Subgroups — C₂×C₄×#208 #82044

Order 384.

656 conjugacy classes of subgroups.

The subgroup lattice has 138 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	C₄×#208	C₂	1	8	yes	maximal, direct product factor
4	C₂×#803	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×M₁₆	—	3	1	no	maximal, Sylow
7	C₂³×C₃×C₄	C₄	1	2	yes	—
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²	C₄	1	4	yes	—
10	C₂×C₃×C₄²	C₄	1	2	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²	C₂²	1	4	yes	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×#208	C₄	1	16	yes	direct product factor
15	C₂×#208	C₂²	1	8	yes	—
16	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
17	#803	C₂²	1	8	yes	complemented
18	C₄×C₃⋊C₈	C₂²	1	8	yes	complemented
19	C₂²×M₁₆	—	3	2	no	—
20	C₂²×C₄²	D₃	1	1	yes	characteristic
21	C₄×M₁₆	—	3	8	no	—
22	C₂×#97	—	3	2	no	—
23	C₂×C₄×C₈	—	3	2	no	—
24	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	—	2	8	no	—
26	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
29	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂³	1	2	yes	—
34	C₃×C₄²	C₂×C₄	1	2	yes	—
35	C₃×C₄²	C₂×C₄	1	8	yes	—
36	C₃×C₄²	C₂×C₄	1	2	yes	—
37	C₃×C₄²	C₂³	1	2	yes	—
38	C₃×C₄²	C₂³	1	2	yes	—
39	#208	C₂×C₄	1	32	yes	direct product factor
40	C₂×C₃⋊C₈	C₂×C₄	1	16	yes	complemented
41	C₂×C₃⋊C₈	C₂³	1	8	yes	—
42	C₂³×C₄	Dic₃	1	2	yes	—
43	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂×M₁₆	—	3	16	no	—
45	C₂×M₁₆	—	3	8	no	—
46	C₂²×C₈	—	3	4	no	—
47	C₂×C₄²	Dic₃	1	2	yes	—
48	C₂×C₄²	Dic₃	1	4	yes	—
49	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₄²	C₂×D₃	1	4	yes	—
51	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
52	#97	—	3	8	no	—
53	C₄×C₈	—	3	8	no	—
54	C₂³×C₃	—	2	4	no	—
55	C₂³×C₃	C₄²	1	4	yes	—
56	C₂³×C₃	C₂²×C₄	1	2	yes	—
57	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄	—	2	4	no	—
59	C₂×C₃×C₄	—	2	16	no	—
60	C₂×C₃×C₄	C₄²	1	8	yes	—
61	C₂×C₃×C₄	C₄²	1	4	yes	—
62	C₂×C₃×C₄	M₁₆	1	8	yes	complemented
63	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
64	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
65	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
66	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
68	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
69	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
70	C₃⋊C₈	C₂²×C₄	1	16	yes	complemented
71	C₂⁴	C₂×Dic₃	1	1	yes	characteristic
72	C₂²×C₄	—	2	8	no	—
73	C₂²×C₄	C₄×D₃	1	8	yes	—
74	C₂²×C₄	C₂×Dic₃	1	2	yes	—
75	C₂²×C₄	C₂×Dic₃	1	8	yes	—
76	C₂²×C₄	C₂×Dic₃	1	2	yes	—
77	C₂²×C₄	C₂×Dic₃	1	2	yes	—
78	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₃	1	2	yes	—
80	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
81	M₁₆	—	3	32	no	—
82	C₂×C₈	—	3	16	no	—
83	C₂×C₈	—	3	8	no	—
84	C₄²	C₂×Dic₃	1	2	yes	—
85	C₄²	C₂×Dic₃	1	8	yes	—
86	C₄²	C₂×Dic₃	1	2	yes	—
87	C₄²	C₂²×D₃	1	2	yes	—
88	C₄²	C₂²×D₃	1	2	yes	—
89	C₂²×C₃	—	2	4	no	—
90	C₂²×C₃	—	2	8	no	—
91	C₂²×C₃	C₂×C₄²	1	4	yes	—
92	C₂²×C₃	C₂×C₄²	1	2	yes	—
93	C₂²×C₃	C₂×M₁₆	1	4	yes	—
94	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
95	C₃×C₄	—	2	4	no	—
96	C₃×C₄	C₂×C₄²	1	2	yes	—
97	C₃×C₄	C₂×C₄²	1	4	yes	—
98	C₃×C₄	C₂×M₁₆	1	8	yes	complemented
99	C₃×C₄	C₂³×C₄	1	2	yes	—
100	C₂³	—	2	4	no	—
101	C₂³	C₄×Dic₃	1	4	yes	—
102	C₂³	C₂²×Dic₃	1	2	yes	—
103	C₂³	C₂²×Dic₃	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	16	no	—
106	C₂×C₄	#208	1	8	yes	direct product factor
107	C₂×C₄	C₄×Dic₃	1	8	yes	—
108	C₂×C₄	C₄×Dic₃	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
110	C₂×C₄	C₂×C₄×D₃	1	8	yes	—
111	C₂×C₄	C₂²×Dic₃	1	4	yes	—
112	C₂×C₄	C₂²×Dic₃	1	4	yes	—
113	C₂×C₄	C₂²×Dic₃	1	2	yes	—
114	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
115	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
116	C₈	—	3	16	no	—
117	C₂×C₃	—	2	4	no	—
118	C₂×C₃	C₄×M₁₆	1	4	yes	complemented
119	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
120	C₂×C₃	C₂²×M₁₆	1	2	yes	—
121	C₂²	—	2	8	no	—
122	C₂²	—	2	4	no	—
123	C₂²	C₂×#208	1	4	yes	—
124	C₂²	C₂×C₄×Dic₃	1	4	yes	—
125	C₂²	C₂×C₄×Dic₃	1	2	yes	—
126	C₂²	C₂³×Dic₃	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	C₂×#208	1	8	yes	direct product factor
129	C₄	C₂×C₄×Dic₃	1	2	yes	—
130	C₄	C₂×C₄×Dic₃	1	4	yes	—
131	C₄	C₂²×C₄×D₃	1	2	yes	—
132	C₃	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
133	C₂	—	2	4	no	—
134	C₂	C₄×#208	1	4	yes	direct product factor
135	C₂	C₂²×#208	1	2	yes	—
136	C₂	C₂²×C₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₃
commutator	C₂×C₃	C₂²×C₄²
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×C₄×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×#208	C₂	direct (N × H)
4. C₂×#803	C₂	semidirect
5. C₂×C₄×C₃⋊C₈	C₂	semidirect
14. C₂×#208	C₄	direct (N × H)
17. #803	C₂²	semidirect
18. C₄×C₃⋊C₈	C₂²	semidirect
39. #208	C₂×C₄	direct (N × H)
40. C₂×C₃⋊C₈	C₂×C₄	semidirect
62. C₂×C₃×C₄	M₁₆	semidirect
70. C₃⋊C₈	C₂²×C₄	semidirect
98. C₃×C₄	C₂×M₁₆	semidirect
106. C₂×C₄	#208	direct (N × H)
118. C₂×C₃	C₄×M₁₆	semidirect
128. C₄	C₂×#208	direct (N × H)
132. C₃	C₂×C₄×M₁₆	semidirect
134. C₂	C₄×#208	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
C₄×#208	2
C₂×#803	2
C₂×C₄×C₃⋊C₈	2
C₂×C₄×M₁₆	3

Subgroups — C2×C4×#208 #82044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#208 #82044