Finite Group Databasebeta

← #30640746

Subgroups — #30640746

Order 16256.

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

normal subgroups only

This group has 248 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#22550618C211yesmaximal, characteristic, complemented
2#22550049C212yesmaximal, complemented
3#22549673C211yesmaximal, characteristic, complemented
4#22549672C212yesmaximal, complemented
5#454×C127C211yesmaximal, characteristic, complemented
6#16861234—22no—
7#16861198C2212yes—
8#16861197—21no—
9#16861195—22no—
10#16861179—22no—
11#16861177—21no—
12#16861136—22no—
13#16861128C2212yescomplemented
142−1+4×C127C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C127—21no—
16#124×C127C2211yescharacteristic
17#104×C127—21no—
18#101×C127C2211yescharacteristic, complemented
19C23×D127—21no—
20C2×Dic254—21no—
21C2×D508—21no—
22C2×D508D412yes—
23C2×C22⋊D127—22no—
24#14523235—42no—
25#14523233—42no—
26#14523232—22no—
27C22⋊Dic127—21no—
28(C2×D127)⋊C4—22no—
29(C2×D127)⋊C4—22no—
30(C2×D127)⋊C4—22no—
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
43C22×D127—22no—
44C22×D127—82no—
45C22×D127—81no—
46Dic254—81no—
47D508—42no—
48D508—81no—
49C22⋊D127—82no—
50C2×Dic127—22no—
51C127⋊C8—42no—
52C23×C127—21no—
53Q8×C127—21no—
54Q8×C127—22no—
55Q8×C127—41no—
56D4×C127—21no—
57D4×C127—22no—
58D4×C127—41no—
59D4×C127—41no—
60C2×C4×C127—21no—
61C2×C4×C127—41no—
62C2×C4×C127—42no—
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D411yescharacteristic
65C8×C127—41no—
66C22×C127—42no—
67C22×C127—41no—
68C22×C127—41no—
69C22×C127#12011yescharacteristic
70C2×D127—42no—
71C2×D127—42no—
72C2×D127—42no—
73C2×D127—84no—
74C4×C127—21no—
75C4×C127—22no—
76C4×C127—41no—
77C4×C127—41no—
78Dic127—42no—
79C2×C127—21no—
80C2×C127—81no—
81C2×C127—81no—
82C2×C127B(4,𝔽2)11yescharacteristic
83D127—82no—
84#2202—1271nomaximal, Sylow
85C127#220211yescharacteristic, complemented, Sylow
86H(3,𝔽4)—1271no—
87#454—1272no—
88#454D12711yescharacteristic, complemented
89#355—1271no—
90#355—1272no—
912−1+4C2×D12711yescharacteristic, complemented
92AGL(1,ℤ/8)—21no—
93AGL(1,ℤ/8)—2542no—
94#124—1272no—
95#124—2542no—
96#124—2541no—
97#124C2×D12711yescharacteristic
98#120—2542no—
99#120—2541no—
100#104—21no—
101#104—2542no—
102#101—1272no—
103#101C2×D12711yescharacteristic, complemented
104C24—2541no—
105#41—22no—
106#41—21no—
107#41—41no—
108C2×Q8—41no—
109C2×Q8—2541no—
110C2×Q8C22×D12711yescharacteristic
111C2×D4—1272no—
112C2×D4—2541no—
113C2×D4—2542no—
114C2×D4C22⋊D12711yescharacteristic
115SD16—41no—
116SD16—5082no—
117D8—41no—
118D8—5082no—
119M16—21no—
120M16—2542no—
121C22⋊C4—22no—
122C22⋊C4—2542no—
123C22⋊C4—2541no—
124C22⋊C4—2542no—
125C22⋊C4—2542no—
126C42—1272no—
127C42C22⋊D12711yescharacteristic
128C23—21no—
129C23—10162no—
130C23—10161no—
131C23—2542no—
132Q8—21no—
133Q8—22no—
134Q8—41no—
135Q8—10161no—
136D4—21no—
137D4—22no—
138D4—41no—
139D4—41no—
140D4—10161no—
141D4—10162no—
142D4—5082no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—2542no—
147C2×C4D4×D12712yes—
148C2×C4C2×C22⋊D12711yescharacteristic
149C8—41no—
150C8—5082no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—10164no—
155C22—5082no—
156C22—5082no—
157C22—5082no—
158C22#1686117711yescharacteristic
159C4—21no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—5082no—
164C2—21no—
165C2—81no—
166C2—81no—
167C2—10162no—
168C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting#454×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#454D127
hypocenterC127#2202
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. #22550618C2semidirect
2. #22550049C2semidirect
3. #22549673C2semidirect
4. #22549672C2semidirect
5. #454×C127C2semidirect
13. #16861128C22semidirect
14. 2−1+4×C127C22semidirect
18. #101×C127C22semidirect
85. C127#2202semidirect
88. #454D127semidirect
91. 2−1+4C2×D127semidirect
103. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506182
#225500492
#225496732
#225496722
#454×C1272
#2202127