Finite Group Databasebeta

← #30637379

Subgroups — #30637379

Order 16256.

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

normal subgroups only

This group has 256 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#22549929C211yesmaximal, characteristic, complemented
2C2×C4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
3#22549622C212yesmaximal, complemented
4#22549620C212yesmaximal, complemented
5#532×C127C211yesmaximal, characteristic
6C22×C127⋊C8—21no—
7C4⋊(C127⋊C8)—21no—
8C4⋊(C127⋊C8)C2212yes—
9C4⋊(C127⋊C8)C2212yes—
10C22⋊(C127⋊C8)—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—
23C2×C127⋊C8—42no—
24#41×C127—42no—
25#41×C127—42no—
26C2×Q8×C127—21no—
27C2×D4×C127—22no—
28C2×D4×C127—21no—
29C2×D4×C127—21no—
30C2×D4×C127C2×C412yes—
31C22×C4×C127—22no—
32C22×C4×C127C2×C411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C4⋊C4×C127—21no—
36C4⋊C4×C127C2×C412yes—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C42×C127C2×C411yescharacteristic
40C42×C127D411yescharacteristic
41C42×C127D411yescharacteristic
42C42×C127C2311yescharacteristic
43C127⋊C8—42no—
44C127⋊C8—42no—
45C23×C127—22no—
46C23×C127C22⋊C411yescharacteristic
47Q8×C127—41no—
48Q8×C127—41no—
49D4×C127—42no—
50D4×C127—44no—
51D4×C127—41no—
52D4×C127—41no—
53D4×C127—42no—
54C2×C4×C127—22no—
55C2×C4×C127—22no—
56C2×C4×C127—41no—
57C2×C4×C127—44no—
58C2×C4×C127C22⋊C411yescharacteristic
59C2×C4×C127C22⋊C411yescharacteristic
60C2×C4×C127C22⋊C411yescharacteristic
61C2×C4×C127D812yes—
62C2×C4×C127SD1612yes—
63C2×C4×C127C22×C411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C22×C127—21no—
67C22×C127—21no—
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—42no—
71C22×C127#10212yes—
72C22×C127C2×C22⋊C411yescharacteristic
73C4×C127—21no—
74C4×C127—21no—
75C4×C127—21no—
76C4×C127—42no—
77C4×C127#10212yes—
78C4×C127C2×D811yescharacteristic
79C4×C127C2×SD1611yescharacteristic
80C4×C127#13512yes—
81C2×C127—22no—
82C2×C127—82no—
83C2×C127#41311yescharacteristic
84C2×C127C2×#10211yescharacteristic
85C2×C127#41611yescharacteristic
86#1540—1271nomaximal, Sylow
87C127#154011yescharacteristic, complemented, Sylow
88#532D12711yescharacteristic
89#431—1271no—
90C2×C4⋊C8—1271no—
91#332—1272no—
92#331—1272no—
93C2×#41—21no—
94C22×C8—2541no—
95C4⋊Q8C2×D12711yescharacteristic
96#127C2×D12711yescharacteristic
97#124—21no—
98#121—21no—
99#121Dic12712yes—
100C4×D4—21no—
101C4×D4Dic12712yes—
102C2×C42C2×D12711yescharacteristic
103C4⋊C8—1272no—
104C4⋊C8—1272no—
105C4⋊C8—2541no—
106C22⋊C8—2541no—
107C4×C8—2541no—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4C2×Dic12712yes—
115C22×C4—22no—
116C22×C4C2×Dic12711yescharacteristic
117C22×C4C22⋊D12711yescharacteristic
118C22×C4C22⋊D12711yescharacteristic
119C2×C8—5082no—
120C2×C8—2542no—
121C2×C8—2542no—
122C4⋊C4—21no—
123C4⋊C4C2×Dic12712yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C42C2×Dic12711yescharacteristic
127C42C22⋊D12711yescharacteristic
128C42C22⋊D12711yescharacteristic
129C42C22×D12711yescharacteristic
130C23—22no—
131C23C22⋊Dic12711yescharacteristic
132Q8—41no—
133Q8—41no—
134D4—42no—
135D4—44no—
136D4—41no—
137D4—41no—
138D4—42no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—44no—
143C2×C4C22⋊Dic12711yescharacteristic
144C2×C4C22⋊Dic12711yescharacteristic
145C2×C4C22⋊Dic12711yescharacteristic
146C2×C4#1452323312yes—
147C2×C4#1452323612yes—
148C2×C4C22×Dic12711yescharacteristic
149C2×C4C2×C22⋊D12711yescharacteristic
150C2×C4C2×C22⋊D12711yescharacteristic
151C8—5082no—
152C8—5082no—
153C22—21no—
154C22—21no—
155C22—42no—
156C22—42no—
157C22—42no—
158C22#1686113012yes—
159C22C2×C22⋊Dic12711yescharacteristic
160C4—21no—
161C4—21no—
162C4—21no—
163C4—42no—
164C4#1686113012yes—
165C4C2×#1452323311yescharacteristic
166C4C2×#1452323611yescharacteristic
167C4#1686122912yes—
168C2—22no—
169C2—82no—
170C2#2254986011yescharacteristic
171C2C2×#1686113011yescharacteristic
172C2#2254987211yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting, prime radical#532×C127C2
socleC22×C127C2×C22⋊C4
cosocleC42×C127C23
hypercenter#532D127
hypocenterC127#1540
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #22549929C2semidirect
2. C2×C4⋊(C127⋊C8)C2semidirect
3. #22549622C2semidirect
4. #22549620C2semidirect
87. C127#1540semidirect

Maximal subgroups

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

subgroupindex
#225499292
C2×C4⋊(C127⋊C8)2
#225496222
#225496202
#532×C1272
#1540127