Subgroups — C₃×C₅×D₄×Q₈ #11756638

Order 960.

560 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₈	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×C₂²⋊Q₈	C₂	1	6	yes	maximal, complemented
4	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₅×D₄	C₂	1	3	yes	maximal
6	C₅×D₄×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₈	—	2	8	no	—
8	C₂×C₃×C₅×Q₈	C₂²	1	6	yes	—
9	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	6	yes	—
12	C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
13	C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
14	C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
15	C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄²×C₅	C₂²	1	3	yes	—
17	C₃×D₄×Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₅×Q₈	C₂×C₃	1	2	yes	complemented
19	C₅×C₄⋊Q₈	C₂×C₃	1	3	yes	complemented
20	C₅×C₂²⋊Q₈	C₂×C₃	1	6	yes	complemented
21	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
22	C₄×C₅×D₄	C₂×C₃	1	3	yes	—
23	C₂³×C₃×C₅	C₂³	1	2	yes	—
24	C₃×C₅×Q₈	—	2	12	no	—
25	C₃×C₅×Q₈	D₄	1	4	yes	direct product factor
26	C₃×C₅×D₄	Q₈	1	4	yes	direct product factor
27	C₂×C₃×C₄×C₅	—	2	12	no	—
28	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
29	C₂×C₃×C₄×C₅	C₂³	1	6	yes	—
30	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
31	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Q₈	C₂×C₅	1	2	yes	complemented
33	C₃×C₄⋊Q₈	C₂×C₅	1	3	yes	complemented
34	C₃×C₂²⋊Q₈	C₂×C₅	1	6	yes	complemented
35	C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
36	C₃×C₄×D₄	C₂×C₅	1	3	yes	—
37	C₂×C₅×Q₈	—	2	8	no	—
38	C₂×C₅×Q₈	C₂²×C₃	1	6	yes	—
39	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
40	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂²×C₃	1	6	yes	—
42	C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
43	C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
44	C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
45	C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
46	C₄²×C₅	C₂²×C₃	1	3	yes	—
47	D₄×Q₈	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₃×C₅	—	2	2	no	—
49	C₂²×C₃×C₅	—	2	2	no	—
50	C₂²×C₃×C₅	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
52	C₃×C₄×C₅	—	2	6	no	—
53	C₃×C₄×C₅	—	2	3	no	—
54	C₃×C₄×C₅	C₂×D₄	1	6	yes	—
55	C₃×C₄×C₅	C₂×Q₈	1	2	yes	complemented
56	C₂×C₃×Q₈	—	2	8	no	—
57	C₂×C₃×Q₈	C₂²×C₅	1	6	yes	—
58	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
59	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂²×C₅	1	6	yes	—
61	C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
62	C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
63	C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
64	C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
65	C₃×C₄²	C₂²×C₅	1	3	yes	—
66	C₂³×C₅	C₂³×C₃	1	2	yes	—
67	C₅×Q₈	—	2	12	no	—
68	C₅×Q₈	C₃×D₄	1	4	yes	direct product factor
69	C₅×D₄	C₃×Q₈	1	4	yes	direct product factor
70	C₂×C₄×C₅	—	2	12	no	—
71	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
72	C₂×C₄×C₅	C₂³×C₃	1	6	yes	—
73	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
74	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
75	C₂²×Q₈	C₂×C₃×C₅	1	2	yes	complemented
76	C₄⋊Q₈	C₂×C₃×C₅	1	3	yes	complemented
77	C₂²⋊Q₈	C₂×C₃×C₅	1	6	yes	complemented
78	C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
79	C₄×D₄	C₂×C₃×C₅	1	3	yes	—
80	C₂×C₃×C₅	—	2	4	no	—
81	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂³×C₃	C₂³×C₅	1	2	yes	—
85	C₃×Q₈	—	2	12	no	—
86	C₃×Q₈	C₅×D₄	1	4	yes	direct product factor
87	C₃×D₄	C₅×Q₈	1	4	yes	direct product factor
88	C₂×C₃×C₄	—	2	12	no	—
89	C₂×C₃×C₄	C₂³×C₅	1	3	yes	—
90	C₂×C₃×C₄	C₂³×C₅	1	6	yes	—
91	C₂×C₃×C₄	C₂³×C₅	1	3	yes	—
92	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
93	C₂²×C₅	—	2	2	no	—
94	C₂²×C₅	—	2	2	no	—
95	C₂²×C₅	C₂×C₃×Q₈	1	4	yes	complemented
