Finite Group Databasebeta

Subgroups — #33857

Order 256.

323 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3416	C₂	1	2	yes	maximal, complemented
2	#3290	C₂	1	2	yes	maximal, complemented
3	#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	#2223	C₂	1	2	yes	maximal, complemented
5	#2212	C₂	1	2	yes	maximal, complemented
6	#2195	C₂	1	2	yes	maximal, complemented
7	#2192	C₂	1	2	yes	maximal, complemented
8	#2178	C₂	1	1	yes	maximal, characteristic, complemented
9	#2170	C₂	1	1	yes	maximal, characteristic, complemented
10	#578	C₂²	1	2	yes	—
11	#577	—	2	2	no	—
12	#576	C₂²	1	2	yes	complemented
13	#510	—	2	2	no	—
14	#510	C₂²	1	2	yes	complemented
15	#509	—	2	2	no	—
16	#509	C₂²	1	2	yes	—
17	#508	—	2	2	no	—
18	#508	C₂²	1	1	yes	characteristic
19	C₂×Q₃₂	C₂²	1	1	yes	characteristic, complemented
20	C₂×SD₃₂	—	2	2	no	—
21	C₂×SD₃₂	C₂²	1	2	yes	complemented
22	C₂×D₁₆	C₂²	1	1	yes	characteristic
23	#504	C₂²	1	2	yes	complemented
24	#495	C₂²	1	2	yes	complemented
25	#473	C₂²	1	2	yes	complemented
26	#472	C₂²	1	2	yes	—
27	#471	—	2	2	no	—
28	#455	—	2	2	no	—
29	#454	—	2	2	no	—
30	#443	C₂²	1	1	yes	characteristic
31	#443	C₂²	1	2	yes	complemented
32	#368	C₂²	1	1	yes	characteristic, complemented
33	#364	C₂²	1	1	yes	characteristic, complemented
34	#362	C₂²	1	2	yes	complemented
35	#361	C₂²	1	2	yes	complemented
36	#359	C₂²	1	2	yes	complemented
37	#350	C₂²	1	2	yes	complemented
38	C₄×C₁₆	C₂²	1	1	yes	characteristic, complemented
39	2₋¹⁺⁴	—	2	2	no	—
40	H(5,𝔽₂)	—	2	2	no	—
41	#137	—	2	2	no	—
42	#137	—	2	2	no	—
43	#137	—	4	2	no	—
44	#137	—	4	2	no	—
45	AGL(1,ℤ/8)	—	2	2	no	—
46	AGL(1,ℤ/8)	—	2	2	no	—
47	AGL(1,ℤ/8)	—	4	2	no	—
48	AGL(1,ℤ/8)	—	4	2	no	—
49	#135	—	2	2	no	—
50	#135	—	2	2	no	—
51	#135	—	4	2	no	—
52	#135	D₄	1	1	yes	characteristic
53	#135	C₂³	1	2	yes	—
54	C₂×Q₁₆	—	2	2	no	—
55	C₂×Q₁₆	C₂³	1	2	yes	—
56	C₂×SD₁₆	—	2	2	no	—
57	C₂×SD₁₆	—	4	2	no	—
58	C₂×D₈	—	2	2	no	—
59	C₂×D₈	C₂³	1	2	yes	—
60	#131	—	2	2	no	—
61	#131	C₂³	1	2	yes	—
62	#124	—	2	2	no	—
63	Q₃₂	—	2	2	no	—
64	Q₃₂	—	4	1	no	—
65	Q₃₂	D₄	1	1	yes	characteristic
66	SD₃₂	—	2	2	no	—
67	SD₃₂	—	2	2	no	—
68	SD₃₂	—	2	2	no	—
69	SD₃₂	—	2	2	no	—
70	SD₃₂	D₄	1	2	yes	—
71	D₁₆	—	2	2	no	—
72	D₁₆	—	4	1	no	—
73	D₁₆	D₄	1	1	yes	characteristic
74	M₃₂	—	2	2	no	—
75	M₃₂	C₂³	1	2	yes	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	C₂³	1	1	yes	characteristic
78	C₂×C₁₆	C₂³	1	1	yes	characteristic
79	#108	D₄	1	1	yes	characteristic
80	#108	C₂³	1	2	yes	—
81	#104	—	2	2	no	—
82	#104	D₄	1	2	yes	—
83	#101	—	2	2	no	—
84	#100	—	2	2	no	—
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	#41	—	2	2	no	—
87	#41	—	4	2	no	—
88	#41	—	4	2	no	—
89	#41	—	4	2	no	—
90	#41	—	4	2	no	—
91	#41	—	8	2	no	—
