Finite Group Databasebeta

Subgroups — #33471

Order 256.

334 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×#509	C₂	1	1	yes	maximal, characteristic
2	#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	AGL(1,ℤ/16)	C₂	1	8	yes	maximal, complemented
5	#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#361	C₂	1	1	yes	maximal, characteristic, complemented
7	#2145	C₂	1	1	yes	maximal, characteristic, complemented
8	#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	C₂²	1	1	yes	characteristic
10	C₂²×D₈	C₂²	1	1	yes	characteristic
11	#568	—	2	1	no	—
12	#518	—	2	1	no	—
13	#509	C₄	1	4	yes	complemented
14	#509	C₂²	1	4	yes	—
15	C₂×SD₃₂	C₄	1	2	yes	complemented
16	C₂×D₁₆	C₄	1	2	yes	complemented
17	C₂×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×M₃₂	C₂²	1	1	yes	characteristic
19	#444	C₂²	1	4	yes	complemented
20	#443	—	2	2	no	—
21	#442	C₂²	1	4	yes	complemented
22	C₄×D₈	—	2	2	no	—
23	C₂×#108	C₂²	1	1	yes	characteristic
24	#427	C₂²	1	1	yes	characteristic
25	#421	—	2	1	no	—
26	#418	—	2	1	no	—
27	#405	C₂²	1	1	yes	characteristic
28	#365	C₂²	1	4	yes	complemented
29	#361	C₂²	1	4	yes	complemented
30	#360	C₂²	1	4	yes	complemented
31	#347	C₂²	1	4	yes	complemented
32	C₂×#41	—	2	1	no	—
33	C₂²×D₄	—	2	1	no	—
34	#135	—	2	1	no	—
35	#135	—	2	1	no	—
36	#135	C₂×C₄	1	2	yes	complemented
37	#135	C₂³	1	2	yes	—
38	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×SD₁₆	—	2	1	no	—
40	C₂×D₈	—	2	1	no	—
41	C₂×D₈	—	2	1	no	—
42	C₂×D₈	—	2	1	no	—
43	C₂×D₈	C₂×C₄	1	2	yes	—
44	C₂×D₈	C₂×C₄	1	2	yes	—
45	C₂×D₈	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₈	C₂³	1	2	yes	—
47	#131	—	2	4	no	—
48	#131	—	4	2	no	—
49	C₂×M₁₆	—	2	1	no	—
50	C₂×M₁₆	—	4	2	no	—
51	C₂×M₁₆	D₄	1	1	yes	characteristic
52	C₂²×C₈	C₂³	1	1	yes	characteristic
53	C₄×D₄	—	2	4	no	—
54	C₄×D₄	—	4	2	no	—
55	#117	—	2	1	no	—
56	#117	D₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄	—	4	2	no	—
58	SD₃₂	C₂×C₄	1	8	yes	complemented
59	D₁₆	C₂×C₄	1	8	yes	complemented
60	M₃₂	—	2	1	no	—
61	M₃₂	C₂×C₄	1	2	yes	complemented
62	M₃₂	C₂³	1	2	yes	—
63	M₃₂	C₂³	1	2	yes	—
64	C₂×C₁₆	—	2	1	no	—
65	C₂×C₁₆	C₂×C₄	1	2	yes	complemented
66	#108	—	2	1	no	—
67	#108	C₂³	1	2	yes	—
68	#107	—	2	1	no	—
69	#106	C₂³	1	2	yes	—
70	#104	—	2	4	no	—
71	#102	—	2	4	no	—
72	#97	C₂³	1	2	yes	—
73	C₄×C₈	D₄	1	2	yes	—
74	C₂⁴	—	4	1	no	—
75	#41	—	2	2	no	—
76	#41	—	2	2	no	—
77	#41	—	4	1	no	—
78	#41	—	4	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂×D₄	—	4	1	no	—
86	C₂×D₄	—	4	1	no	—
87	C₂×D₄	—	4	1	no	—
88	C₂²×C₄	—	4	4	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₂²×C₄	1	2	yes	—
93	SD₁₆	—	2	2	no	—
94	SD₁₆	—	2	2	no	—
95	D₈	—	2	2	no	—
96	D₈	—	2	2	no	—
97	D₈	—	2	1	no	—
98	D₈	—	2	2	no	—
99	D₈	C₂²×C₄	1	2	yes	—
100	D₈	C₂²×C₄	1	4	yes	—
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	4	4	no	—
105	M₁₆	—	4	1	no	—
106	C₂×C₈	—	4	4	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈	#41	1	1	yes	characteristic
113	C₂×C₈	#41	1	1	yes	characteristic
114	C₂×C₈	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	—	4	4	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₄²	—	4	1	no	—
121	C₄²	C₂×D₄	1	2	yes	—
122	C₁₆	—	2	2	no	—
123	C₁₆	C₂²×C₄	1	4	yes	—
124	C₂³	—	4	2	no	—
125	C₂³	—	4	1	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	16	2	no	—
129	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
130	Q₈	—	2	2	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	4	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	4	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	16	4	no	—
146	C₂×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	C₄×D₄	1	1	yes	characteristic
155	C₈	C₄×D₄	1	1	yes	characteristic
156	C₈	C₂³×C₄	1	1	yes	characteristic
157	C₈	C₂×#41	1	1	yes	characteristic
158	C₂²	—	4	1	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	4	no	—
161	C₂²	—	8	4	no	—
162	C₂²	—	16	1	no	—
163	C₂²	—	16	2	no	—
164	C₂²	—	16	2	no	—
165	C₂²	#442	1	1	yes	characteristic
166	C₂²	#442	1	1	yes	characteristic
167	C₂²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	8	2	no	—
171	C₄	#442	1	1	yes	characteristic
172	C₄	#442	1	1	yes	characteristic
173	C₄	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	—	8	4	no	—
179	C₂	—	8	2	no	—
180	C₂	C₂×#442	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#442
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#509	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2957	C₂	semidirect
3. #2948	C₂	semidirect
4. AGL(1,ℤ/16)	C₂	semidirect
5. #2155	C₂	semidirect
6. C₂×#361	C₂	semidirect
7. #2145	C₂	semidirect
8. #2110	C₂	semidirect
13. #509	C₄	semidirect
15. C₂×SD₃₂	C₄	semidirect
16. C₂×D₁₆	C₄	semidirect
19. #444	C₂²	semidirect
21. #442	C₂²	semidirect
28. #365	C₂²	semidirect
29. #361	C₂²	semidirect
30. #360	C₂²	semidirect
31. #347	C₂²	semidirect
36. #135	C₂×C₄	semidirect
38. C₂×Q₁₆	C₂×C₄	semidirect
45. C₂×D₈	C₂×C₄	semidirect
58. SD₃₂	C₂×C₄	semidirect
59. D₁₆	C₂×C₄	semidirect
61. M₃₂	C₂×C₄	semidirect
65. C₂×C₁₆	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509	2
#2957	2
#2948	2
AGL(1,ℤ/16)	2
#2155	2
C₂×#361	2
#2145	2
#2110	2