Subgroups — C₂×C₃×C₄×Q₁₆ #84802

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	2	yes	maximal
3	C₃×C₄×Q₁₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#107	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#103	C₂	1	2	yes	maximal
6	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×Q₈	C₂²	1	2	yes	—
9	C₂×C₃×Q₁₆	C₄	1	8	yes	direct product factor
10	C₂×C₃×Q₁₆	C₂²	1	4	yes	—
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×Q₈	—	2	4	no	—
14	C₃×C₄×Q₈	C₂²	1	8	yes	—
15	C₂×C₃×C₄⋊C₄	—	2	2	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	—	2	2	no	—
18	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₃×#107	C₂²	1	4	yes	—
20	C₃×#103	C₂²	1	8	yes	—
21	C₃×C₄×C₈	C₂²	1	4	yes	—
22	C₂²×Q₁₆	C₂×C₃	1	1	yes	characteristic
23	C₂×C₄×Q₈	C₂×C₃	1	2	yes	—
24	C₄×Q₁₆	C₂×C₃	1	8	yes	direct product factor
25	C₂×#107	C₂×C₃	1	1	yes	characteristic
26	C₂×#103	C₂×C₃	1	2	yes	—
27	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×Q₈	—	2	2	no	—
29	C₂×C₃×Q₈	—	2	4	no	—
30	C₂×C₃×Q₈	C₂×C₄	1	8	yes	—
31	C₂×C₃×Q₈	C₂³	1	4	yes	—
32	C₂²×C₃×C₄	—	2	2	no	—
33	C₂²×C₃×C₄	—	2	2	no	—
34	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₃×Q₁₆	C₂×C₄	1	16	yes	direct product factor
38	C₂×C₃×C₈	—	2	2	no	—
39	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₈	C₂³	1	2	yes	—
41	C₂×C₃×C₈	C₂³	1	2	yes	—
42	C₃×C₄⋊C₄	—	2	8	no	—
43	C₃×C₄⋊C₄	—	2	2	no	—
44	C₃×C₄⋊C₄	C₂³	1	4	yes	—
45	C₃×C₄²	—	2	8	no	—
46	C₃×C₄²	D₄	1	2	yes	—
47	C₃×C₄²	C₂³	1	2	yes	—
48	C₂²×Q₈	C₂²×C₃	1	2	yes	—
49	C₂×Q₁₆	C₃×C₄	1	8	yes	direct product factor
50	C₂×Q₁₆	C₂²×C₃	1	4	yes	—
51	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₄×Q₈	—	2	4	no	—
54	C₄×Q₈	C₂²×C₃	1	8	yes	—
55	C₂×C₄⋊C₄	—	2	2	no	—
56	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₂×C₄²	—	2	2	no	—
58	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
59	#107	C₂²×C₃	1	4	yes	—
60	#103	C₂²×C₃	1	8	yes	—
61	C₄×C₈	C₂²×C₃	1	4	yes	—
62	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
63	C₃×Q₈	—	2	8	no	—
64	C₃×Q₈	—	2	4	no	—
65	C₃×Q₈	C₂²×C₄	1	8	yes	—
66	C₂×C₃×C₄	—	2	8	no	—
67	C₂×C₃×C₄	—	2	2	no	—
68	C₂×C₃×C₄	—	2	4	no	—
69	C₂×C₃×C₄	—	2	4	no	—
70	C₂×C₃×C₄	—	4	4	no	—
71	C₂×C₃×C₄	Q₁₆	1	4	yes	direct product factor
72	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
73	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
74	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄	#41	1	2	yes	—
77	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
78	C₃×C₈	—	2	2	no	—
79	C₃×C₈	C₂²×C₄	1	4	yes	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	C₂×C₃×C₄	1	8	yes	—
83	C₂×Q₈	C₂³×C₃	1	4	yes	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
89	Q₁₆	C₂×C₃×C₄	1	16	yes	direct product factor
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	C₂×C₃×C₄	1	4	yes	—
92	C₂×C₈	C₂³×C₃	1	2	yes	—
93	C₂×C₈	C₂³×C₃	1	2	yes	—
94	C₄⋊C₄	—	2	8	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	C₂³×C₃	1	4	yes	—
97	C₄²	—	2	8	no	—
98	C₄²	C₃×D₄	1	2	yes	—
99	C₄²	C₂³×C₃	1	2	yes	—
100	C₂²×C₃	C₄×D₄	1	2	yes	—
101	C₂²×C₃	C₂×Q₁₆	1	2	yes	—
102	C₂²×C₃	#135	1	2	yes	—
103	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
104	C₃×C₄	—	2	8	no	—
105	C₃×C₄	—	2	2	no	—
106	C₃×C₄	—	4	4	no	—
107	C₃×C₄	C₄×D₄	1	2	yes	—
108	C₃×C₄	C₂×Q₁₆	1	4	yes	direct product factor
109	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
110	C₃×C₄	C₂×#41	1	1	yes	characteristic
111	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
112	Q₈	—	2	8	no	—
113	Q₈	—	2	4	no	—
114	Q₈	C₂²×C₃×C₄	1	8	yes	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	C₃×Q₁₆	1	4	yes	direct product factor
121	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
122	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
123	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
124	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
125	C₂×C₄	C₃×#41	1	2	yes	—
126	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	C₂²×C₃×C₄	1	4	yes	—
129	C₂×C₃	C₄×Q₁₆	1	4	yes	direct product factor
130	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂×C₃	C₂²×Q₁₆	1	1	yes	characteristic
132	C₂×C₃	C₂×#135	1	1	yes	characteristic
133	C₂²	C₃×C₄×D₄	1	2	yes	—
134	C₂²	C₂×C₃×Q₁₆	1	2	yes	—
135	C₂²	C₃×#135	1	2	yes	—
136	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	8	no	—
139	C₄	—	4	4	no	—
140	C₄	C₃×C₄×D₄	1	2	yes	—
141	C₄	C₂×C₃×Q₁₆	1	4	yes	direct product factor
142	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
143	C₄	C₂×C₃×#41	1	1	yes	characteristic
144	C₃	C₂×C₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
145	C₂	C₃×C₄×Q₁₆	1	4	yes	direct product factor
146	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
147	C₂	C₂²×C₃×Q₁₆	1	1	yes	characteristic
148	C₂	C₂×C₃×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×Q₁₆	C₂	direct (N × H)
7. C₂×C₄×Q₁₆	C₃	direct (N × H)
9. C₂×C₃×Q₁₆	C₄	direct (N × H)
24. C₄×Q₁₆	C₂×C₃	direct (N × H)
37. C₃×Q₁₆	C₂×C₄	direct (N × H)
49. C₂×Q₁₆	C₃×C₄	direct (N × H)
71. C₂×C₃×C₄	Q₁₆	direct (N × H)
89. Q₁₆	C₂×C₃×C₄	direct (N × H)
108. C₃×C₄	C₂×Q₁₆	direct (N × H)
120. C₂×C₄	C₃×Q₁₆	direct (N × H)
129. C₂×C₃	C₄×Q₁₆	direct (N × H)
141. C₄	C₂×C₃×Q₁₆	direct (N × H)
144. C₃	C₂×C₄×Q₁₆	direct (N × H)
145. C₂	C₃×C₄×Q₁₆	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×Q₁₆	2
C₂×C₃×C₄×Q₈	2
C₃×C₄×Q₁₆	2
C₂×C₃×#107	2
C₂×C₃×#103	2
C₂×C₃×C₄×C₈	2
C₂×C₄×Q₁₆	3

Subgroups — C2×C3×C4×Q16 #84802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×Q₁₆ #84802