Finite Group Databasebeta

← #30639954

Subgroups — #30639954

Order 16256.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#22550457C211yesmaximal, characteristic
2C2×#16861137C212yesmaximal, complemented
3#22549875C212yesmaximal, complemented
4#22549645C211yesmaximal, characteristic, complemented
5#384×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7C2×#14523232C2212yes—
8#16861189—22no—
9#16861184—22no—
10#16861183—22no—
11#16861137—24no—
12#16861132—22no—
13#16861129—22no—
14C2×C4⋊C4×C127—22no—
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C42×C127C2211yescharacteristic
18#14523248—22no—
19#14523248—22no—
20#14523248—22no—
21#14523248—22no—
22C2×Dic254D412yes—
23C2×D508—22no—
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26C2×C4×D127—22no—
27#14523232—22no—
28#14523232—22no—
29C2×C127⋊C8—22no—
30C4⋊Dic127—22no—
31(C2×D127)⋊C4—24no—
32C22×C4×C127—21no—
33C22×C4×C127—21no—
34C22×C4×C127D412yes—
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—22no—
37C4⋊C4×C127—22no—
38C4⋊C4×C127—44no—
39C42×C127—21no—
40C42×C127D412yescomplemented
41C22×D127—22no—
42Dic254—22no—
43Dic254—22no—
44D508—22no—
45D508—22no—
46D508—82no—
47C22⋊D127—42no—
48C22⋊D127—42no—
49C2×Dic127—22no—
50C4×D127—42no—
51C4×D127—42no—
52C127⋊C8—42no—
53C23×C127C2×D411yescharacteristic
54C2×C4×C127—22no—
55C2×C4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127—42no—
59C2×C4×C127—42no—
60C2×C4×C127—41no—
61C2×C4×C127C2×D412yes—
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127#4111yescharacteristic
64C2×C4×C127#4112yes—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127#12011yescharacteristic
68C22×C127#12111yescharacteristic
69C22×C127#12411yescharacteristic
70C2×D127—42no—
71C2×D127—42no—
72C2×D127—42no—
73C4×C127—24no—
74C4×C127—42no—
75C4×C127—81no—
76C4×C127#12112yes—
77C4×C127#12312yes—
78Dic127—42no—
79C2×C127—21no—
80C2×C127—21no—
81C2×C127#39411yescharacteristic
82C2×C127#45311yescharacteristic
83C2×C127#45611yescharacteristic
84D127—82no—
85#2039—1271nomaximal, Sylow
86C127#203911yescharacteristic, complemented, Sylow
87#537—1271no—
88C2×#104—1272no—
89#417—1272no—
90#384D12711yescharacteristic, complemented
91#343—1271no—
92C2×#41—1272no—
93C2×M16—1272no—
94C22⋊Q8—2542no—
95#121—2542no—
96#121—2542no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D12711yescharacteristic
99C2×C4⋊C4C2×D12711yescharacteristic
100C2×C42C2×D12711yescharacteristic
101#104—2544no—
102#103—2542no—
103#102—2542no—
104#41—2542no—
105#41—2542no—
106#41—2542no—
107#41—2542no—
108C2×Q8—1272no—
109C2×D4—1272no—
110C2×D4—2542no—
111C2×D4—2542no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—2542no—
115C22×C4C22⋊D12712yes—
116C22×C4C22×D12711yescharacteristic
117M16—2542no—
118M16—2542no—
119C2×C8—2542no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—2542no—
124C22⋊C4—2544no—
125C42—21no—
126C42D50812yescomplemented
127C23—2542no—
128C23C2×C22⋊D12711yescharacteristic
129Q8—2542no—
130Q8—2542no—
131D4—10162no—
132D4—5082no—
133D4—5082no—
134D4—2542no—
135D4—2542no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—5082no—
144C2×C4—5082no—
145C2×C4—2542no—
146C2×C4D4×D12712yes—
147C2×C4C2×D50811yescharacteristic
148C2×C4#1452324811yescharacteristic
149C2×C4#1452324912yes—
150C8—5082no—
151C22—21no—
152C22—21no—
153C22—5082no—
154C22—5082no—
155C22—5082no—
156C22#1686117711yescharacteristic
157C22#1686118311yescharacteristic
158C22#1686119811yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—5082no—
163C4#1686118412yes—
164C4#1686119412yes—
165C2—21no—
166C2—21no—
167C2—10162no—
168C2#2254979311yescharacteristic
169C2#2255004311yescharacteristic
170C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#384×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#384D127
hypocenterC127#2039
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#16861137C2semidirect
3. #22549875C2semidirect
4. #22549645C2semidirect
5. #384×C127C2semidirect
40. C42×C127D4semidirect
86. C127#2039semidirect
90. #384D127semidirect
126. C42D508semidirect

Maximal subgroups

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

subgroupindex
#225504572
C2×#168611372
#225498752
#225496452
#384×C1272
#2039127