—	2	840	no	—
144	C₂²×C₄	C₂×#334	1	840	yes	—
145	C₂²×C₄	C₂²×C₄⋊C₈	1	140	yes	—
146	C₂²×C₄	C₂²×#106	1	210	yes	—
147	C₂²×C₄	C₂²×#106	1	105	yes	—
148	C₂²×C₄	C₂²×#108	1	140	yes	—
149	C₂²×C₄	C₂³×C₄⋊C₄	1	15	yes	—
150	C₂²×C₄	C₂³×SD₁₆	1	30	yes	—
151	C₂×C₈	—	2	210	no	—
152	C₂×C₈	—	2	30	no	—
153	C₂×C₈	—	4	32	no	—
154	C₂×C₈	—	8	480	no	—
155	C₂×C₈	C₂²×C₄⋊C₈	1	420	yes	—
156	C₂×C₈	C₂³×C₄⋊C₄	1	15	yes	—
157	C₂×C₈	C₂³×C₄⋊C₄	1	15	yes	—
158	C₂×C₈	C₂⁴×C₈	1	30	yes	—
159	C₂×C₈	C₂³×M₁₆	1	30	yes	—
160	C₂×C₈	C₂⁵×C₄	1	1	yes	characteristic
161	C₂×C₈	C₂⁴×Q₈	1	1	yes	characteristic
162	C₄²	C₂²×#106	1	210	yes	—
163	C₄²	C₂³×C₄⋊C₄	1	15	yes	—
164	C₄²	C₂³×SD₁₆	1	30	yes	—
165	C₄²	C₂⁴×D₄	1	1	yes	characteristic
166	C₁₆	—	2	16	no	—
167	C₁₆	C₂⁴×C₈	1	32	yes	complemented
168	C₂³	C₂×#1368	1	960	yes	direct product factor
169	C₂³	C₂²×#334	1	140	yes	—
170	C₂³	C₂²×#365	1	280	yes	—
171	C₂³	C₂³×#106	1	15	yes	—
172	C₂×C₄	—	2	60	no	—
173	C₂×C₄	—	2	420	no	—
174	C₂×C₄	C₂²×#334	1	420	yes	—
175	C₂×C₄	C₂³×C₄⋊C₈	1	30	yes	—
176	C₂×C₄	C₂³×#106	1	30	yes	—
177	C₂×C₄	C₂³×#106	1	15	yes	—
178	C₂×C₄	C₂³×#108	1	30	yes	—
179	C₂×C₄	C₂⁴×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₂⁴×SD₁₆	1	2	yes	—
181	C₈	—	2	1	no	—
182	C₈	—	2	15	no	—
183	C₈	—	8	32	no	—
184	C₈	C₂³×C₄⋊C₈	1	30	yes	—
185	C₈	C₂⁵×C₈	1	1	yes	characteristic
186	C₈	C₂⁴×M₁₆	1	1	yes	characteristic
187	C₂²	C₂²×#1368	1	560	yes	direct product factor
188	C₂²	C₂³×#334	1	30	yes	—
189	C₂²	C₂³×#365	1	60	yes	—
190	C₂²	C₂⁴×#106	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	30	no	—
193	C₄	C₂³×#334	1	30	yes	—
194	C₄	C₂⁴×#108	1	1	yes	characteristic
195	C₄	C₂⁴×C₄⋊C₈	1	1	yes	characteristic
196	C₂	C₂³×#1368	1	60	yes	direct product factor
197	C₂	C₂⁴×#365	1	2	yes	—
198	C₂	C₂⁴×#334	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁶	#106
commutator	C₈	C₂⁵×C₈
Frattini, cosocle	C₄×C₈	C₂⁶
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁴×C₄² ◁ C₂⁴×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂³×#1368	C₂	direct (N × H)
12. C₂²×#1368	C₂²	direct (N × H)
18. C₂⁴×C₁₆	C₈	semidirect
31. C₂×#1368	C₂³	direct (N × H)
47. C₂³×C₁₆	C₂×C₈	semidirect
61. #1368	C₂⁴	direct (N × H)
86. C₂²×C₁₆	C₂²×C₈	semidirect
128. C₂×C₁₆	C₂³×C₈	semidirect
138. C₂⁴	#1368	direct (N × H)
167. C₁₆	C₂⁴×C₈	semidirect
168. C₂³	C₂×#1368	direct (N × H)
187. C₂²	C₂²×#1368	direct (N × H)
196. C₂	C₂³×#1368	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×#346	2
C₂⁴×#334	2
C₂³×#1368	2

Subgroups — C24×#1368 #72287465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#1368 #72287465