Subgroups — C₂×Q₈×#6090 #16016401

Order 3328.

582 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6090	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#6090	C₂	1	3	yes	maximal
3	C₂×C₄×#6090	C₂	1	3	yes	maximal
4	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₈×Q₈×C₁₃	C₄	1	4	yes	—
7	C₈×Q₈×C₁₃	C₂²	1	4	yes	—
8	C₂×C₁₃×C₄⋊C₈	C₄	1	3	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
10	C₂²×#6090	C₂²	1	3	yes	—
11	C₂²×#6090	C₂²	1	1	yes	characteristic
12	C₄⋊#6090	C₂²	1	12	yes	—
13	C₄×#6090	C₂²	1	12	yes	—
14	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₄×Q₈×C₁₃	C₈	1	4	yes	—
18	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₈	1	3	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
21	C₁₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
22	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
23	C₄×C₈×C₁₃	C₂³	1	6	yes	—
24	C₂×#6090	—	2	6	no	—
25	C₂×#6090	Q₈	1	4	yes	direct product factor
26	C₂×#6090	C₂³	1	6	yes	—
27	C₂×#6090	C₂³	1	2	yes	—
28	C₂×Q₈×C₁₆	—	13	1	no	maximal, Sylow
29	C₂×Q₈×C₁₃	C₁₆	1	8	yes	complemented
30	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
31	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
32	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	—	2	6	no	—
35	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
36	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
38	C₂×C₈×C₁₃	#41	1	2	yes	—
39	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
40	C₁₃×C₄⋊C₄	C₂×C₈	1	12	yes	—
41	C₄²×C₁₃	C₂×C₈	1	6	yes	—
42	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
43	#6090	—	2	6	no	—
44	#6090	C₂×Q₈	1	4	yes	direct product factor
45	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
46	Q₈×C₁₆	—	13	8	no	—
47	C₂×C₄⋊C₁₆	—	13	3	no	—
48	C₂×C₄×C₁₆	—	13	3	no	—
49	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
50	Q₈×C₁₃	C₂×C₁₆	1	16	yes	complemented
51	C₂×C₄×C₁₃	—	2	6	no	—
52	C₂×C₄×C₁₃	C₂×C₁₆	1	12	yes	—
53	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
54	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
55	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
57	C₂×C₄×C₁₃	#131	1	2	yes	—
58	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
59	C₈×C₁₃	—	2	6	no	—
60	C₈×C₁₃	C₄×Q₈	1	2	yes	—
61	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
62	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
63	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
64	C₂²×C₁₆	—	13	1	no	—
65	C₂²×C₁₆	—	13	3	no	—
66	C₈×Q₈	Dic₁₃	1	4	yes	—
67	C₈×Q₈	C₂×D₁₃	1	4	yes	—
68	C₂×C₄⋊C₈	Dic₁₃	1	3	yes	—
69	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
70	C₄⋊C₁₆	—	13	12	no	—
71	C₄×C₁₆	—	13	12	no	—
72	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
73	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
74	C₂²×C₁₃	#504	1	2	yes	—
75	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
76	C₄×C₁₃	—	2	6	no	—
77	C₄×C₁₃	C₈×Q₈	1	2	yes	—
78	C₄×C₁₃	C₂²×C₁₆	1	12	yes	—
79	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
80	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
81	C₂²×Q₈	#1055	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
83	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
84	C₄×Q₈	#1055	1	4	yes	—
85	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
86	C₂×C₄⋊C₄	#1055	1	3	yes	—
87	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
88	C₂×C₁₆	—	13	4	no	—
89	C₂×C₁₆	—	13	2	no	—
90	C₂×C₁₆	—	13	6	no	—
91	C₂×C₁₆	—	26	6	no	—
92	C₄⋊C₈	C₂×Dic₁₃	1	12	yes	—
93	C₄×C₈	C₂×Dic₁₃	1	6	yes	—
94	C₄×C₈	C₂²×D₁₃	1	6	yes	—
95	C₂×C₁₃	Q₈×C₁₆	1	4	yes	complemented
96	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
97	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
98	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
99	C₂×Q₈	#6090	1	8	yes	direct product factor
100	C₂×Q₈	C₂×#1055	1	4	yes	—
101	C₂²×C₄	C₂×#1055	1	3	yes	—
102	C₂²×C₄	C₂×#1055	1	3	yes	—
103	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
104	C₂×C₈	—	2	6	no	—
105	C₂×C₈	Q₈×D₁₃	1	2	yes	—
106	C₂×C₈	#6131	1	2	yes	—
107	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
108	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
109	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
110	C₄⋊C₄	C₂×#1055	1	12	yes	—
111	C₄²	C₂×#1055	1	6	yes	—
112	C₄²	C₂²×Dic₁₃	1	6	yes	—
113	C₁₆	—	13	4	no	—
114	C₁₆	—	26	6	no	—
115	C₁₃	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
116	C₂³	C₂²×#1055	1	1	yes	characteristic
117	Q₈	C₂×#6090	1	16	yes	direct product factor
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	C₂×#6090	1	12	yes	—
120	C₂×C₄	C₂²×#1055	1	6	yes	—
121	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
122	C₂×C₄	C₂²×#1055	1	6	yes	—
123	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
124	C₂×C₄	#88202	1	2	yes	—
125	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
126	C₈	—	2	6	no	—
127	C₈	Q₈×Dic₁₃	1	2	yes	—
128	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
129	C₈	C₂×#6131	1	1	yes	characteristic
130	C₂²	Q₈×#1055	1	2	yes	—
131	C₂²	C₂²×#6090	1	2	yes	—
132	C₂²	#11718801	1	2	yes	—
133	C₂²	C₂³×#1055	1	1	yes	characteristic
134	C₄	—	2	6	no	—
135	C₄	Q₈×#1055	1	2	yes	—
136	C₄	C₂²×#6090	1	12	yes	—
137	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
138	C₄	C₂×#88202	1	1	yes	characteristic
139	C₂	Q₈×#6090	1	4	yes	direct product factor
140	C₂	C₂×Q₈×#1055	1	1	yes	characteristic
141	C₂	C₂³×#6090	1	1	yes	characteristic
142	C₂	C₂×#11718801	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#6090	C₂	direct (N × H)
25. C₂×#6090	Q₈	direct (N × H)
29. C₂×Q₈×C₁₃	C₁₆	semidirect
44. #6090	C₂×Q₈	direct (N × H)
50. Q₈×C₁₃	C₂×C₁₆	semidirect
95. C₂×C₁₃	Q₈×C₁₆	semidirect
99. C₂×Q₈	#6090	direct (N × H)
115. C₁₃	C₂×Q₈×C₁₆	semidirect
117. Q₈	C₂×#6090	direct (N × H)
139. C₂	Q₈×#6090	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#6090	2
C₂×C₄⋊#6090	2
C₂×C₄×#6090	2
C₂×C₈×Q₈×C₁₃	2
C₂×Q₈×C₁₆	13

Subgroups — C2×Q8×#6090 #16016401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#6090 #16016401