Finite Group Databasebeta

Subgroups — #27599

Order 256.

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

normal subgroups only

This group has 360 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	#2923	C₂	1	1	yes	maximal, characteristic, complemented
2	#2901	C₂	1	1	yes	maximal, characteristic
3	#2098	C₂	1	2	yes	maximal, complemented
4	#2080	C₂	1	2	yes	maximal, complemented
5	#1938	C₂	1	2	yes	maximal, complemented
6	#1932	C₂	1	2	yes	maximal, complemented
7	#1868	C₂	1	1	yes	maximal, characteristic, complemented
8	#1858	C₂	1	1	yes	maximal, characteristic, complemented
9	#1827	C₂	1	1	yes	maximal, characteristic, complemented
10	#1794	C₂	1	1	yes	maximal, characteristic
11	#1791	C₂	1	1	yes	maximal, characteristic
12	#585	C₂²	1	1	yes	characteristic
13	#568	C₂²	1	1	yes	characteristic
14	C₄×#41	C₂²	1	1	yes	characteristic
15	#433	—	2	2	no	—
16	#432	—	2	2	no	—
17	#430	C₂²	1	2	yes	complemented
18	#424	C₂²	1	1	yes	characteristic, complemented
19	#424	C₂²	1	1	yes	characteristic, complemented
20	#424	C₂²	1	2	yes	complemented
21	#421	—	2	2	no	—
22	#421	C₂²	1	2	yes	—
23	#421	C₂²	1	2	yes	—
24	C₂×#104	—	2	2	no	—
25	C₂×#104	C₂²	1	2	yes	—
26	#416	C₂²	1	2	yes	—
27	#413	C₂²	1	2	yes	—
28	#413	C₂²	1	1	yes	characteristic
29	#410	C₂²	1	1	yes	characteristic
30	#408	C₂²	1	1	yes	characteristic
31	#408	C₂²	1	1	yes	characteristic
32	#408	C₂²	1	2	yes	—
33	C₄×M₁₆	C₂²	1	1	yes	characteristic, complemented
34	C₂×#97	C₂²	1	1	yes	characteristic, complemented
35	#344	C₂²	1	2	yes	complemented
36	#341	C₂²	1	2	yes	complemented
37	#339	C₂²	1	2	yes	complemented
38	H(3,ℤ/4)	C₂²	1	2	yes	complemented
39	2₋¹⁺⁴	—	2	2	no	—
40	2₋¹⁺⁴	—	2	1	no	—
41	H(5,𝔽₂)	—	2	2	no	—
42	H(5,𝔽₂)	—	2	2	no	—
43	H(5,𝔽₂)	—	2	1	no	—
44	C₂×#41	—	2	2	no	—
45	C₂×#41	—	4	2	no	—
46	C₂×#41	C₂³	1	2	yes	—
47	C₂×#41	C₂³	1	1	yes	characteristic
48	#131	—	4	2	no	—
49	C₂×M₁₆	—	2	2	no	—
50	C₂×M₁₆	C₂³	1	2	yes	—
51	C₂×M₁₆	C₂³	1	2	yes	—
52	C₂×M₁₆	C₂³	1	1	yes	characteristic
53	C₂×M₁₆	C₂³	1	1	yes	characteristic
54	C₂²×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₈	C₂³	1	1	yes	characteristic
56	C₄×Q₈	—	2	1	no	—
57	C₄×D₄	—	2	2	no	—
58	C₄×D₄	—	2	1	no	—
59	#117	D₄	1	2	yes	—
60	#117	C₂³	1	2	yes	—
61	#117	C₂³	1	1	yes	characteristic
62	C₂×C₄²	D₄	1	2	yes	—
63	C₂×C₄²	C₂³	1	1	yes	characteristic
64	#108	—	4	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	#104	—	2	2	no	—
69	#104	—	2	4	no	—
70	#104	—	2	2	no	—
71	#104	—	4	2	no	—
72	#103	—	2	2	no	—
73	#102	—	2	2	no	—
74	#101	—	2	2	no	—
75	#101	—	2	1	no	—
76	#100	—	2	2	no	—
77	#100	—	2	1	no	—
78	#99	—	2	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	2	2	no	—
81	C₂²⋊C₈	—	4	2	no	—
82	#97	—	2	1	no	—
83	#97	—	2	1	no	—
84	#97	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	#41	—	2	2	no	—
87	#41	—	2	2	no	—
