Finite Group Databasebeta

← #30640659

Subgroups — #30640659

Order 16256.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×#16861153C212yesmaximal, complemented
2#22550117C211yesmaximal, characteristic, complemented
3#22549677C212yesmaximal, complemented
4C22⋊(C127⋊C16)C211yesmaximal, characteristic, complemented
5C22×D8×C127C211yesmaximal, characteristic
6C2×Dic508C2211yescharacteristic, complemented
7#16861188—21no—
8#16861153—24no—
9C2×C127⋊C16C2212yescomplemented
10#16861143C2211yescharacteristic, complemented
11#16861134—21no—
12C22×D4×C127—21no—
13C2×D8×C127—24no—
14C2×D8×C127—21no—
15C2×D8×C127C2212yes—
16C22×C8×C127C2211yescharacteristic
17C2×Dic254—21no—
18Dic508—22no—
19C4⋊Dic127—21no—
20Dic127⋊C4—41no—
21C22⋊Dic127—41no—
22C127⋊C16—22no—
23C24×C127—41no—
24C2×D4×C127—24no—
25C2×D4×C127—22no—
26C2×D4×C127—42no—
27C2×D4×C127—41no—
28C22×C4×C127D411yescharacteristic
29D8×C127—24no—
30D8×C127—22no—
31D8×C127D414yes—
32C2×C8×C127—22no—
33C2×C8×C127D411yescharacteristic
34C2×C8×C127C2311yescharacteristic
35Dic254—41no—
36Dic254—41no—
37C2×Dic127—41no—
38C2×Dic127—41no—
39C23×C127—44no—
40C23×C127—42no—
41C23×C127—82no—
42C23×C127—82no—
43C23×C127D811yescharacteristic
44D4×C127—24no—
45D4×C127—44no—
46D4×C127—42no—
47C2×C4×C127—21no—
48C2×C4×C127—21no—
49C2×C4×C127D811yescharacteristic
50C2×C4×C127C2×D411yescharacteristic
51C8×C127—21no—
52C8×C127C2×D412yes—
53C22×C127—21no—
54C22×C127—21no—
55C22×C127—44no—
56C22×C127—88no—
57C22×C127—82no—
58C22×C127—82no—
59C22×C127SD3212yescomplemented
60C22×C127C2×D811yescharacteristic
61C4×C127—21no—
62C4×C127#12011yescharacteristic
63C4×C127AGL(1,ℤ/8)11yescharacteristic
64Dic127—81no—
65Dic127—81no—
66C2×C127—22no—
67C2×C127—84no—
68C2×C127#44711yescharacteristic
69C2×C127C2×SD3211yescharacteristic
70C2×C127#50911yescharacteristic
71#2187—1271nomaximal, Sylow
72C127#218711yescharacteristic, complemented, Sylow
73C22×D8D12711yescharacteristic
74C2×SD32—1272no—
75#467—1271no—
76#357—1272no—
77C22⋊C16—1271no—
78C22×D4—21no—
79C2×Q16—1271no—
80C2×D8—24no—
81C2×D8—21no—
82C2×D8C2×D12712yes—
83C22×C8C2×D12711yescharacteristic
84C22⋊Q8—2541no—
85SD32—2544no—
86C2×C16—1272no—
87#107—1271no—
88#103—2541no—
89C24—41no—
90C2×Q8—2541no—
91C2×D4—24no—
92C2×D4—22no—
93C2×D4—42no—
94C2×D4—41no—
95C22×C4C22⋊D12711yescharacteristic
96Q16—2542no—
97D8—24no—
98D8—22no—
99D8C22⋊D12714yes—
100C2×C8—22no—
101C2×C8C22⋊D12711yescharacteristic
102C2×C8C22×D12711yescharacteristic
103C4⋊C4—5081no—
104C4⋊C4—2541no—
105C22⋊C4—5081no—
106C16—2542no—
107C23—44no—
108C23—42no—
109C23—82no—
110C23—82no—
111C23#1452323311yescharacteristic
112Q8—5081no—
113Q8—5081no—
114D4—24no—
115D4—44no—
116D4—42no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—5081no—
120C2×C4—5081no—
121C2×C4#1452323311yescharacteristic
122C2×C4C2×C22⋊D12711yescharacteristic
123C8—21no—
124C8C2×C22⋊D12712yes—
125C22—21no—
126C22—21no—
127C22—44no—
128C22—88no—
129C22—82no—
130C22—82no—
131C22#1686115312yescomplemented
132C22C2×#1452323311yescharacteristic
133C4—21no—
134C4—10161no—
135C4—10161no—
136C4#1686117811yescharacteristic
137C4#1686123511yescharacteristic
138C2—22no—
139C2—84no—
140C2#2255001011yescharacteristic
141C2C2×#1686115311yescharacteristic
142C2#2255031211yescharacteristic
subgroupG/H
centerC22C2×#14523233
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting, prime radicalC22×D8×C127C2
socleC22×C127C2×D8
hypercenterC22×D8D127
hypocenterC127#2187
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861153C2semidirect
2. #22550117C2semidirect
3. #22549677C2semidirect
4. C22⋊(C127⋊C16)C2semidirect
6. C2×Dic508C22semidirect
9. C2×C127⋊C16C22semidirect
10. #16861143C22semidirect
59. C22×C127SD32semidirect
72. C127#2187semidirect
131. C22#16861153semidirect

Maximal subgroups

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

subgroupindex
C2×#168611532
#225501172
#225496772
C22⋊(C127⋊C16)2
C22×D8×C1272
#2187127