Finite Group Databasebeta

← #30639262

Subgroups — #30639262

Order 16256.

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

normal subgroups only

This group has 362 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#22550443C211yesmaximal, characteristic
2#22549894C212yesmaximal, complemented
3C2×#16861126C211yesmaximal, characteristic, complemented
4#22549632C212yesmaximal, complemented
5#518×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2211yescharacteristic
7C2×#14523232C2212yes—
8#16861205C412yescomplemented
9#16861196C412yescomplemented
10#16861180—22no—
11#16861178—22no—
12#16861168C2211yescharacteristic
13#16861137—24no—
14#16861126—22no—
15C22×D4×C127C2211yescharacteristic
16C4×D4×C127—24no—
17#117×C127C2212yes—
18C2×C22⋊C4×C127—22no—
19#14523248—22no—
20#14523248—22no—
21C2×Dic254C2×C411yescharacteristic, complemented
22C2×D508C2×C411yescharacteristic, complemented
23C2×C22⋊D127—24no—
24C2×C22⋊D127—21no—
25C2×C4×D127—21no—
26#14523232—22no—
27#14523232—22no—
28C2×C127⋊C8—22no—
29C4⋊Dic127—21no—
30C22⋊Dic127—24no—
31C22⋊Dic127—21no—
32C4×Dic127C2×C412yescomplemented
33C24×C127D412yes—
34C2×D4×C127—22no—
35C2×D4×C127—22no—
36C2×D4×C127C2×C412yes—
37C2×D4×C127D412yes—
38C22×C4×C127—24no—
39C22×C4×C127C2311yescharacteristic
40C4⋊C4×C127—22no—
41C22⋊C4×C127—24no—
42C22⋊C4×C127—22no—
43C42×C127D414yescomplemented
44C22×D127—21no—
45Dic254—22no—
46D508—22no—
47C22⋊D127—41no—
48C22⋊D127—41no—
49C22⋊D127—84no—
50C2×Dic127—22no—
51C2×Dic127—21no—
52C4×D127—42no—
53C127⋊C8—42no—
54C23×C127—22no—
55C23×C127—22no—
56C23×C127—82no—
57C23×C127C22⋊C414yes—
58C23×C127C2×D411yescharacteristic
59D4×C127—24no—
60D4×C127—24no—
61C2×C4×C127—24no—
62C2×C4×C127—84no—
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—82no—
72C22×C127C2×C22⋊C411yescharacteristic
73C22×C127#12011yescharacteristic
74C22×C127#12311yescharacteristic
75C2×D127—42no—
76C2×D127—41no—
77C4×C127—44no—
78C4×C127C4×D412yes—
79C4×C127#12112yes—
80Dic127—42no—
81Dic127—41no—
82C2×C127—21no—
83C2×C127—21no—
84C2×C127—21no—
85C2×C127—44no—
86C2×C127#38611yescharacteristic
87D127—81no—
88#1911—1271nomaximal, Sylow
89C127#191111yescharacteristic, complemented, Sylow
90#535—1271no—
91#518D12711yescharacteristic, complemented
92#421—1272no—
93C2×#100—1271no—
94H(3,ℤ/4)—1272no—
95C2×#41—1271no—
96C22×D4C2×D12711yescharacteristic
97C2×M16—1272no—
98#127—1272no—
99#124—1272no—
100#121—2542no—
101#120—2542no—
102C4×D4—24no—
103#117—1271no—
104#117C2×D12712yes—
105C2×C22⋊C4—22no—
106#104—2544no—
107#100—2542no—
108C24C22⋊D12712yes—
109#41—2542no—
110#41—2542no—
111C2×Q8—1271no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—1271no—
115C2×D4—2544no—
116C2×D4—2541no—
117C2×D4C4×D12712yes—
118C2×D4D50812yes—
119C22×C4—24no—
120C22×C4—2541no—
121C22×C4C22×D12711yescharacteristic
122M16—2542no—
123M16—2542no—
124C2×C8—2542no—
125C4⋊C4—22no—
126C4⋊C4—2541no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—2544no—
130C22⋊C4—2541no—
131C42—1272no—
132C42C22⋊D12714yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—2541no—
137C23(C2×D127)⋊C414yes—
138C23C2×C22⋊D12711yescharacteristic
139Q8—2542no—
140D4—24no—
141D4—24no—
142D4—10164no—
143D4—5081no—
144D4—5081no—
145D4—2542no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—5082no—
149C2×C4—2542no—
150C2×C4—2541no—
151C2×C4C2×C4×D12711yescharacteristic
152C2×C4C2×C22⋊D12712yes—
153C2×C4C2×D50811yescharacteristic
154C2×C4#1452324812yes—
155C8—5082no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—5082no—
162C22—5081no—
163C22C2×(C2×D127)⋊C411yescharacteristic
164C22#1686117811yescharacteristic
165C22#1686119211yescharacteristic
166C4—44no—
167C4—5082no—
168C4—5081no—
169C4C4×C22⋊D12712yes—
170C4#1686118312yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—10161no—
176C2#2254975511yescharacteristic
subgroupG/H
centerC2#22549755
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#518×C127C2
socleC2×C127#386
prime radical#22550443C2
cosocleC22×C4×C127C23
hypercenter#518D127
hypocenterC127#1911
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #22549894C2semidirect
3. C2×#16861126C2semidirect
4. #22549632C2semidirect
5. #518×C127C2semidirect
8. #16861205C4semidirect
9. #16861196C4semidirect
21. C2×Dic254C2×C4semidirect
22. C2×D508C2×C4semidirect
32. C4×Dic127C2×C4semidirect
43. C42×C127D4semidirect
89. C127#1911semidirect
91. #518D127semidirect
132. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504432
#225498942
C2×#168611262
#225496322
#518×C1272
#1911127