Finite Group Databasebeta

← #30638789

Subgroups — #30638789

Order 16256.

264 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16861145C211yesmaximal, characteristic, complemented
2C2×#16861141C211yesmaximal, characteristic, complemented
3#22549877C212yesmaximal
4#22549644C212yesmaximal, complemented
5C2×#131×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×C4⋊Dic127C2212yes—
8#16861145—22no—
9#16861141—21no—
10#16861141—21no—
11#16861133—22no—
12#16861130—22no—
13C2×#41×C127C2211yescharacteristic
14#131×C127—22no—
15#131×C127C414yescomplemented
16C2×M16×C127—21no—
17C2×M16×C127C2211yescharacteristic
18C22×C8×C127—21no—
19C22×C8×C127C2211yescharacteristic
20C22×Dic127—22no—
21#14523232—22no—
22#14523232—22no—
23C2×C127⋊C8—22no—
24C4⋊Dic127—22no—
25C4⋊Dic127—22no—
26#41×C127—21no—
27#41×C127—21no—
28#41×C127C2×C412yes—
29#41×C127Q812yes—
30C2×Q8×C127D411yescharacteristic
31C2×D4×C127—21no—
32C2×D4×C127D411yescharacteristic
33C22×C4×C127—21no—
34C22×C4×C127C2311yescharacteristic
35M16×C127—21no—
36M16×C127—24no—
37M16×C127C2×C412yes—
38C2×C8×C127—21no—
39C2×C8×C127—24no—
40C2×C8×C127C2×C412yes—
41C2×C8×C127D411yescharacteristic
42C2×C8×C127D411yescharacteristic
43C2×C8×C127D412yes—
44C2×Dic127—42no—
45C2×Dic127—82no—
46C127⋊C8—42no—
47C23×C127—21no—
48C23×C127#4111yescharacteristic
49Q8×C127—21no—
50Q8×C127C4⋊C412yes—
51D4×C127—22no—
52D4×C127—22no—
53D4×C127—21no—
54D4×C127C4⋊C412yes—
55C2×C4×C127—21no—
56C2×C4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127C22×C411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127C2×D411yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×Q811yescharacteristic
63C2×C4×C127#4111yescharacteristic
64C8×C127—22no—
65C8×C127C22⋊C412yes—
66C8×C127C22⋊C412yes—
67C22×C127—22no—
68C22×C127—22no—
69C22×C127—42no—
70C22×C127#11711yescharacteristic
71C22×C127#12111yescharacteristic
72C22×C127C22⋊Q811yescharacteristic
73C4×C127—22no—
74C4×C127C2×C22⋊C411yescharacteristic
75C4×C127C2×C4⋊C411yescharacteristic
76C4×C127#12111yescharacteristic
77C4×C127C22⋊Q811yescharacteristic
78Dic127—82no—
79C2×C127—21no—
80C2×C127—21no—
81C2×C127—42no—
82C2×C127#38011yescharacteristic
83C2×C127#47112yes—
84#1815—1271nomaximal, Sylow
85C127#181511yescharacteristic, complemented, Sylow
86C2×#131D12711yescharacteristic
87C2×#108—1271no—
88C2×#106—1271no—
89#417—1272no—
90#343—1272no—
91C2×#41C2×D12711yescharacteristic
92#131—22no—
93#131Dic12714yescomplemented
94C2×M16—21no—
95C2×M16—1272no—
96C2×M16C2×D12711yescharacteristic
97C22×C8—21no—
98C22×C8C2×D12711yescharacteristic
99C2×C4⋊C4—1272no—
100#108—2542no—
101#106—2541no—
102#106—2541no—
103#103—2542no—
104#102—2542no—
105#41—21no—
106#41—21no—
107#41C2×Dic12712yes—
108#41Dic25412yes—
109C2×Q8D50811yescharacteristic
110C2×D4—21no—
111C2×D4D50811yescharacteristic
112C22×C4—21no—
113C22×C4—2542no—
114C22×C4C22×D12711yescharacteristic
115M16—21no—
116M16—24no—
117M16—2542no—
118M16—2542no—
119M16C2×Dic12712yes—
120C2×C8—21no—
121C2×C8—24no—
122C2×C8—2542no—
123C2×C8C2×Dic12712yes—
124C2×C8C22⋊D12711yescharacteristic
125C2×C8C22⋊D12711yescharacteristic
126C2×C8C22⋊D12712yes—
127C4⋊C4—2542no—
128C4⋊C4—2542no—
129C23—21no—
130C23#1452324811yescharacteristic
131Q8—21no—
132Q8C4⋊Dic12712yes—
133D4—22no—
134D4—22no—
135D4—21no—
136D4C4⋊Dic12712yes—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—10162no—
141C2×C4—5082no—
142C2×C4C22×Dic12711yescharacteristic
143C2×C4C2×C22⋊D12711yescharacteristic
144C2×C4C2×C22⋊D12711yescharacteristic
145C2×C4C2×D50811yescharacteristic
146C2×C4C2×Dic25411yescharacteristic
147C2×C4#1452324811yescharacteristic
148C8—22no—
149C8—5082no—
150C8C22⋊Dic12712yes—
151C8C22⋊Dic12712yes—
152C22—22no—
153C22—22no—
154C22—42no—
155C22#1686116811yescharacteristic
156C22#1686118311yescharacteristic
157C22#1686118811yescharacteristic
158C4—22no—
159C4—10162no—
160C4C2×C22⋊Dic12711yescharacteristic
161C4C2×C4⋊Dic12711yescharacteristic
162C4#1686118311yescharacteristic
163C4#1686118811yescharacteristic
164C2—21no—
165C2—21no—
166C2—42no—
167C2#2254973611yescharacteristic
168C2#2255013912yes—
subgroupG/H
centerC22#16861183
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#131×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#131D127
hypocenterC127#1815
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#16861145C2semidirect
2. C2×#16861141C2semidirect
4. #22549644C2semidirect
15. #131×C127C4semidirect
85. C127#1815semidirect
93. #131Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611452
C2×#168611412
#225498772
#225496442
C2×#131×C1272
#1815127