Subgroups — C₂×#4924 #82046

Order 384.

576 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂²×#208	C₂	1	2	yes	maximal
3	#4924	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₂×#423	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₄	1	2	yes	—
7	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄²	C₄	1	4	yes	—
10	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄²	C₂²	1	2	yes	—
12	C₂×C₃×C₄²	C₂²	1	4	yes	—
13	C₂×#208	—	2	8	no	—
14	C₂×#208	C₂²	1	8	yes	—
15	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
16	C₄⋊(C₃⋊C₈)	C₂²	1	16	yes	complemented
17	C₂²×M₁₆	—	3	2	no	—
18	C₂²×C₄²	D₃	1	1	yes	characteristic
19	#423	—	3	8	no	—
20	C₂×C₄⋊C₈	—	3	4	no	—
21	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄	—	2	8	no	—
23	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
25	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	D₄	1	4	yes	—
29	C₂²×C₃×C₄	Q₈	1	4	yes	—
30	C₂²×C₃×C₄	C₂³	1	2	yes	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₃×C₄²	C₂×C₄	1	4	yes	—
33	C₃×C₄²	C₂×C₄	1	4	yes	—
34	C₃×C₄²	C₂×C₄	1	4	yes	—
35	C₃×C₄²	C₂³	1	4	yes	—
36	#208	—	2	16	no	—
37	C₂×C₃⋊C₈	—	2	8	no	—
38	C₂×C₃⋊C₈	C₂³	1	8	yes	—
39	C₂³×C₄	Dic₃	1	2	yes	—
40	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂×M₁₆	—	3	8	no	—
42	C₂×M₁₆	—	6	8	no	—
43	C₂²×C₈	—	3	4	no	—
44	C₂×C₄²	Dic₃	1	1	yes	characteristic
45	C₂×C₄²	Dic₃	1	4	yes	—
46	C₂×C₄²	Dic₃	1	1	yes	characteristic
47	C₂×C₄²	C₂×D₃	1	4	yes	—
48	C₂×C₄²	C₂×D₃	1	2	yes	—
49	C₄⋊C₈	—	3	16	no	—
50	C₂³×C₃	—	2	4	no	—
51	C₂³×C₃	C₄⋊C₄	1	4	yes	—
52	C₂³×C₃	C₂²×C₄	1	2	yes	—
53	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄	—	2	4	no	—
55	C₂×C₃×C₄	—	2	16	no	—
56	C₂×C₃×C₄	C₄⋊C₄	1	8	yes	—
57	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
58	C₂×C₃×C₄	M₁₆	1	8	yes	complemented
59	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
60	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
61	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
63	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
65	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
66	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
68	C₃⋊C₈	—	2	8	no	—
69	C₂⁴	C₂×Dic₃	1	1	yes	characteristic
70	C₂²×C₄	—	2	8	no	—
71	C₂²×C₄	Dic₆	1	4	yes	—
72	C₂²×C₄	D₁₂	1	4	yes	—
73	C₂²×C₄	C₂×Dic₃	1	2	yes	—
74	C₂²×C₄	C₂×Dic₃	1	8	yes	—
75	C₂²×C₄	C₂×Dic₃	1	2	yes	—
76	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×Dic₃	1	2	yes	—
78	C₂²×C₄	C₂²×D₃	1	2	yes	—
79	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
80	M₁₆	—	6	16	no	—
81	C₂×C₈	—	3	8	no	—
82	C₂×C₈	—	6	8	no	—
83	C₄²	C₂×Dic₃	1	4	yes	—
84	C₄²	C₂×Dic₃	1	4	yes	—
85	C₄²	C₂×Dic₃	1	4	yes	—
86	C₄²	C₂²×D₃	1	4	yes	—
87	C₂²×C₃	—	2	4	no	—
88	C₂²×C₃	—	2	8	no	—
89	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
90	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₃	C₂×M₁₆	1	4	yes	—
92	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
93	C₃×C₄	—	2	4	no	—
94	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
95	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
96	C₃×C₄	C₂×M₁₆	1	8	yes	complemented
97	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
98	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
99	C₂³	—	2	4	no	—
100	C₂³	C₄⋊Dic₃	1	4	yes	—
101	C₂³	C₂²×Dic₃	1	2	yes	—
102	C₂³	C₂²×Dic₃	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	16	no	—
105	C₂×C₄	#208	1	8	yes	complemented
106	C₂×C₄	C₄⋊Dic₃	1	8	yes	—
107	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
108	C₂×C₄	C₂×Dic₆	1	4	yes	—
109	C₂×C₄	C₂×Dic₆	1	2	yes	—
110	C₂×C₄	C₂×D₁₂	1	2	yes	—
111	C₂×C₄	C₂×D₁₂	1	4	yes	—
112	C₂×C₄	C₂²×Dic₃	1	4	yes	—
113	C₂×C₄	C₂²×Dic₃	1	4	yes	—
114	C₂×C₄	C₂²×Dic₃	1	2	yes	—
115	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
117	C₈	—	6	8	no	—
118	C₂×C₃	—	2	4	no	—
119	C₂×C₃	#423	1	4	yes	complemented
120	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₃	C₂²×M₁₆	1	2	yes	—
122	C₂²	—	2	8	no	—
123	C₂²	—	2	4	no	—
124	C₂²	C₂×#208	1	4	yes	—
125	C₂²	C₂×C₄⋊Dic₃	1	4	yes	—
126	C₂²	C₂×C₄⋊Dic₃	1	2	yes	—
127	C₂²	C₂³×Dic₃	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	C₂×#208	1	8	yes	complemented
130	C₄	C₂×C₄⋊Dic₃	1	4	yes	—
131	C₄	C₂×C₄⋊Dic₃	1	2	yes	—
132	C₄	C₂²×Dic₆	1	1	yes	characteristic
133	C₄	C₂²×D₁₂	1	1	yes	characteristic
134	C₃	C₂×#423	1	1	yes	characteristic, complemented, Sylow
135	C₂	—	2	4	no	—
136	C₂	#4924	1	4	yes	direct product factor
137	C₂	C₂²×#208	1	2	yes	—
138	C₂	C₂²×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	C₂×#423
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. #4924	C₂	direct (N × H)
4. C₂×C₄⋊(C₃⋊C₈)	C₂	semidirect
16. C₄⋊(C₃⋊C₈)	C₂²	semidirect
58. C₂×C₃×C₄	M₁₆	semidirect
96. C₃×C₄	C₂×M₁₆	semidirect
105. C₂×C₄	#208	semidirect
119. C₂×C₃	#423	semidirect
129. C₄	C₂×#208	semidirect
134. C₃	C₂×#423	semidirect
136. C₂	#4924	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×C₄²	2
C₂²×#208	2
#4924	2
C₂×C₄⋊(C₃⋊C₈)	2
C₂×#423	3

Subgroups — C2×#4924 #82046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4924 #82046