Finite Group Databasebeta

← #30639837

Subgroups — #30639837

Order 16256.

382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#16861235C212yesmaximal, complemented
2C2×#16861126C212yesmaximal, complemented
3#22549842C211yesmaximal, characteristic, complemented
4#22549641C211yesmaximal, characteristic, complemented
5C22×D8×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2211yescharacteristic
7#16861235—24no—
8#16861235—24no—
9C2×#14523236—22no—
10C2×#14523233—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2212yes—
13#16861126—24no—
14(C2×D127)⋊C8—22no—
15C22×D4×C127C2212yes—
16C2×D8×C127—24no—
17C2×D8×C127—22no—
18C22×C8×C127C2211yescharacteristic
19#14523248—22no—
20#14523248—22no—
21C2×Dic254D411yescharacteristic
22C2×D508D411yescharacteristic
23C2×C22⋊D127—21no—
24C2×C4×D127—21no—
25#14523236—28no—
26#14523233—28no—
27#14523232—22no—
28#14523232—22no—
29#14523232D414yes—
30C2×C127⋊C8—22no—
31C2×C127⋊C8—21no—
32C24×C127—22no—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127—22no—
36C2×D4×C127—22no—
37C2×D4×C127—22no—
38C2×D4×C127D414yes—
39C22×C4×C127C2311yescharacteristic
40D8×C127—28no—
41C2×C8×C127—22no—
42C2×C8×C127D412yes—
43C22×D127—21no—
44Dic254—22no—
45D508—22no—
46C22⋊D127—42no—
47C2×Dic127—21no—
48C4×D127—41no—
49C4×D127—41no—
50C127⋊C8—24no—
51C127⋊C8—41no—
52C23×C127—24no—
53C23×C127—42no—
54C23×C127—42no—
55C23×C127—84no—
56C23×C127#4111yescharacteristic
57D4×C127—24no—
58D4×C127—24no—
59D4×C127—42no—
60D4×C127—42no—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D411yescharacteristic
65C8×C127—22no—
66C22×C127—22no—
67C22×C127—44no—
68C22×C127—84no—
69C22×C127—84no—
70C22×C127—84no—
71C22×C127#12111yescharacteristic
72C22×C127#12112yes—
73C2×D127—41no—
74C2×D127—42no—
75C4×C127#12012yes—
76C4×C127#12011yescharacteristic
77C4×C127#12711yescharacteristic
78Dic127—41no—
79C2×C127—22no—
80C2×C127—84no—
81C2×C127#39211yescharacteristic
82C2×C127#47212yes—
83D127—81no—
84#2018—1271nomaximal, Sylow
85C127#201811yescharacteristic, complemented, Sylow
86C2×AGL(1,ℤ/8)—1272no—
87C22×D8D12711yescharacteristic, complemented
88C2×#100—1272no—
89#408—1271no—
90#341—1271no—
91C2×#41—1271no—
92C22×D4C2×D12712yes—
93AGL(1,ℤ/8)—2544no—
94AGL(1,ℤ/8)—2544no—
95C2×SD16—2542no—
96C2×D8—24no—
97C2×D8—22no—
98C2×D8—2542no—
99C2×M16—1272no—
100C2×M16—1271no—
101C22×C8C2×D12711yescharacteristic
102#100—2544no—
103C22⋊C8—2542no—
104C24—22no—
105#41—2542no—
106#41—2542no—
107C2×Q8—1271no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—1271no—
114C2×D4—2541no—
115C2×D4C22⋊D12714yes—
116C22×C4—2541no—
117C22×C4C22×D12711yescharacteristic
118SD16—2548no—
119D8—28no—
120D8—2548no—
121M16—1274no—
122M16—2542no—
123M16—2542no—
124C2×C8—22no—
125C2×C8—2541no—
126C2×C8—2542no—
127C2×C8C22⋊D12712yes—
128C23—24no—
129C23—42no—
130C23—42no—
131C23—84no—
132C23—2541no—
133C23#1452324711yescharacteristic
134Q8—2542no—
135D4—24no—
136D4—24no—
137D4—42no—
138D4—42no—
139D4—5082no—
140D4—2542no—
141C2×C4—5081no—
142C2×C4—5081no—
143C2×C4—2541no—
144C2×C4D4×D12712yes—
145C2×C4D4×D12711yescharacteristic
146C2×C4C2×C22⋊D12711yescharacteristic
147C2×C4C2×C22⋊D12712yes—
148C8—22no—
149C8—5081no—
150C8—2544no—
151C22—22no—
152C22—44no—
153C22—84no—
154C22—84no—
155C22—84no—
156C22—5081no—
157C22—5082no—
158C22#1686118011yescharacteristic
159C22#1686118112yes—
160C4—5081no—
161C4#1686117712yes—
162C4#1686117811yescharacteristic
163C4#1686120511yescharacteristic
164C2—22no—
165C2—84no—
166C2—10161no—
167C2#2254978411yescharacteristic
168C2#2255014312yes—
subgroupG/H
centerC22#16861180
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC22×D8×C127C2
socleC22×C127#121
hypercenterC22×D8D127
hypocenterC127#2018
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. C2×#16861126C2semidirect
3. #22549842C2semidirect
4. #22549641C2semidirect
5. C22×D8×C127C2semidirect
85. C127#2018semidirect
87. C22×D8D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C2×#168611262
#225498422
#225496412
C22×D8×C1272
#2018127