Finite Group Databasebeta

← #30638341

Subgroups — #30638341

Order 16256.

178 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550228C211yesmaximal, characteristic, complemented
2#22550227C211yesmaximal, characteristic, complemented
3#22550226C211yesmaximal, characteristic, complemented
4#22550222C211yesmaximal, characteristic
5#22550222C211yesmaximal, characteristic
6C8×C127⋊C8C211yesmaximal, characteristic, complemented
7#486×C127C211yesmaximal, characteristic
8#16861209C2211yescharacteristic
9#16861209C2211yescharacteristic
10#16861134—22no—
11#16861133—22no—
12#16861132—22no—
13#16861130—22no—
14#16861130—22no—
15C4×C127⋊C8C2211yescharacteristic
16C4×C127⋊C8C2211yescharacteristic
17C4⋊Q8×C127C2211yescharacteristic
18#127×C127C2211yescharacteristic
19#102×C127—22no—
20C4×C8×C127C2211yescharacteristic
21C2×Dic254—21no—
22C2×Dic254—21no—
23C2×C127⋊C8D412yes—
24C2×C127⋊C8D412yes—
25C4⋊Dic127—21no—
26C4⋊Dic127—21no—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29C2×Q8×C127—21no—
30C2×D4×C127—21no—
31C2×D4×C127—21no—
32C2×D4×C127—21no—
33C2×C8×C127D412yes—
34C4⋊C4×C127—21no—
35C4⋊C4×C127—21no—
36C42×C127C2311yescharacteristic
37Dic254—42no—
38Dic254—42no—
39C2×Dic127—41no—
40C2×Dic127—41no—
41C127⋊C8—22no—
42C127⋊C8—22no—
43C23×C127—41no—
44Q8×C127—42no—
45D4×C127—42no—
46D4×C127—42no—
47D4×C127—42no—
48C2×C4×C127—41no—
49C2×C4×C127C2×D411yescharacteristic
50C2×C4×C127C2×D411yescharacteristic
51C2×C4×C127C2×D411yescharacteristic
52C8×C127—22no—
53C22×C127—81no—
54C22×C127—81no—
55C22×C127—81no—
56C22×C127#12711yescharacteristic
57C4×C127—81no—
58C4×C127#13512yes—
59C4×C127#13512yes—
60C4×C127#13512yes—
61Dic127—81no—
62Dic127—81no—
63C2×C127—161no—
64C2×C127#49511yescharacteristic
65C2×C127#49511yescharacteristic
66C2×C127#49511yescharacteristic
67#1711—1271nomaximal, Sylow
68C127#171111yescharacteristic, complemented, Sylow
69#487—1271no—
70#487—1271no—
71#487—1271no—
72#486—1271no—
73#486—1271no—
74#486D12711yescharacteristic
75C82—1271no—
76C4⋊Q8—1271no—
77C4⋊Q8—1271no—
78C4⋊Q8C2×D12711yescharacteristic
79#127C2×D12711yescharacteristic
80#103—2542no—
81#103—2542no—
82#103—2542no—
83#102—22no—
84#102—2542no—
85#102—2542no—
86C4×C8—1271no—
87C4×C8—1271no—
88C4×C8C2×D12711yescharacteristic
89C2×Q8—21no—
90C2×Q8—2541no—
91C2×Q8—2541no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×C8—1272no—
96C2×C8—1272no—
97C2×C8C22⋊D12712yes—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—2541no—
101C4⋊C4—2541no—
102C4⋊C4—2541no—
103C4⋊C4—2541no—
104C42C22×D12711yescharacteristic
105C23—41no—
106Q8—42no—
107Q8—5082no—
108Q8—5082no—
109D4—42no—
110D4—42no—
111D4—42no—
112C2×C4—41no—
113C2×C4—5081no—
114C2×C4—5081no—
115C2×C4D4×D12711yescharacteristic
116C2×C4D4×D12711yescharacteristic
117C2×C4C2×C22⋊D12711yescharacteristic
118C8—22no—
119C8—2542no—
120C8—2542no—
121C22—81no—
122C22—81no—
123C22—81no—
124C22#1686120511yescharacteristic
125C4—81no—
126C4—10161no—
127C4—10161no—
128C4#1686122812yes—
129C4#1686122912yes—
130C4#1686123012yes—
131C2—161no—
132C2#2255025711yescharacteristic
133C2#2255025811yescharacteristic
134C2#2255025911yescharacteristic
subgroupG/H
centerC22#16861205
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#486×C127C2
socleC22×C127#127
prime radical#127×C127C22
hypercenter#486D127
hypocenterC127#1711
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #22550228C2semidirect
2. #22550227C2semidirect
3. #22550226C2semidirect
6. C8×C127⋊C8C2semidirect
68. C127#1711semidirect

Maximal subgroups

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

subgroupindex
#225502282
#225502272
#225502262
#225502222
C8×C127⋊C82
#486×C1272
#1711127