Finite Group Databasebeta

Subgroups — #19283

Order 256.

353 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 109 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	#3109	C₂	1	2	yes	maximal, complemented
3	C₂×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	#1697	C₂	1	4	yes	maximal, complemented
5	#1695	C₂	1	2	yes	maximal, complemented
6	#1687	C₂	1	2	yes	maximal, complemented
7	#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	#1551	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆	—	2	1	no	—
10	#532	C₂²	1	2	yes	—
11	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
12	#500	C₂²	1	1	yes	characteristic, complemented
13	#493	C₂²	1	1	yes	characteristic, complemented
14	#486	C₂²	1	2	yes	complemented
15	#481	—	2	2	no	—
16	#476	C₂²	1	4	yes	—
17	#466	—	2	1	no	—
18	#466	—	2	1	no	—
19	#459	C₂²	1	4	yes	—
20	#447	—	2	2	no	—
21	#442	—	2	4	no	—
22	C₄×D₈	—	2	1	no	—
23	C₄×D₈	—	2	1	no	—
24	#436	—	2	1	no	—
25	#436	—	2	1	no	—
26	#436	C₂²	1	2	yes	—
27	#436	C₂²	1	2	yes	—
28	#432	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	2	yes	complemented
33	#334	C₂²	1	2	yes	complemented
34	#333	C₂²	1	4	yes	complemented
35	#331	C₂²	1	4	yes	complemented
36	C₂×#41	—	2	2	no	—
37	C₂²×D₄	D₄	1	1	yes	characteristic
38	C₂³×C₄	—	2	1	no	—
39	C₂×D₈	—	2	4	no	—
40	C₂×D₈	—	2	1	no	—
41	C₂×D₈	—	2	1	no	—
42	C₂×M₁₆	—	2	2	no	—
43	C₂×M₁₆	—	2	2	no	—
44	C₂×M₁₆	—	4	4	no	—
45	C₂²×C₈	—	2	1	no	—
46	C₂²×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₈	D₄	1	1	yes	characteristic
48	C₄⋊Q₈	C₂³	1	2	yes	—
49	#127	C₂³	1	2	yes	—
50	#124	—	2	2	no	—
51	#121	—	2	2	no	—
52	#121	—	2	2	no	—
53	C₄×D₄	—	2	1	no	—
54	C₄×D₄	—	2	2	no	—
55	C₄×D₄	—	2	2	no	—
56	C₄×D₄	—	2	1	no	—
57	C₄×D₄	D₄	1	2	yes	—
58	C₄×D₄	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄	—	2	1	no	—
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	#107	—	2	1	no	—
63	#107	—	2	1	no	—
64	#106	—	2	4	no	—
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	C₂³	1	2	yes	—
67	C₄⋊C₈	C₂³	1	4	yes	—
68	#102	—	2	4	no	—
69	#102	—	2	2	no	—
70	#102	—	2	2	no	—
71	#102	—	2	4	no	—
72	C₂²⋊C₈	—	2	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	2	2	no	—
75	#97	—	2	2	no	—
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	D₄	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	C₂⁴	—	2	1	no	—
81	#41	—	8	4	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	4	2	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	C₂×D₄	1	2	yes	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	—	4	2	no	—
96	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄	#41	1	1	yes	characteristic
99	D₈	—	4	2	no	—
100	D₈	—	4	8	no	—
101	D₈	—	4	2	no	—
102	M₁₆	—	4	4	no	—
103	M₁₆	—	4	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	C₂×D₄	1	2	yes	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	4	2	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	#41	1	1	yes	characteristic
124	C₄²	C₂⁴	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	4	2	no	—
129	C₂³	—	4	2	no	—
130	C₂³	#121	1	1	yes	characteristic
131	Q₈	—	8	2	no	—
132	D₄	—	2	4	no	—
133	D₄	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	4	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	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	#121	1	1	yes	characteristic
151	C₂×C₄	#121	1	1	yes	characteristic
152	C₂×C₄	#121	1	1	yes	characteristic
153	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄	C₂×#41	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	4	2	no	—
160	C₈	—	4	4	no	—
161	C₈	AGL(1,ℤ/8)	1	2	yes	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	8	4	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	8	2	no	—
171	C₂²	#472	1	2	yes	—
172	C₂²	C₂×#121	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	4	2	no	—
178	C₄	—	8	2	no	—
179	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₄	#575	1	1	yes	characteristic
181	C₄	#575	1	2	yes	—
182	C₂	—	2	2	no	—
183	C₂	—	8	2	no	—
184	C₂	—	16	2	no	—
185	C₂	#3036	1	1	yes	characteristic
186	C₂	#3054	1	1	yes	characteristic
187	C₂	C₂×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3169	C₂	semidirect
2. #3109	C₂	semidirect
3. C₂×#436	C₂	semidirect
4. #1697	C₂	semidirect
5. #1695	C₂	semidirect
6. #1687	C₂	semidirect
7. #1568	C₂	semidirect
8. #1551	C₂	semidirect
12. #500	C₂²	semidirect
13. #493	C₂²	semidirect
14. #486	C₂²	semidirect
32. #335	C₂²	semidirect
33. #334	C₂²	semidirect
34. #333	C₂²	semidirect
35. #331	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
#3109	2
C₂×#436	2
#1697	2
#1695	2
#1687	2
#1568	2
#1551	2