Finite Group Databasebeta

← #30639257

Subgroups — #30639257

Order 16256.

344 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550440C211yesmaximal, characteristic
2#22549882C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic, complemented
4#22549633C212yesmaximal, complemented
5#568×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7#16861205C412yescomplemented
8#16861198C412yescomplemented
9#16861183—22no—
10#16861177—22no—
11#16861168C2212yes—
12#16861168C2211yescharacteristic
13#16861131—24no—
14#16861124—22no—
15C2×#41×C127C2211yescharacteristic
16#131×C127—24no—
17C2×M16×C127—22no—
18C2×M16×C127C2212yes—
19C23×D127—21no—
20C2×D508—22no—
21C2×D508—21no—
22C2×D508—22no—
23C2×D508C2×C412yes—
24C2×C22⋊D127—24no—
25C4⋊Dic127—21no—
26C4⋊Dic127—22no—
27C22⋊Dic127—21no—
28C22⋊Dic127—22no—
29(C2×D127)⋊C4—24no—
30C4×Dic127—22no—
31C4×Dic127C2×C412yes—
32#41×C127—22no—
33#41×C127—22no—
34C2×Q8×C127C2×C411yescharacteristic
35C2×D4×C127C2×C411yescharacteristic
36C2×D4×C127D412yes—
37C22×C4×C127D412yes—
38C22×C4×C127C2311yescharacteristic
39M16×C127—24no—
40M16×C127—22no—
41M16×C127—22no—
42C2×C8×C127—24no—
43C2×C8×C127D414yescomplemented
44C22×D127—22no—
45C22×D127—42no—
46C22×D127—82no—
47D508—24no—
48D508—41no—
49D508—41no—
50C22⋊D127—84no—
51C2×Dic127—22no—
52C2×Dic127—42no—
53C23×C127C22⋊C412yes—
54C23×C127C2×D411yescharacteristic
55Q8×C127—22no—
56D4×C127—24no—
57D4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127C22⋊C412yes—
61C2×C4×C127C22×C411yescharacteristic
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127#4112yes—
65C8×C127—24no—
66C22×C127—22no—
67C22×C127—41no—
68C22×C127—42no—
69C22×C127C2×C22⋊C411yescharacteristic
70C22×C127#12112yes—
71C2×D127—44no—
72C2×D127—42no—
73C2×D127—84no—
74C4×C127—22no—
75C4×C127C4×D412yes—
76C4×C127#12011yescharacteristic
77C4×C127#12311yescharacteristic
78Dic127—42no—
79Dic127—82no—
80C2×C127—21no—
81C2×C127—22no—
82C2×C127—42no—
83C2×C127#38611yescharacteristic
84D127—82no—
85#1910—1271nomaximal, Sylow
86C127#191011yescharacteristic, complemented, Sylow
87#568D12711yescharacteristic, complemented
88#535—1271no—
89#418—1272no—
90#410—1271no—
91H(3,ℤ/4)—1272no—
92C2×#41C2×D12711yescharacteristic
93C22×D4—1271no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D12712yes—
97#127—1272no—
98#124—1272no—
99#121—2542no—
100#120—2542no—
101#117—1271no—
102#117—1272no—
103#102—2544no—
104#99—2542no—
105C24—2541no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D12711yescharacteristic
109C2×D4—1272no—
110C2×D4—2544no—
111C2×D4—2541no—
112C2×D4—2542no—
113C2×D4—2542no—
114C2×D4C4×D12711yescharacteristic
115C2×D4D50812yes—
116C22×C4C22⋊D12712yes—
117C22×C4C22×D12711yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D12714yescomplemented
123C4⋊C4—2542no—
124C4⋊C4—2541no—
125C22⋊C4—2542no—
126C22⋊C4—2541no—
127C22⋊C4—2544no—
128C42—1272no—
129C42—2542no—
130C23—10162no—
131C23—5082no—
132C23—2542no—
133C23(C2×D127)⋊C412yes—
134C23C2×D50811yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—10164no—
139D4—5081no—
140D4—5081no—
141D4—2544no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—5082no—
145C2×C4—2542no—
146C2×C4(C2×D127)⋊C412yes—
147C2×C4C2×C4×D12711yescharacteristic
148C2×C4C2×C22⋊D12711yescharacteristic
149C2×C4C2×C22⋊D12712yes—
150C2×C4#1452324812yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—10164no—
156C22—5084no—
157C22—5082no—
158C22C2×(C2×D127)⋊C411yescharacteristic
159C22#1686118312yes—
160C4—22no—
161C4—10162no—
162C4—5082no—
163C4C4×C22⋊D12712yes—
164C4#1686117811yescharacteristic
165C4#1686119211yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—10162no—
170C2#2254975511yescharacteristic
subgroupG/H
centerC2#22549755
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#386
prime radical#22550440C2
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1910
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22549882C2semidirect
3. #22549850C2semidirect
4. #22549633C2semidirect
5. #568×C127C2semidirect
7. #16861205C4semidirect
8. #16861198C4semidirect
43. C2×C8×C127D4semidirect
86. C127#1910semidirect
87. #568D127semidirect
122. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
#225498822
#225498502
#225496332
#568×C1272
#1910127