88	#41	—	2	4	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	#41	—	2	2	no	—
92	#41	—	2	2	no	—
93	#41	—	4	4	no	—
94	#41	—	4	2	no	—
95	#41	—	4	2	no	—
96	#41	—	4	2	no	—
97	#41	—	4	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	#41	1	2	yes	—
101	C₂×Q₈	#41	1	1	yes	characteristic
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	4	2	no	—
105	C₂×D₄	—	4	4	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	#41	1	2	yes	—
108	C₂×D₄	#41	1	2	yes	—
109	C₂×D₄	#41	1	1	yes	characteristic
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	4	2	no	—
114	C₂²×C₄	C₂×D₄	1	2	yes	—
115	C₂²×C₄	C₂⁴	1	1	yes	characteristic
116	M₁₆	—	2	2	no	—
117	M₁₆	—	2	2	no	—
118	M₁₆	—	2	2	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	2	no	—
121	M₁₆	—	2	2	no	—
122	M₁₆	—	4	2	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₂²⋊C₄	—	2	1	no	—
137	C₂²⋊C₄	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	2	no	—
141	C₄²	C₂×D₄	1	1	yes	characteristic
142	C₄²	C₂×D₄	1	1	yes	characteristic
143	C₄²	C₂×D₄	1	2	yes	—
144	C₂³	—	2	2	no	—
145	C₂³	—	4	2	no	—
146	C₂³	#123	1	2	yes	—
147	C₂³	C₂×#41	1	1	yes	characteristic
148	Q₈	—	2	2	no	—
149	Q₈	—	2	2	no	—
150	Q₈	—	2	2	no	—
151	Q₈	—	4	2	no	—
152	D₄	—	2	4	no	—
153	D₄	—	2	2	no	—
154	D₄	—	2	2	no	—
155	D₄	—	2	2	no	—
156	D₄	—	4	2	no	—
157	D₄	—	4	4	no	—
158	D₄	—	4	2	no	—
159	D₄	—	4	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	8	1	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	#123	1	2	yes	—
174	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂×#41	1	2	yes	—
176	C₂×C₄	C₂×#41	1	1	yes	characteristic
177	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
178	C₈	—	4	1	no	—
179	C₈	—	4	1	no	—
180	C₈	—	4	1	no	—
181	C₈	—	4	2	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	8	2	no	—
189	C₂²	—	16	2	no	—
190	C₂²	C₂×#123	1	1	yes	characteristic
191	C₂²	#551	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	8	2	no	—
198	C₄	#525	1	1	yes	characteristic
199	C₄	#535	1	1	yes	characteristic
200	C₄	#551	1	2	yes	—
201	C₂	—	2	1	no	—
202	C₂	—	2	2	no	—
203	C₂	—	4	2	no	—
204	C₂	—	8	2	no	—
205	C₂	#2482	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2482
prime radical	#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2923	C₂	semidirect
3. #2098	C₂	semidirect
4. #2080	C₂	semidirect
5. #1938	C₂	semidirect
6. #1932	C₂	semidirect
7. #1868	C₂	semidirect
8. #1858	C₂	semidirect
9. #1827	C₂	semidirect
17. #430	C₂²	semidirect
18. #424	C₂²	semidirect
19. #424	C₂²	semidirect
20. #424	C₂²	semidirect
33. C₄×M₁₆	C₂²	semidirect
34. C₂×#97	C₂²	semidirect
35. #344	C₂²	semidirect
36. #341	C₂²	semidirect
37. #339	C₂²	semidirect
38. H(3,ℤ/4)	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#2923	2
#2901	2
#2098	2
#2080	2
#1938	2
#1932	2
#1868	2
#1858	2
#1827	2
#1794	2
#1791	2