Finite Group Databasebeta

← #30640018

Subgroups — #30640018

Order 16256.

266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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#22550676C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic, complemented
3C2×#16861135C212yesmaximal, complemented
4#22549858C212yesmaximal, complemented
5#338×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7#16861242—22no—
8#16861237—22no—
9#16861231—22no—
10C2×#14523232C2211yescharacteristic
11C2×C4×Dic127C2211yescharacteristic
12Dic127⋊C8—22no—
13#16861135—22no—
14#16861135—22no—
15#16861127—22no—
16#16861125—22no—
17C2×M16×C127C2212yes—
18C2×M16×C127C2211yescharacteristic
19#14523248—22no—
20#14523248—22no—
21#14523248—22no—
22#14523248—22no—
23C2×Dic254D412yes—
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26C22×Dic127—21no—
27C2×C4×D127—22no—
28Dic508—22no—
29#14523237—24no—
30D1016—22no—
31#14523232—22no—
32C2×C127⋊C8—21no—
33C4×Dic127—21no—
34C4×Dic127D411yescharacteristic
35C4×Dic127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37M16×C127—22no—
38M16×C127—22no—
39M16×C127—22no—
40C2×C8×C127—22no—
41C2×C8×C127D412yes—
42C22×D127—22no—
43Dic254—22no—
44Dic254—22no—
45D508—22no—
46D508—22no—
47C22⋊D127—42no—
48C22⋊D127—42no—
49C2×Dic127—21no—
50C2×Dic127—21no—
51C2×Dic127—22no—
52C2×Dic127—42no—
53C4×D127—42no—
54C4×D127—42no—
55C127⋊C8—41no—
56C23×C127#4111yescharacteristic
57C2×C4×C127C2×D412yes—
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127#4112yes—
61C8×C127—22no—
62C8×C127—42no—
63C22×C127—41no—
64C22×C127#12111yescharacteristic
65C22×C127#12312yes—
66C2×D127—42no—
67C2×D127—82no—
68C4×C127#12011yescharacteristic
69C4×C127#12112yes—
70C4×C127#12411yescharacteristic
71Dic127—22no—
72Dic127—41no—
73Dic127—42no—
74C2×C127—21no—
75C2×C127—22no—
76C2×C127#39411yescharacteristic
77D127—82no—
78#2050—1271nomaximal, Sylow
79C127#205011yescharacteristic, complemented, Sylow
80#575—1271no—
81#423—1271no—
82C2×#104—1272no—
83#413—1272no—
84#338D12711yescharacteristic, complemented
85C2×#41—1272no—
86#137—2542no—
87AGL(1,ℤ/8)—2542no—
88#135—2542no—
89C2×M16—1271no—
90C2×M16C2×D12712yes—
91C2×M16C2×D12711yescharacteristic
92C2×C42—1271no—
93C4⋊C8—2542no—
94#104—2542no—
95#104—2542no—
96#101—2542no—
97#100—2542no—
98#41—2542no—
99#41—2542no—
100#41—2542no—
101#41—2542no—
102C2×Q8—1272no—
103C2×D4—1272no—
104C2×D4—2542no—
105C22×C4—2541no—
106C22×C4—2542no—
107C22×C4C22×D12711yescharacteristic
108Q16—2542no—
109SD16—2544no—
110D8—2542no—
111M16—22no—
112M16—22no—
113M16—22no—
114M16—2542no—
115C2×C8—22no—
116C2×C8—2541no—
117C2×C8D50812yes—
118C42—1271no—
119C42—1271no—
120C42—2541no—
121C23—2542no—
122C23#1452324911yescharacteristic
123Q8—2542no—
124Q8—2542no—
125D4—5082no—
126D4—5082no—
127D4—2542no—
128D4—2542no—
129C2×C4—5082no—
130C2×C4—5082no—
131C2×C4—5082no—
132C2×C4—2541no—
133C2×C4—2542no—
134C2×C4—2541no—
135C2×C4D4×D12712yes—
136C2×C4D4×D12711yescharacteristic
137C2×C4C2×D50811yescharacteristic
138C2×C4#1452324812yes—
139C8—22no—
140C8—42no—
141C8—5081no—
142C22—41no—
143C22—10162no—
144C22—5082no—
145C22#1686118411yescharacteristic
146C22#1686119412yes—
147C4—5081no—
148C4—5082no—
149C4—2542no—
150C4#1686117911yescharacteristic
151C4#1686118212yes—
152C4#1686119711yescharacteristic
153C2—21no—
154C2—22no—
155C2—10162no—
156C2#2254979411yescharacteristic
subgroupG/H
centerC4#16861197
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#338×C127C2
socleC2×C127#394
prime radical#22550676C2
hypercenter#338D127
hypocenterC127#2050
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #338

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
3. C2×#16861135C2semidirect
4. #22549858C2semidirect
5. #338×C127C2semidirect
79. C127#2050semidirect
84. #338D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225498992
C2×#168611352
#225498582
#338×C1272
#2050127