Finite Group Databasebeta

← #30637225

Subgroups — #30637225

Order 16256.

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

normal subgroups only

This group has 284 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
2C4×C4⋊Dic127C211yesmaximal, characteristic, complemented
3#22549610C214yesmaximal, complemented
4#532×C127C211yesmaximal, characteristic
5C2×C4⋊Dic127—21no—
6C2×C4×Dic127—21no—
7C4⋊(C127⋊C8)—21no—
8C22⋊(C127⋊C8)C2212yes—
9C4×C127⋊C8—21no—
10C127⋊#95C2212yes—
11C2×#41×C127—21no—
12C4⋊Q8×C127C411yescharacteristic, complemented
13#127×C127C411yescharacteristic, complemented
14#124×C127—21no—
15#121×C127—21no—
16#121×C127C2212yes—
17C4×D4×C127—21no—
18C4×D4×C127C412yescomplemented
19C2×C42×C127C2211yescharacteristic
20C22×Dic127—22no—
21C2×C127⋊C8—22no—
22C4⋊Dic127—24no—
23C4×Dic127—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×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127C2311yescharacteristic
35C4⋊C4×C127—21no—
36C4⋊C4×C127C2×C412yes—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C42×C127C2×C411yescharacteristic
40C42×C127C2×C411yescharacteristic
41C42×C127D411yescharacteristic
42C42×C127D411yescharacteristic
43C2×Dic127—24no—
44C2×Dic127—42no—
45C2×Dic127—42no—
46C127⋊C8—42no—
47C23×C127—22no—
48C23×C127C2×D411yescharacteristic
49Q8×C127—42no—
50D4×C127—42no—
51D4×C127—42no—
52D4×C127—42no—
53D4×C127—42no—
54D4×C127—42no—
55C2×C4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—41no—
58C2×C4×C127—44no—
59C2×C4×C127C22⋊C411yescharacteristic
60C2×C4×C127C22⋊C411yescharacteristic
61C2×C4×C127C22⋊C411yescharacteristic
62C2×C4×C127C22⋊C411yescharacteristic
63C2×C4×C127D812yes—
64C2×C4×C127SD1612yes—
65C2×C4×C127C22×C411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C22×C127—21no—
68C22×C127—21no—
69C22×C127—42no—
70C22×C127—42no—
71C22×C127—42no—
72C22×C127C2×C22⋊C411yescharacteristic
73C22×C127C2×D811yescharacteristic
74C22×C127C2×SD1611yescharacteristic
75C4×C127—21no—
76C4×C127—21no—
77C4×C127—21no—
78C4×C127—42no—
79C4×C127#10212yes—
80C4×C127#10212yes—
81C4×C127#10412yes—
82Dic127—44no—
83C2×C127—21no—
84C2×C127—21no—
85C2×C127—82no—
86C2×C127#41011yescharacteristic
87C2×C127C2×#10211yescharacteristic
88C2×C127C2×#10411yescharacteristic
89#1502—1271nomaximal, Sylow
90C127#150211yescharacteristic, complemented, Sylow
91#532D12711yescharacteristic
92#431—1271no—
93C4×C4⋊C4—1271no—
94#327—1274no—
95C2×#41—21no—
96C4⋊Q8Dic12711yescharacteristic, complemented
97#127Dic12711yescharacteristic, complemented
98#124—21no—
99#121—21no—
100#121C2×D12712yes—
101C4×D4—21no—
102C4×D4Dic12712yescomplemented
103C2×C4⋊C4—2541no—
104C2×C42—2541no—
105C2×C42C2×D12711yescharacteristic
106C4⋊C8—2541no—
107C22⋊C8—1272no—
108C4×C8—2541no—
109#95—1272no—
110#41—42no—
111#41—42no—
112C2×Q8—21no—
113C2×D4—22no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4C2×Dic12712yes—
117C22×C4—22no—
118C22×C4—2542no—
119C22×C4C22⋊D12711yescharacteristic
120C22×C4C22⋊D12711yescharacteristic
121C22×C4C22×D12711yescharacteristic
122C2×C8—2542no—
123C4⋊C4—21no—
124C4⋊C4—2544no—
125C4⋊C4C2×Dic12712yes—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C42—5082no—
129C42C2×Dic12711yescharacteristic
130C42C2×Dic12711yescharacteristic
131C42C22⋊D12711yescharacteristic
132C42C22⋊D12711yescharacteristic
133C23—22no—
134C23C2×C22⋊D12711yescharacteristic
135Q8—42no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—42no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—44no—
145C2×C4—5082no—
146C2×C4—5082no—
147C2×C4—2544no—
148C2×C4C22⋊Dic12711yescharacteristic
149C2×C4C22⋊Dic12711yescharacteristic
150C2×C4C22⋊Dic12711yescharacteristic
151C2×C4C22⋊Dic12711yescharacteristic
152C2×C4#1452323312yes—
153C2×C4#1452323612yes—
154C2×C4C22×Dic12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C8—5082no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22C2×C22⋊Dic12711yescharacteristic
163C22C2×#1452323311yescharacteristic
164C22C2×#1452323611yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—42no—
169C4—5084no—
170C4#1686113012yes—
171C4#1686113012yes—
172C4#1686113612yes—
173C2—21no—
174C2—21no—
175C2—82no—
176C2#2254985011yescharacteristic
177C2C2×#1686113011yescharacteristic
178C2C2×#1686113611yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#532×C127C2
socleC22×C127C2×C22⋊C4
cosocleC22×C4×C127C23
hypercenter#532D127
hypocenterC127#1502
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. C4×C4⋊Dic127C2semidirect
3. #22549610C2semidirect
12. C4⋊Q8×C127C4semidirect
13. #127×C127C4semidirect
18. C4×D4×C127C4semidirect
90. C127#1502semidirect
96. C4⋊Q8Dic127semidirect
97. #127Dic127semidirect
102. C4×D4Dic127semidirect

Maximal subgroups

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

subgroupindex
#225499292
C4×C4⋊Dic1272
#225496102
#532×C1272
#1502127