Finite Group Databasebeta

Subgroups — #33779

Order 256.

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

normal subgroups only

This group has 321 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	#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#495	C₂	1	1	yes	maximal, characteristic, complemented
3	#2250	C₂	1	2	yes	maximal, complemented
4	#2243	C₂	1	2	yes	maximal, complemented
5	#2210	C₂	1	4	yes	maximal, complemented
6	#2185	C₂	1	2	yes	maximal, complemented
7	#2184	C₂	1	2	yes	maximal, complemented
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	#532	—	2	1	no	—
13	C₂×#124	—	2	1	no	—
14	C₂×SD₃₂	C₂²	1	4	yes	complemented
15	C₂×D₁₆	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	#493	C₂²	1	1	yes	characteristic, complemented
22	#486	—	2	1	no	—
23	#466	—	2	1	no	—
24	#466	C₂²	1	2	yes	—
25	C₄×D₈	—	2	1	no	—
26	C₄×D₈	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	C₂×#41	—	4	1	no	—
34	C₂²×Q₈	—	2	1	no	—
35	C₂²×D₄	—	2	1	no	—
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	—	4	2	no	—
38	C₂×Q₁₆	C₂³	1	2	yes	—
39	C₂×SD₁₆	—	2	2	no	—
40	C₂×SD₁₆	—	2	2	no	—
41	C₂×SD₁₆	—	4	4	no	—
42	C₂×D₈	—	2	1	no	—
43	C₂×D₈	—	2	1	no	—
44	C₂×D₈	—	4	2	no	—
45	C₂×D₈	C₂³	1	2	yes	—
46	C₂×D₈	C₂³	1	2	yes	—
47	C₂²×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈	—	2	1	no	—
50	#127	—	2	1	no	—
51	#124	—	2	2	no	—
52	#124	—	2	2	no	—
53	#124	—	4	1	no	—
54	#124	—	4	1	no	—
55	#124	—	4	1	no	—
56	#121	—	2	2	no	—
57	#121	—	4	1	no	—
58	C₄×D₄	—	2	2	no	—
59	C₄×D₄	—	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	SD₃₂	—	4	4	no	—
64	D₁₆	—	4	4	no	—
65	C₂×C₁₆	C₂³	1	4	yes	—
66	#107	—	2	1	no	—
67	#107	C₂³	1	2	yes	—
68	#102	—	2	2	no	—
69	#102	—	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	#41	—	8	4	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	—	2	2	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	4	2	no	—
85	C₂×D₄	—	4	2	no	—
86	C₂×D₄	—	4	1	no	—
87	C₂×D₄	—	4	1	no	—
88	C₂×D₄	—	4	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	D₈	1	1	yes	characteristic
92	C₂²×C₄	D₈	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
94	Q₁₆	—	2	4	no	—
95	Q₁₆	—	4	2	no	—
96	SD₁₆	—	4	4	no	—
97	SD₁₆	—	4	4	no	—
98	D₈	—	2	4	no	—
99	D₈	—	2	4	no	—
100	D₈	—	4	2	no	—
101	D₈	—	4	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	4	1	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	C₂²⋊C₄	—	4	2	no	—
115	C₂²⋊C₄	—	4	2	no	—
116	C₄²	D₈	1	1	yes	characteristic
117	C₄²	D₈	1	1	yes	characteristic
118	C₄²	C₂×D₄	1	1	yes	characteristic
119	C₄²	C₂×D₄	1	1	yes	characteristic
120	C₁₆	—	2	4	no	—
121	C₂³	—	4	2	no	—
122	C₂³	—	4	2	no	—
123	C₂³	—	8	2	no	—
124	C₂³	—	8	2	no	—
125	C₂³	—	8	2	no	—
126	C₂³	C₂×D₈	1	1	yes	characteristic
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	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	8	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	8	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	AGL(1,ℤ/8)	1	2	yes	—
156	C₈	H(5,𝔽₂)	1	2	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	2	no	—
163	C₂²	—	8	2	no	—
164	C₂²	—	8	2	no	—
165	C₂²	—	16	4	no	—
166	C₂²	#509	1	2	yes	—
167	C₂²	C₂²×D₈	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	8	2	no	—
174	C₄	—	8	2	no	—
175	C₄	#535	1	1	yes	characteristic
176	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	—	16	2	no	—
179	C₂	—	16	2	no	—
180	C₂	#3205	1	1	yes	characteristic
181	C₂	C₂×#509	1	1	yes	characteristic
182	C₂	#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
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. #3169	C₂	semidirect
2. C₂×#495	C₂	semidirect
3. #2250	C₂	semidirect
4. #2243	C₂	semidirect
5. #2210	C₂	semidirect
6. #2185	C₂	semidirect
7. #2184	C₂	semidirect
8. #2163	C₂	semidirect
14. C₂×SD₃₂	C₂²	semidirect
16. #500	C₂²	semidirect
21. #493	C₂²	semidirect
26. C₄×D₈	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
#3169	2
C₂×#495	2
#2250	2
#2243	2
#2210	2
#2185	2
#2184	2
#2163	2