Subgroups — C₂⁴×D₄×Q₈ #72339120

Order 1024.

858725 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×Q₈	C₂	1	2	yes	maximal, complemented
2	C₂³×D₄×Q₈	C₂	1	240	yes	maximal, direct product factor
3	C₂⁴×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
4	C₂⁴×C₂²⋊Q₈	C₂	1	6	yes	maximal, complemented
5	C₂⁴×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₄×D₄	C₂	1	3	yes	maximal
7	C₂⁵×Q₈	—	2	128	no	—
8	C₂⁵×Q₈	C₂²	1	6	yes	—
9	C₂⁵×Q₈	C₂²	1	1	yes	characteristic
10	C₂⁵×Q₈	C₂²	1	240	yes	complemented
11	C₂⁵×D₄	C₂²	1	1	yes	characteristic
12	C₂⁶×C₄	C₂²	1	6	yes	—
13	C₂²×D₄×Q₈	C₂²	1	8960	yes	direct product factor
14	C₂³×C₄⋊Q₈	C₂²	1	360	yes	complemented
15	C₂³×C₂²⋊Q₈	C₂²	1	720	yes	complemented
16	C₂³×C₄×Q₈	C₂²	1	120	yes	complemented
17	C₂³×C₄×D₄	C₂²	1	360	yes	—
18	C₂⁴×C₄⋊C₄	C₂²	1	3	yes	—
19	C₂⁴×C₄⋊C₄	C₂²	1	3	yes	—
20	C₂⁴×C₄⋊C₄	C₂²	1	6	yes	—
21	C₂⁴×C₂²⋊C₄	C₂²	1	6	yes	—
22	C₂⁴×C₄²	C₂²	1	3	yes	—
23	C₂⁷	C₂³	1	2	yes	—
24	C₂⁴×Q₈	—	2	7680	no	—
25	C₂⁴×Q₈	—	2	192	no	—
26	C₂⁴×Q₈	D₄	1	64	yes	direct product factor
27	C₂⁴×Q₈	C₂³	1	360	yes	—
28	C₂⁴×Q₈	C₂³	1	60	yes	—
29	C₂⁴×Q₈	C₂³	1	4480	yes	complemented
30	C₂⁴×D₄	Q₈	1	64	yes	direct product factor
31	C₂⁴×D₄	C₂³	1	60	yes	—
32	C₂⁵×C₄	—	2	192	no	—
33	C₂⁵×C₄	C₂³	1	360	yes	—
34	C₂⁵×C₄	C₂³	1	6	yes	—
35	C₂⁵×C₄	C₂³	1	3	yes	—
36	C₂⁵×C₄	C₂³	1	3	yes	—
37	C₂⁵×C₄	C₂³	1	1	yes	characteristic
38	C₂×D₄×Q₈	C₂³	1	61440	yes	direct product factor
39	C₂²×C₄⋊Q₈	C₂³	1	6720	yes	complemented
40	C₂²×C₂²⋊Q₈	C₂³	1	13440	yes	complemented
41	C₂²×C₄×Q₈	C₂³	1	2240	yes	complemented
42	C₂²×C₄×D₄	C₂³	1	6720	yes	—
43	C₂³×C₄⋊C₄	C₂³	1	180	yes	—
44	C₂³×C₄⋊C₄	C₂³	1	180	yes	—
45	C₂³×C₄⋊C₄	C₂³	1	360	yes	—
46	C₂³×C₂²⋊C₄	C₂³	1	360	yes	—
47	C₂³×C₄²	C₂³	1	180	yes	—
48	C₂⁶	—	2	32	no	—
49	C₂⁶	—	2	32	no	—
50	C₂⁶	C₂×Q₈	1	64	yes	complemented
51	C₂⁶	C₂⁴	1	60	yes	—
52	C₂⁶	C₂⁴	1	1	yes	characteristic
53	C₂³×Q₈	—	2	71680	no	—
54	C₂³×Q₈	—	2	5760	no	—
