Finite Group Databasebeta

← #30638944

Subgroups — #30638944

Order 16256.

272 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550354C211yesmaximal, characteristic
2#22549916C212yesmaximal
3C2×#16861128C211yesmaximal, characteristic
4#22549636C212yesmaximal
5#391×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7Q8×Dic127—22no—
8Q8×Dic127—22no—
9#16861168—22no—
10#16861168C2212yes—
11#16861142—22no—
12#16861140—22no—
13#16861128C414yescomplemented
14C22×Q8×C127C2211yescharacteristic
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C42×C127C2211yescharacteristic
17#95×C127—22no—
18#14523232—22no—
19#14523232C2×C414yes—
20C2×C127⋊C8—22no—
21C4⋊Dic127—22no—
22C4⋊Dic127—22no—
23C4⋊Dic127D412yes—
24C4⋊Dic127D412yes—
25C22⋊Dic127—22no—
26C4×Dic127—24no—
27C4×Dic127—22no—
28C2×Q8×C127—22no—
29C2×Q8×C127—22no—
30C2×Q8×C127C2×C412yes—
31C2×Q8×C127Q812yes—
32C22×C4×C127—21no—
33C22×C4×C127—21no—
34C22×C4×C127D412yes—
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—22no—
37C42×C127—22no—
38C2×Dic127—22no—
39C2×Dic127—22no—
40C127⋊C8—24no—
41C23×C127C2×D411yescharacteristic
42Q8×C127—24no—
43Q8×C127—24no—
44C2×C4×C127—22no—
45C2×C4×C127—22no—
46C2×C4×C127—22no—
47C2×C4×C127—41no—
48C2×C4×C127—41no—
49C2×C4×C127—42no—
50C2×C4×C127—41no—
51C2×C4×C127C4⋊C414yes—
52C2×C4×C127C22×C411yescharacteristic
53C2×C4×C127C2×D412yes—
54C2×C4×C127C2×Q811yescharacteristic
55C2×C4×C127#4112yes—
56C22×C127—21no—
57C22×C127—21no—
58C22×C127C2×C4⋊C411yescharacteristic
59C22×C127#12011yescharacteristic
60C22×C127#12311yescharacteristic
61C4×C127—24no—
62C4×C127—42no—
63C4×C127—81no—
64C4×C127C4×D412yes—
65C4×C127C22⋊Q812yes—
66Dic127—42no—
67Dic127—42no—
68C2×C127—21no—
69C2×C127—21no—
70C2×C127#38511yescharacteristic
71C2×C127#45411yescharacteristic
72C2×C127#45611yescharacteristic
73#1857—1271nomaximal, Sylow
74C127#185711yescharacteristic, complemented, Sylow
75#519—1271no—
76#428—1272no—
77C2×#101—1271no—
78#391D12711yescharacteristic
79#339—1272no—
80C22×Q8C2×D12711yescharacteristic
81C2×M16—1272no—
82C4×Q8—2542no—
83C4×Q8—2542no—
84#117—1272no—
85#117—2542no—
86C2×C4⋊C4C2×D12711yescharacteristic
87C2×C42C2×D12711yescharacteristic
88#107—2542no—
89#106—2542no—
90#101—1274no—
91#95—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8C4×D12712yes—
95C2×Q8Dic25412yes—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D12712yes—
99C22×C4C22×D12711yescharacteristic
100M16—1274no—
101M16—2542no—
102C2×C8—2542no—
103C4⋊C4—22no—
104C4⋊C4—1272no—
105C4⋊C4—1272no—
106C4⋊C4—2542no—
107C4⋊C4—2542no—
108C22⋊C4—2542no—
109C42—22no—
110C42—2544no—
111C42—2542no—
112C23C2×C22⋊D12711yescharacteristic
113Q8—24no—
114Q8—24no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—2542no—
123C2×C4—2542no—
124C2×C4Dic127⋊C414yes—
125C2×C4C2×C4×D12711yescharacteristic
126C2×C4D4×D12712yes—
127C2×C4C2×Dic25411yescharacteristic
128C2×C4#1452324712yes—
129C8—2544no—
130C22—21no—
131C22—21no—
132C22C2×Dic127⋊C411yescharacteristic
133C22#1686117711yescharacteristic
134C22#1686119011yescharacteristic
135C4—24no—
136C4—42no—
137C4—81no—
138C4—5082no—
139C4—5082no—
140C4(C22⋊D127)⋊C412yes—
141C4#1686118612yes—
142C2—21no—
143C2—21no—
144C2#2254974811yescharacteristic
145C2#2255004811yescharacteristic
146C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#391D127
hypocenterC127#1857
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
13. #16861128C4semidirect
74. C127#1857semidirect

Maximal subgroups

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

subgroupindex
#225503542
#225499162
C2×#168611282
#225496362
#391×C1272
#1857127