Finite Group Databasebeta

← #30639134

Subgroups — #30639134

Order 16256.

344 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 97 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#22550354C211yesmaximal, characteristic, complemented
2#22549881C212yesmaximal
3C2×#16861128C211yesmaximal, characteristic, complemented
4#22549636C212yesmaximal, complemented
5C2×#124×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7Q8×Dic127—24no—
8#16861168—22no—
9#16861168C2212yes—
10#16861130—24no—
11#16861128—22no—
12C22×Q8×C127C2211yescharacteristic
13C22×D4×C127C2211yescharacteristic
14#124×C127—22no—
15#124×C127C414yescomplemented
16C2×C22⋊C4×C127—22no—
17C2×C42×C127C2211yescharacteristic
18#14523232—22no—
19#14523232—22no—
20C2×C127⋊C8—22no—
21C4⋊Dic127—24no—
22C4⋊Dic127D414yes—
23C22⋊Dic127—22no—
24C4×Dic127—24no—
25C4×Dic127—22no—
26C24×C127—21no—
27C2×Q8×C127—22no—
28C2×Q8×C127—22no—
29C2×Q8×C127C2×C412yes—
30C2×Q8×C127D412yes—
31C2×D4×C127—22no—
32C2×D4×C127—21no—
33C2×D4×C127—22no—
34C2×D4×C127C2×C412yes—
35C22×C4×C127—21no—
36C22×C4×C127D412yes—
37C22×C4×C127C2311yescharacteristic
38C22⋊C4×C127—28no—
39C42×C127—21no—
40C42×C127C2×C412yes—
41C2×Dic127—24no—
42C127⋊C8—42no—
43C23×C127—22no—
44C23×C127—44no—
45C23×C127—41no—
46C23×C127—41no—
47C23×C127C2×D411yescharacteristic
48Q8×C127—24no—
49Q8×C127—24no—
50D4×C127—24no—
51D4×C127—42no—
52C2×C4×C127—22no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—42no—
56C2×C4×C127C22⋊C414yes—
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D412yes—
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127#4112yes—
61C22×C127—22no—
62C22×C127—44no—
63C22×C127—42no—
64C22×C127—82no—
65C22×C127—82no—
66C22×C127C2×C22⋊C411yescharacteristic
67C22×C127#12011yescharacteristic
68C22×C127#12311yescharacteristic
69C4×C127—24no—
70C4×C127—42no—
71C4×C127C4×D412yes—
72C4×C127#12112yes—
73Dic127—44no—
74C2×C127—21no—
75C2×C127—21no—
76C2×C127—82no—
77C2×C127#38611yescharacteristic
78C2×C127#45412yes—
79#1888—1271nomaximal, Sylow
80C127#188811yescharacteristic, complemented, Sylow
81C2×#124D12711yescharacteristic
82#519—1271no—
83#418—1272no—
84C2×#101—1271no—
85#339—1272no—
86C22×Q8C2×D12711yescharacteristic
87C22×D4C2×D12711yescharacteristic
88C2×M16—1272no—
89#124—22no—
90#124Dic12714yescomplemented
91C4×Q8—2544no—
92#117—1272no—
93#117—2542no—
94C2×C22⋊C4—22no—
95C2×C42C2×D12711yescharacteristic
96#102—2544no—
97#101—2542no—
98C24—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8C2×Dic12712yes—
102C2×Q8C22⋊D12712yes—
103C2×D4—22no—
104C2×D4—21no—
105C2×D4—22no—
106C2×D4C2×Dic12712yes—
107C22×C4—21no—
108C22×C4C22⋊D12712yes—
109C22×C4C22×D12711yescharacteristic
110M16—2542no—
111M16—2542no—
112C2×C8—2542no—
113C4⋊C4—1274no—
114C4⋊C4—2544no—
115C22⋊C4—28no—
116C22⋊C4—2542no—
117C42—21no—
118C42—2544no—
119C42—2542no—
120C42C2×Dic12712yes—
121C23—22no—
122C23—44no—
123C23—41no—
124C23—41no—
125C23C2×C22⋊D12711yescharacteristic
126Q8—24no—
127Q8—24no—
128D4—24no—
129D4—42no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—42no—
134C2×C4—2544no—
135C2×C4C22⋊Dic12714yes—
136C2×C4C22×Dic12711yescharacteristic
137C2×C4D4×D12712yes—
138C2×C4C2×C22⋊D12711yescharacteristic
139C2×C4#1452324712yes—
140C8—5082no—
141C22—22no—
142C22—44no—
143C22—42no—
144C22—82no—
145C22—82no—
146C22C2×C22⋊Dic12711yescharacteristic
147C22#1686117711yescharacteristic
148C22#1686119011yescharacteristic
149C4—24no—
150C4—42no—
151C4—5084no—
152C4D4×Dic12712yes—
153C4#1686118112yes—
154C2—21no—
155C2—21no—
156C2—82no—
157C2#2254975611yescharacteristic
158C2#2255004812yes—
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#124×C127C2
socleC22×C127#120
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#124D127
hypocenterC127#1888
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #22550354C2semidirect
3. C2×#16861128C2semidirect
4. #22549636C2semidirect
15. #124×C127C4semidirect
80. C127#1888semidirect
90. #124Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503542
#225498812
C2×#168611282
#225496362
C2×#124×C1272
#1888127