55	C₂³×Q₈	C₂×D₄	1	1920	yes	direct product factor
56	C₂³×Q₈	C₂⁴	1	3360	yes	—
57	C₂³×Q₈	C₂⁴	1	560	yes	—
58	C₂³×Q₈	C₂⁴	1	15360	yes	complemented
59	C₂³×D₄	C₂×Q₈	1	1920	yes	direct product factor
60	C₂³×D₄	C₂⁴	1	560	yes	—
61	C₂⁴×C₄	—	2	96	no	—
62	C₂⁴×C₄	—	2	48	no	—
63	C₂⁴×C₄	—	2	5760	no	—
64	C₂⁴×C₄	C₂×D₄	1	96	yes	—
65	C₂⁴×C₄	C₂×Q₈	1	32	yes	complemented
66	C₂⁴×C₄	C₂⁴	1	180	yes	—
67	C₂⁴×C₄	C₂⁴	1	3360	yes	—
68	C₂⁴×C₄	C₂⁴	1	90	yes	—
69	C₂⁴×C₄	C₂⁴	1	90	yes	—
70	C₂⁴×C₄	C₂⁴	1	30	yes	—
71	D₄×Q₈	C₂⁴	1	65536	yes	direct product factor
72	C₂×C₄⋊Q₈	C₂⁴	1	23040	yes	complemented
73	C₂×C₂²⋊Q₈	C₂⁴	1	46080	yes	complemented
74	C₂×C₄×Q₈	C₂⁴	1	7680	yes	complemented
75	C₂×C₄×D₄	C₂⁴	1	23040	yes	—
76	C₂²×C₄⋊C₄	C₂⁴	1	1680	yes	—
77	C₂²×C₄⋊C₄	C₂⁴	1	1680	yes	—
78	C₂²×C₄⋊C₄	C₂⁴	1	3360	yes	—
79	C₂²×C₂²⋊C₄	C₂⁴	1	3360	yes	—
80	C₂²×C₄²	C₂⁴	1	1680	yes	—
81	C₂⁵	—	2	1024	no	—
82	C₂⁵	—	2	480	no	—
83	C₂⁵	—	2	480	no	—
84	C₂⁵	C₂²×D₄	1	16	yes	—
85	C₂⁵	C₂²×Q₈	1	16	yes	—
86	C₂⁵	C₂²×Q₈	1	960	yes	complemented
87	C₂⁵	2₋¹⁺⁴	1	16	yes	—
88	C₂⁵	C₂⁵	1	280	yes	—
89	C₂⁵	C₂⁵	1	15	yes	—
90	C₂²×Q₈	—	2	122880	no	—
91	C₂²×Q₈	—	2	26880	no	—
92	C₂²×Q₈	C₂²×D₄	1	8960	yes	direct product factor
93	C₂²×Q₈	C₂⁵	1	5760	yes	—
94	C₂²×Q₈	C₂⁵	1	960	yes	—
95	C₂²×Q₈	C₂⁵	1	8192	yes	complemented
96	C₂²×D₄	C₂²×Q₈	1	8960	yes	direct product factor
97	C₂²×D₄	C₂⁵	1	960	yes	—
98	C₂³×C₄	—	2	1440	no	—
99	C₂³×C₄	—	2	720	no	—
100	C₂³×C₄	—	2	26880	no	—
101	C₂³×C₄	C₂²×D₄	1	1440	yes	—
102	C₂³×C₄	C₂²×Q₈	1	480	yes	complemented
103	C₂³×C₄	C₂⁵	1	420	yes	—
104	C₂³×C₄	C₂⁵	1	840	yes	—
105	C₂³×C₄	C₂⁵	1	5760	yes	—
106	C₂³×C₄	C₂⁵	1	420	yes	—
107	C₂³×C₄	C₂⁵	1	140	yes	—
108	C₄⋊Q₈	C₂⁵	1	12288	yes	complemented
109	C₂²⋊Q₈	C₂⁵	1	24576	yes	complemented
110	C₄×Q₈	C₂⁵	1	4096	yes	complemented
111	C₄×D₄	C₂⁵	1	12288	yes	—
112	C₂×C₄⋊C₄	C₂⁵	1	2880	yes	—
113	C₂×C₄⋊C₄	C₂⁵	1	2880	yes	—
114	C₂×C₄⋊C₄	C₂⁵	1	5760	yes	—
115	C₂×C₂²⋊C₄	C₂⁵	1	5760	yes	—
116	C₂×C₄²	C₂⁵	1	2880	yes	—
