Finite Group Databasebeta

← C₂×#1841 #20775

Subgroups — C₂×#1841 #20775

Order 256.

365 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#1841	C₂	1	8	yes	maximal, direct product factor
3	C₂×#344	C₂	1	2	yes	maximal
4	C₂×#339	C₂	1	2	yes	maximal
5	C₂×#338	C₂	1	1	yes	maximal, characteristic
6	C₂×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆	C₂²	1	2	yes	—
8	C₂²×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×#117	C₂²	1	2	yes	—
11	C₂²×C₄²	C₂²	1	1	yes	characteristic
12	C₄×M₁₆	C₄	1	4	yes	complemented
13	C₄×M₁₆	C₂²	1	4	yes	—
14	C₂×#97	C₄	1	2	yes	complemented
15	C₂×C₄×C₈	C₄	1	2	yes	complemented
16	#344	C₂²	1	8	yes	—
17	#339	C₂²	1	8	yes	—
18	#338	C₂²	1	4	yes	—
19	H(3,ℤ/4)	C₂²	1	4	yes	—
20	C₂³×C₄	—	2	1	no	—
21	C₂³×C₄	C₂³	1	1	yes	characteristic
22	C₂×M₁₆	—	2	2	no	—
23	C₂×M₁₆	—	2	2	no	—
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	—	2	2	no	—
26	C₂×M₁₆	—	2	2	no	—
27	C₂×M₁₆	—	2	2	no	—
28	C₂×M₁₆	—	2	2	no	—
29	C₂×M₁₆	—	2	2	no	—
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×M₁₆	C₂×C₄	1	2	yes	—
33	C₂×M₁₆	C₂³	1	4	yes	—
34	C₂×M₁₆	C₂³	1	2	yes	—
35	C₂×M₁₆	C₂³	1	2	yes	—
36	C₂²×C₈	—	2	2	no	—
37	C₂²×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₈	C₂×C₄	1	2	yes	—
39	#117	—	2	2	no	—
40	#117	—	2	2	no	—
41	#117	—	2	2	no	—
42	#117	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄	—	2	2	no	—
44	C₂×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₄²	—	2	1	no	—
46	C₂×C₄²	—	2	1	no	—
47	C₂×C₄²	—	2	1	no	—
48	C₂×C₄²	—	2	2	no	—
49	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₄	1	2	yes	—
52	C₂×C₄²	C₂³	1	2	yes	—
53	#97	C₂×C₄	1	8	yes	complemented
54	C₄×C₈	C₂×C₄	1	8	yes	complemented
55	C₂⁴	#41	1	1	yes	characteristic
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	—	2	2	no	—
60	C₂²×C₄	—	4	4	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄	#41	1	1	yes	characteristic
65	C₂²×C₄	#41	1	2	yes	—
66	C₂²×C₄	#41	1	2	yes	—
67	C₂²×C₄	#41	1	2	yes	—
68	C₂²×C₄	#41	1	2	yes	—
69	C₂²×C₄	#41	1	1	yes	characteristic
70	C₂²×C₄	#41	1	1	yes	characteristic
71	C₂²×C₄	C₂⁴	1	1	yes	characteristic
72	M₁₆	—	2	4	no	—
73	M₁₆	—	2	4	no	—
74	M₁₆	—	2	4	no	—
75	M₁₆	—	2	4	no	—
76	M₁₆	—	2	4	no	—
77	M₁₆	—	2	4	no	—
78	M₁₆	—	4	2	no	—
79	M₁₆	—	4	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₈	C₂²×C₄	1	4	yes	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₂²⋊C₄	—	2	4	no	—
92	C₂²⋊C₄	—	4	2	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	2	no	—
95	C₄²	—	2	1	no	—
96	C₄²	—	2	2	no	—
97	C₄²	—	2	4	no	—
98	C₄²	—	4	2	no	—
99	C₄²	C₂²×C₄	1	2	yes	—
100	C₄²	C₂²×C₄	1	2	yes	—
101	C₂³	—	4	2	no	—
102	C₂³	#117	1	1	yes	characteristic
103	C₂³	#117	1	1	yes	characteristic
104	C₂³	#126	1	2	yes	—
105	C₂³	#126	1	2	yes	—
106	C₂³	C₂×#41	1	1	yes	characteristic
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	8	4	no	—
115	C₂×C₄	#117	1	1	yes	characteristic
116	C₂×C₄	#117	1	1	yes	characteristic
117	C₂×C₄	#117	1	1	yes	characteristic
118	C₂×C₄	#117	1	2	yes	—
119	C₂×C₄	#117	1	2	yes	—
120	C₂×C₄	#117	1	1	yes	characteristic
121	C₂×C₄	#117	1	1	yes	characteristic
122	C₂×C₄	#117	1	1	yes	characteristic
123	C₂×C₄	#126	1	2	yes	—
124	C₂×C₄	#126	1	2	yes	—
125	C₂×C₄	#126	1	2	yes	—
126	C₂×C₄	#126	1	2	yes	—
127	C₂×C₄	#126	1	2	yes	—
128	C₂×C₄	#126	1	2	yes	—
129	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄	C₂×#41	1	2	yes	—
131	C₂×C₄	C₂×#41	1	2	yes	—
132	C₂×C₄	C₂×#41	1	1	yes	characteristic
133	C₈	—	2	4	no	—
134	C₈	—	2	4	no	—
135	C₈	—	4	4	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	4	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	2	no	—
141	C₂²	#383	1	1	yes	characteristic
142	C₂²	#383	1	2	yes	—
143	C₂²	#383	1	1	yes	characteristic
144	C₂²	C₂×#117	1	1	yes	characteristic
145	C₂²	C₂×#126	1	2	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	—	8	4	no	—
149	C₄	#383	1	1	yes	characteristic
150	C₄	#383	1	2	yes	—
151	C₄	#383	1	1	yes	characteristic
152	C₄	C₂×#117	1	1	yes	characteristic
153	C₄	C₂×#117	1	1	yes	characteristic
154	C₄	C₂×#126	1	2	yes	—
155	C₂	—	2	1	no	—
156	C₂	—	2	2	no	—
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	#1841	1	2	yes	direct product factor
160	C₂	C₂×#383	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#117
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	#383
prime radical	C₂⁴	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1841	C₂	direct (N × H)
12. C₄×M₁₆	C₄	semidirect
14. C₂×#97	C₄	semidirect
15. C₂×C₄×C₈	C₄	semidirect
53. #97	C₂×C₄	semidirect
54. C₄×C₈	C₂×C₄	semidirect
159. C₂	#1841	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆	2
#1841	2
C₂×#344	2
C₂×#339	2
C₂×#338	2
C₂×H(3,ℤ/4)	2