Finite Group Databasebeta

← #30640690

Subgroups — #30640690

Order 16256.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550677C211yesmaximal, characteristic, complemented
2#22550314C211yesmaximal, characteristic, complemented
3C2×#16861150C211yesmaximal, characteristic, complemented
4#22550147C211yesmaximal, characteristic, complemented
5#22549682C211yesmaximal, characteristic, complemented
6#22549661C211yesmaximal, characteristic, complemented
7#361×C127C211yesmaximal, characteristic, complemented
8#16861254—21no—
9#16861237—21no—
10#16861235—21no—
11#16861233—22no—
12#16861231C2211yescharacteristic
13#16861228—21no—
14C2×D1016C2211yescharacteristic, complemented
15D8×D127—21no—
16#16861215C2211yescharacteristic
17#16861154—21no—
18D2032—21no—
19#16861150—22no—
20C2×C127⋊C16C2211yescharacteristic, complemented
21#16861125—21no—
22C2×D8×C127C2211yescharacteristic
23M32×C127C2211yescharacteristic, complemented
24#108×C127C2211yescharacteristic, complemented
25#14523248—21no—
26#14523248D411yescharacteristic
27#14523247—22no—
28#14523247—41no—
29C2×D508—21no—
30C2×C22⋊D127—42no—
31D4×D127—22no—
32D4×D127—41no—
33Dic508D411yescharacteristic
34#14523237—21no—
35#14523237—41no—
36#14523236—22no—
37D1016—21no—
38D1016—21no—
39D1016—41no—
40D1016D411yescharacteristic
41#14523233—22no—
42#14523232D411yescharacteristic
43#14523231—21no—
44C8×D127—21no—
45C127⋊C16—21no—
46C2×D4×C127—21no—
47D8×C127—21no—
48D8×C127D412yes—
49M16×C127—21no—
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52C22×D127—42no—
53C22×D127—41no—
54Dic254—21no—
55Dic254D811yescharacteristic
56D508—21no—
57D508—41no—
58D508—41no—
59D508D811yescharacteristic
60C22⋊D127—21no—
61C22⋊D127—42no—
62C22⋊D127—42no—
63C22⋊D127—41no—
64C2×Dic127—42no—
65C4×D127—21no—
66C4×D127—41no—
67C127⋊C8—21no—
68C23×C127—41no—
69D4×C127—22no—
70D4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—42no—
76C22×C127—161no—
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C2×D127—21no—
79C2×D127—41no—
80C2×D127—81no—
81C2×D127—162no—
82C2×D127—161no—
83C4×C127#12011yescharacteristic
84C4×C127C2×D811yescharacteristic
85Dic127—21no—
86Dic127—41no—
87C2×C127—21no—
88C2×C127—82no—
89C2×C127#44711yescharacteristic
90D127—41no—
91D127—81no—
92D127—161no—
93#2192—1271nomaximal, Sylow
94C127#219211yescharacteristic, complemented, Sylow
95#576—1271no—
96#509—1271no—
97C2×D16—1271no—
98#472—1271no—
99#361D12711yescharacteristic, complemented
100#359—1271no—
101#350—1271no—
102H(5,𝔽2)—2541no—
103AGL(1,ℤ/8)—2541no—
104AGL(1,ℤ/8)—2541no—
105AGL(1,ℤ/8)—2542no—
106#135—1271no—
107#135—2541no—
108C2×D8—1271no—
109C2×D8—2541no—
110C2×D8C2×D12711yescharacteristic
111#131—1271no—
112SD32—2541no—
113D16—2541no—
114D16—2542no—
115M32C2×D12711yescharacteristic, complemented
116C2×C16—1271no—
117#108C2×D12711yescharacteristic, complemented
118#100—2541no—
119#41—1271no—
120#41—5081no—
121#41—2542no—
122#41—2541no—
123C2×D4—21no—
124C2×D4—5081no—
125C2×D4—5082no—
126C2×D4—2542no—
127C2×D4—2541no—
128Q16—1271no—
129SD16—5081no—
130SD16—2542no—
131SD16—2541no—
132D8—21no—
133D8—1271no—
134D8—5081no—
135D8—2542no—
136D8—2541no—
137D8—2541no—
138D8D50812yes—
139M16—21no—
140M16—1271no—
141M16—2541no—
142C2×C8—2541no—
143C2×C8C22×D12711yescharacteristic
144C16—21no—
145C16—2541no—
146C23—41no—
147C23—5082no—
148C23—5081no—
149Q8—1271no—
150Q8—2541no—
151D4—22no—
152D4—41no—
153D4—1271no—
154D4—5082no—
155D4—5081no—
156D4—5082no—
157D4—5081no—
158D4—5081no—
159D4—2541no—
160D4—2541no—
161C2×C4—5082no—
162C2×C4—5081no—
163C2×C4—2541no—
164C2×C4D4×D12711yescharacteristic
165C8—41no—
166C8—2541no—
167C8D4×D12711yescharacteristic
168C8C2×D50811yescharacteristic
169C22—42no—
170C22—161no—
171C22—20321no—
172C22—20322no—
173C22—10161no—
174C22—5081no—
175C22—2541no—
176C22#1686123611yescharacteristic
177C4—5081no—
178C4—2541no—
179C4#1686117911yescharacteristic
180C4D8×D12711yescharacteristic
181C2—21no—
182C2—82no—
183C2—20321no—
184C2—10161no—
185C2—5081no—
186C2#2255001111yescharacteristic
subgroupG/H
centerC2#22550011
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#361×C127C2
socleC2×C127#447
hypercenter#361D127
hypocenterC127#2192
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
1. #22550677C2semidirect
2. #22550314C2semidirect
3. C2×#16861150C2semidirect
4. #22550147C2semidirect
5. #22549682C2semidirect
6. #22549661C2semidirect
7. #361×C127C2semidirect
14. C2×D1016C22semidirect
20. C2×C127⋊C16C22semidirect
23. M32×C127C22semidirect
24. #108×C127C22semidirect
94. C127#2192semidirect
99. #361D127semidirect
115. M32C2×D127semidirect
117. #108C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506772
#225503142
C2×#168611502
#225501472
#225496822
#225496612
#361×C1272
#2192127