Finite Group Databasebeta

Subgroups — #19647

Order 256.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3207	C₂	1	2	yes	maximal, complemented
2	#3184	C₂	1	2	yes	maximal, complemented
3	#3169	C₂	1	1	yes	maximal, characteristic, complemented
4	#3147	C₂	1	1	yes	maximal, characteristic, complemented
5	#1717	C₂	1	2	yes	maximal, complemented
6	#1715	C₂	1	2	yes	maximal, complemented
7	#1712	C₂	1	1	yes	maximal, characteristic, complemented
8	#1709	C₂	1	1	yes	maximal, characteristic, complemented
9	#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	—	2	2	no	—
13	#532	C₂²	1	2	yes	—
14	#532	C₂²	1	2	yes	—
15	#500	C₂²	1	1	yes	characteristic
16	#498	C₂²	1	2	yes	—
17	#495	—	2	2	no	—
18	#494	C₂²	1	1	yes	characteristic
19	#493	C₂²	1	1	yes	characteristic
20	#493	C₂²	1	1	yes	characteristic
21	#493	C₂²	1	2	yes	—
22	#493	C₂²	1	2	yes	—
23	#492	C₂²	1	2	yes	—
24	#492	C₂²	1	2	yes	—
25	#487	C₂²	1	2	yes	—
26	#486	C₂²	1	2	yes	—
27	#486	C₂²	1	2	yes	—
28	#486	C₂²	1	2	yes	—
29	#486	C₂²	1	2	yes	—
30	#469	—	2	4	no	—
31	#466	—	2	2	no	—
32	#463	—	2	4	no	—
33	#449	—	2	4	no	—
34	#442	—	2	4	no	—
35	C₄×D₈	—	2	2	no	—
36	#431	C₂²	1	1	yes	characteristic
37	C₄×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#322	C₂²	1	1	yes	characteristic
40	#322	C₂²	1	1	yes	characteristic
41	C₈²	C₂²	1	1	yes	characteristic
42	C₈²	C₂²	1	1	yes	characteristic
43	C₂×#41	—	2	2	no	—
44	C₂×#41	—	2	2	no	—
45	#135	—	4	8	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂×SD₁₆	—	2	4	no	—
48	C₂×SD₁₆	—	2	4	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	2	2	no	—
51	C₂×D₈	—	2	4	no	—
52	C₂×D₈	—	2	4	no	—
53	C₂×M₁₆	D₄	1	2	yes	—
54	C₂²×C₈	D₄	1	2	yes	—
55	C₄⋊Q₈	C₂³	1	2	yes	—
56	C₄⋊Q₈	C₂³	1	2	yes	—
57	#127	C₂³	1	2	yes	—
58	#127	C₂³	1	2	yes	—
59	#124	—	2	2	no	—
60	#124	—	2	2	no	—
61	#121	—	2	2	no	—
62	#121	—	2	2	no	—
63	#121	—	2	2	no	—
64	#121	—	2	2	no	—
65	C₄×D₄	—	2	2	no	—
66	C₄×D₄	—	2	2	no	—
67	C₄×D₄	—	2	2	no	—
68	C₄×D₄	—	2	2	no	—
69	C₂×C₄²	C₂³	1	1	yes	characteristic
70	#107	—	2	2	no	—
71	#106	—	2	4	no	—
72	C₄⋊C₈	D₄	1	2	yes	—
73	#103	—	2	4	no	—
74	#102	—	2	4	no	—
75	#102	—	2	4	no	—
76	#102	—	2	4	no	—
77	#102	—	2	4	no	—
78	C₂²⋊C₈	D₄	1	2	yes	—
79	#97	D₄	1	2	yes	—
80	C₄×C₈	D₄	1	2	yes	—
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₈	C₂³	1	1	yes	characteristic
87	#41	—	4	8	no	—
88	#41	—	8	4	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	2	no	—
96	C₂×D₄	—	2	2	no	—
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
104	Q₁₆	—	4	4	no	—
105	SD₁₆	—	4	8	no	—
106	SD₁₆	—	8	4	no	—
107	D₈	—	4	8	no	—
108	D₈	—	4	4	no	—
109	D₈	—	4	4	no	—
110	D₈	—	8	4	no	—
111	M₁₆	—	2	4	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	C₂×D₄	1	2	yes	—
114	C₂×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₈	C₂×D₄	1	2	yes	—
117	C₂×C₈	C₂×D₄	1	2	yes	—
118	C₂×C₈	C₂×D₄	1	2	yes	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	2	no	—
125	C₂²⋊C₄	—	4	2	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₄²	C₂×D₄	1	1	yes	characteristic
128	C₄²	C₂×D₄	1	1	yes	characteristic
129	C₄²	C₂×D₄	1	1	yes	characteristic
130	C₄²	C₂⁴	1	1	yes	characteristic
131	C₂³	—	4	2	no	—
132	C₂³	—	4	2	no	—
133	C₂³	#127	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	Q₈	—	4	4	no	—
136	D₄	—	4	4	no	—
137	D₄	—	4	4	no	—
138	D₄	—	4	4	no	—
139	D₄	—	4	4	no	—
140	D₄	—	4	4	no	—
141	D₄	—	4	4	no	—
142	D₄	—	8	4	no	—
143	D₄	—	8	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	#127	1	1	yes	characteristic
152	C₂×C₄	#127	1	1	yes	characteristic
153	C₂×C₄	#127	1	1	yes	characteristic
154	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	#135	1	4	yes	complemented
162	C₈	AGL(1,ℤ/8)	1	4	yes	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	8	2	no	—
167	C₂²	—	8	2	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	8	2	no	—
171	C₂²	—	8	2	no	—
172	C₂²	C₂×#127	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	8	2	no	—
177	C₄	—	8	2	no	—
178	C₄	C₂×#135	1	2	yes	—
179	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	C₄	#576	1	2	yes	—
181	C₂	—	4	1	no	—
182	C₂	—	16	2	no	—
183	C₂	—	16	2	no	—
184	C₂	#3147	1	1	yes	characteristic
185	C₂	#3151	1	1	yes	characteristic
186	C₂	#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
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. #3207	C₂	semidirect
2. #3184	C₂	semidirect
3. #3169	C₂	semidirect
4. #3147	C₂	semidirect
5. #1717	C₂	semidirect
6. #1715	C₂	semidirect
7. #1712	C₂	semidirect
8. #1709	C₂	semidirect
9. #1707	C₂	semidirect
10. #1706	C₂	semidirect
11. C₈×M₁₆	C₂	semidirect
161. C₈	#135	semidirect

Maximal subgroups

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

subgroup	index
#3207	2
#3184	2
#3169	2
#3147	2
#1717	2
#1715	2
#1712	2
#1709	2
#1707	2
#1706	2
C₈×M₁₆	2