Finite Group Databasebeta

Subgroups — #33783

Order 256.

344 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3168	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	#2250	C₂	1	2	yes	maximal, complemented
4	#2244	C₂	1	2	yes	maximal, complemented
5	#2207	C₂	1	2	yes	maximal, complemented
6	#2206	C₂	1	2	yes	maximal, complemented
7	#2187	C₂	1	2	yes	maximal, complemented
8	#2184	C₂	1	2	yes	maximal, complemented
9	#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈	—	2	1	no	—
11	C₂²×D₈	C₂²	1	2	yes	—
12	C₂×#127	—	2	1	no	—
13	#529	—	2	1	no	—
14	C₂×SD₃₂	C₂²	1	4	yes	complemented
15	C₂×D₁₆	C₂²	1	4	yes	complemented
16	#499	C₂²	1	1	yes	characteristic, complemented
17	#495	C₂²	1	1	yes	characteristic, complemented
18	#493	—	2	1	no	—
19	#493	—	2	1	no	—
20	#493	C₂²	1	2	yes	—
21	#493	C₂²	1	2	yes	—
22	#488	—	2	1	no	—
23	#467	C₂²	1	1	yes	characteristic
24	#466	C₂²	1	1	yes	characteristic
25	#465	—	2	1	no	—
26	C₄×Q₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₄×SD₁₆	—	2	1	no	—
28	C₄×D₈	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₄⋊C₁₆	C₂²	1	2	yes	complemented
31	#357	C₂²	1	4	yes	complemented
32	#357	C₂²	1	4	yes	complemented
33	C₂²⋊C₁₆	C₂²	1	2	yes	—
34	#346	C₂²	1	2	yes	complemented
35	C₂²×D₄	—	2	2	no	—
36	C₂²×D₄	—	4	1	no	—
37	C₂²×D₄	—	4	1	no	—
38	C₂×Q₁₆	C₂³	1	1	yes	characteristic
39	C₂×SD₁₆	—	2	1	no	—
40	C₂×D₈	—	2	2	no	—
41	C₂×D₈	—	2	2	no	—
42	C₂×D₈	—	2	2	no	—
43	C₂×D₈	—	4	4	no	—
44	C₂×D₈	—	4	4	no	—
45	C₂×D₈	C₂³	1	4	yes	—
46	C₂×D₈	C₂³	1	1	yes	characteristic
47	C₂²×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	#127	—	2	2	no	—
50	#127	—	2	2	no	—
51	#127	—	4	1	no	—
52	#127	—	4	1	no	—
53	#126	—	4	1	no	—
54	#125	—	2	1	no	—
55	#124	—	2	1	no	—
56	#123	—	4	1	no	—
57	C₂²⋊Q₈	—	2	1	no	—
58	#121	—	2	1	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₄×D₄	—	2	1	no	—
61	C₄×D₄	—	4	1	no	—
62	#117	—	4	1	no	—
63	C₂×C₄²	D₄	1	1	yes	characteristic
64	SD₃₂	—	4	4	no	—
65	D₁₆	—	4	4	no	—
66	C₂×C₁₆	C₂³	1	4	yes	—
67	#107	C₂³	1	1	yes	characteristic
68	#107	C₂³	1	1	yes	characteristic
69	#106	—	2	1	no	—
70	#103	—	2	1	no	—
71	#103	—	2	1	no	—
72	#102	—	2	1	no	—
73	#102	—	2	1	no	—
74	C₄×C₈	D₄	1	1	yes	characteristic
75	C₄×C₈	D₄	1	1	yes	characteristic
76	C₄×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₂⁴	—	4	2	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×D₄	—	2	1	no	—
81	C₂×D₄	—	2	4	no	—
82	C₂×D₄	—	4	4	no	—
83	C₂×D₄	—	4	4	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂×D₄	—	4	2	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	4	2	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	—	8	4	no	—
91	C₂×D₄	—	8	4	no	—
92	C₂²×C₄	—	4	1	no	—
93	C₂²×C₄	D₈	1	1	yes	characteristic
94	C₂²×C₄	D₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	Q₁₆	—	2	2	no	—
97	SD₁₆	—	4	2	no	—
98	D₈	—	2	2	no	—
99	D₈	—	2	4	no	—
100	D₈	—	2	4	no	—
101	D₈	—	4	4	no	—
102	D₈	—	4	4	no	—
103	D₈	—	4	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₂²⋊C₄	—	4	1	no	—
117	C₂²⋊C₄	—	4	1	no	—
118	C₂²⋊C₄	—	4	1	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₂²⋊C₄	—	4	1	no	—
121	C₄²	—	4	1	no	—
122	C₄²	D₈	1	1	yes	characteristic
123	C₄²	D₈	1	1	yes	characteristic
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₄²	C₂×D₄	1	1	yes	characteristic
126	C₁₆	—	2	4	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	4	4	no	—
129	C₂³	—	8	4	no	—
130	C₂³	—	8	4	no	—
131	C₂³	—	8	4	no	—
132	C₂³	C₂×D₈	1	1	yes	characteristic
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	D₄	—	4	4	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	4	no	—
138	D₄	—	4	1	no	—
139	D₄	—	8	4	no	—
140	D₄	—	8	4	no	—
141	D₄	—	8	4	no	—
142	D₄	—	8	4	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	AGL(1,ℤ/8)	1	2	yes	—
162	C₈	H(5,𝔽₂)	1	2	yes	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	8	4	no	—
167	C₂²	—	8	4	no	—
168	C₂²	—	8	1	no	—
169	C₂²	—	8	4	no	—
170	C₂²	—	8	1	no	—
171	C₂²	—	16	8	no	—
172	C₂²	#509	1	2	yes	—
173	C₂²	C₂²×D₈	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	1	no	—
179	C₄	—	8	1	no	—
180	C₄	—	8	1	no	—
181	C₄	#509	1	2	yes	—
182	C₄	#535	1	1	yes	characteristic
183	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	—	16	1	no	—
186	C₂	—	16	4	no	—
187	C₂	#3205	1	1	yes	characteristic
188	C₂	C₂×#509	1	1	yes	characteristic
189	C₂	C₂×#509	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. #3168	C₂	semidirect
2. C₂×#493	C₂	semidirect
3. #2250	C₂	semidirect
4. #2244	C₂	semidirect
5. #2207	C₂	semidirect
6. #2206	C₂	semidirect
7. #2187	C₂	semidirect
8. #2184	C₂	semidirect
9. #2163	C₂	semidirect
14. C₂×SD₃₂	C₂²	semidirect
15. C₂×D₁₆	C₂²	semidirect
16. #499	C₂²	semidirect
17. #495	C₂²	semidirect
26. C₄×Q₁₆	C₂²	semidirect
28. C₄×D₈	C₂²	semidirect
30. C₄⋊C₁₆	C₂²	semidirect
31. #357	C₂²	semidirect
32. #357	C₂²	semidirect
34. #346	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3168	2
C₂×#493	2
#2250	2
#2244	2
#2207	2
#2206	2
#2187	2
#2184	2
#2163	2