Finite Group Databasebeta

← #30639278

Subgroups — #30639278

Order 16256.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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#22549860C211yesmaximal, characteristic, complemented
3C2×#16861126C212yesmaximal, complemented
4#22549646C212yesmaximal, complemented
5#535×C127C211yesmaximal, characteristic
6#16861216—24no—
7C2×#14523232—22no—
8C2×#14523232C2212yes—
9C2×#14523232C2212yes—
10#16861128—21no—
11#16861126—24no—
12#16861126—21no—
13C2×#41×C127C2211yescharacteristic
14C22×D4×C127C2211yescharacteristic
15#127×C127—21no—
16#124×C127—21no—
17#121×C127—22no—
18#120×C127C414yes—
19#117×C127C2211yescharacteristic
20#14523232—24no—
21#14523232—22no—
22#14523232—22no—
23#14523232—22no—
24#14523232D414yescomplemented
25C2×C127⋊C8—24no—
26C2×C127⋊C8—22no—
27C2×C127⋊C8—22no—
28C24×C127C2×C412yes—
29#41×C127—22no—
30#41×C127—22no—
31C2×Q8×C127D411yescharacteristic
32C2×D4×C127—24no—
33C2×D4×C127—21no—
34C2×D4×C127—21no—
35C2×D4×C127—22no—
36C2×D4×C127—22no—
37C2×D4×C127C2×C412yes—
38C2×D4×C127D411yescharacteristic
39C22×C4×C127D412yes—
40C22×C4×C127C2311yescharacteristic
41C4⋊C4×C127—21no—
42C22⋊C4×C127—24no—
43C22⋊C4×C127C2×C412yes—
44C42×C127—21no—
45C127⋊C8—24no—
46C127⋊C8—42no—
47C23×C127—22no—
48C23×C127—24no—
49C23×C127—82no—
50C23×C127C22⋊C412yes—
51C23×C127C22×C411yescharacteristic
52Q8×C127—22no—
53D4×C127—24no—
54D4×C127—22no—
55D4×C127—41no—
56D4×C127—42no—
57D4×C127—41no—
58D4×C127—84no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127C22⋊C412yes—
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127#4112yes—
67C22×C127—22no—
68C22×C127—41no—
69C22×C127—42no—
70C22×C127—44no—
71C22×C127—42no—
72C22×C127—84no—
73C22×C127C2×C22⋊C411yescharacteristic
74C22×C127C4×D412yes—
75C4×C127—22no—
76C4×C127—42no—
77C4×C127#12011yescharacteristic
78C4×C127#12112yes—
79C4×C127#12311yescharacteristic
80C2×C127—21no—
81C2×C127—22no—
82C2×C127—42no—
83C2×C127—82no—
84C2×C127#38611yescharacteristic
85#1914—1271nomaximal, Sylow
86C127#191411yescharacteristic, complemented, Sylow
87#568—1271no—
88#535D12711yescharacteristic
89#413—1271no—
90C2×#100—1272no—
91#344—1272no—
92C2×#41C2×D12711yescharacteristic
93C22×D4C2×D12711yescharacteristic
94#131—2544no—
95C2×M16—1272no—
96C2×M16—1272no—
97C2×M16—2542no—
98#127—21no—
99#124—21no—
100#121—22no—
101#120Dic12714yes—
102#117C2×D12711yescharacteristic
103#101—2541no—
104#100—2544no—
105#100—2541no—
106C24C2×Dic12712yes—
107#41—22no—
108#41—22no—
109C2×Q8C22⋊D12711yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4C2×Dic12712yes—
116C2×D4C22⋊D12711yescharacteristic
117C22×C4C22⋊D12712yes—
118C22×C4C22×D12711yescharacteristic
119M16—1274no—
120M16—2542no—
121M16—2542no—
122M16—2544no—
123M16—2542no—
124C2×C8—2542no—
125C2×C8—2544no—
126C2×C8—2542no—
127C4⋊C4—21no—
128C22⋊C4—24no—
129C22⋊C4C2×Dic12712yes—
130C42—21no—
131C23—22no—
132C23—24no—
133C23—82no—
134C23C22⋊Dic12712yes—
135C23C22×Dic12711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—41no—
140D4—41no—
141D4—42no—
142D4—84no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4C22⋊Dic12712yes—
147C2×C4D4×D12712yes—
148C2×C4C2×C22⋊D12711yescharacteristic
149C2×C4C2×C22⋊D12711yescharacteristic
150C2×C4#1452324712yes—
151C8—5082no—
152C8—2544no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—44no—
157C22—42no—
158C22—84no—
159C22C2×C22⋊Dic12711yescharacteristic
160C22D4×Dic12712yes—
161C4—22no—
162C4—42no—
163C4#1686117711yescharacteristic
164C4#1686118112yes—
165C4#1686119011yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2254975611yescharacteristic
subgroupG/H
centerC2#22549756
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#535×C127C2
socleC2×C127#386
cosocleC22×C4×C127C23
hypercenter#535D127
hypocenterC127#1914
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22550643C2semidirect
2. #22549860C2semidirect
3. C2×#16861126C2semidirect
4. #22549646C2semidirect
24. #14523232D4semidirect
86. C127#1914semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225498602
C2×#168611262
#225496462
#535×C1272
#1914127