96	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
97	C₄×C₅	—	2	6	no	—
98	C₄×C₅	—	2	3	no	—
99	C₄×C₅	C₂×C₃×D₄	1	6	yes	—
100	C₄×C₅	C₂×C₃×Q₈	1	2	yes	complemented
101	C₂×Q₈	—	2	8	no	—
102	C₂×Q₈	C₂²×C₃×C₅	1	6	yes	—
103	C₂×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
104	C₂×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
105	C₂²×C₄	C₂²×C₃×C₅	1	6	yes	—
106	C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
107	C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
108	C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
109	C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
110	C₄²	C₂²×C₃×C₅	1	3	yes	—
111	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, direct product factor, Hall
112	C₂²×C₃	—	2	2	no	—
113	C₂²×C₃	—	2	2	no	—
114	C₂²×C₃	C₂×C₅×Q₈	1	4	yes	complemented
115	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
116	C₃×C₄	—	2	6	no	—
117	C₃×C₄	—	2	3	no	—
118	C₃×C₄	C₂×C₅×D₄	1	6	yes	—
119	C₃×C₄	C₂×C₅×Q₈	1	2	yes	complemented
120	C₂×C₅	—	2	4	no	—
121	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
122	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
123	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
124	C₂³	C₂³×C₃×C₅	1	2	yes	—
125	Q₈	—	2	12	no	—
126	Q₈	C₃×C₅×D₄	1	4	yes	direct product factor
127	D₄	C₃×C₅×Q₈	1	4	yes	direct product factor
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	C₂³×C₃×C₅	1	3	yes	—
130	C₂×C₄	C₂³×C₃×C₅	1	6	yes	—
131	C₂×C₄	C₂³×C₃×C₅	1	3	yes	—
132	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
133	C₂×C₃	—	2	4	no	—
134	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
135	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
136	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
137	C₅	C₃×D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	C₂×C₃×C₅×Q₈	1	4	yes	complemented
141	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
142	C₄	—	2	6	no	—
143	C₄	—	2	3	no	—
144	C₄	C₂×C₃×C₅×D₄	1	6	yes	—
145	C₄	C₂×C₃×C₅×Q₈	1	2	yes	complemented
146	C₃	C₅×D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
147	C₂	—	2	4	no	—
148	C₂	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
149	C₂	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
150	C₂	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
prime radical	C₂×C₃×C₅×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×Q₈	C₂	semidirect
2. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
4. C₃×C₄×C₅×Q₈	C₂	semidirect
6. C₅×D₄×Q₈	C₃	direct (N × H)
17. C₃×D₄×Q₈	C₅	direct (N × H)
18. C₂²×C₅×Q₈	C₂×C₃	semidirect
19. C₅×C₄⋊Q₈	C₂×C₃	semidirect
20. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
21. C₄×C₅×Q₈	C₂×C₃	semidirect
25. C₃×C₅×Q₈	D₄	direct (N × H)
26. C₃×C₅×D₄	Q₈	direct (N × H)
32. C₂²×C₃×Q₈	C₂×C₅	semidirect
33. C₃×C₄⋊Q₈	C₂×C₅	semidirect
34. C₃×C₂²⋊Q₈	C₂×C₅	semidirect
35. C₃×C₄×Q₈	C₂×C₅	semidirect
47. D₄×Q₈	C₃×C₅	direct (N × H)
50. C₂²×C₃×C₅	C₂×Q₈	semidirect
55. C₃×C₄×C₅	C₂×Q₈	semidirect
68. C₅×Q₈	C₃×D₄	direct (N × H)
69. C₅×D₄	C₃×Q₈	direct (N × H)
75. C₂²×Q₈	C₂×C₃×C₅	semidirect
76. C₄⋊Q₈	C₂×C₃×C₅	semidirect
77. C₂²⋊Q₈	C₂×C₃×C₅	semidirect
78. C₄×Q₈	C₂×C₃×C₅	semidirect
86. C₃×Q₈	C₅×D₄	direct (N × H)
87. C₃×D₄	C₅×Q₈	direct (N × H)
95. C₂²×C₅	C₂×C₃×Q₈	semidirect
100. C₄×C₅	C₂×C₃×Q₈	semidirect
111. C₃×C₅	D₄×Q₈	direct (N × H)
114. C₂²×C₃	C₂×C₅×Q₈	semidirect
119. C₃×C₄	C₂×C₅×Q₈	semidirect
126. Q₈	C₃×C₅×D₄	direct (N × H)
127. D₄	C₃×C₅×Q₈	direct (N × H)
137. C₅	C₃×D₄×Q₈	direct (N × H)
140. C₂²	C₂×C₃×C₅×Q₈	semidirect
145. C₄	C₂×C₃×C₅×Q₈	semidirect
146. C₃	C₅×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
C₂²×C₃×C₅×Q₈	2
C₃×C₅×C₄⋊Q₈	2
C₃×C₅×C₂²⋊Q₈	2
C₃×C₄×C₅×Q₈	2
C₃×C₄×C₅×D₄	2
C₅×D₄×Q₈	3
C₃×D₄×Q₈	5

Subgroups — C3×C5×D4×Q8 #11756638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×D₄×Q₈ #11756638