Finite Group Databasebeta

← #30637047

Subgroups — #30637047

Order 16256.

332 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 81 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×#16861171C211yesmaximal, characteristic
2C22⋊C8×D127C212yesmaximal, complemented
3#22549602C211yesmaximal, characteristic, complemented
4#22549599C212yesmaximal, complemented
5#324×C127C211yesmaximal, characteristic, complemented
6C22×D508C411yescharacteristic
7C22×C4×D127C411yescharacteristic
8C22×C4×D127C2211yescharacteristic
9C2×C8×D127—22no—
10#16861171—22no—
11#16861171—22no—
12C2×(C2×D127)⋊C4C411yescharacteristic
13C2×(C2×D127)⋊C4C2211yescharacteristic
14C2×C4×Dic127C411yescharacteristic
15C22⋊(C127⋊C8)C2212yes—
16(C2×D127)⋊C8—22no—
17C2×C4⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C23×D127C2×C411yescharacteristic
20C23×D127C2×C411yescharacteristic
21C2×D508—22no—
22C2×D508—22no—
23C2×D508C814yescomplemented
24C22×Dic127C2×C411yescharacteristic
25C22×Dic127C2×C411yescharacteristic
26C2×C4×D127—21no—
27C2×C4×D127—21no—
28C2×C4×D127—42no—
29C2×C4×D127—42no—
30C2×C4×D127D412yes—
31C2×C4×D127D412yes—
32C2×C127⋊C8—22no—
33C8×D127—44no—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—22no—
36C4×Dic127—21no—
37C4×Dic127—21no—
38C22×C4×C127C2×C411yescharacteristic
39C22×C4×C127C2×C411yescharacteristic
40C22×C4×C127C2311yescharacteristic
41C2×C8×C127—22no—
42C4⋊C4×C127—22no—
43C22×D127—21no—
44C22×D127—21no—
45C22×D127—41no—
46C22×D127—41no—
47C22×D127—41no—
48C22×D127—41no—
49C22×D127C22⋊C412yes—
50C22×D127C2×C812yes—
51C22×D127C2×C812yescomplemented
52C22×D127M1612yes—
53D508—28no—
54C2×Dic127—22no—
55C2×Dic127—21no—
56C2×Dic127—41no—
57C2×Dic127C22⋊C411yescharacteristic
58C2×Dic127C22⋊C411yescharacteristic
59C4×D127—24no—
60C4×D127—42no—
61C4×D127—42no—
62C4×D127—42no—
63C127⋊C8—42no—
64C23×C127C22×C411yescharacteristic
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127C2×C812yescomplemented
71C2×C4×C127C2×D412yes—
72C8×C127—42no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127C2×C22⋊C411yescharacteristic
76C22×C127C22×C811yescharacteristic
77C22×C127C2×M1611yescharacteristic
78C2×D127—22no—
79C2×D127—22no—
80C2×D127—22no—
81C2×D127—42no—
82C2×D127—42no—
83C2×D127—42no—
84C2×D127—41no—
85C2×D127—41no—
86C2×D127—42no—
87C2×D127C22⋊C814yes—
88C4×C127—22no—
89C4×C127—22no—
90C4×C127—41no—
91Dic127—21no—
92Dic127—41no—
93Dic127#9912yes—
94C2×C127—21no—
95C2×C127—21no—
96C2×C127C2×C22⋊C811yescharacteristic
97C2×C127C2×#9911yescharacteristic
98C2×C127#41311yescharacteristic
99D127—24no—
100D127—42no—
101#1459—1271nomaximal, Sylow
102C127#145911yescharacteristic, complemented, Sylow
103C2×C4×D4—1271no—
104C2×C22⋊C8—1272no—
105#324—1271no—
106#324D12711yescharacteristic, complemented
107#323—1272no—
108C22×D4—1271no—
109C23×C4—1271no—
110C23×C4—1271no—
111C22×C8—2542no—
112C4×D4—2542no—
113C4×D4—2542no—
114C2×C4⋊C4C2×D12711yescharacteristic
115C2×C22⋊C4—1271no—
116C2×C22⋊C4—1271no—
117C2×C42—1271no—
118C22⋊C8—1272no—
119C22⋊C8—2542no—
120C22⋊C8C2×D12712yes—
121C24—1271no—
122C24—1271no—
123C2×D4—1274no—
124C2×D4—2542no—
125C2×D4—2542no—
126C22×C4—1271no—
127C22×C4—1272no—
128C22×C4—1272no—
129C22×C4—1271no—
130C22×C4—5082no—
131C22×C4—5082no—
132C22×C4—2541no—
133C22×C4—2541no—
134C22×C4C4×D12711yescharacteristic
135C22×C4C4×D12711yescharacteristic
136C22×C4C22×D12711yescharacteristic
137C2×C8—22no—
138C2×C8—5084no—
139C2×C8—2542no—
140C4⋊C4—22no—
141C22⋊C4—2542no—
142C22⋊C4—2542no—
143C42—2541no—
144C42—2541no—
145C23—1272no—
146C23—1272no—
147C23—1272no—
148C23—1272no—
149C23—5081no—
150C23—5081no—
151C23—5081no—
152C23—5081no—
153C23—2541no—
154C23—2541no—
155C23C2×C4×D12711yescharacteristic
156D4—2548no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—1271no—
163C2×C4—1271no—
164C2×C4—5082no—
165C2×C4—5082no—
166C2×C4—5082no—
167C2×C4—5081no—
168C2×C4—2544no—
169C2×C4—2542no—
170C2×C4—2541no—
171C2×C4C8×D12712yescomplemented
172C2×C4D4×D12712yes—
173C8—42no—
174C8—5082no—
175C22—21no—
176C22—21no—
177C22—1274no—
178C22—5082no—
179C22—5082no—
180C22—5082no—
181C22—5082no—
182C22—5081no—
183C22—5081no—
184C22—2542no—
185C22—2542no—
186C22—2542no—
187C22C22⋊C4×D12711yescharacteristic
188C22C2×C8×D12711yescharacteristic
189C22M16×D12711yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—1272no—
194C4—5081no—
195C4—2541no—
196C2—21no—
197C2—21no—
198C2—5082no—
199C2—2544no—
200C2C22⋊C8×D12711yescharacteristic
201C2#99×D12711yescharacteristic
202C2#2254985911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC22×C127C22×C8
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D508C4
cosocleC22×C4×C127C23
hypercenter#324D127
hypocenterC127#1459
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D127C2semidirect
3. #22549602C2semidirect
4. #22549599C2semidirect
5. #324×C127C2semidirect
23. C2×D508C8semidirect
51. C22×D127C2×C8semidirect
70. C2×C4×C127C2×C8semidirect
102. C127#1459semidirect
106. #324D127semidirect
171. C2×C4C8×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611712
C22⋊C8×D1272
#225496022
#225495992
#324×C1272
#1459127