Finite Group Databasebeta

← #30639263

Subgroups — #30639263

Order 16256.

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

normal subgroups only

This group has 362 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#22550351C211yesmaximal, characteristic, complemented
2#22549893C212yesmaximal, complemented
3C2×#16861126C211yesmaximal, characteristic, complemented
4#22549632C212yesmaximal, complemented
5#535×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7D4×Dic127—24no—
8#16861168C2212yes—
9C2×C22⋊Dic127—22no—
10#16861136—24no—
11#16861126—22no—
12C2×#41×C127C2211yescharacteristic
13C22×D4×C127C2211yescharacteristic
14#127×C127C412yescomplemented
15#124×C127C412yescomplemented
16#121×C127—22no—
17#120×C127—22no—
18#117×C127C2211yescharacteristic
19C22×Dic127—24no—
20#14523232—22no—
21#14523232—22no—
22C2×C127⋊C8—22no—
23C4⋊Dic127—22no—
24C22⋊Dic127—24no—
25C22⋊Dic127—22no—
26C4×Dic127D414yescomplemented
27C24×C127D412yes—
28#41×C127—22no—
29#41×C127—22no—
30C2×Q8×C127C2×C411yescharacteristic, complemented
31C2×D4×C127—21no—
32C2×D4×C127—24no—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127C2×C412yes—
36C2×D4×C127C2×C411yescharacteristic, complemented
37C2×D4×C127D412yes—
38C22×C4×C127—21no—
39C22×C4×C127C2311yescharacteristic
40C4⋊C4×C127—21no—
41C22⋊C4×C127—24no—
42C22⋊C4×C127—21no—
43C42×C127C2×C412yescomplemented
44C2×Dic127—24no—
45C2×Dic127—84no—
46C127⋊C8—42no—
47C23×C127—22no—
48C23×C127—22no—
49C23×C127—21no—
50C23×C127—82no—
51C23×C127C22⋊C414yes—
52C23×C127C2×D411yescharacteristic
53Q8×C127—22no—
54D4×C127—24no—
55D4×C127—22no—
56D4×C127—24no—
57D4×C127—41no—
58D4×C127—41no—
59D4×C127—84no—
60C2×C4×C127—22no—
61C2×C4×C127—21no—
62C2×C4×C127—42no—
63C2×C4×C127C22×C411yescharacteristic
64C2×C4×C127C2×D412yes—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127#4112yes—
67C22×C127—24no—
68C22×C127—41no—
69C22×C127—44no—
70C22×C127—44no—
71C22×C127—42no—
72C22×C127—41no—
73C22×C127—82no—
74C22×C127C2×C22⋊C411yescharacteristic
75C22×C127#12011yescharacteristic
76C22×C127#12311yescharacteristic
77C4×C127—42no—
78C4×C127—41no—
79C4×C127C4×D412yes—
80C4×C127#12112yes—
81Dic127—44no—
82C2×C127—21no—
83C2×C127—21no—
84C2×C127—21no—
85C2×C127—44no—
86C2×C127—81no—
87C2×C127#38611yescharacteristic
88#1911—1271nomaximal, Sylow
89C127#191111yescharacteristic, complemented, Sylow
90#535D12711yescharacteristic
91#518—1271no—
92#421—1272no—
93C2×#100—1271no—
94H(3,ℤ/4)—1272no—
95C2×#41C2×D12711yescharacteristic
96C22×D4C2×D12711yescharacteristic
97C2×M16—1272no—
98#127Dic12712yescomplemented
99#124Dic12712yescomplemented
100#121—22no—
101#120—22no—
102C4×D4—2544no—
103#117—1272no—
104#117C2×D12711yescharacteristic
105C2×C22⋊C4—2542no—
106#104—2544no—
107#100—2542no—
108C24C22⋊D12712yes—
109#41—22no—
110#41—22no—
111C2×Q8C2×Dic12711yescharacteristic, complemented
112C2×D4—21no—
113C2×D4—24no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4C2×Dic12712yes—
117C2×D4C2×Dic12711yescharacteristic, complemented
118C2×D4C22⋊D12712yes—
119C22×C4—21no—
120C22×C4—2544no—
121C22×C4C22×D12711yescharacteristic
122M16—2542no—
123M16—2542no—
124C2×C8—2542no—
125C4⋊C4—21no—
126C4⋊C4—2542no—
127C22⋊C4—24no—
128C22⋊C4—21no—
129C22⋊C4—2544no—
130C22⋊C4—2542no—
131C42—1274no—
132C42C2×Dic12712yescomplemented
133C23—22no—
134C23—22no—
135C23—21no—
136C23—82no—
137C23C22⋊Dic12714yes—
138C23C2×C22⋊D12711yescharacteristic
139Q8—22no—
140D4—24no—
141D4—22no—
142D4—24no—
143D4—41no—
144D4—41no—
145D4—84no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—42no—
149C2×C4—10164no—
150C2×C4—2544no—
151C2×C4C22×Dic12711yescharacteristic
152C2×C4D4×D12712yes—
153C2×C4C2×C22⋊D12711yescharacteristic
154C2×C4#1452324712yes—
155C8—5082no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—42no—
161C22—41no—
162C22—82no—
163C22C2×C22⋊Dic12711yescharacteristic
164C22#1686117711yescharacteristic
165C22#1686119011yescharacteristic
166C4—42no—
167C4—41no—
168C4—5084no—
169C4D4×Dic12712yes—
170C4#1686118112yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—81no—
176C2#2254975611yescharacteristic
subgroupG/H
centerC2#22549756
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#535×C127C2
socleC2×C127#386
cosocleC22×C4×C127C23
hypercenter#535D127
hypocenterC127#1911
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22550351C2semidirect
2. #22549893C2semidirect
3. C2×#16861126C2semidirect
4. #22549632C2semidirect
14. #127×C127C4semidirect
15. #124×C127C4semidirect
26. C4×Dic127D4semidirect
30. C2×Q8×C127C2×C4semidirect
36. C2×D4×C127C2×C4semidirect
43. C42×C127C2×C4semidirect
89. C127#1911semidirect
98. #127Dic127semidirect
99. #124Dic127semidirect
111. C2×Q8C2×Dic127semidirect
117. C2×D4C2×Dic127semidirect
132. C42C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503512
#225498932
C2×#168611262
#225496322
#535×C1272
#1911127