Subgroups — C₂⁴×C₄×D₈ #72286600

Order 1024.

158389 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₄	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄×D₈	C₂	1	120	yes	maximal, direct product factor
4	C₂⁴×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×#102	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₄	C₂²	1	2	yes	—
8	C₂⁶×C₄	—	2	2	no	—
9	C₂⁴×D₈	C₄	1	64	yes	direct product factor
10	C₂⁴×D₈	C₂²	1	60	yes	—
11	C₂⁵×C₈	C₂²	1	1	yes	characteristic
12	C₂⁵×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₄×D₄	—	2	60	no	—
14	C₂³×C₄×D₄	C₂²	1	120	yes	complemented
15	C₂⁴×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂⁴×C₂²⋊C₄	—	2	2	no	—
17	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₈	C₂²	1	2240	yes	direct product factor
19	C₂³×#107	C₂²	1	60	yes	complemented
20	C₂³×#102	C₂²	1	120	yes	complemented
21	C₂³×C₄×C₈	C₂²	1	60	yes	complemented
22	C₂⁷	—	2	2	no	—
23	C₂⁴×D₄	—	2	30	no	—
24	C₂⁴×D₄	—	2	32	no	—
25	C₂⁴×D₄	C₂×C₄	1	64	yes	complemented
26	C₂⁴×D₄	C₂³	1	60	yes	—
27	C₂⁵×C₄	—	2	2	no	—
28	C₂⁵×C₄	—	2	120	no	—
29	C₂⁵×C₄	—	4	64	no	—
30	C₂⁵×C₄	D₄	1	1	yes	characteristic
31	C₂⁵×C₄	D₄	1	1	yes	characteristic
32	C₂⁵×C₄	C₂³	1	1	yes	characteristic
33	C₂³×D₈	C₂×C₄	1	1920	yes	direct product factor
34	C₂³×D₈	C₂³	1	560	yes	—
35	C₂⁴×C₈	—	2	16	no	—
36	C₂⁴×C₈	C₂×C₄	1	32	yes	complemented
37	C₂⁴×C₈	C₂³	1	30	yes	—
38	C₂⁴×C₈	C₂³	1	30	yes	—
39	C₂²×C₄×D₄	—	2	1680	no	—
40	C₂²×C₄×D₄	C₂³	1	1120	yes	complemented
41	C₂³×C₄⋊C₄	—	2	30	no	—
42	C₂³×C₄⋊C₄	C₂³	1	60	yes	—
43	C₂³×C₂²⋊C₄	—	2	120	no	—
44	C₂³×C₄²	D₄	1	30	yes	—
45	C₂³×C₄²	C₂³	1	30	yes	—
46	C₂×C₄×D₈	C₂³	1	7680	yes	direct product factor
47	C₂²×#107	C₂³	1	560	yes	complemented
48	C₂²×#102	C₂³	1	1120	yes	complemented
49	C₂²×C₄×C₈	C₂³	1	560	yes	complemented
50	C₂⁶	—	2	64	no	—
51	C₂⁶	—	2	60	no	—
52	C₂⁶	—	4	32	no	—
53	C₂⁶	—	4	32	no	—
54	C₂⁶	C₂×D₄	1	1	yes	characteristic
55	C₂³×D₄	—	2	960	no	—
56	C₂³×D₄	—	2	420	no	—
57	C₂³×D₄	—	2	480	no	—
58	C₂³×D₄	C₂²×C₄	1	960	yes	complemented
59	C₂³×D₄	C₂⁴	1	280	yes	—
60	C₂⁴×C₄	—	2	16	no	—
61	C₂⁴×C₄	—	2	60	no	—
62	C₂⁴×C₄	—	2	1120	no	—
63	C₂⁴×C₄	—	4	32	no	—
64	C₂⁴×C₄	—	4	1920	no	—
