Finite Group Databasebeta

Subgroups — #33781

Order 256.

299 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#495	C₂	1	1	yes	maximal, characteristic
3	#2251	C₂	1	2	yes	maximal, complemented
4	#2243	C₂	1	2	yes	maximal, complemented
5	#2211	C₂	1	4	yes	maximal, complemented
6	#2188	C₂	1	2	yes	maximal, complemented
7	#2187	C₂	1	2	yes	maximal
8	#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆	—	2	1	no	—
11	C₂²×D₈	C₂²	1	1	yes	characteristic
12	#533	—	2	1	no	—
13	C₂×#124	—	2	1	no	—
14	C₂×Q₃₂	C₂²	1	4	yes	complemented
15	C₂×SD₃₂	C₂²	1	4	yes	—
16	#500	C₂²	1	1	yes	characteristic, complemented
17	#495	—	2	1	no	—
18	#495	—	2	1	no	—
19	#495	C₂²	1	2	yes	—
20	#495	C₂²	1	2	yes	—
21	#494	C₂²	1	1	yes	characteristic, complemented
22	#487	—	2	1	no	—
23	#467	—	2	1	no	—
24	#467	C₂²	1	2	yes	—
25	C₄×Q₁₆	—	2	1	no	—
26	C₄×Q₁₆	C₂²	1	2	yes	complemented
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	#358	C₂²	1	4	yes	complemented
30	#357	C₂²	1	4	yes	—
31	C₂²⋊C₁₆	C₂²	1	2	yes	—
32	#346	C₂²	1	2	yes	complemented
33	C₂²×Q₈	—	2	1	no	—
34	C₂²×D₄	—	2	1	no	—
35	C₂×Q₁₆	—	2	1	no	—
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	—	4	2	no	—
38	C₂×Q₁₆	C₂³	1	2	yes	—
39	C₂×Q₁₆	C₂³	1	2	yes	—
40	C₂×SD₁₆	—	2	2	no	—
41	C₂×SD₁₆	—	2	2	no	—
42	C₂×SD₁₆	—	4	4	no	—
43	C₂×D₈	—	2	1	no	—
44	C₂×D₈	—	4	2	no	—
45	C₂×D₈	C₂³	1	2	yes	—
46	C₂²×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	2	1	no	—
50	#125	—	4	1	no	—
51	#124	—	2	2	no	—
52	#124	—	2	2	no	—
53	#124	—	4	1	no	—
54	#124	—	4	1	no	—
55	C₂²⋊Q₈	—	2	2	no	—
56	C₂²⋊Q₈	—	4	1	no	—
57	C₄×Q₈	—	2	2	no	—
58	C₄×Q₈	—	4	1	no	—
59	#117	—	4	1	no	—
60	C₂×C₂²⋊C₄	—	4	1	no	—
61	C₂×C₂²⋊C₄	—	4	1	no	—
62	C₂×C₄²	D₄	1	1	yes	characteristic
63	Q₃₂	—	4	4	no	—
64	SD₃₂	—	4	4	no	—
65	C₂×C₁₆	C₂³	1	4	yes	—
66	#107	—	2	1	no	—
67	#107	C₂³	1	2	yes	—
68	#103	—	2	2	no	—
69	#103	—	2	2	no	—
70	C₄×C₈	D₄	1	1	yes	characteristic
71	C₄×C₈	D₄	1	1	yes	characteristic
72	C₄×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	C₂⁴	—	4	1	no	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	4	2	no	—
78	C₂×Q₈	—	4	2	no	—
79	C₂×Q₈	—	4	1	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	4	2	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	D₈	1	1	yes	characteristic
87	C₂²×C₄	D₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	Q₁₆	—	2	4	no	—
90	Q₁₆	—	2	4	no	—
91	Q₁₆	—	4	2	no	—
92	Q₁₆	—	4	2	no	—
93	SD₁₆	—	4	4	no	—
94	SD₁₆	—	4	4	no	—
95	D₈	—	2	4	no	—
96	D₈	—	4	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂²⋊C₄	—	4	2	no	—
112	C₂²⋊C₄	—	4	2	no	—
113	C₂²⋊C₄	—	4	2	no	—
114	C₄²	—	4	2	no	—
115	C₄²	D₈	1	1	yes	characteristic
116	C₄²	D₈	1	1	yes	characteristic
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₄²	C₂×D₄	1	1	yes	characteristic
119	C₁₆	—	2	4	no	—
120	C₂³	—	4	2	no	—
121	C₂³	—	8	2	no	—
122	C₂³	—	8	2	no	—
123	C₂³	—	8	2	no	—
124	C₂³	C₂×D₈	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	8	2	no	—
130	Q₈	—	8	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	8	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	AGL(1,ℤ/8)	1	2	yes	—
151	C₈	H(5,𝔽₂)	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	8	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	16	4	no	—
158	C₂²	#510	1	2	yes	—
159	C₂²	C₂²×D₈	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	8	2	no	—
166	C₄	—	8	2	no	—
167	C₄	—	8	2	no	—
168	C₄	#535	1	1	yes	characteristic
169	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	—	16	2	no	—
172	C₂	#3205	1	1	yes	characteristic
173	C₂	C₂×#510	1	1	yes	characteristic
174	C₂	#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂²×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3170	C₂	semidirect
3. #2251	C₂	semidirect
4. #2243	C₂	semidirect
5. #2211	C₂	semidirect
6. #2188	C₂	semidirect
8. #2163	C₂	semidirect
14. C₂×Q₃₂	C₂²	semidirect
16. #500	C₂²	semidirect
21. #494	C₂²	semidirect
26. C₄×Q₁₆	C₂²	semidirect
28. C₄⋊C₁₆	C₂²	semidirect
29. #358	C₂²	semidirect
32. #346	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3170	2
C₂×#495	2
#2251	2
#2243	2
#2211	2
#2188	2
#2187	2
#2163	2