Finite Group Databasebeta

Subgroups — #33859

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	#3417	C₂	1	1	yes	maximal, characteristic, complemented
2	#3415	C₂	1	1	yes	maximal, characteristic, complemented
3	#3290	C₂	1	2	yes	maximal, complemented
4	#2249	C₂	1	1	yes	maximal, characteristic, complemented
5	#2222	C₂	1	1	yes	maximal, characteristic, complemented
6	#2221	C₂	1	1	yes	maximal, characteristic, complemented
7	#2212	C₂	1	2	yes	maximal, complemented
8	#2194	C₂	1	2	yes	maximal, complemented
9	#2193	C₂	1	2	yes	maximal, complemented
10	#2178	C₂	1	1	yes	maximal, characteristic, complemented
11	#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	#578	C₂²	1	2	yes	complemented
13	#577	—	2	2	no	—
14	#576	C₂²	1	2	yes	—
15	#510	—	2	2	no	—
16	#510	C₂²	1	2	yes	complemented
17	#509	—	2	2	no	—
18	#509	C₂²	1	2	yes	—
19	#508	—	2	1	no	—
20	#508	—	2	1	no	—
21	#508	C₂²	1	1	yes	characteristic
22	C₂×Q₃₂	—	2	1	no	—
23	C₂×Q₃₂	C₂²	1	1	yes	characteristic, complemented
24	C₂×SD₃₂	C₂²	1	2	yes	complemented
25	C₂×D₁₆	—	2	1	no	—
26	C₂×D₁₆	C₂²	1	1	yes	characteristic
27	#504	C₂²	1	1	yes	characteristic
28	#504	C₂²	1	1	yes	characteristic, complemented
29	#495	C₂²	1	2	yes	complemented
30	#473	C₂²	1	2	yes	—
31	#472	C₂²	1	2	yes	complemented
32	#471	—	2	2	no	—
33	#455	—	2	2	no	—
34	#454	—	2	2	no	—
35	#443	C₂²	1	1	yes	characteristic
36	#443	C₂²	1	2	yes	complemented
37	#368	C₂²	1	1	yes	characteristic, complemented
38	#364	C₂²	1	1	yes	characteristic, complemented
39	#362	C₂²	1	2	yes	complemented
40	#361	C₂²	1	2	yes	complemented
41	#359	C₂²	1	2	yes	complemented
42	#350	C₂²	1	1	yes	characteristic
43	#350	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₁₆	C₂²	1	1	yes	characteristic, complemented
45	2₋¹⁺⁴	—	2	2	no	—
46	H(5,𝔽₂)	—	2	2	no	—
47	#137	—	2	2	no	—
48	#137	—	2	2	no	—
49	#137	—	4	2	no	—
50	#137	—	4	2	no	—
51	AGL(1,ℤ/8)	—	2	2	no	—
52	AGL(1,ℤ/8)	—	2	2	no	—
53	AGL(1,ℤ/8)	—	4	2	no	—
54	AGL(1,ℤ/8)	—	4	2	no	—
55	#135	—	2	2	no	—
56	#135	—	2	2	no	—
57	#135	—	4	2	no	—
58	#135	D₄	1	1	yes	characteristic
59	#135	C₂³	1	2	yes	—
60	C₂×Q₁₆	—	2	2	no	—
61	C₂×Q₁₆	C₂³	1	2	yes	—
62	C₂×SD₁₆	—	2	2	no	—
63	C₂×SD₁₆	—	4	2	no	—
64	C₂×D₈	—	2	2	no	—
65	C₂×D₈	C₂³	1	2	yes	—
66	#131	—	2	2	no	—
67	#131	C₂³	1	1	yes	characteristic
68	#131	C₂³	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	Q₃₂	—	2	2	no	—
71	Q₃₂	—	2	1	no	—
72	Q₃₂	—	2	1	no	—
73	Q₃₂	D₄	1	1	yes	characteristic
74	SD₃₂	—	2	2	no	—
75	SD₃₂	—	2	2	no	—
76	SD₃₂	—	4	2	no	—
77	SD₃₂	D₄	1	2	yes	—
78	D₁₆	—	2	2	no	—
79	D₁₆	—	2	1	no	—
80	D₁₆	—	2	1	no	—
81	D₁₆	D₄	1	1	yes	characteristic
82	M₃₂	—	2	1	no	—
83	M₃₂	—	2	1	no	—
84	M₃₂	C₂³	1	1	yes	characteristic
85	M₃₂	C₂³	1	1	yes	characteristic
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₁₆	C₂³	1	1	yes	characteristic
90	#108	D₄	1	1	yes	characteristic
91	#108	C₂³	1	2	yes	—
92	#104	—	2	2	no	—
93	#104	D₄	1	1	yes	characteristic
94	#104	D₄	1	1	yes	characteristic
95	#101	—	2	2	no	—
96	#100	—	2	2	no	—
97	C₄×C₈	C₂³	1	1	yes	characteristic
98	#41	—	2	2	no	—
