Finite Group Databasebeta

Subgroups — #13073

Order 256.

239 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 75 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×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472	C₂	1	1	yes	maximal, characteristic
3	C₂×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	#1816	C₂	1	1	yes	maximal, characteristic, complemented
6	#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂²×D₈	C₂²	1	1	yes	characteristic
10	C₂×#131	C₂²	1	1	yes	characteristic
11	#504	—	2	2	no	—
12	#504	—	2	2	no	—
13	C₂×M₃₂	—	2	1	no	—
14	C₂×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆	—	2	1	no	—
16	C₂²×C₁₆	C₂²	1	1	yes	characteristic
17	#472	—	2	2	no	—
18	#472	C₄	1	4	yes	complemented
19	C₂×#108	C₂²	1	1	yes	characteristic
20	C₂×#108	C₂²	1	1	yes	characteristic
21	C₂×#107	C₂²	1	1	yes	characteristic
22	#417	—	2	1	no	—
23	C₂×#100	—	2	1	no	—
24	#361	—	2	2	no	—
25	#357	—	2	1	no	—
26	#357	—	2	1	no	—
27	#343	—	2	1	no	—
28	C₂×#41	D₄	1	1	yes	characteristic
29	C₂²×D₄	—	2	1	no	—
30	AGL(1,ℤ/8)	—	2	4	no	—
31	AGL(1,ℤ/8)	—	4	2	no	—
32	C₂×SD₁₆	—	4	1	no	—
33	C₂×D₈	—	2	1	no	—
34	C₂×D₈	—	2	1	no	—
35	C₂×D₈	—	2	1	no	—
36	C₂×D₈	—	2	1	no	—
37	C₂×D₈	—	2	1	no	—
38	C₂×D₈	—	4	1	no	—
39	C₂×D₈	C₂×C₄	1	2	yes	—
40	#131	—	2	1	no	—
41	#131	—	2	1	no	—
42	#131	C₂×C₄	1	2	yes	—
43	#131	D₄	1	2	yes	—
44	C₂×M₁₆	—	2	1	no	—
45	C₂×M₁₆	—	2	1	no	—
46	C₂×M₁₆	—	2	1	no	—
47	C₂×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈	—	2	1	no	—
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄	—	2	1	no	—
51	M₃₂	—	2	4	no	—
52	M₃₂	—	2	1	no	—
53	M₃₂	D₄	1	2	yes	—
54	C₂×C₁₆	—	2	4	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	D₄	1	1	yes	characteristic
59	C₂×C₁₆	D₄	1	1	yes	characteristic
60	#108	—	2	1	no	—
61	#108	—	2	1	no	—
62	#108	—	2	1	no	—
63	#108	C₂×C₄	1	2	yes	—
64	#107	—	2	1	no	—
65	#107	—	2	1	no	—
66	#103	—	4	1	no	—
67	#102	—	4	1	no	—
68	#100	—	2	4	no	—
69	C₂⁴	—	4	1	no	—
70	#41	—	2	1	no	—
71	#41	—	2	1	no	—
72	#41	C₂²⋊C₄	1	2	yes	—
73	#41	SD₁₆	1	2	yes	—
74	C₂×Q₈	D₈	1	1	yes	characteristic
75	C₂×D₄	—	2	1	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	4	1	no	—
78	C₂×D₄	—	4	1	no	—
79	C₂×D₄	—	4	1	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	1	no	—
82	C₂×D₄	D₈	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	SD₁₆	—	4	4	no	—
87	D₈	—	2	2	no	—
88	D₈	—	2	2	no	—
89	D₈	—	4	4	no	—
90	D₈	—	4	1	no	—
91	D₈	—	4	1	no	—
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	2	2	no	—
96	M₁₆	—	4	1	no	—
97	M₁₆	—	4	1	no	—
98	M₁₆	—	4	1	no	—
99	M₁₆	C₂²⋊C₄	1	2	yes	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	#41	1	1	yes	characteristic
110	C₂×C₈	#41	1	1	yes	characteristic
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	2	2	no	—
115	C₁₆	—	2	1	no	—
116	C₂³	—	2	1	no	—
117	C₂³	—	4	2	no	—
118	C₂³	—	8	1	no	—
119	C₂³	—	8	1	no	—
120	C₂³	—	16	1	no	—
121	C₂³	—	16	1	no	—
122	C₂³	#135	1	1	yes	characteristic
123	Q₈	—	2	1	no	—
124	Q₈	#102	1	2	yes	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	D₄	—	2	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	8	1	no	—
131	D₄	—	8	1	no	—
132	D₄	#102	1	2	yes	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	8	1	no	—
137	C₂×C₄	—	16	1	no	—
138	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄	#121	1	1	yes	characteristic
140	C₂×C₄	#121	1	1	yes	characteristic
141	C₂×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄	#135	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	4	2	no	—
146	C₈	—	8	1	no	—
147	C₈	C₄×D₄	1	1	yes	characteristic
148	C₈	C₄×D₄	1	1	yes	characteristic
149	C₈	#120	1	1	yes	characteristic
150	C₈	#123	1	1	yes	characteristic
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	16	2	no	—
158	C₂²	—	16	2	no	—
159	C₂²	—	16	1	no	—
160	C₂²	—	16	1	no	—
161	C₂²	#416	1	1	yes	characteristic
162	C₂²	#465	1	1	yes	characteristic
163	C₂²	#466	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	16	1	no	—
166	C₄	#386	1	1	yes	characteristic
167	C₄	C₂×#102	1	1	yes	characteristic
168	C₄	#465	1	1	yes	characteristic
169	C₄	#466	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	—	4	2	no	—
173	C₂	—	16	2	no	—
174	C₂	#1893	1	1	yes	characteristic
175	C₂	#2221	1	1	yes	characteristic
176	C₂	#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#466
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#504	C₂	semidirect
3. C₂×#361	C₂	semidirect
4. C₂×#357	C₂	semidirect
5. #1816	C₂	semidirect
6. #1387	C₂	semidirect
7. #1381	C₂	semidirect
18. #472	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504	2
C₂×#472	2
C₂×#361	2
C₂×#357	2
#1816	2
#1387	2
#1381	2