65	C₂⁴×C₄	D₈	1	32	yes	direct product factor
66	C₂⁴×C₄	C₂²×C₄	1	16	yes	—
67	C₂⁴×C₄	C₂×D₄	1	30	yes	—
68	C₂⁴×C₄	C₂×D₄	1	15	yes	—
69	C₂⁴×C₄	C₂×D₄	1	30	yes	—
70	C₂⁴×C₄	#41	1	16	yes	—
71	C₂⁴×C₄	C₂⁴	1	15	yes	—
72	C₂²×D₈	C₂²×C₄	1	8960	yes	direct product factor
73	C₂²×D₈	C₂⁴	1	960	yes	—
74	C₂³×C₈	—	2	240	no	—
75	C₂³×C₈	C₂²×C₄	1	480	yes	complemented
76	C₂³×C₈	C₂⁴	1	140	yes	—
77	C₂³×C₈	C₂⁴	1	140	yes	—
78	C₂×C₄×D₄	—	2	6720	no	—
79	C₂×C₄×D₄	C₂⁴	1	1920	yes	complemented
80	C₂²×C₄⋊C₄	—	2	420	no	—
81	C₂²×C₄⋊C₄	C₂⁴	1	280	yes	—
82	C₂²×C₂²⋊C₄	—	2	1120	no	—
83	C₂²×C₄²	C₂×D₄	1	420	yes	—
84	C₂²×C₄²	C₂⁴	1	140	yes	—
85	C₄×D₈	C₂⁴	1	4096	yes	direct product factor
86	C₂×#107	C₂⁴	1	960	yes	complemented
87	C₂×#102	C₂⁴	1	1920	yes	complemented
88	C₂×C₄×C₈	C₂⁴	1	960	yes	complemented
89	C₂⁵	—	2	960	no	—
90	C₂⁵	—	2	280	no	—
91	C₂⁵	—	4	480	no	—
92	C₂⁵	—	4	480	no	—
93	C₂⁵	—	4	1024	no	—
94	C₂⁵	C₄×D₄	1	16	yes	—
95	C₂⁵	C₂×D₈	1	16	yes	—
96	C₂⁵	#135	1	16	yes	—
97	C₂⁵	C₂²×D₄	1	15	yes	—
98	C₂²×D₄	—	2	6720	no	—
99	C₂²×D₄	—	2	840	no	—
100	C₂²×D₄	—	2	1120	no	—
101	C₂²×D₄	C₂³×C₄	1	2240	yes	complemented
102	C₂²×D₄	C₂⁵	1	240	yes	—
103	C₂³×C₄	—	2	240	no	—
104	C₂³×C₄	—	2	280	no	—
105	C₂³×C₄	—	2	1920	no	—
106	C₂³×C₄	—	4	480	no	—
107	C₂³×C₄	—	4	8960	no	—
108	C₂³×C₄	C₄×D₄	1	240	yes	—
109	C₂³×C₄	C₂×D₈	1	480	yes	direct product factor
110	C₂³×C₄	C₂³×C₄	1	120	yes	—
111	C₂³×C₄	C₂²×D₄	1	140	yes	—
112	C₂³×C₄	C₂²×D₄	1	105	yes	—
113	C₂³×C₄	C₂²×D₄	1	140	yes	—
114	C₂³×C₄	C₂×#41	1	120	yes	—
115	C₂³×C₄	C₂⁵	1	35	yes	—
116	C₂×D₈	C₂³×C₄	1	7680	yes	direct product factor
117	C₂×D₈	C₂⁵	1	256	yes	—
118	C₂²×C₈	—	2	560	no	—
119	C₂²×C₈	C₂³×C₄	1	1120	yes	complemented
120	C₂²×C₈	C₂⁵	1	120	yes	—
121	C₂²×C₈	C₂⁵	1	120	yes	—
122	C₄×D₄	—	2	3840	no	—
123	C₄×D₄	C₂⁵	1	512	yes	complemented
124	C₂×C₄⋊C₄	—	2	840	no	—
125	C₂×C₄⋊C₄	C₂⁵	1	240	yes	—
126	C₂×C₂²⋊C₄	—	2	1920	no	—
127	C₂×C₄²	C₂²×D₄	1	840	yes	—
128	C₂×C₄²	C₂⁵	1	120	yes	—
129	#107	C₂⁵	1	256	yes	complemented
130	#102	C₂⁵	1	512	yes	complemented
131	C₄×C₈	C₂⁵	1	256	yes	complemented
132	C₂⁴	—	2	240	no	—
133	C₂⁴	—	2	2240	no	—
134	C₂⁴	—	4	7680	no	—
135	C₂⁴	—	4	1120	no	—
