Finite Group Databasebeta

← #30638940

Subgroups — #30638940

Order 16256.

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

normal subgroups only

This group has 324 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#22549919C212yesmaximal, complemented
2C2×#16861126C211yesmaximal, characteristic
3#22549779C211yesmaximal, characteristic
4#22549635C212yesmaximal, complemented
5#518×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×C4⋊Dic127C2211yescharacteristic
8C2×C22⋊Dic127—22no—
9C2×C4×Dic127C2211yescharacteristic
10#16861142—22no—
11#16861140—22no—
12#16861126C414yescomplemented
13C22×D4×C127C2211yescharacteristic
14C4×D4×C127—22no—
15C4×D4×C127—22no—
16#117×C127C2212yes—
17C2×C22⋊C4×C127—22no—
18C22×Dic127—21no—
19C22×Dic127—21no—
20#14523232—22no—
21#14523232C2×C414yescomplemented
22C2×C127⋊C8—22no—
23C4⋊Dic127—22no—
24C22⋊Dic127—44no—
25C4×Dic127—22no—
26C24×C127D412yes—
27C2×D4×C127—22no—
28C2×D4×C127—22no—
29C2×D4×C127C2×C412yes—
30C2×D4×C127Q812yes—
31C22×C4×C127—22no—
32C22×C4×C127—22no—
33C22×C4×C127C2311yescharacteristic
34C4⋊C4×C127D412yes—
35C4⋊C4×C127D412yes—
36C22⋊C4×C127—24no—
37C22⋊C4×C127—22no—
38C42×C127—22no—
39C2×Dic127—22no—
40C2×Dic127—41no—
41C2×Dic127—41no—
42C2×Dic127—42no—
43C2×Dic127—41no—
44C127⋊C8—24no—
45C23×C127—22no—
46C23×C127—22no—
47C23×C127—42no—
48C23×C127—42no—
49C23×C127C4⋊C414yes—
50C23×C127C2×D411yescharacteristic
51D4×C127—24no—
52D4×C127—24no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—82no—
56C2×C4×C127—82no—
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D412yes—
59C2×C4×C127C2×Q811yescharacteristic
60C2×C4×C127#4112yes—
61C22×C127—21no—
62C22×C127—21no—
63C22×C127—24no—
64C22×C127—44no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127—82no—
68C22×C127C2×C4⋊C411yescharacteristic
69C22×C127#12011yescharacteristic
70C22×C127#12311yescharacteristic
71C4×C127—42no—
72C4×C127—42no—
73C4×C127C4×D412yes—
74C4×C127C22⋊Q812yes—
75Dic127—42no—
76Dic127—81no—
77C2×C127—21no—
78C2×C127—21no—
79C2×C127—44no—
80C2×C127#38511yescharacteristic
81C2×C127#45311yescharacteristic
82C2×C127#45511yescharacteristic
83#1856—1271nomaximal, Sylow
84C127#185611yescharacteristic, complemented, Sylow
85#518D12711yescharacteristic
86#428—1272no—
87C2×#100—1271no—
88#390—1271no—
89#339—1272no—
90C22×D4C2×D12711yescharacteristic
91C2×M16—1272no—
92C4×D4—22no—
93C4×D4—22no—
94#117C2×D12712yes—
95C2×C4⋊C4—1271no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—2542no—
98C2×C42—1271no—
99#107—2542no—
100#106—2542no—
101#100—1274no—
102C24C22⋊D12712yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4C4×D12712yes—
106C2×D4Dic25412yes—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—2541no—
110C22×C4—2541no—
111C22×C4C22×D12711yescharacteristic
112M16—1274no—
113M16—2542no—
114C2×C8—2542no—
115C4⋊C4—2542no—
116C4⋊C4C22⋊D12712yes—
117C4⋊C4C22⋊D12712yes—
118C22⋊C4—24no—
119C22⋊C4—22no—
120C22⋊C4—5084no—
121C42—22no—
122C42—2542no—
123C23—22no—
124C23—22no—
125C23—42no—
126C23—42no—
127C23Dic127⋊C414yes—
128C23C2×C22⋊D12711yescharacteristic
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—5082no—
136C2×C4—5081no—
137C2×C4—5081no—
138C2×C4—5081no—
139C2×C4—2542no—
140C2×C4C2×C4×D12711yescharacteristic
141C2×C4C2×C22⋊D12712yes—
142C2×C4C2×Dic25411yescharacteristic
143C2×C4#1452324812yes—
144C8—2544no—
145C22—21no—
146C22—21no—
147C22—24no—
148C22—44no—
149C22—42no—
150C22—42no—
151C22—82no—
152C22C2×Dic127⋊C411yescharacteristic
153C22#1686117811yescharacteristic
154C22#1686119211yescharacteristic
155C4—42no—
156C4—42no—
157C4—10161no—
158C4—5082no—
159C4C4×C22⋊D12712yes—
160C4#1686118812yes—
161C2—21no—
162C2—21no—
163C2—44no—
164C2#2254974911yescharacteristic
165C2#2255004411yescharacteristic
166C2#2255005411yescharacteristic
subgroupG/H
centerC22#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#518×C127C2
socleC22×C127#120
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenter#518D127
hypocenterC127#1856
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #22549919C2semidirect
4. #22549635C2semidirect
12. #16861126C4semidirect
21. #14523232C2×C4semidirect
84. C127#1856semidirect

Maximal subgroups

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

subgroupindex
#225499192
C2×#168611262
#225497792
#225496352
#518×C1272
#1856127