Finite Group Databasebeta

← #30640253

Subgroups — #30640253

Order 16256.

274 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16861142C211yesmaximal, characteristic, complemented
2C2×#16861141C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic
4#22549779C211yesmaximal, characteristic
5#22549741C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C2×C4⋊Dic127C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C22⋊Dic127—22no—
12C2×C4×Dic127C2211yescharacteristic
13#16861142—22no—
14#16861141—22no—
15#16861130—22no—
16C127⋊#95—22no—
17C22×D4×C127C2211yescharacteristic
18C22×C8×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20#102×C127—22no—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C22×Dic127D412yes—
24C2×C127⋊C8—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8D412yes—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C22⋊Dic127—44no—
32C4×Dic127—21no—
33C4×Dic127—21no—
34C24×C127—21no—
35C2×D4×C127—21no—
36C2×D4×C127—21no—
37C2×D4×C127—21no—
38C2×D4×C127—21no—
39C2×D4×C127—21no—
40C2×D4×C127Q812yes—
41C22×C4×C127—21no—
42C22×C4×C127C2311yescharacteristic
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127D412yes—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C2×Dic127—22no—
49C2×Dic127—22no—
50C2×Dic127—22no—
51C2×Dic127—41no—
52C2×Dic127—41no—
53C2×Dic127—42no—
54C2×Dic127—42no—
55C127⋊C8—22no—
56C23×C127—22no—
57C23×C127—41no—
58C23×C127—41no—
59C23×C127—42no—
60C23×C127—42no—
61C23×C127C2×D411yescharacteristic
62D4×C127—22no—
63D4×C127—22no—
64D4×C127—41no—
65D4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—42no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×Q811yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C2×C4×C127#4111yescharacteristic
74C8×C127—22no—
75C22×C127—42no—
76C22×C127—44no—
77C22×C127—82no—
78C22×C127—81no—
79C22×C127—81no—
80C22×C127C22⋊Q811yescharacteristic
81C22×C127#12311yescharacteristic
82C22×C127#12711yescharacteristic
83C22×C127#13512yes—
84C22×C127AGL(1,ℤ/8)12yes—
85C4×C127—81no—
86C4×C127C22⋊Q811yescharacteristic
87C4×C127C22⋊Q811yescharacteristic
88C4×C127#12311yescharacteristic
89C4×C127#12311yescharacteristic
90Dic127—42no—
91Dic127—81no—
92Dic127—81no—
93C2×C127—82no—
94C2×C127#39911yescharacteristic
95C2×C127#47812yes—
96C2×C127#48212yes—
97C2×C127#49511yescharacteristic
98C2×C127#49611yescharacteristic
99#2092—1271nomaximal, Sylow
100C127#209211yescharacteristic, complemented, Sylow
101C2×#107—1271no—
102C2×#106—1271no—
103C2×#102—1271no—
104C2×#102D12711yescharacteristic
105#390—1271no—
106#382—1271no—
107#336—1271no—
108C22×D4C2×D12711yescharacteristic
109C22×C8—1271no—
110C22×C8C2×D12711yescharacteristic
111C2×C4⋊C4—1271no—
112C2×C4⋊C4—1271no—
113C2×C4⋊C4C2×D12711yescharacteristic
114C2×C22⋊C4—2542no—
115C2×C42—1271no—
116#107—2542no—
117#106—2542no—
118#102—22no—
119#102—2542no—
120#95—2542no—
121C24—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4Dic25412yes—
128C22×C4—21no—
129C22×C4—1272no—
130C22×C4—2541no—
131C22×C4—2541no—
132C22×C4C22×D12711yescharacteristic
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—1272no—
136C2×C8—2541no—
137C2×C8—2541no—
138C2×C8C22⋊D12712yes—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C4⋊C4—2541no—
144C4⋊C4—2541no—
145C22⋊C4—5084no—
146C42—2541no—
147C42—2541no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23D4×D12711yescharacteristic
154D4—22no—
155D4—22no—
156D4—41no—
157D4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—5081no—
161C2×C4—5081no—
162C2×C4—5082no—
163C2×C4—5082no—
164C2×C4—2542no—
165C2×C4—2542no—
166C2×C4—2542no—
167C2×C4D4×D12711yescharacteristic
168C2×C4C2×C22⋊D12711yescharacteristic
169C2×C4C2×Dic25411yescharacteristic
170C2×C4#1452324711yescharacteristic
171C2×C4#1452324711yescharacteristic
172C2×C4#1452324811yescharacteristic
173C8—22no—
174C8—2542no—
175C22—42no—
176C22—44no—
177C22—82no—
178C22—81no—
179C22—81no—
180C22#1686118611yescharacteristic
181C22#1686119111yescharacteristic
182C22#1686120511yescharacteristic
183C22#1686123012yes—
184C22#1686123312yes—
185C4—81no—
186C4—10161no—
187C4—10161no—
188C4—5082no—
189C4#1686118611yescharacteristic
190C4#1686118811yescharacteristic
191C4#1686119011yescharacteristic
192C4#1686119111yescharacteristic
193C2—82no—
194C2#2254981211yescharacteristic
195C2#2255017812yes—
196C2#2255019912yes—
197C2#2255025711yescharacteristic
198C2#2255026411yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC2×#102D127
hypocenterC127#2092
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16861142C2semidirect
2. C2×#16861141C2semidirect
5. #22549741C2semidirect
6. #22549630C2semidirect
100. C127#2092semidirect

Maximal subgroups

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

subgroupindex
C2×#168611422
C2×#168611412
C2×#168611302
#225497792
#225497412
#225496302
C2×#102×C1272
#2092127