Finite Group Databasebeta

← #30637378

Subgroups — #30637378

Order 16256.

256 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 73 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
1#22550427C211yesmaximal, characteristic
2#22549928C211yesmaximal, characteristic, complemented
3C4⋊C8×D127C211yesmaximal, characteristic, complemented
4#22549622C211yesmaximal, characteristic, complemented
5#22549619C211yesmaximal, characteristic, complemented
6#22549619C211yesmaximal, characteristic, complemented
7#332×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9C2×C8×D127—21no—
10#16861206C2211yescharacteristic
11#16861198—21no—
12#16861184—21no—
13#16861184C412yes—
14#16861171—21no—
15#16861171C412yes—
16C42×D127C2211yescharacteristic
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19Dic127⋊C8—21no—
20(C2×D127)⋊C8—21no—
21C8×Dic127—21no—
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523249—42no—
26#14523249—42no—
27C2×D508—22no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508C2×C412yes—
31C2×C4×D127—22no—
32C2×C4×D127C2×C411yescharacteristic
33C2×C4×D127D411yescharacteristic
34C2×C4×D127D411yescharacteristic
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C8×D127—42no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—22no—
40C4×Dic127C2×C411yescharacteristic
41C4×Dic127D411yescharacteristic
42C4×Dic127D411yescharacteristic
43C2×Q8×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127C2×C412yes—
48C42×C127C2311yescharacteristic
49C22×D127—22no—
50C22×D127C22⋊C411yescharacteristic
51D508—42no—
52D508—44no—
53D508—41no—
54D508—41no—
55D508—41no—
56D508—41no—
57C2×Dic127C22⋊C411yescharacteristic
58C2×Dic127C22⋊C411yescharacteristic
59C2×Dic127C22⋊C411yescharacteristic
60C4×D127—21no—
61C4×D127—21no—
62C4×D127—41no—
63C4×D127—44no—
64C4×D127D812yes—
65C4×D127SD1612yes—
66C127⋊C8—41no—
67C127⋊C8—41no—
68Q8×C127—41no—
69Q8×C127—41no—
70C2×C4×C127—22no—
71C2×C4×C127C22×C411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—41no—
75C8×C127—41no—
76C22×C127C2×C22⋊C411yescharacteristic
77C2×D127—21no—
78C2×D127—21no—
79C2×D127—42no—
80C2×D127—42no—
81C2×D127—42no—
82C2×D127#10212yes—
83C4×C127—21no—
84C4×C127—42no—
85C4×C127C2×D811yescharacteristic
86C4×C127C2×SD1611yescharacteristic
87C4×C127#13511yescharacteristic
88C4×C127#13511yescharacteristic
89Dic127—21no—
90Dic127—21no—
91Dic127#10212yes—
92C2×C127#41311yescharacteristic
93C2×C127C2×#10211yescharacteristic
94C2×C127#41611yescharacteristic
95D127—22no—
96D127—82no—
97#1540—1271nomaximal, Sylow
98C127#154011yescharacteristic, complemented, Sylow
99#532—1271no—
100#431—1271no—
101C2×C4⋊C8—1271no—
102#332—1271no—
103#332D12711yescharacteristic, complemented
104#331—1271no—
105#331—1271no—
106C2×#41—2541no—
107C22×C8—2541no—
108C4⋊Q8C2×D12711yescharacteristic
109#127—1271no—
110#124—2541no—
111#121—1272no—
112#121—2541no—
113C4×D4—1272no—
114C4×D4—2541no—
115C2×C42—1271no—
116C4⋊C8—1271no—
117C4⋊C8—1271no—
118C4⋊C8—2541no—
119C4⋊C8C2×D12711yescharacteristic
120C4⋊C8C2×D12711yescharacteristic
121C22⋊C8—2541no—
122C4×C8—2541no—
123#41—5082no—
124#41—5082no—
125C2×Q8—21no—
126C2×D4—1272no—
127C2×D4—2542no—
128C2×D4—2541no—
129C2×D4—2541no—
130C22×C4—1271no—
131C22×C4—1271no—
132C22×C4—1271no—
133C22×C4—2542no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—5082no—
137C2×C8—2541no—
138C2×C8—2541no—
139C4⋊C4—21no—
140C4⋊C4C4×D12712yes—
141C22⋊C4—2542no—
142C22⋊C4—2542no—
143C42—1271no—
144C42—1271no—
145C42—1271no—
146C42C22×D12711yescharacteristic
147C23—1271no—
148C23—2542no—
149Q8—41no—
150Q8—41no—
151D4—5082no—
152D4—5084no—
153D4—5081no—
154D4—5081no—
155D4—5081no—
156D4—5081no—
157C2×C4—22no—
158C2×C4—1271no—
159C2×C4—1272no—
160C2×C4—1272no—
161C2×C4—1271no—
162C2×C4—1271no—
163C2×C4—5081no—
164C2×C4—5084no—
165C2×C4—2541no—
166C2×C4—2541no—
167C2×C4C2×C4×D12711yescharacteristic
168C2×C4D4×D12711yescharacteristic
169C2×C4D4×D12711yescharacteristic
170C8—41no—
171C8—41no—
172C8—5081no—
173C8—5081no—
174C22—1272no—
175C22—5082no—
176C22—5082no—
177C22—5082no—
178C22—2541no—
179C22—2541no—
180C22C22⋊C4×D12711yescharacteristic
181C4—21no—
182C4—42no—
183C4—1272no—
184C4—2541no—
185C4—2541no—
186C4D8×D12711yescharacteristic
187C4SD16×D12711yescharacteristic
188C4#1686123011yescharacteristic
189C4#1686123211yescharacteristic
190C2—10162no—
191C2—2542no—
192C2#2254985911yescharacteristic
193C2#102×D12711yescharacteristic
194C2#2254987111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550427C2
cosocleC42×C127C23
hypercenter#332D127
hypocenterC127#1540
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22549928C2semidirect
3. C4⋊C8×D127C2semidirect
4. #22549622C2semidirect
5. #22549619C2semidirect
6. #22549619C2semidirect
7. #332×C127C2semidirect
98. C127#1540semidirect
103. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225504272
#225499282
C4⋊C8×D1272
#225496222
#225496192
#332×C1272
#1540127