117	C₂⁴	—	2	1120	no	—
118	C₂⁴	—	2	1120	no	—
119	C₂⁴	—	2	7680	no	—
120	C₂⁴	D₄×Q₈	1	256	yes	direct product factor
121	C₂⁴	C₂³×D₄	1	120	yes	—
122	C₂⁴	C₂³×Q₈	1	120	yes	—
123	C₂⁴	C₂³×Q₈	1	2240	yes	complemented
124	C₂⁴	C₂×2₋¹⁺⁴	1	120	yes	—
125	C₂⁴	C₂⁶	1	240	yes	—
126	C₂⁴	C₂⁶	1	35	yes	—
127	C₂×Q₈	—	2	32768	no	—
128	C₂×Q₈	—	2	23040	no	—
129	C₂×Q₈	C₂³×D₄	1	7680	yes	direct product factor
130	C₂×Q₈	C₂⁶	1	1536	yes	—
131	C₂×Q₈	C₂⁶	1	256	yes	—
132	C₂×D₄	C₂³×Q₈	1	7680	yes	direct product factor
133	C₂×D₄	C₂⁶	1	256	yes	—
134	C₂²×C₄	—	2	3360	no	—
135	C₂²×C₄	—	2	1680	no	—
136	C₂²×C₄	—	2	23040	no	—
137	C₂²×C₄	C₂³×D₄	1	3360	yes	—
138	C₂²×C₄	C₂³×Q₈	1	1120	yes	complemented
139	C₂²×C₄	C₂⁶	1	360	yes	—
140	C₂²×C₄	C₂⁶	1	720	yes	—
141	C₂²×C₄	C₂⁶	1	1536	yes	—
142	C₂²×C₄	C₂⁶	1	360	yes	—
143	C₂²×C₄	C₂⁶	1	120	yes	—
144	C₄⋊C₄	C₂⁶	1	768	yes	—
145	C₄⋊C₄	C₂⁶	1	768	yes	—
146	C₄⋊C₄	C₂⁶	1	1536	yes	—
147	C₂²⋊C₄	C₂⁶	1	1536	yes	—
148	C₄²	C₂⁶	1	768	yes	—
149	C₂³	—	2	480	no	—
150	C₂³	—	2	480	no	—
151	C₂³	—	2	8960	no	—
152	C₂³	C₂×D₄×Q₈	1	960	yes	direct product factor
153	C₂³	C₂⁴×D₄	1	140	yes	—
154	C₂³	C₂⁴×Q₈	1	140	yes	—
155	C₂³	C₂⁴×Q₈	1	960	yes	complemented
156	C₂³	C₂²×2₋¹⁺⁴	1	140	yes	—
157	C₂³	C₂⁷	1	32	yes	—
158	C₂³	C₂⁷	1	15	yes	—
159	Q₈	—	2	3072	no	—
160	Q₈	C₂⁴×D₄	1	1024	yes	direct product factor
161	D₄	C₂⁴×Q₈	1	1024	yes	direct product factor
162	C₂×C₄	—	2	1440	no	—
163	C₂×C₄	—	2	720	no	—
164	C₂×C₄	—	2	3072	no	—
165	C₂×C₄	C₂⁴×D₄	1	1440	yes	—
166	C₂×C₄	C₂⁴×Q₈	1	480	yes	complemented
167	C₂×C₄	C₂⁷	1	48	yes	—
168	C₂×C₄	C₂⁷	1	96	yes	—
169	C₂×C₄	C₂⁷	1	48	yes	—
170	C₂×C₄	C₂⁷	1	16	yes	—
171	C₂²	—	2	1920	no	—
172	C₂²	—	2	32	no	—
173	C₂²	—	2	32	no	—
174	C₂²	C₂²×D₄×Q₈	1	560	yes	direct product factor
175	C₂²	C₂⁵×D₄	1	30	yes	—
176	C₂²	C₂⁵×Q₈	1	30	yes	—
177	C₂²	C₂⁵×Q₈	1	64	yes	complemented
178	C₂²	C₂³×2₋¹⁺⁴	1	30	yes	—
179	C₂²	C₂⁸	1	1	yes	characteristic
180	C₄	—	2	96	no	—
181	C₄	—	2	48	no	—
