Finite Group Databasebeta

Subgroups — #22695

Order 256.

404 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 197 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	#2957	C₂	1	2	yes	maximal
2	C₂×#444	C₂	1	1	yes	maximal, characteristic
3	#2916	C₂	1	2	yes	maximal, complemented
4	#1953	C₂	1	2	yes	maximal, complemented
5	#1861	C₂	1	4	yes	maximal, complemented
6	#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	#1814	C₂	1	2	yes	maximal, complemented
8	#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	C₂²	1	1	yes	characteristic
10	#568	C₂²	1	4	yes	—
11	C₂×#131	C₂²	1	2	yes	—
12	#444	—	2	4	no	—
13	#444	C₄	1	8	yes	complemented
14	#443	—	2	4	no	—
15	#430	—	2	4	no	—
16	#430	C₂²	1	4	yes	—
17	C₂×#108	—	2	4	no	—
18	C₂×#108	C₂²	1	2	yes	—
19	C₂×#108	C₂²	1	2	yes	—
20	C₂×#108	C₂²	1	1	yes	characteristic
21	C₂×#108	C₂²	1	1	yes	characteristic
22	#427	C₂²	1	1	yes	characteristic
23	#423	C₂²	1	2	yes	—
24	#421	C₂²	1	4	yes	—
25	C₂×#104	C₂²	1	2	yes	—
26	#405	C₂²	1	2	yes	—
27	C₂×#97	C₂²	1	1	yes	characteristic
28	#344	C₂²	1	4	yes	—
29	#339	C₂²	1	2	yes	—
30	C₂×#41	C₂³	1	2	yes	—
31	#135	C₂×C₄	1	4	yes	—
32	#135	D₄	1	4	yes	—
33	C₂×Q₁₆	Q₈	1	1	yes	characteristic
34	C₂×SD₁₆	Q₈	1	2	yes	—
35	C₂×D₈	Q₈	1	1	yes	characteristic
36	#131	—	2	2	no	—
37	#131	—	2	2	no	—
38	#131	—	2	8	no	—
39	#131	—	4	4	no	—
40	#131	C₂×C₄	1	4	yes	—
41	#131	C₂×C₄	1	4	yes	—
42	C₂×M₁₆	—	2	2	no	—
43	C₂×M₁₆	—	2	8	no	—
44	C₂×M₁₆	C₂³	1	4	yes	—
45	C₂×M₁₆	C₂³	1	2	yes	—
46	C₂×M₁₆	C₂³	1	2	yes	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	#117	C₂³	1	2	yes	—
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	#108	—	2	8	no	—
53	#108	—	2	2	no	—
54	#108	—	2	2	no	—
55	#108	—	2	2	no	—
56	#108	—	2	2	no	—
57	#108	C₂×C₄	1	4	yes	—
58	#108	D₄	1	4	yes	—
59	#107	—	2	1	no	—
60	#106	—	2	1	no	—
61	C₄⋊C₈	—	4	2	no	—
62	#104	—	2	8	no	—
63	#104	C₂×C₄	1	8	yes	—
64	#97	—	2	2	no	—
65	#97	C₂×C₄	1	4	yes	—
66	C₄×C₈	—	2	2	no	—
67	#41	—	2	2	no	—
68	#41	—	2	2	no	—
69	#41	C₂²×C₄	1	4	yes	—
70	#41	C₂×D₄	1	4	yes	—
71	C₂×Q₈	C₂×Q₈	1	2	yes	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	C₂×Q₈	1	2	yes	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	C₂⁴	1	1	yes	characteristic
77	Q₁₆	C₄⋊C₄	1	4	yes	—
78	SD₁₆	C₄⋊C₄	1	8	yes	—
79	D₈	C₄⋊C₄	1	4	yes	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	M₁₆	—	2	4	no	—
83	M₁₆	—	2	4	no	—
84	M₁₆	—	2	4	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	4	8	no	—
87	M₁₆	C₂²×C₄	1	4	yes	—
88	M₁₆	C₂×D₄	1	4	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	4	8	no	—
95	C₂×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈	#41	1	1	yes	characteristic
102	C₂×C₈	#41	1	1	yes	characteristic
103	C₄⋊C₄	—	2	2	no	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	2	no	—
107	C₄²	C₂²×C₄	1	2	yes	—
108	C₂³	—	2	2	no	—
109	C₂³	2₋¹⁺⁴	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	C₂×C₄⋊C₄	1	4	yes	—
112	D₄	—	2	4	no	—
113	D₄	—	2	4	no	—
114	D₄	—	2	2	no	—
115	D₄	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄	C₂×#41	1	1	yes	characteristic
127	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
128	C₈	—	2	4	no	—
129	C₈	—	2	4	no	—
130	C₈	—	4	4	no	—
131	C₈	C₂×C₄⋊C₄	1	2	yes	—
132	C₈	C₂×C₄⋊C₄	1	2	yes	—
133	C₈	C₄×D₄	1	2	yes	—
134	C₈	C₄×D₄	1	2	yes	—
135	C₂²	—	2	4	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	8	2	no	—
138	C₂²	#520	1	1	yes	characteristic
139	C₂²	#547	1	1	yes	characteristic
140	C₂²	D₄×Q₈	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	4	2	no	—
143	C₄	—	8	2	no	—
144	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄	D₄²	1	1	yes	characteristic
147	C₄	#554	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	4	4	no	—
152	C₂	D₄×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄²
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	D₄×C₄⋊C₄
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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. #2916	C₂	semidirect
4. #1953	C₂	semidirect
5. #1861	C₂	semidirect
6. #1849	C₂	semidirect
7. #1814	C₂	semidirect
8. #1773	C₂	semidirect
13. #444	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2957	2
C₂×#444	2
#2916	2
#1953	2
#1861	2
#1849	2
#1814	2
#1773	2