Finite Group Databasebeta

← #30640750

Subgroups — #30640750

Order 16256.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550626C211yesmaximal, characteristic, complemented
2#22550059C212yesmaximal, complemented
3#22549674C212yesmaximal, complemented
4#22549673C211yesmaximal, characteristic, complemented
5#454×C127C211yesmaximal, characteristic
6#16861239—22no—
7#16861207C2212yes—
8#16861202—22no—
9#16861201—21no—
10#16861190—21no—
11#16861186—22no—
12#16861136—22no—
13#16861128C2212yes—
142−1+4×C127C2211yescharacteristic
15AGL(1,ℤ/8)×C127—21no—
16#124×C127C2211yescharacteristic
17#104×C127—21no—
18#101×C127C2211yescharacteristic
19C2×Dic254D412yes—
20C22×Dic127—21no—
21#14523238—42no—
22#14523236—42no—
23#14523232—22no—
24C4⋊Dic127—21no—
25C4⋊Dic127—21no—
26Dic127⋊C4—22no—
27Dic127⋊C4—22no—
28Dic127⋊C4—22no—
29C22⋊Dic127—22no—
30C22⋊Dic127—21no—
31C4×Dic127D412yes—
32#41×C127—22no—
33#41×C127—21no—
34#41×C127—41no—
35C2×Q8×C127—41no—
36C2×Q8×C127C2311yescharacteristic
37C2×D4×C127D411yescharacteristic
38SD16×C127—41no—
39D8×C127—41no—
40M16×C127—21no—
41C22⋊C4×C127—22no—
42C42×C127D411yescharacteristic
43Dic254—42no—
44C2×Dic127—22no—
45C2×Dic127—22no—
46C2×Dic127—81no—
47C127⋊C8—42no—
48C23×C127—21no—
49Q8×C127—21no—
50Q8×C127—22no—
51Q8×C127—41no—
52D4×C127—21no—
53D4×C127—22no—
54D4×C127—41no—
55D4×C127—41no—
56C2×C4×C127—21no—
57C2×C4×C127—41no—
58C2×C4×C127—42no—
59C2×C4×C127C2×D412yes—
60C2×C4×C127C2×D411yescharacteristic
61C8×C127—41no—
62C22×C127—42no—
63C22×C127—41no—
64C22×C127—41no—
65C22×C127#12011yescharacteristic
66C4×C127—21no—
67C4×C127—22no—
68C4×C127—41no—
69C4×C127—41no—
70Dic127—42no—
71Dic127—42no—
72C2×C127—21no—
73C2×C127—81no—
74C2×C127—81no—
75C2×C127B(4,𝔽2)11yescharacteristic
76#2203—1271nomaximal, Sylow
77C127#220311yescharacteristic, complemented, Sylow
78#563—1271no—
79#456—1272no—
80#454D12711yescharacteristic
81#356—1272no—
82#355—1271no—
832−1+4C2×D12711yescharacteristic
84#137—2542no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—1272no—
87#126—2542no—
88#125—2541no—
89#124C2×D12711yescharacteristic
90#123—2541no—
91C22⋊Q8—2542no—
92#104—21no—
93#104—2542no—
94#101—1272no—
95#101C2×D12711yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—1272no—
101C2×Q8C22×D12711yescharacteristic
102C2×D4C22⋊D12711yescharacteristic
103C22×C4—2541no—
104Q16—5082no—
105SD16—41no—
106SD16—5082no—
107D8—41no—
108M16—21no—
109M16—2542no—
110C4⋊C4—2542no—
111C4⋊C4—2541no—
112C4⋊C4—2541no—
113C4⋊C4—2542no—
114C4⋊C4—2542no—
115C22⋊C4—22no—
116C22⋊C4—2542no—
117C22⋊C4—2541no—
118C42—1272no—
119C42C22⋊D12711yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—5082no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—10161no—
133C2×C4—2542no—
134C2×C4—2542no—
135C2×C4D4×D12712yes—
136C2×C4C2×C22⋊D12711yescharacteristic
137C8—41no—
138C8—5082no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#1686117711yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—5082no—
148C4—5082no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting, prime radical#454×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#454D127
hypocenterC127#2203
LCSG ▷ C2×Q8×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #22550626C2semidirect
2. #22550059C2semidirect
3. #22549674C2semidirect
4. #22549673C2semidirect
77. C127#2203semidirect

Maximal subgroups

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

subgroupindex
#225506262
#225500592
#225496742
#225496732
#454×C1272
#2203127