Finite Group Databasebeta

Subgroups — #20513

Order 256.

499 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 291 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×#444	C₂	1	2	yes	maximal
2	C₄×#135	C₂	1	1	yes	maximal, characteristic
3	#2875	C₂	1	2	yes	maximal, complemented
4	C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	#1773	C₂	1	2	yes	maximal, complemented
6	#1764	C₂	1	4	yes	maximal, complemented
7	#1759	C₂	1	2	yes	maximal, complemented
8	#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	C₂²	1	1	yes	characteristic
10	C₂×#131	C₂²	1	4	yes	—
11	C₄×#41	C₂²	1	2	yes	—
12	#444	C₄	1	16	yes	complemented
13	C₄×Q₁₆	C₄	1	2	yes	complemented
14	C₄×SD₁₆	C₄	1	4	yes	complemented
15	C₄×D₈	C₄	1	2	yes	complemented
16	C₂×#108	C₂²	1	2	yes	—
17	#427	C₂²	1	1	yes	characteristic
18	C₂×#104	C₂²	1	4	yes	—
19	#416	C₂²	1	2	yes	—
20	#405	—	2	4	no	—
21	#405	C₂²	1	4	yes	—
22	C₄×M₁₆	—	2	2	no	—
23	C₄×M₁₆	C₂²	1	2	yes	—
24	C₂×#97	—	2	2	no	—
25	C₂×#97	C₂²	1	2	yes	—
26	C₂×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂²	1	2	yes	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#376	C₂²	1	1	yes	characteristic
30	#341	C₂²	1	2	yes	—
31	#339	C₂²	1	4	yes	—
32	C₂×#41	C₂³	1	2	yes	—
33	#135	C₂×C₄	1	8	yes	—
34	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×SD₁₆	C₂×C₄	1	2	yes	—
36	C₂×D₈	C₂×C₄	1	1	yes	characteristic
37	#131	—	2	8	no	—
38	#131	C₂×C₄	1	16	yes	complemented
39	C₂×M₁₆	—	2	4	no	—
40	C₂×M₁₆	C₂³	1	4	yes	—
41	C₂²×C₈	—	2	4	no	—
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	2	yes	—
45	C₄×Q₈	C₂×C₄	1	4	yes	—
46	C₄×D₄	—	2	4	no	—
47	C₄×D₄	C₂×C₄	1	4	yes	—
48	#117	—	2	2	no	—
49	#117	C₂³	1	2	yes	—
50	C₂×C₄²	—	2	2	no	—
51	C₂×C₄²	C₂³	1	2	yes	—
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	#108	C₂×C₄	1	8	yes	complemented
54	#107	C₂×C₄	1	2	yes	—
55	#106	C₂×C₄	1	2	yes	—
56	#104	C₂×C₄	1	16	yes	complemented
57	#103	C₂×C₄	1	4	yes	—
58	#102	C₂×C₄	1	4	yes	—
59	#97	—	2	4	no	—
60	#97	—	2	2	no	—
61	#97	C₂×C₄	1	8	yes	complemented
62	#97	D₄	1	2	yes	—
63	#97	D₄	1	2	yes	—
64	C₄×C₈	—	2	4	no	—
65	C₄×C₈	—	2	4	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	C₂×C₄	1	2	yes	—
68	C₄×C₈	C₂×C₄	1	2	yes	—
69	#95	—	2	2	no	—
70	#41	—	2	4	no	—
71	#41	C₂²×C₄	1	8	yes	—
72	C₂×Q₈	C₂²×C₄	1	2	yes	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	C₂²×C₄	1	2	yes	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	—	2	4	no	—
78	C₂²×C₄	#41	1	2	yes	—
79	C₂²×C₄	C₂⁴	1	1	yes	characteristic
80	Q₁₆	C₄²	1	4	yes	—
81	SD₁₆	C₄²	1	8	yes	—
82	D₈	C₄²	1	4	yes	—
83	M₁₆	—	2	4	no	—
84	M₁₆	—	2	16	no	—
85	M₁₆	C₂²×C₄	1	8	yes	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	16	no	—
89	C₂×C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈	C₂²×C₄	1	4	yes	—
93	C₂×C₈	C₂²×C₄	1	4	yes	—
94	C₂×C₈	C₂×D₄	1	4	yes	—
95	C₂×C₈	#41	1	4	yes	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	C₂²×C₄	1	4	yes	—
104	C₄²	C₂²×C₄	1	2	yes	—
105	C₄²	C₂×D₄	1	1	yes	characteristic
106	C₄²	C₂×D₄	1	1	yes	characteristic
107	C₂³	—	2	2	no	—
108	C₂³	C₂×#41	1	1	yes	characteristic
109	Q₈	—	2	2	no	—
110	Q₈	C₂×C₄²	1	4	yes	—
111	D₄	—	2	8	no	—
112	D₄	—	2	2	no	—
113	D₄	C₂×C₄²	1	4	yes	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	C₄×D₄	1	2	yes	—
122	C₂×C₄	C₄×D₄	1	2	yes	—
123	C₂×C₄	C₂³×C₄	1	2	yes	—
124	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄	C₂×#41	1	2	yes	—
127	C₈	—	2	8	no	—
128	C₈	—	2	2	no	—
129	C₈	C₂×C₄²	1	2	yes	—
130	C₈	C₂×C₄²	1	2	yes	—
131	C₈	C₄×D₄	1	8	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	4	4	no	—
135	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²	C₄×#41	1	2	yes	—
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	4	no	—
140	C₄	—	4	4	no	—
141	C₄	—	4	4	no	—
142	C₄	C₂²×C₄²	1	1	yes	characteristic
143	C₄	C₂×C₄×D₄	1	2	yes	—
144	C₄	C₄×#41	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	4	4	no	—
147	C₂	C₄²×D₄	1	1	yes	characteristic
148	C₂	#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×#135	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2875	C₂	semidirect
4. C₄×#108	C₂	semidirect
5. #1773	C₂	semidirect
6. #1764	C₂	semidirect
7. #1759	C₂	semidirect
8. #1726	C₂	semidirect
12. #444	C₄	semidirect
13. C₄×Q₁₆	C₄	semidirect
14. C₄×SD₁₆	C₄	semidirect
15. C₄×D₈	C₄	semidirect
38. #131	C₂×C₄	semidirect
53. #108	C₂×C₄	semidirect
56. #104	C₂×C₄	semidirect
61. #97	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#444	2
C₄×#135	2
#2875	2
C₄×#108	2
#1773	2
#1764	2
#1759	2
#1726	2