Finite Group Databasebeta

← #30640748

Subgroups — #30640748

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#22550051C212yesmaximal, complemented
3#22550048C211yesmaximal, characteristic, complemented
4#22549672C212yesmaximal, complemented
5#355×C127C211yesmaximal, characteristic, complemented
6#16861257C2211yescharacteristic, complemented
7#16861237—22no—
8#16861235—21no—
9#16861198C2212yes—
10#16861197—21no—
11#16861195—22no—
12#16861179—22no—
13#16861177—21no—
14#16861137—21no—
15#16861135—22no—
16#16861128C2211yescharacteristic, complemented
17#124×C127C2211yescharacteristic
18#101×C127C2212yescomplemented
19C23×D127—21no—
20#14523249—41no—
21#14523248—21no—
22#14523248—22no—
23C2×Dic254—21no—
24Q8×D127—41no—
25C2×D508—21no—
26C2×D508D412yes—
27C2×C22⋊D127—22no—
28#14523237—42no—
29#14523236—41no—
30D1016—42no—
31#14523233—41no—
32#14523232—21no—
33C22⋊Dic127—21no—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—22no—
37C4×Dic127D412yes—
38C2×Q8×C127C2311yescharacteristic
39C2×D4×C127D411yescharacteristic
40M16×C127—22no—
41C22⋊C4×C127—22no—
42C42×C127D411yescharacteristic
43C22×D127—22no—
44C22×D127—82no—
45C22×D127—81no—
46Dic254—21no—
47Dic254—22no—
48Dic254—81no—
49D508—21no—
50D508—22no—
51D508—42no—
52D508—81no—
53C22⋊D127—41no—
54C22⋊D127—82no—
55C2×Dic127—22no—
56C4×D127—42no—
57C4×D127—41no—
58C127⋊C8—41no—
59C23×C127—21no—
60Q8×C127—41no—
61D4×C127—41no—
62C2×C4×C127—21no—
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D411yescharacteristic
65C8×C127—42no—
66C22×C127—42no—
67C22×C127—41no—
68C22×C127#12011yescharacteristic
69C2×D127—42no—
70C2×D127—42no—
71C2×D127—41no—
72C2×D127—42no—
73C2×D127—84no—
74C4×C127—22no—
75C4×C127—21no—
76C4×C127—41no—
77Dic127—42no—
78Dic127—41no—
79C2×C127—21no—
80C2×C127—81no—
81C2×C127B(4,𝔽2)11yescharacteristic
82D127—81no—
83D127—82no—
84#2202—1271nomaximal, Sylow
85C127#220211yescharacteristic, complemented, Sylow
86H(3,𝔽4)—1271no—
87#454—1271no—
88#454—1272no—
89#355—1272no—
90#355D12711yescharacteristic, complemented
912−1+4—1271no—
92AGL(1,ℤ/8)—2542no—
93AGL(1,ℤ/8)—2541no—
94#124—1272no—
95#124—2542no—
96#124—2541no—
97#124C2×D12711yescharacteristic
98#120—2542no—
99#120—2541no—
100#104—2542no—
101#104—2541no—
102#101—1271no—
103#101C2×D12712yescomplemented
104C24—2541no—
105#41—5081no—
106#41—2542no—
107#41—2541no—
108C2×Q8—5081no—
109C2×Q8—2541no—
110C2×Q8C22×D12711yescharacteristic
111C2×D4—1272no—
112C2×D4—2541no—
113C2×D4—2542no—
114C2×D4D50811yescharacteristic
115SD16—5082no—
116SD16—5081no—
117D8—5082no—
118D8—5081no—
119M16—22no—
120M16—2541no—
121C22⋊C4—22no—
122C22⋊C4—2542no—
123C22⋊C4—2541no—
124C22⋊C4—2542no—
125C22⋊C4—2542no—
126C42—1272no—
127C42D50811yescharacteristic
128C23—21no—
129C23—10162no—
130C23—10161no—
131C23—2542no—
132Q8—41no—
133Q8—10161no—
134Q8—2542no—
135Q8—2541no—
136D4—41no—
137D4—10161no—
138D4—10162no—
139D4—5082no—
140D4—5081no—
141D4—2542no—
142D4—2541no—
143C2×C4—21no—
144C2×C4—5082no—
145C2×C4—5081no—
146C2×C4—2542no—
147C2×C4D4×D12712yes—
148C2×C4C2×D50811yescharacteristic
149C8—42no—
150C8—5081no—
151C22—42no—
152C22—41no—
153C22—10164no—
154C22—5082no—
155C22—5082no—
156C22—5081no—
157C22—5082no—
158C22#1686117911yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—5082no—
163C4—5081no—
164C2—21no—
165C2—81no—
166C2—10162no—
167C2—10161no—
168C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting#355×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#355D127
hypocenterC127#2202
LCSG ▷ C2×Q8×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #22550618C2semidirect
2. #22550051C2semidirect
3. #22550048C2semidirect
4. #22549672C2semidirect
5. #355×C127C2semidirect
6. #16861257C22semidirect
16. #16861128C22semidirect
18. #101×C127C22semidirect
85. C127#2202semidirect
90. #355D127semidirect
103. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506182
#225500512
#225500482
#225496722
#355×C1272
#2202127