Finite Group Databasebeta

← #30640751

Subgroups — #30640751

Order 16256.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550058C212yesmaximal, complemented
2#22550048C211yesmaximal, characteristic
3#22549675C212yesmaximal, complemented
4#22549673C211yesmaximal, characteristic, complemented
5#563×C127C211yesmaximal, characteristic, complemented
6#16861257C2211yescharacteristic
7#16861240—22no—
8#16861235—21no—
9#16861137—21no—
10#16861137—22no—
11#16861128C2211yescharacteristic
12#16861128C2212yes—
13C4⋊Q8×C127C2212yes—
14#126×C127—22no—
15#125×C127—21no—
16#124×C127C2211yescharacteristic
17#123×C127—21no—
18C22⋊Q8×C127—22no—
19#14523249—41no—
20#14523248—21no—
21#14523248—22no—
22Q8×D127—41no—
23#14523238—42no—
24#14523236—41no—
25#14523235—42no—
26#14523233—41no—
27#14523232—21no—
28#14523232—22no—
29C2×Q8×C127D412yes—
30C2×Q8×C127C2311yescharacteristic
31C2×D4×C127D411yescharacteristic
32C22×C4×C127—21no—
33C4⋊C4×C127—21no—
34C4⋊C4×C127—22no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—22no—
37C4⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C22⋊C4×C127—21no—
40C22⋊C4×C127—22no—
41C42×C127D411yescharacteristic
42C42×C127D412yes—
43Dic254—21no—
44Dic254—22no—
45D508—21no—
46D508—22no—
47C22⋊D127—41no—
48C4×D127—41no—
49C4×D127—42no—
50C127⋊C8—41no—
51C127⋊C8—42no—
52C23×C127—21no—
53Q8×C127—41no—
54Q8×C127—42no—
55D4×C127—41no—
56C2×C4×C127—21no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—81no—
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C22×C127—42no—
63C22×C127—41no—
64C22×C127#12011yescharacteristic
65C2×D127—41no—
66C4×C127—21no—
67C4×C127—22no—
68C4×C127—41no—
69C4×C127—42no—
70C4×C127—42no—
71Dic127—41no—
72C2×C127—21no—
73C2×C127—81no—
74C2×C127B(4,𝔽2)11yescharacteristic
75D127—81no—
76#2203—1271nomaximal, Sylow
77C127#220311yescharacteristic, complemented, Sylow
78#563D12711yescharacteristic, complemented
79#456—1272no—
80#454—1271no—
81#356—1272no—
82#355—1271no—
832−1+4—1271no—
84#137—2542no—
85AGL(1,ℤ/8)—2541no—
86C4⋊Q8C2×D12712yes—
87#126—22no—
88#125—21no—
89#124C2×D12711yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—2541no—
93#104—2542no—
94#101—1272no—
95#101—1271no—
96#41—5081no—
97#41—2541no—
98#41—2542no—
99C2×Q8—5081no—
100C2×Q8C22⋊D12712yes—
101C2×Q8C22×D12711yescharacteristic
102C2×D4C22⋊D12711yescharacteristic
103C22×C4—21no—
104Q16—5082no—
105SD16—5081no—
106SD16—5082no—
107D8—5081no—
108M16—2541no—
109M16—2542no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D12711yescharacteristic
119C42C22⋊D12712yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—2541no—
124Q8—2542no—
125D4—41no—
126D4—5081no—
127D4—2541no—
128D4—2542no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—5081no—
134C2×C4—5082no—
135C2×C4C2×C22⋊D12712yes—
136C2×C4C2×C22⋊D12711yescharacteristic
137C8—5081no—
138C8—5082no—
139C22—42no—
140C22—41no—
141C22—5081no—
142C22#1686117811yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—5081no—
149C2—21no—
150C2—81no—
151C2—10161no—
152C2#2255006311yescharacteristic
subgroupG/H
centerC2#22550063
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting#563×C127C2
socleC2×C127B(4,𝔽2)
prime radical#22550048C2
hypercenter#563D127
hypocenterC127#2203
LCSG ▷ C2×Q8×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #22550058C2semidirect
3. #22549675C2semidirect
4. #22549673C2semidirect
5. #563×C127C2semidirect
77. C127#2203semidirect
78. #563D127semidirect

Maximal subgroups

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

subgroupindex
#225500582
#225500482
#225496752
#225496732
#563×C1272
#2203127