99	#41	—	4	2	no	—
100	#41	—	4	2	no	—
101	#41	—	4	2	no	—
102	#41	—	4	2	no	—
103	#41	—	8	2	no	—
104	#41	C₂×D₄	1	1	yes	characteristic
105	#41	C₂×D₄	1	1	yes	characteristic
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	—	4	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂×D₄	—	4	2	no	—
112	C₂×D₄	—	8	2	no	—
113	Q₁₆	—	2	2	no	—
114	Q₁₆	—	2	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	4	2	no	—
117	Q₁₆	D₈	1	1	yes	characteristic
118	Q₁₆	C₂×D₄	1	2	yes	—
119	SD₁₆	—	2	2	no	—
120	SD₁₆	—	4	2	no	—
121	SD₁₆	—	4	2	no	—
122	SD₁₆	—	4	2	no	—
123	SD₁₆	—	4	2	no	—
124	SD₁₆	—	4	2	no	—
125	SD₁₆	—	4	2	no	—
126	SD₁₆	D₈	1	1	yes	characteristic
127	SD₁₆	D₈	1	1	yes	characteristic
128	D₈	—	2	2	no	—
129	D₈	—	2	2	no	—
130	D₈	—	4	2	no	—
131	D₈	—	4	2	no	—
132	D₈	D₈	1	1	yes	characteristic
133	D₈	C₂×D₄	1	2	yes	—
134	M₁₆	—	2	1	no	—
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	C₂×D₄	1	1	yes	characteristic
139	M₁₆	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈	C₂⁴	1	1	yes	characteristic
145	C₂²⋊C₄	—	8	2	no	—
146	C₄²	C₂×D₄	1	1	yes	characteristic
147	C₁₆	—	2	1	no	—
148	C₁₆	—	2	1	no	—
149	C₁₆	—	2	1	no	—
150	C₁₆	C₂×D₄	1	1	yes	characteristic
151	C₁₆	C₂×D₄	1	1	yes	characteristic
152	C₂³	—	4	2	no	—
153	C₂³	—	8	2	no	—
154	Q₈	—	2	2	no	—
155	Q₈	—	4	2	no	—
156	Q₈	—	4	2	no	—
157	Q₈	—	8	2	no	—
158	Q₈	C₂×D₈	1	1	yes	characteristic
159	Q₈	C₂×D₈	1	1	yes	characteristic
160	D₄	—	2	1	no	—
161	D₄	—	2	1	no	—
162	D₄	—	2	2	no	—
163	D₄	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	4	2	no	—
166	D₄	—	8	2	no	—
167	D₄	—	8	2	no	—
168	D₄	—	8	2	no	—
169	D₄	C₂×D₈	1	1	yes	characteristic
170	D₄	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	4	2	no	—
183	C₈	C₂×D₈	1	1	yes	characteristic
184	C₈	C₂×D₈	1	1	yes	characteristic
185	C₈	C₂²×D₄	1	1	yes	characteristic
186	C₈	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	8	2	no	—
191	C₂²	—	16	2	no	—
192	C₂²	—	32	2	no	—
193	C₂²	#576	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	4	1	no	—
197	C₄	—	4	2	no	—
198	C₄	—	8	2	no	—
199	C₄	D₄²	1	1	yes	characteristic
200	C₄	C₂²×D₈	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	4	1	no	—
203	C₂	—	4	1	no	—
204	C₂	—	8	2	no	—
205	C₂	—	16	2	no	—
206	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. #3417	C₂	semidirect
2. #3415	C₂	semidirect
3. #3290	C₂	semidirect
4. #2249	C₂	semidirect
5. #2222	C₂	semidirect
6. #2221	C₂	semidirect
7. #2212	C₂	semidirect
8. #2194	C₂	semidirect
9. #2193	C₂	semidirect
10. #2178	C₂	semidirect
11. #2170	C₂	semidirect
12. #578	C₂²	semidirect
16. #510	C₂²	semidirect
23. C₂×Q₃₂	C₂²	semidirect
24. C₂×SD₃₂	C₂²	semidirect
28. #504	C₂²	semidirect
29. #495	C₂²	semidirect
31. #472	C₂²	semidirect
36. #443	C₂²	semidirect
37. #368	C₂²	semidirect
38. #364	C₂²	semidirect
39. #362	C₂²	semidirect
40. #361	C₂²	semidirect
41. #359	C₂²	semidirect
43. #350	C₂²	semidirect
44. 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
#3417	2
#3415	2
#3290	2
#2249	2
#2222	2
#2221	2
#2212	2
#2194	2
#2193	2
#2178	2
#2170	2