Finite Group Databasebeta

← #30639898

Subgroups — #30639898

Order 16256.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550502C211yesmaximal, characteristic
2#22549737C212yesmaximal
3#22549737C211yesmaximal, characteristic
4#22549642C212yesmaximal
5#342×C127C211yesmaximal, characteristic
6C23×Dic127C2211yescharacteristic
7#16861188—22no—
8#16861186—24no—
9C2×C22⋊Dic127C2211yescharacteristic
10C2×C22⋊Dic127C2212yes—
11C127⋊#95—24no—
12C127⋊#95—22no—
13C2×C22⋊C4×C127C2212yes—
14C2×C22⋊C4×C127C2211yescharacteristic
15C2×Dic254—22no—
16C22×Dic127—22no—
17C22×Dic127—21no—
18C22×Dic127—21no—
19C22×Dic127—21no—
20C22×Dic127—22no—
21C22×Dic127D414yes—
22C22×Dic127Q812yescomplemented
23C4⋊Dic127—22no—
24Dic127⋊C4—24no—
25C22⋊Dic127—22no—
26C22⋊Dic127—22no—
27C22⋊Dic127—22no—
28C24×C127C2311yescharacteristic
29C22×C4×C127—22no—
30C22×C4×C127D412yes—
31C22⋊C4×C127—22no—
32C22⋊C4×C127—22no—
33C22⋊C4×C127—22no—
34Dic254—82no—
35C2×Dic127—24no—
36C2×Dic127—24no—
37C2×Dic127—22no—
38C2×Dic127—41no—
39C2×Dic127—42no—
40C2×Dic127—41no—
41C2×Dic127—44no—
42C2×Dic127—42no—
43C2×Dic127—41no—
44C23×C127—42no—
45C23×C127C2×D412yes—
46C23×C127C2×D411yescharacteristic
47C23×C127C2×Q811yescharacteristic
48C23×C127#4111yescharacteristic
49C23×C127#4112yes—
50C2×C4×C127—22no—
51C2×C4×C127—42no—
52C2×C4×C127—44no—
53C2×C4×C127—42no—
54C22×C127—24no—
55C22×C127—22no—
56C22×C127—41no—
57C22×C127—41no—
58C22×C127—42no—
59C22×C127#12011yescharacteristic
60C22×C127C22⋊Q812yes—
61C22×C127C22⋊Q811yescharacteristic
62C22×C127#12412yes—
63C22×C127#12411yescharacteristic
64C4×C127—42no—
65C4×C127—82no—
66Dic127—42no—
67Dic127—44no—
68Dic127—81no—
69C2×C127—21no—
70C2×C127—22no—
71C2×C127—22no—
72C2×C127—21no—
73C2×C127#39311yescharacteristic
74C2×C127#45812yes—
75#2030—1271nomaximal, Sylow
76C127#203011yescharacteristic, complemented, Sylow
77#543—1271no—
78#381—1272no—
79#381—1271no—
80#342—1272no—
81#342D12711yescharacteristic
82C23×C4—1271no—
83C22⋊Q8—2544no—
84C22⋊Q8—2542no—
85C2×C22⋊C4—1272no—
86C2×C22⋊C4—1271no—
87C2×C22⋊C4C2×D12712yes—
88C2×C22⋊C4C2×D12711yescharacteristic
89#95—2544no—
90#95—2542no—
91C24C22×D12711yescharacteristic
92C2×Q8—2542no—
93C22×C4—22no—
94C22×C4—1274no—
95C22×C4—1272no—
96C22×C4—2541no—
97C22×C4—2542no—
98C22×C4—2541no—
99C22×C4—2541no—
100C22×C4—2542no—
101C22×C4D50812yes—
102C4⋊C4—2544no—
103C4⋊C4—2542no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—2542no—
108C22⋊C4—2542no—
109C22⋊C4—2542no—
110C23—42no—
111C23D4×D12712yes—
112C23C2×D50811yescharacteristic
113C23Q8×D12711yescharacteristic
114C23#1452324711yescharacteristic
115C23#1452324812yes—
116Q8—10162no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—5082no—
122C2×C4—5081no—
123C2×C4—5082no—
124C2×C4—5081no—
125C2×C4—5081no—
126C2×C4—5084no—
127C2×C4—2544no—
128C2×C4—2542no—
129C2×C4—2544no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#1686117911yescharacteristic
136C22#1686118712yes—
137C22#1686118911yescharacteristic
138C22#1686119512yes—
139C22#1686119711yescharacteristic
140C4—42no—
141C4—82no—
142C4—10161no—
143C4—5082no—
144C4—5084no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#2254978911yescharacteristic
150C2#2255006712yes—
subgroupG/H
centerC22#16861179
commutator, prime radical, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127#120
hypercenter#342D127
hypocenterC127#2030
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic127Q8semidirect
76. C127#2030semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225497372
#225496422
#342×C1272
#2030127