182	C₄	C₂⁵×D₄	1	96	yes	—
183	C₄	C₂⁵×Q₈	1	32	yes	complemented
184	C₂	—	2	64	no	—
185	C₂	C₂³×D₄×Q₈	1	60	yes	direct product factor
186	C₂	C₂⁶×D₄	1	1	yes	characteristic
187	C₂	C₂⁶×Q₈	1	1	yes	characteristic
188	C₂	C₂⁴×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁶×Q₈	C₂	semidirect
2. C₂³×D₄×Q₈	C₂	direct (N × H)
3. C₂⁴×C₄⋊Q₈	C₂	semidirect
4. C₂⁴×C₂²⋊Q₈	C₂	semidirect
5. C₂⁴×C₄×Q₈	C₂	semidirect
10. C₂⁵×Q₈	C₂²	semidirect
13. C₂²×D₄×Q₈	C₂²	direct (N × H)
14. C₂³×C₄⋊Q₈	C₂²	semidirect
15. C₂³×C₂²⋊Q₈	C₂²	semidirect
16. C₂³×C₄×Q₈	C₂²	semidirect
26. C₂⁴×Q₈	D₄	direct (N × H)
29. C₂⁴×Q₈	C₂³	semidirect
30. C₂⁴×D₄	Q₈	direct (N × H)
38. C₂×D₄×Q₈	C₂³	direct (N × H)
39. C₂²×C₄⋊Q₈	C₂³	semidirect
40. C₂²×C₂²⋊Q₈	C₂³	semidirect
41. C₂²×C₄×Q₈	C₂³	semidirect
50. C₂⁶	C₂×Q₈	semidirect
55. C₂³×Q₈	C₂×D₄	direct (N × H)
58. C₂³×Q₈	C₂⁴	semidirect
59. C₂³×D₄	C₂×Q₈	direct (N × H)
65. C₂⁴×C₄	C₂×Q₈	semidirect
71. D₄×Q₈	C₂⁴	direct (N × H)
72. C₂×C₄⋊Q₈	C₂⁴	semidirect
73. C₂×C₂²⋊Q₈	C₂⁴	semidirect
74. C₂×C₄×Q₈	C₂⁴	semidirect
86. C₂⁵	C₂²×Q₈	semidirect
92. C₂²×Q₈	C₂²×D₄	direct (N × H)
95. C₂²×Q₈	C₂⁵	semidirect
96. C₂²×D₄	C₂²×Q₈	direct (N × H)
102. C₂³×C₄	C₂²×Q₈	semidirect
108. C₄⋊Q₈	C₂⁵	semidirect
109. C₂²⋊Q₈	C₂⁵	semidirect
110. C₄×Q₈	C₂⁵	semidirect
120. C₂⁴	D₄×Q₈	direct (N × H)
123. C₂⁴	C₂³×Q₈	semidirect
129. C₂×Q₈	C₂³×D₄	direct (N × H)
132. C₂×D₄	C₂³×Q₈	direct (N × H)
138. C₂²×C₄	C₂³×Q₈	semidirect
152. C₂³	C₂×D₄×Q₈	direct (N × H)
155. C₂³	C₂⁴×Q₈	semidirect
160. Q₈	C₂⁴×D₄	direct (N × H)
161. D₄	C₂⁴×Q₈	direct (N × H)
166. C₂×C₄	C₂⁴×Q₈	semidirect
174. C₂²	C₂²×D₄×Q₈	direct (N × H)
177. C₂²	C₂⁵×Q₈	semidirect
183. C₄	C₂⁵×Q₈	semidirect
185. C₂	C₂³×D₄×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×Q₈	2
C₂³×D₄×Q₈	2
C₂⁴×C₄⋊Q₈	2
C₂⁴×C₂²⋊Q₈	2
C₂⁴×C₄×Q₈	2
C₂⁴×C₄×D₄	2

Subgroups — C24×D4×Q8 #72339120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₄×Q₈ #72339120