Finite Group Databasebeta

Subgroups — #22689

Order 256.

431 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 199 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×#443	C₂	1	1	yes	maximal, characteristic
2	#2952	C₂	1	2	yes	maximal
3	#2912	C₂	1	2	yes	maximal, complemented
4	#1951	C₂	1	2	yes	maximal, complemented
5	#1859	C₂	1	4	yes	maximal, complemented
6	#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	#1815	C₂	1	2	yes	maximal, complemented
8	#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	C₂²	1	1	yes	characteristic
10	C₂×#131	C₂²	1	2	yes	—
11	#520	C₂²	1	4	yes	—
12	#443	C₄	1	8	yes	complemented
13	#442	—	2	2	no	—
14	#441	—	2	2	no	—
15	C₄×SD₁₆	—	2	4	no	—
16	C₂×#108	C₂²	1	1	yes	characteristic
17	C₂×#108	C₂²	1	1	yes	characteristic
18	#428	—	2	4	no	—
19	#428	C₂²	1	4	yes	—
20	#427	—	2	2	no	—
21	#427	C₂²	1	1	yes	characteristic
22	C₂×#106	—	2	2	no	—
23	C₂×#106	C₂²	1	2	yes	—
24	C₂×#106	C₂²	1	2	yes	—
25	C₂×#104	C₂²	1	2	yes	—
26	#417	C₂²	1	4	yes	—
27	#405	C₂²	1	2	yes	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#384	C₂²	1	2	yes	—
30	#343	C₂²	1	4	yes	—
31	#339	C₂²	1	2	yes	—
32	C₂×#41	C₂³	1	2	yes	—
33	#135	C₂×C₄	1	4	yes	—
34	#135	Q₈	1	4	yes	—
35	C₂×Q₁₆	D₄	1	1	yes	characteristic
36	C₂×SD₁₆	D₄	1	2	yes	—
37	C₂×D₈	D₄	1	1	yes	characteristic
38	#131	—	2	4	no	—
39	#131	C₂×C₄	1	8	yes	complemented
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	C₂³	1	2	yes	—
42	C₂×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₄×Q₈	—	2	4	no	—
47	C₄×D₄	—	2	4	no	—
48	C₄×D₄	—	4	4	no	—
49	#117	—	2	4	no	—
50	#117	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄	—	2	4	no	—
52	C₂×C₄⋊C₄	—	4	2	no	—
53	C₂×C₄⋊C₄	C₂³	1	4	yes	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#108	—	2	2	no	—
56	#108	C₂×C₄	1	4	yes	complemented
57	#107	—	2	4	no	—
58	#107	D₄	1	2	yes	—
59	#106	—	2	2	no	—
60	#106	—	2	2	no	—
61	#106	—	2	2	no	—
62	#106	—	2	2	no	—
63	#106	—	2	4	no	—
64	#106	D₄	1	2	yes	—
65	#104	C₂×C₄	1	8	yes	complemented
66	#103	—	2	4	no	—
67	#102	—	2	4	no	—
68	#97	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	C₂×C₄	1	4	yes	complemented
71	#41	—	2	2	no	—
72	#41	—	2	2	no	—
73	#41	C₂²×C₄	1	4	yes	—
74	#41	C₂×Q₈	1	4	yes	—
75	C₂×Q₈	C₂×D₄	1	2	yes	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	C₂×D₄	1	2	yes	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	4	4	no	—
82	C₂²×C₄	C₂⁴	1	1	yes	characteristic
83	Q₁₆	C₄⋊C₄	1	4	yes	complemented
84	SD₁₆	C₄⋊C₄	1	8	yes	complemented
85	D₈	C₄⋊C₄	1	4	yes	complemented
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	C₂²×C₄	1	4	yes	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	8	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×Q₈	1	2	yes	—
101	C₂×C₈	#41	1	2	yes	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	8	4	no	—
106	C₄⋊C₄	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	4	4	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	4	4	no	—
112	C₄²	C₂²×C₄	1	2	yes	—
113	C₂³	—	2	2	no	—
114	C₂³	H(5,𝔽₂)	1	1	yes	characteristic
115	Q₈	—	2	2	no	—
116	Q₈	C₂×C₄⋊C₄	1	4	yes	—
117	D₄	—	2	4	no	—
118	D₄	—	2	4	no	—
119	D₄	—	2	2	no	—
120	D₄	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	—	16	4	no	—
130	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄	C₂×#41	1	1	yes	characteristic
135	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
136	C₈	—	2	4	no	—
137	C₈	—	4	2	no	—
138	C₈	C₂×C₄⋊C₄	1	2	yes	—
139	C₈	C₂×C₄⋊C₄	1	2	yes	—
140	C₈	C₄×D₄	1	2	yes	—
141	C₈	C₄×D₄	1	2	yes	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	4	no	—
144	C₂²	—	4	4	no	—
145	C₂²	#520	1	1	yes	characteristic
146	C₂²	D₄²	1	1	yes	characteristic
147	C₂²	#554	1	1	yes	characteristic
148	C₄	—	2	4	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	4	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄	#547	1	1	yes	characteristic
155	C₄	D₄×Q₈	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	—	4	4	no	—
159	C₂	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂	#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	D₄²
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2912	C₂	semidirect
4. #1951	C₂	semidirect
5. #1859	C₂	semidirect
6. #1844	C₂	semidirect
7. #1815	C₂	semidirect
8. #1772	C₂	semidirect
12. #443	C₄	semidirect
39. #131	C₂×C₄	semidirect
56. #108	C₂×C₄	semidirect
65. #104	C₂×C₄	semidirect
70. C₄×C₈	C₂×C₄	semidirect
83. Q₁₆	C₄⋊C₄	semidirect
84. SD₁₆	C₄⋊C₄	semidirect
85. D₈	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#443	2
#2952	2
#2912	2
#1951	2
#1859	2
#1844	2
#1815	2
#1772	2