Finite Group Databasebeta

← #30640653

Subgroups — #30640653

Order 16256.

230 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 57 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×#16861152C211yesmaximal, characteristic, complemented
2C2×#16861150C211yesmaximal, characteristic, complemented
3#22550112C211yesmaximal, characteristic, complemented
4#22549680C211yesmaximal, characteristic, complemented
5#22549677C211yesmaximal, characteristic, complemented
6C22⋊(C127⋊C16)C211yesmaximal, characteristic, complemented
7C2×#135×C127C211yesmaximal, characteristic, complemented
8C2×D1016C2211yescharacteristic
9#16861183—21no—
10#16861152—22no—
11#16861150—22no—
12C2×C127⋊C16C2211yescharacteristic
13C2×C127⋊C16C2211yescharacteristic
14#16861143C2211yescharacteristic
15#16861131—21no—
16C2×#41×C127—21no—
17#135×C127—22no—
18#135×C127—22no—
19C2×Q16×C127C2211yescharacteristic
20C2×SD16×C127—21no—
21C2×D8×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C2×D508—21no—
24C2×C22⋊D127—41no—
25D1016—21no—
26D1016—21no—
27C4⋊Dic127—21no—
28(C2×D127)⋊C4—41no—
29C127⋊C16—21no—
30C127⋊C16—21no—
31#41×C127—24no—
32#41×C127—42no—
33C2×Q8×C127—21no—
34C2×D4×C127—21no—
35C2×D4×C127—41no—
36C22×C4×C127—41no—
37C22×C4×C127D411yescharacteristic
38Q16×C127—21no—
39Q16×C127D412yescomplemented
40SD16×C127—24no—
41D8×C127—21no—
42D8×C127D412yescomplemented
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127D411yescharacteristic
46C2×C8×C127C2311yescharacteristic
47C22×D127—41no—
48D508—41no—
49D508—41no—
50C22⋊D127—82no—
51C2×Dic127—41no—
52C23×C127—41no—
53C23×C127D811yescharacteristic
54Q8×C127—22no—
55Q8×C127—41no—
56D4×C127—22no—
57D4×C127—44no—
58D4×C127—41no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127—44no—
62C2×C4×C127—41no—
63C2×C4×C127D811yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C8×C127—21no—
66C8×C127C2×D411yescharacteristic
67C8×C127C2×D411yescharacteristic
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—21no—
71C22×C127—42no—
72C22×C127—81no—
73C22×C127—81no—
74C22×C127C2×D811yescharacteristic
75C2×D127—81no—
76C2×D127—81no—
77C2×D127—81no—
78C4×C127—21no—
79C4×C127—42no—
80C4×C127#12011yescharacteristic
81C4×C127AGL(1,ℤ/8)11yescharacteristic
82Dic127—81no—
83C2×C127—41no—
84C2×C127—82no—
85C2×C127#44711yescharacteristic
86C2×C127#50811yescharacteristic
87C2×C127#50911yescharacteristic
88D127—161no—
89#2186—1271nomaximal, Sylow
90C127#218611yescharacteristic, complemented, Sylow
91C2×#135D12711yescharacteristic, complemented
92C2×SD32—1271no—
93C2×D16—1271no—
94#466—1271no—
95#358—1271no—
96#357—1271no—
97C22⋊C16—1271no—
98C2×#41—21no—
99#135—22no—
100#135—22no—
101C2×Q16C2×D12711yescharacteristic
102C2×SD16—21no—
103C2×D8—1271no—
104C2×D8C2×D12711yescharacteristic
105C22×C8C2×D12711yescharacteristic
106#121—2541no—
107SD32—2542no—
108D16—2542no—
109C2×C16—1271no—
110C2×C16—1271no—
111#107—1271no—
112#102—2541no—
113#41—24no—
114#41—42no—
115C2×Q8—21no—
116C2×D4—21no—
117C2×D4—41no—
118C2×D4—5081no—
119C2×D4—2541no—
120C22×C4—41no—
121C22×C4C22⋊D12711yescharacteristic
122Q16—21no—
123Q16C22⋊D12712yescomplemented
124SD16—24no—
125D8—21no—
126D8—2541no—
127D8—2541no—
128D8C22⋊D12712yescomplemented
129C2×C8—21no—
130C2×C8—21no—
131C2×C8C22⋊D12711yescharacteristic
132C2×C8C22×D12711yescharacteristic
133C4⋊C4—2541no—
134C22⋊C4—5081no—
135C16—2541no—
136C16—2541no—
137C23—41no—
138C23—5081no—
139C23#1452323311yescharacteristic
140Q8—22no—
141Q8—41no—
142D4—22no—
143D4—44no—
144D4—41no—
145D4—10162no—
146D4—5081no—
147D4—5081no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—44no—
151C2×C4—41no—
152C2×C4—5081no—
153C2×C4#1452323311yescharacteristic
154C2×C4C2×C22⋊D12711yescharacteristic
155C8—21no—
156C8C2×C22⋊D12711yescharacteristic
157C8C2×C22⋊D12711yescharacteristic
158C22—21no—
159C22—21no—
160C22—21no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—10161no—
165C22—10161no—
166C22—10161no—
167C22C2×#1452323311yescharacteristic
168C4—21no—
169C4—42no—
170C4—10161no—
171C4#1686117811yescharacteristic
172C4#1686123511yescharacteristic
173C2—41no—
174C2—82no—
175C2—20321no—
176C2#2255001011yescharacteristic
177C2#2255030611yescharacteristic
178C2#2255031111yescharacteristic
subgroupG/H
centerC22C2×#14523233
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
FittingC2×#135×C127C2
socleC22×C127C2×D8
hypercenterC2×#135D127
hypocenterC127#2186
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16861152C2semidirect
2. C2×#16861150C2semidirect
3. #22550112C2semidirect
4. #22549680C2semidirect
5. #22549677C2semidirect
6. C22⋊(C127⋊C16)C2semidirect
7. C2×#135×C127C2semidirect
39. Q16×C127D4semidirect
42. D8×C127D4semidirect
90. C127#2186semidirect
91. C2×#135D127semidirect
123. Q16C22⋊D127semidirect
128. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611522
C2×#168611502
#225501122
#225496802
#225496772
C22⋊(C127⋊C16)2
C2×#135×C1272
#2186127