136	C₂⁴	—	4	1120	no	—
137	C₂⁴	C₄×D₈	1	256	yes	direct product factor
138	C₂⁴	C₂×C₄×D₄	1	120	yes	—
139	C₂⁴	C₂²×D₈	1	120	yes	—
140	C₂⁴	C₂×#135	1	120	yes	—
141	C₂⁴	C₂³×D₄	1	35	yes	—
142	C₂×D₄	—	2	240	no	—
143	C₂×D₄	—	2	480	no	—
144	C₂×D₄	—	2	6720	no	—
145	C₂×D₄	C₂⁴×C₄	1	960	yes	complemented
146	C₂×D₄	C₂⁶	1	32	yes	—
147	C₂²×C₄	—	2	560	no	—
148	C₂²×C₄	—	2	240	no	—
149	C₂²×C₄	—	2	512	no	—
150	C₂²×C₄	—	4	1120	no	—
151	C₂²×C₄	—	4	7680	no	—
152	C₂²×C₄	C₂×C₄×D₄	1	840	yes	—
153	C₂²×C₄	C₂²×D₈	1	1120	yes	direct product factor
154	C₂²×C₄	C₂⁴×C₄	1	140	yes	—
155	C₂²×C₄	C₂³×D₄	1	120	yes	—
156	C₂²×C₄	C₂³×D₄	1	105	yes	—
157	C₂²×C₄	C₂³×D₄	1	120	yes	—
158	C₂²×C₄	C₂²×#41	1	140	yes	—
159	C₂²×C₄	C₂⁶	1	15	yes	—
160	D₈	C₂⁴×C₄	1	1024	yes	direct product factor
161	C₂×C₈	—	2	240	no	—
162	C₂×C₈	C₂⁴×C₄	1	480	yes	complemented
163	C₂×C₈	C₂⁶	1	16	yes	—
164	C₂×C₈	C₂⁶	1	16	yes	—
165	C₄⋊C₄	—	2	240	no	—
166	C₄⋊C₄	C₂⁶	1	32	yes	—
167	C₂²⋊C₄	—	2	512	no	—
168	C₄²	C₂³×D₄	1	240	yes	—
169	C₄²	C₂⁶	1	16	yes	—
170	C₂³	—	2	32	no	—
171	C₂³	—	2	960	no	—
172	C₂³	—	4	480	no	—
173	C₂³	—	4	480	no	—
174	C₂³	—	4	8960	no	—
175	C₂³	C₂×C₄×D₈	1	960	yes	direct product factor
176	C₂³	C₂²×C₄×D₄	1	140	yes	—
177	C₂³	C₂³×D₈	1	140	yes	—
178	C₂³	C₂²×#135	1	140	yes	—
179	C₂³	C₂⁴×D₄	1	15	yes	—
180	D₄	—	2	32	no	—
181	D₄	—	2	960	no	—
182	D₄	C₂⁵×C₄	1	64	yes	complemented
183	C₂×C₄	—	2	240	no	—
184	C₂×C₄	—	2	32	no	—
185	C₂×C₄	—	4	480	no	—
186	C₂×C₄	—	4	1024	no	—
187	C₂×C₄	C₂²×C₄×D₄	1	420	yes	—
188	C₂×C₄	C₂³×D₈	1	480	yes	direct product factor
189	C₂×C₄	C₂⁵×C₄	1	30	yes	—
190	C₂×C₄	C₂⁴×D₄	1	16	yes	—
191	C₂×C₄	C₂⁴×D₄	1	16	yes	—
192	C₂×C₄	C₂⁴×D₄	1	15	yes	—
193	C₂×C₄	C₂³×#41	1	30	yes	—
194	C₂×C₄	C₂⁷	1	1	yes	characteristic
195	C₈	—	2	16	no	—
196	C₈	C₂⁵×C₄	1	32	yes	complemented
197	C₂²	—	2	64	no	—
198	C₂²	—	4	1920	no	—
199	C₂²	—	4	32	no	—
200	C₂²	—	4	32	no	—
201	C₂²	C₂²×C₄×D₈	1	560	yes	direct product factor
202	C₂²	C₂³×C₄×D₄	1	30	yes	—
203	C₂²	C₂⁴×D₈	1	30	yes	—
204	C₂²	C₂³×#135	1	30	yes	—
205	C₂²	C₂⁵×D₄	1	1	yes	characteristic
206	C₄	—	2	16	no	—
207	C₄	—	4	32	no	—
208	C₄	C₂³×C₄×D₄	1	30	yes	—
209	C₄	C₂⁴×D₈	1	32	yes	direct product factor
