Finite Group Databasebeta

← #30638647

Subgroups — #30638647

Order 16256.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14523248C211yesmaximal, characteristic
2C2×#16861123C214yesmaximal, complemented
3#22549739C211yesmaximal, characteristic, complemented
4#381×C127C211yesmaximal, characteristic, complemented
5C2×#14523248—28no—
6C2×#14523248—22no—
7C2×#14523248—22no—
8C22×Dic254C411yescharacteristic, complemented
9C22×D508C411yescharacteristic, complemented
10C22×C22⋊D127C2212yes—
11C22×C4×D127C412yescomplemented
12C2×C22⋊Dic127C2212yes—
13#16861123—28no—
14C127⋊#95—22no—
15C23×C4×C127C2211yescharacteristic
16C2×C22⋊C4×C127C2212yes—
17#95×C127—22no—
18C23×D127C2×C412yes—
19#14523248—216no—
20#14523248—216no—
21C2×Dic254—24no—
22C2×Dic254—22no—
23C2×D508—24no—
24C2×D508—22no—
25C2×C22⋊D127—24no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127—22no—
28C2×C22⋊D127D418yes—
29C22×Dic127—22no—
30C22×Dic127C2×C412yes—
31C2×C4×D127—24no—
32C2×C4×D127—24no—
33C2×C4×D127—24no—
34C22⋊Dic127—24no—
35C24×C127C2311yescharacteristic
36C22×C4×C127—24no—
37C22×C4×C127—22no—
38C22×C4×C127C2×C411yescharacteristic
39C22×C4×C127C2×C411yescharacteristic
40C22×C4×C127D412yes—
41C22×C4×C127D412yes—
42C22⋊C4×C127—24no—
43C22×D127—22no—
44C22×D127—22no—
45C22×D127—44no—
46C22×D127C22⋊C414yes—
47Dic254—28no—
48D508—28no—
49C22⋊D127—28no—
50C22⋊D127—28no—
51C2×Dic127—22no—
52C2×Dic127—22no—
53C2×Dic127—44no—
54C2×Dic127—42no—
55C2×Dic127C22⋊C414yes—
56C4×D127—216no—
57C23×C127—42no—
58C23×C127C22×C411yescharacteristic
59C23×C127C2×D414yes—
60C23×C127C2×D411yescharacteristic
61C23×C127C2×D411yescharacteristic
62C2×C4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127—24no—
65C2×C4×C127—24no—
66C2×C4×C127—44no—
67C2×C4×C127—42no—
68C2×C4×C127C22⋊C412yes—
69C2×C4×C127C22⋊C412yes—
70C22×C127—22no—
71C22×C127—22no—
72C22×C127—22no—
73C22×C127—44no—
74C22×C127C2×C22⋊C412yes—
75C22×C127C2×C22⋊C411yescharacteristic
76C22×C127#12012yes—
77C22×C127#12012yes—
78C2×D127—24no—
79C2×D127—44no—
80C2×D127—44no—
81C2×D127—44no—
82C4×C127—22no—
83C4×C127—22no—
84C4×C127—82no—
85Dic127—24no—
86Dic127—82no—
87C2×C127—22no—
88C2×C127—22no—
89C2×C127—22no—
90C2×C127#37911yescharacteristic
91C2×C127#41012yes—
92D127—44no—
93#1782—1271nomaximal, Sylow
94C127#178211yescharacteristic, complemented, Sylow
95C22×#41—1271no—
96C2×#99—1274no—
97#381—1271no—
98#381D12711yescharacteristic, complemented
99C2×#41—2548no—
100C2×#41—2542no—
101C2×#41—2542no—
102C22×Q8—1271no—
103C22×D4—1272no—
104C22×D4—1271no—
105C23×C4—1272no—
106C23×C4C2×D12711yescharacteristic
107C2×C22⋊C4—1272no—
108C2×C22⋊C4C2×D12712yes—
109#99—2548no—
110#95—22no—
111#95—2542no—
112C24—1272no—
113C24C22×D12711yescharacteristic
114#41—25416no—
115#41—25416no—
116C2×Q8—2544no—
117C2×Q8—2542no—
118C2×D4—1278no—
119C2×D4—2544no—
120C2×D4—2544no—
121C2×D4—2542no—
122C2×D4—2542no—
123C2×D4—2542no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—1272no—
127C22×C4—2544no—
128C22×C4—2544no—
129C22×C4—2542no—
130C22×C4—2544no—
131C22×C4C4×D12711yescharacteristic
132C22×C4C4×D12711yescharacteristic
133C22×C4C22⋊D12712yes—
134C22×C4D50812yes—
135C22⋊C4—24no—
136C22⋊C4—2544no—
137C23—42no—
138C23—1274no—
139C23—5084no—
140C23—2542no—
141C23—2542no—
142C23C2×C4×D12711yescharacteristic
143C23D4×D12714yes—
144C23C2×C22⋊D12711yescharacteristic
145C23C2×D50811yescharacteristic
146Q8—2548no—
147D4—2548no—
148D4—2548no—
149D4—2548no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—1274no—
157C2×C4—5084no—
158C2×C4—5082no—
159C2×C4—25416no—
160C2×C4—2542no—
161C2×C4—2542no—
162C2×C4(C2×D127)⋊C412yes—
163C2×C4(C2×D127)⋊C412yes—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—5084no—
169C22—5084no—
170C22—5084no—
171C22—2544no—
172C22C22⋊C4×D12712yes—
173C22C2×(C2×D127)⋊C411yescharacteristic
174C22#1686117712yes—
175C22#1686117912yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—10162no—
180C4—2544no—
181C2—22no—
182C2—22no—
183C2—22no—
184C2—5084no—
185C2#2254973111yescharacteristic
186C2#2254984912yes—
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#381×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×#14523248C2
cosocleC24×C127C23
hypercenter#381D127
hypocenterC127#1782
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#16861123C2semidirect
3. #22549739C2semidirect
4. #381×C127C2semidirect
8. C22×Dic254C4semidirect
9. C22×D508C4semidirect
11. C22×C4×D127C4semidirect
94. C127#1782semidirect
98. #381D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168611232
#225497392
#381×C1272
#1782127