Finite Group Databasebeta

← #30640695

Subgroups — #30640695

Order 16256.

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

normal subgroups only

This group has 210 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#22550690C211yesmaximal, characteristic, complemented
2#22550311C211yesmaximal, characteristic, complemented
3C2×#16861154C211yesmaximal, characteristic, complemented
4#22550147C211yesmaximal, characteristic, complemented
5#22549691C211yesmaximal, characteristic, complemented
6#22549659C211yesmaximal, characteristic, complemented
7#359×C127C211yesmaximal, characteristic, complemented
8#16861256—21no—
9#16861239—21no—
10#16861238—22no—
11#16861237—21no—
12#16861228—21no—
13C2×Dic508C2211yescharacteristic
14Q16×D127—21no—
15C2×D1016C2211yescharacteristic
16#16861215C2211yescharacteristic
17#16861154—22no—
18#16861152—21no—
19#16861150—21no—
20#16861149C2211yescharacteristic
21#16861125—21no—
22#135×C127C2211yescharacteristic
23C2×C16×C127C2211yescharacteristic
24#108×C127C2211yescharacteristic
25#14523248—41no—
26#14523248D411yescharacteristic
27#14523247—22no—
28#14523247—42no—
29C2×Dic254—21no—
30C2×Dic254—41no—
31Q8×D127—22no—
32C2×D508—21no—
33Dic508—21no—
34Dic508D412yes—
35#14523238—22no—
36#14523237—41no—
37#14523236—22no—
38D1016—21no—
39D1016—21no—
40D1016—41no—
41#14523232D411yescharacteristic
42#14523231—21no—
43C8×D127—21no—
44C127⋊C16—21no—
45#41×C127—21no—
46Q16×C127D411yescharacteristic
47SD16×C127—21no—
48D8×C127D411yescharacteristic
49M16×C127—21no—
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52C22×D127—41no—
53Dic254—22no—
54Dic254—42no—
55Dic254—41no—
56Dic254D811yescharacteristic
57D508—41no—
58D508—41no—
59D508D811yescharacteristic
60C22⋊D127—21no—
61C22⋊D127—42no—
62C2×Dic127—42no—
63C2×Dic127—41no—
64C4×D127—21no—
65C4×D127—42no—
66C127⋊C8—21no—
67Q8×C127—21no—
68D4×C127—21no—
69D4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—41no—
76C22×C127AGL(1,ℤ/8)11yescharacteristic
77C2×D127—21no—
78C2×D127—81no—
79C2×D127—161no—
80C4×C127—41no—
81C4×C127#12011yescharacteristic
82C4×C127C2×D811yescharacteristic
83Dic127—21no—
84Dic127—42no—
85C2×C127—21no—
86C2×C127—81no—
87C2×C127#44711yescharacteristic
88D127—41no—
89D127—161no—
90#2193—1271nomaximal, Sylow
91C127#219311yescharacteristic, complemented, Sylow
92#578—1271no—
93#509—1271no—
94C2×SD32—1271no—
95#472—1271no—
96#362—1271no—
97#359D12711yescharacteristic, complemented
98#350—1271no—
992−1+4—2541no—
100#137—2541no—
101#137—2542no—
102AGL(1,ℤ/8)—2541no—
103#135—2541no—
104#135C2×D12711yescharacteristic
105C2×Q16—1271no—
106C2×Q16—2541no—
107C2×D8—1271no—
108#131—1271no—
109SD32—2542no—
110SD32—2541no—
111D16—2541no—
112M32—1271no—
113C2×C16C2×D12711yescharacteristic
114#108C2×D12711yescharacteristic
115#100—2541no—
116#41—21no—
117#41—1271no—
118#41—5081no—
119#41—5082no—
120#41—2542no—
121C2×Q8—5081no—
122C2×Q8—2542no—
123C2×Q8—2541no—
124C2×D4—2541no—
125Q16—1272no—
126Q16—2542no—
127Q16—2541no—
128Q16D50811yescharacteristic
129SD16—21no—
130SD16—5081no—
131SD16—2542no—
132D8—5081no—
133D8—2541no—
134D8—2541no—
135D8D50811yescharacteristic
136M16—21no—
137M16—1271no—
138M16—2541no—
139C2×C8—2541no—
140C2×C8C22×D12711yescharacteristic
141C16—21no—
142C16—2541no—
143C23—5081no—
144Q8—21no—
145Q8—1271no—
146Q8—5082no—
147Q8—5081no—
148Q8—2542no—
149D4—21no—
150D4—41no—
151D4—1271no—
152D4—5082no—
153D4—5081no—
154D4—5081no—
155D4—2541no—
156C2×C4—41no—
157C2×C4—5082no—
158C2×C4—5081no—
159C2×C4—5082no—
160C2×C4—2541no—
161C2×C4D4×D12711yescharacteristic
162C8—41no—
163C8—2541no—
164C8D4×D12711yescharacteristic
165C8C2×D50811yescharacteristic
166C22—41no—
167C22—20321no—
168C22—10161no—
169C22—2541no—
170C22#1686123611yescharacteristic
171C4—41no—
172C4—5082no—
173C4—2541no—
174C4#1686117911yescharacteristic
175C4D8×D12711yescharacteristic
176C2—21no—
177C2—81no—
178C2—20321no—
179C2—5081no—
180C2#2255001111yescharacteristic
subgroupG/H
centerC2#22550011
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#359×C127C2
socleC2×C127#447
hypercenter#359D127
hypocenterC127#2193
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #359

Complements

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

NH ≅ G/Ntype
1. #22550690C2semidirect
2. #22550311C2semidirect
3. C2×#16861154C2semidirect
4. #22550147C2semidirect
5. #22549691C2semidirect
6. #22549659C2semidirect
7. #359×C127C2semidirect
91. C127#2193semidirect
97. #359D127semidirect

Maximal subgroups

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

subgroupindex
#225506902
#225503112
C2×#168611542
#225501472
#225496912
#225496592
#359×C1272
#2193127