210	C₄	C₂⁶×C₄	1	1	yes	characteristic
211	C₄	C₂⁴×#41	1	1	yes	characteristic
212	C₂	—	4	64	no	—
213	C₂	C₂³×C₄×D₈	1	60	yes	direct product factor
214	C₂	C₂⁴×C₄×D₄	1	1	yes	characteristic
215	C₂	C₂⁵×D₈	1	1	yes	characteristic
216	C₂	C₂⁴×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵×C₄	D₄
commutator	C₄	C₂⁶×C₄
Frattini, cosocle	C₂×C₄	C₂⁷
socle	C₂⁶	C₂×D₄
prime radical	C₂⁵×D₈	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₄ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₄	C₂	semidirect
3. C₂³×C₄×D₈	C₂	direct (N × H)
4. C₂⁴×#107	C₂	semidirect
5. C₂⁴×#102	C₂	semidirect
6. C₂⁴×C₄×C₈	C₂	semidirect
9. C₂⁴×D₈	C₄	direct (N × H)
14. C₂³×C₄×D₄	C₂²	semidirect
18. C₂²×C₄×D₈	C₂²	direct (N × H)
19. C₂³×#107	C₂²	semidirect
20. C₂³×#102	C₂²	semidirect
21. C₂³×C₄×C₈	C₂²	semidirect
25. C₂⁴×D₄	C₂×C₄	semidirect
33. C₂³×D₈	C₂×C₄	direct (N × H)
36. C₂⁴×C₈	C₂×C₄	semidirect
40. C₂²×C₄×D₄	C₂³	semidirect
46. C₂×C₄×D₈	C₂³	direct (N × H)
47. C₂²×#107	C₂³	semidirect
48. C₂²×#102	C₂³	semidirect
49. C₂²×C₄×C₈	C₂³	semidirect
58. C₂³×D₄	C₂²×C₄	semidirect
65. C₂⁴×C₄	D₈	direct (N × H)
72. C₂²×D₈	C₂²×C₄	direct (N × H)
75. C₂³×C₈	C₂²×C₄	semidirect
79. C₂×C₄×D₄	C₂⁴	semidirect
85. C₄×D₈	C₂⁴	direct (N × H)
86. C₂×#107	C₂⁴	semidirect
87. C₂×#102	C₂⁴	semidirect
88. C₂×C₄×C₈	C₂⁴	semidirect
101. C₂²×D₄	C₂³×C₄	semidirect
109. C₂³×C₄	C₂×D₈	direct (N × H)
116. C₂×D₈	C₂³×C₄	direct (N × H)
119. C₂²×C₈	C₂³×C₄	semidirect
123. C₄×D₄	C₂⁵	semidirect
129. #107	C₂⁵	semidirect
130. #102	C₂⁵	semidirect
131. C₄×C₈	C₂⁵	semidirect
137. C₂⁴	C₄×D₈	direct (N × H)
145. C₂×D₄	C₂⁴×C₄	semidirect
153. C₂²×C₄	C₂²×D₈	direct (N × H)
160. D₈	C₂⁴×C₄	direct (N × H)
162. C₂×C₈	C₂⁴×C₄	semidirect
175. C₂³	C₂×C₄×D₈	direct (N × H)
182. D₄	C₂⁵×C₄	semidirect
188. C₂×C₄	C₂³×D₈	direct (N × H)
196. C₈	C₂⁵×C₄	semidirect
201. C₂²	C₂²×C₄×D₈	direct (N × H)
209. C₄	C₂⁴×D₈	direct (N × H)
213. C₂	C₂³×C₄×D₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×D₈	2
C₂⁴×C₄×D₄	2
C₂³×C₄×D₈	2
C₂⁴×#107	2
C₂⁴×#102	2
C₂⁴×C₄×C₈	2

Subgroups — C24×C4×D8 #72286600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄×D₈ #72286600