Finite Group Databasebeta

← #30639886

Subgroups — #30639886

Order 16256.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#22550507C211yesmaximal, characteristic, complemented
2#22549881C212yesmaximal
3#22549645C211yesmaximal, characteristic, complemented
4#22549636C212yesmaximal, complemented
5#390×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7#16861208—22no—
8#16861200—22no—
9#16861188—22no—
10#16861168C2212yes—
11#16861130—22no—
12#16861130—22no—
13C22×D4×C127C2211yescharacteristic
14C2×C4⋊C4×C127C2211yescharacteristic
15C2×C22⋊C4×C127—22no—
16C2×C42×C127C2211yescharacteristic
17C2×Dic254—22no—
18#14523232—22no—
19#14523232—22no—
20C2×C127⋊C8—22no—
21C4⋊Dic127—22no—
22C4⋊Dic127D412yes—
23C4⋊Dic127D412yes—
24Dic127⋊C4—24no—
25C22⋊Dic127—22no—
26C4×Dic127—22no—
27C24×C127—21no—
28C2×D4×C127—22no—
29C2×D4×C127—22no—
30C2×D4×C127—21no—
31C2×D4×C127Q812yescomplemented
32C22×C4×C127—21no—
33C22×C4×C127D412yes—
34C22×C4×C127C2311yescharacteristic
35C4⋊C4×C127—22no—
36C22⋊C4×C127—44no—
37C42×C127—22no—
38Dic254—82no—
39C2×Dic127—22no—
40C2×Dic127—22no—
41C127⋊C8—42no—
42C23×C127—22no—
43C23×C127—42no—
44C23×C127—42no—
45C23×C127—41no—
46C23×C127—41no—
47C23×C127C2×D411yescharacteristic
48D4×C127—24no—
49D4×C127—42no—
50C2×C4×C127—22no—
51C2×C4×C127—22no—
52C2×C4×C127—41no—
53C2×C4×C127—41no—
54C2×C4×C127—42no—
55C2×C4×C127C2×D412yes—
56C2×C4×C127C2×Q811yescharacteristic
57C2×C4×C127#4112yes—
58C2×C4×C127#4111yescharacteristic
59C22×C127—21no—
60C22×C127—21no—
61C22×C127—44no—
62C22×C127—42no—
63C22×C127—82no—
64C22×C127—81no—
65C22×C127—81no—
66C22×C127#12011yescharacteristic
67C22×C127C22⋊Q811yescharacteristic
68C22×C127#12411yescharacteristic
69C4×C127—42no—
70C4×C127—42no—
71C4×C127C22⋊Q812yes—
72C4×C127#12412yes—
73Dic127—42no—
74Dic127—42no—
75C2×C127—21no—
76C2×C127—21no—
77C2×C127—82no—
78C2×C127#39311yescharacteristic
79C2×C127#45311yescharacteristic
80C2×C127#45411yescharacteristic
81#2027—1271nomaximal, Sylow
82C127#202711yescharacteristic, complemented, Sylow
83#544—1271no—
84#418—1272no—
85#390D12711yescharacteristic
86#343—1271no—
87#339—1272no—
88C22×D4C2×D12711yescharacteristic
89C2×M16—1272no—
90C4⋊Q8—2542no—
91#125—2542no—
92C22⋊Q8—2542no—
93#117—1272no—
94C2×C4⋊C4C2×D12711yescharacteristic
95C2×C22⋊C4—22no—
96C2×C42C2×D12711yescharacteristic
97#102—2542no—
98#102—2542no—
99C24—21no—
100C2×Q8—2542no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4Dic25412yescomplemented
105C22×C4—21no—
106C22×C4C22⋊D12712yes—
107C22×C4C22×D12711yescharacteristic
108M16—2542no—
109M16—2542no—
110C2×C8—2542no—
111C4⋊C4—22no—
112C4⋊C4—1272no—
113C4⋊C4—1272no—
114C4⋊C4—2544no—
115C4⋊C4—2542no—
116C22⋊C4—44no—
117C22⋊C4—2542no—
118C42—22no—
119C42—2542no—
120C23—22no—
121C23—42no—
122C23—42no—
123C23—41no—
124C23—41no—
125C23C2×C22⋊D12711yescharacteristic
126Q8—10162no—
127D4—24no—
128D4—42no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—2542no—
135C2×C4—2542no—
136C2×C4D4×D12712yes—
137C2×C4C2×Dic25411yescharacteristic
138C2×C4#1452324712yes—
139C2×C4#1452324811yescharacteristic
140C8—5082no—
141C22—21no—
142C22—21no—
143C22—44no—
144C22—42no—
145C22—82no—
146C22—81no—
147C22—81no—
148C22#1686117711yescharacteristic
149C22#1686118811yescharacteristic
150C22#1686119611yescharacteristic
151C4—42no—
152C4—42no—
153C4—5082no—
154C4—5082no—
155C4#1686118612yes—
156C4#1686119512yes—
157C2—21no—
158C2—21no—
159C2—82no—
160C2#2254978811yescharacteristic
161C2#2255004311yescharacteristic
162C2#2255004811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC22×C127#120
prime radicalC22×D4×C127C22
hypercenter#390D127
hypocenterC127#2027
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #22550507C2semidirect
3. #22549645C2semidirect
4. #22549636C2semidirect
31. C2×D4×C127Q8semidirect
82. C127#2027semidirect
104. C2×D4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225498812
#225496452
#225496362
#390×C1272
#2027127