Finite Group Databasebeta

← #30640260

Subgroups — #30640260

Order 16256.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16861140C211yesmaximal, characteristic, complemented
2C2×#16861131C211yesmaximal, characteristic
3C2×#16861129C211yesmaximal, characteristic
4#22549777C211yesmaximal, characteristic
5#22549745C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#106×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—22no—
12C2×(C2×D127)⋊C4—22no—
13C2×C4×Dic127C2211yescharacteristic
14#16861140—22no—
15#16861131—22no—
16#16861129—22no—
17C22×C8×C127C2211yescharacteristic
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20#106×C127—22no—
21C23×D127—21no—
22C2×D508—21no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508Q812yes—
28C22×Dic127—21no—
29C22×Dic127D412yes—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8D412yes—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35Dic127⋊C4—44no—
36(C2×D127)⋊C4—44no—
37C4×Dic127—21no—
38C4×Dic127—21no—
39C22×C4×C127—21no—
40C22×C4×C127—21no—
41C22×C4×C127C2311yescharacteristic
42C2×C8×C127—21no—
43C2×C8×C127—21no—
44C2×C8×C127D412yes—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C22×D127—22no—
50C22×D127—41no—
51C22×D127—41no—
52C22×D127—42no—
53C22×D127—42no—
54D508—22no—
55D508—22no—
56D508—41no—
57D508—41no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—41no—
61C2×Dic127—42no—
62C2×Dic127SD1614yes—
63C127⋊C8—22no—
64C23×C127C2×D411yescharacteristic
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—42no—
68C2×C4×C127—42no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×Q811yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C2×C4×C127#4111yescharacteristic
74C2×C4×C127#4111yescharacteristic
75C8×C127—22no—
76C22×C127C22⋊Q811yescharacteristic
77C22×C127#12311yescharacteristic
78C22×C127#12711yescharacteristic
79C22×C127C2×SD1612yes—
80C22×C127AGL(1,ℤ/8)12yes—
81C2×D127—42no—
82C2×D127—44no—
83C2×D127—82no—
84C2×D127—81no—
85C2×D127—81no—
86C4×C127—81no—
87C4×C127—81no—
88C4×C127C22⋊Q811yescharacteristic
89C4×C127C22⋊Q811yescharacteristic
90C4×C127#12311yescharacteristic
91C4×C127#12311yescharacteristic
92Dic127—24no—
93Dic127—81no—
94C2×C127#39911yescharacteristic
95C2×C127#47612yes—
96C2×C127#48112yes—
97C2×C127#49211yescharacteristic
98C2×C127#49611yescharacteristic
99D127—82no—
100#2094—1271nomaximal, Sylow
101C127#209411yescharacteristic, complemented, Sylow
102C2×#106—1271no—
103C2×#106D12711yescharacteristic, complemented
104C2×#102—1271no—
105C2×#102—1271no—
106#390—1271no—
107#384—1271no—
108#336—1271no—
109C22×D4—1271no—
110C22×C8—1271no—
111C22×C8C2×D12711yescharacteristic
112C2×C4⋊C4—1271no—
113C2×C4⋊C4—2542no—
114C2×C4⋊C4C2×D12711yescharacteristic
115C2×C4⋊C4C2×D12711yescharacteristic
116C2×C22⋊C4—2542no—
117C2×C42—1271no—
118#106—22no—
119#106—2542no—
120#102—2542no—
121#102—2542no—
122C24—2541no—
123C2×D4—1272no—
124C2×D4—2541no—
125C2×D4—2541no—
126C2×D4—2541no—
127C2×D4—2541no—
128C2×D4—2541no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—1272no—
132C22×C4—2541no—
133C22×C4C22×D12711yescharacteristic
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—1272no—
137C2×C8—2541no—
138C2×C8—2541no—
139C2×C8C22⋊D12712yes—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—5084no—
145C4⋊C4—2541no—
146C4⋊C4—2541no—
147C22⋊C4—5084no—
148C42—2541no—
149C42—2541no—
150C23—5081no—
151C23—5081no—
152C23—5082no—
153C23—5082no—
154C23—2542no—
155C23D4×D12711yescharacteristic
156D4—5081no—
157D4—5081no—
158D4—2542no—
159D4—2542no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—1274no—
165C2×C4—5081no—
166C2×C4—5082no—
167C2×C4—2542no—
168C2×C4—2542no—
169C2×C4D4×D12711yescharacteristic
170C2×C4C2×C22⋊D12711yescharacteristic
171C2×C4Q8×D12711yescharacteristic
172C2×C4#1452324811yescharacteristic
173C2×C4#1452324811yescharacteristic
174C2×C4#1452324911yescharacteristic
175C8—22no—
176C8—2542no—
177C22—10162no—
178C22—10161no—
179C22—10161no—
180C22—5082no—
181C22—5084no—
182C22#1686118711yescharacteristic
183C22#1686119411yescharacteristic
184C22#1686120511yescharacteristic
185C22SD16×D12712yes—
186C22#1686123612yes—
187C4—81no—
188C4—81no—
189C4—10161no—
190C4—2544no—
191C4#1686118511yescharacteristic
192C4#1686118711yescharacteristic
193C4#1686119211yescharacteristic
194C4#1686119411yescharacteristic
195C2—10162no—
196C2#2254981111yescharacteristic
197C2#2255016512yes—
198C2#2255019612yes—
199C2#2255024711yescharacteristic
200C2#2255026211yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#106×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenterC2×#106D127
hypocenterC127#2094
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. C2×#16861140C2semidirect
5. #22549745C2semidirect
6. #22549630C2semidirect
7. C2×#106×C127C2semidirect
101. C127#2094semidirect
103. C2×#106D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
C2×#168611312
C2×#168611292
#225497772
#225497452
#225496302
C2×#106×C1272
#2094127