92	#41	C₂×D₄	1	2	yes	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	2	2	no	—
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	8	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	2	2	no	—
102	Q₁₆	—	4	2	no	—
103	Q₁₆	—	4	2	no	—
104	Q₁₆	D₈	1	1	yes	characteristic
105	Q₁₆	C₂×D₄	1	2	yes	—
106	SD₁₆	—	2	2	no	—
107	SD₁₆	—	4	2	no	—
108	SD₁₆	—	4	2	no	—
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	4	2	no	—
111	SD₁₆	—	4	2	no	—
112	SD₁₆	—	4	2	no	—
113	SD₁₆	D₈	1	2	yes	—
114	D₈	—	2	2	no	—
115	D₈	—	2	2	no	—
116	D₈	—	4	2	no	—
117	D₈	—	4	2	no	—
118	D₈	D₈	1	1	yes	characteristic
119	D₈	C₂×D₄	1	2	yes	—
120	M₁₆	—	2	2	no	—
121	M₁₆	—	2	2	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	C₂×D₄	1	2	yes	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈	C₂⁴	1	1	yes	characteristic
128	C₂²⋊C₄	—	8	2	no	—
129	C₄²	C₂×D₄	1	1	yes	characteristic
130	C₁₆	—	2	2	no	—
131	C₁₆	—	2	1	no	—
132	C₁₆	C₂×D₄	1	1	yes	characteristic
133	C₁₆	C₂×D₄	1	1	yes	characteristic
134	C₂³	—	4	2	no	—
135	C₂³	—	8	2	no	—
136	Q₈	—	2	2	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	2	no	—
139	Q₈	—	8	2	no	—
140	Q₈	C₂×D₈	1	2	yes	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	D₄	—	4	2	no	—
146	D₄	—	8	2	no	—
147	D₄	—	8	2	no	—
148	D₄	—	8	2	no	—
149	D₄	C₂×D₈	1	2	yes	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	4	2	no	—
160	C₈	C₂×D₈	1	1	yes	characteristic
161	C₈	C₂×D₈	1	1	yes	characteristic
162	C₈	C₂²×D₄	1	1	yes	characteristic
163	C₈	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²	—	2	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	8	2	no	—
167	C₂²	—	16	2	no	—
168	C₂²	—	32	2	no	—
169	C₂²	#576	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	2	no	—
173	C₄	—	8	2	no	—
174	C₄	D₄²	1	1	yes	characteristic
175	C₄	C₂²×D₈	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	4	2	no	—
178	C₂	—	8	2	no	—
179	C₂	—	16	2	no	—
180	C₂	D₄×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ 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. #3416	C₂	semidirect
2. #3290	C₂	semidirect
3. #2249	C₂	semidirect
4. #2223	C₂	semidirect
5. #2212	C₂	semidirect
6. #2195	C₂	semidirect
7. #2192	C₂	semidirect
8. #2178	C₂	semidirect
9. #2170	C₂	semidirect
12. #576	C₂²	semidirect
14. #510	C₂²	semidirect
19. C₂×Q₃₂	C₂²	semidirect
21. C₂×SD₃₂	C₂²	semidirect
23. #504	C₂²	semidirect
24. #495	C₂²	semidirect
25. #473	C₂²	semidirect
31. #443	C₂²	semidirect
32. #368	C₂²	semidirect
33. #364	C₂²	semidirect
34. #362	C₂²	semidirect
35. #361	C₂²	semidirect
36. #359	C₂²	semidirect
37. #350	C₂²	semidirect
38. C₄×C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3416	2
#3290	2
#2249	2
#2223	2
#2212	2
#2195	2
#2192	2
#2178	2
#2170	2