Finite Group Databasebeta

← #30639818

Subgroups — #30639818

Order 16256.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×#14523236C212yesmaximal, complemented
2C2×#16861130C212yesmaximal, complemented
3#22549782C211yesmaximal, characteristic, complemented
4#22549631C211yesmaximal, characteristic, complemented
5C2×#127×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7C2×#14523236—24no—
8C2×#14523236—24no—
9C2×#14523236—24no—
10C22×C127⋊C8C2212yes—
11C2×C4⋊Dic127C2211yescharacteristic
12#16861130—24no—
13C127⋊#95—22no—
14C22×D4×C127—22no—
15C22×D4×C127C2212yes—
16#127×C127—24no—
17C2×C42×C127C2211yescharacteristic
18C2×Dic254—22no—
19C2×Dic254—22no—
20C2×Dic254—21no—
21C2×Dic254D412yescomplemented
22C22×Dic127—21no—
23C22×Dic127—21no—
24#14523236—216no—
25C2×C127⋊C8—22no—
26C2×C127⋊C8—22no—
27C2×C127⋊C8D414yescomplemented
28C4⋊Dic127—22no—
29C24×C127—22no—
30C2×D4×C127—28no—
31C2×D4×C127—22no—
32C2×D4×C127—22no—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127—22no—
36C2×D4×C127—44no—
37C2×D4×C127—44no—
38C2×D4×C127D414yes—
39C22×C4×C127D412yes—
40C22×C4×C127C2311yescharacteristic
41C42×C127—22no—
42Dic254—24no—
43Dic254—42no—
44C2×Dic127—22no—
45C2×Dic127—41no—
46C2×Dic127—41no—
47C2×Dic127—41no—
48C2×Dic127—42no—
49C127⋊C8—24no—
50C23×C127—24no—
51C23×C127—42no—
52C23×C127—42no—
53C23×C127—44no—
54C23×C127—44no—
55C23×C127C2×D411yescharacteristic
56D4×C127—24no—
57D4×C127—24no—
58D4×C127—416no—
59D4×C127—42no—
60D4×C127—42no—
61C2×C4×C127—22no—
62C2×C4×C127—22no—
63C2×C4×C127SD1614yescomplemented
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D412yes—
66C2×C4×C127C2×D412yes—
67C2×C4×C127#4111yescharacteristic
68C22×C127—44no—
69C22×C127—48no—
70C22×C127—84no—
71C22×C127—82no—
72C22×C127—82no—
73C22×C127#12011yescharacteristic
74C22×C127#12111yescharacteristic
75C22×C127#12711yescharacteristic
76C22×C127C2×SD1612yes—
77C22×C127AGL(1,ℤ/8)12yes—
78C4×C127—24no—
79C4×C127#12012yes—
80C4×C127#12112yes—
81Dic127—42no—
82Dic127—81no—
83C2×C127—84no—
84C2×C127#39211yescharacteristic
85C2×C127#45012yes—
86C2×C127#46012yes—
87C2×C127#49211yescharacteristic
88C2×C127#49611yescharacteristic
89#2014—1271nomaximal, Sylow
90C127#201411yescharacteristic, complemented, Sylow
91C22×SD16—1272no—
92C2×#127D12711yescharacteristic
93C2×#102—1272no—
94#391—1271no—
95#336—1271no—
96C22×Q8—1271no—
97C22×D4—22no—
98C22×D4C2×D12712yes—
99C2×SD16—2544no—
100C2×SD16—2544no—
101C2×SD16—2544no—
102C22×C8—1272no—
103#127—24no—
104C2×C4⋊C4—1271no—
105C2×C42C2×D12711yescharacteristic
106#102—2544no—
107#95—2542no—
108C24—22no—
109C2×Q8—1272no—
110C2×Q8—2542no—
111C2×Q8—2542no—
112C2×Q8—2541no—
113C2×D4—28no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4C22⋊D12714yes—
122C22×C4—2541no—
123C22×C4—2541no—
124C22×C4C22⋊D12712yes—
125C22×C4C22×D12711yescharacteristic
126SD16—25416no—
127C2×C8—1274no—
128C2×C8—2542no—
129C2×C8—2542no—
130C4⋊C4—2542no—
131C42—22no—
132C23—24no—
133C23—42no—
134C23—44no—
135C23—44no—
136C23—42no—
137C23C2×C22⋊D12711yescharacteristic
138Q8—5082no—
139Q8—2544no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144D4—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—5081no—
148C2×C4—5082no—
149C2×C4—5081no—
150C2×C4—5081no—
151C2×C4—2542no—
152C2×C4#1452323614yescomplemented
153C2×C4D4×D12711yescharacteristic
154C2×C4D4×D12712yes—
155C2×C4C2×C22⋊D12712yes—
156C2×C4#1452324711yescharacteristic
157C8—2544no—
158C22—48no—
159C22—44no—
160C22—84no—
161C22—82no—
162C22—82no—
163C22#1686117811yescharacteristic
164C22#1686118011yescharacteristic
165C22#1686120511yescharacteristic
166C22C2×#1452323612yes—
167C22#1686123512yes—
168C4—24no—
169C4—10161no—
170C4—5082no—
171C4#1686117712yes—
172C4#1686118112yes—
173C2—84no—
174C2#2254978411yescharacteristic
175C2#2255002712yes—
176C2#2255007312yes—
177C2#2255024811yescharacteristic
178C2#2255026311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×#127×C127C2
socleC23×C127C2×D4
hypercenterC2×#127D127
hypocenterC127#2014
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. C2×#16861130C2semidirect
3. #22549782C2semidirect
4. #22549631C2semidirect
21. C2×Dic254D4semidirect
27. C2×C127⋊C8D4semidirect
63. C2×C4×C127SD16semidirect
90. C127#2014semidirect
152. C2×C4#14523236semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168611302
#225497822
#225496312
C2×#127×C1272
#2014127