Finite Group Databasebeta

← #30639255

Subgroups — #30639255

Order 16256.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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#22550643C211yesmaximal, characteristic, complemented
2#22550440C211yesmaximal, characteristic
3#22549880C212yesmaximal, complemented
4#22549632C212yesmaximal, complemented
5#410×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7#16861216—24no—
8C2×#14523232—22no—
9C2×#14523232C2212yes—
10#16861205C412yescomplemented
11#16861198C412yescomplemented
12#16861183—22no—
13#16861177—22no—
14#16861168C2211yescharacteristic
15#16861129—24no—
16C2×#41×C127C2211yescharacteristic
17#117×C127C2212yes—
18#99×C127—22no—
19C23×D127—21no—
20C2×D508—22no—
21C2×D508—21no—
22C2×D508—22no—
23C2×D508C2×C412yes—
24C2×C22⋊D127—24no—
25#14523232—24no—
26#14523232—22no—
27#14523232—22no—
28C2×C127⋊C8—24no—
29C2×C127⋊C8D414yescomplemented
30C4⋊Dic127—21no—
31C22⋊Dic127—21no—
32(C2×D127)⋊C4—24no—
33C4×Dic127C2×C412yes—
34#41×C127—22no—
35#41×C127—22no—
36C2×Q8×C127C2×C411yescharacteristic
37C2×D4×C127C2×C411yescharacteristic
38C2×D4×C127D412yes—
39C22×C4×C127D412yes—
40C22×C4×C127C2311yescharacteristic
41C4⋊C4×C127—22no—
42C22⋊C4×C127—22no—
43C42×C127—22no—
44C22×D127—22no—
45C22×D127—42no—
46C22×D127—82no—
47D508—24no—
48D508—41no—
49D508—41no—
50C22⋊D127—84no—
51C2×Dic127—22no—
52C127⋊C8—24no—
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×C127—42no—
61C2×C4×C127C22⋊C412yes—
62C2×C4×C127C22×C411yescharacteristic
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127#4112yes—
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×C127—82no—
76C4×C127C4×D412yes—
77C4×C127#12011yescharacteristic
78C4×C127#12311yescharacteristic
79Dic127—42no—
80C2×C127—21no—
81C2×C127—22no—
82C2×C127—42no—
83C2×C127#38611yescharacteristic
84D127—82no—
85#1910—1271nomaximal, Sylow
86C127#191011yescharacteristic, complemented, Sylow
87#568—1271no—
88#535—1271no—
89#418—1272no—
90#410D12711yescharacteristic, complemented
91H(3,ℤ/4)—1272no—
92C2×#41C2×D12711yescharacteristic
93C22×D4—1271no—
94#131—2544no—
95C2×M16—1272no—
96C2×M16—2542no—
97#127—1272no—
98#124—1272no—
99#121—2542no—
100#120—2542no—
101#117—1271no—
102#117C2×D12712yes—
103#102—2544no—
104#99—22no—
105C24—2541no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D12711yescharacteristic
109C2×D4—1272no—
110C2×D4—2544no—
111C2×D4—2542no—
112C2×D4—2541no—
113C2×D4—2542no—
114C2×D4C4×D12711yescharacteristic
115C2×D4C22⋊D12712yes—
116C22×C4D50812yes—
117C22×C4C22×D12711yescharacteristic
118M16—2544no—
119M16—2542no—
120M16—2542no—
121C2×C8—1274no—
122C2×C8—2544no—
123C4⋊C4—22no—
124C4⋊C4—2541no—
125C22⋊C4—22no—
126C22⋊C4—2541no—
127C22⋊C4—2544no—
128C42—22no—
129C42—1272no—
130C23—10162no—
131C23—5082no—
132C23—2542no—
133C23(C2×D127)⋊C412yes—
134C23C2×C22⋊D12711yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—10164no—
139D4—5081no—
140D4—5081no—
141D4—2544no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—2542no—
146C2×C4(C2×D127)⋊C412yes—
147C2×C4C2×C4×D12711yescharacteristic
148C2×C4D4×D12712yes—
149C2×C4C2×D50811yescharacteristic
150C2×C4#1452324712yes—
151C8—2544no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—10164no—
156C22—5084no—
157C22—5082no—
158C22C2×(C2×D127)⋊C411yescharacteristic
159C22#1686118112yes—
160C4—22no—
161C4—82no—
162C4—5082no—
163C4(C22⋊D127)⋊C412yes—
164C4#1686117911yescharacteristic
165C4#1686119311yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—10162no—
170C2#2254975311yescharacteristic
subgroupG/H
centerC2#22549753
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#386
prime radical#22550440C2
cosocleC22×C4×C127C23
hypercenter#410D127
hypocenterC127#1910
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #22550643C2semidirect
3. #22549880C2semidirect
4. #22549632C2semidirect
5. #410×C127C2semidirect
10. #16861205C4semidirect
11. #16861198C4semidirect
29. C2×C127⋊C8D4semidirect
86. C127#1910semidirect
90. #410D127semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225504402
#225498802
#225496322
#410×C1272
#1910127