Finite Group Databasebeta

← #30637435

Subgroups — #30637435

Order 16256.

242 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 81 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#22549933C211yesmaximal, characteristic, complemented
2#22549929C211yesmaximal, characteristic, complemented
3#22549622C212yesmaximal, complemented
4#22549620C212yesmaximal, complemented
5#532×C127C211yesmaximal, characteristic
6C4⋊(C127⋊C8)C2212yes—
7C4⋊(C127⋊C8)C2212yes—
8C22⋊(C127⋊C8)—21no—
9C22⋊(C127⋊C8)—21no—
10#16861120—21no—
11C4×C127⋊C8—21no—
12C2×#41×C127—21no—
13C4⋊Q8×C127C2211yescharacteristic
14#127×C127C2211yescharacteristic
15#124×C127—21no—
16#121×C127—21no—
17#121×C127C412yes—
18C4×D4×C127—21no—
19C4×D4×C127C412yes—
20C2×C42×C127C2211yescharacteristic
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23#41×C127—44no—
24C2×Q8×C127—21no—
25C2×D4×C127—22no—
26C2×D4×C127—21no—
27C2×D4×C127—21no—
28C2×D4×C127C2×C412yes—
29C22×C4×C127—22no—
30C22×C4×C127C2×C411yescharacteristic
31C22×C4×C127D411yescharacteristic
32C22×C4×C127D411yescharacteristic
33C4⋊C4×C127—21no—
34C4⋊C4×C127C2×C412yes—
35C22⋊C4×C127—22no—
36C22⋊C4×C127—22no—
37C42×C127C2×C411yescharacteristic
38C42×C127D411yescharacteristic
39C42×C127D411yescharacteristic
40C42×C127C2311yescharacteristic
41C127⋊C8—42no—
42C127⋊C8—42no—
43C23×C127—22no—
44C23×C127C22⋊C411yescharacteristic
45Q8×C127—42no—
46D4×C127—42no—
47D4×C127—44no—
48D4×C127—42no—
49D4×C127—42no—
50C2×C4×C127—22no—
51C2×C4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—41no—
54C2×C4×C127—44no—
55C2×C4×C127C22⋊C411yescharacteristic
56C2×C4×C127C22⋊C411yescharacteristic
57C2×C4×C127C22⋊C411yescharacteristic
58C2×C4×C127C22×C411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127C2×D411yescharacteristic
61C22×C127—21no—
62C22×C127—21no—
63C22×C127—21no—
64C22×C127—42no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127C2×C22⋊C411yescharacteristic
68C4×C127—21no—
69C4×C127—21no—
70C4×C127—21no—
71C4×C127—21no—
72C4×C127—42no—
73C4×C127#13512yes—
74C4×C127AGL(1,ℤ/8)12yes—
75C2×C127—41no—
76C2×C127—82no—
77C2×C127#41311yescharacteristic
78C2×C127#41611yescharacteristic
79C2×C127#41811yescharacteristic
80#1553—1271nomaximal, Sylow
81C127#155311yescharacteristic, complemented, Sylow
82#532D12711yescharacteristic
83#432—1271no—
84#431—1271no—
85#332—1272no—
86#331—1272no—
87C2×#41—21no—
88C4⋊Q8C2×D12711yescharacteristic
89#127C2×D12711yescharacteristic
90#124—21no—
91#121—21no—
92#121Dic12712yes—
93C4×D4—21no—
94C4×D4Dic12712yes—
95C2×C42C2×D12711yescharacteristic
96C4⋊C8—1272no—
97C4⋊C8—1272no—
98C22⋊C8—2541no—
99C22⋊C8—2541no—
100#97—2541no—
101C4×C8—2541no—
102#41—44no—
103C2×Q8—21no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C2×Dic12712yes—
108C22×C4—22no—
109C22×C4C2×Dic12711yescharacteristic
110C22×C4C22⋊D12711yescharacteristic
111C22×C4C22⋊D12711yescharacteristic
112C2×C8—2542no—
113C2×C8—2542no—
114C4⋊C4—21no—
115C4⋊C4C2×Dic12712yes—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C42C2×Dic12711yescharacteristic
119C42C22⋊D12711yescharacteristic
120C42C22⋊D12711yescharacteristic
121C42C22×D12711yescharacteristic
122C23—22no—
123C23C22⋊Dic12711yescharacteristic
124Q8—42no—
125D4—42no—
126D4—44no—
127D4—42no—
128D4—42no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—44no—
134C2×C4C22⋊Dic12711yescharacteristic
135C2×C4C22⋊Dic12711yescharacteristic
136C2×C4C22⋊Dic12711yescharacteristic
137C2×C4C22×Dic12711yescharacteristic
138C2×C4C2×C22⋊D12711yescharacteristic
139C2×C4C2×C22⋊D12711yescharacteristic
140C8—5082no—
141C8—5082no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—42no—
146C22—42no—
147C22—42no—
148C22C2×C22⋊Dic12711yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—21no—
153C4—42no—
154C4#1686122912yes—
155C4#1686123512yes—
156C2—41no—
157C2—82no—
158C2#2254986011yescharacteristic
159C2#2254987211yescharacteristic
160C2#2254988111yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting, prime radical#532×C127C2
socleC22×C127C2×C22⋊C4
cosocleC42×C127C23
hypercenter#532D127
hypocenterC127#1553
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #22549933C2semidirect
2. #22549929C2semidirect
3. #22549622C2semidirect
4. #22549620C2semidirect
81. C127#1553semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225499292
#225496222
#225496202
#532×C1272
#1553127