Finite Group Databasebeta

← #30637137

Subgroups — #30637137

Order 16256.

212 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550413C211yesmaximal, characteristic
2#22549928C211yesmaximal, characteristic, complemented
3#22549927C211yesmaximal, characteristic, complemented
4Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
5#22549605C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7Q8⋊C8×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861200C411yescharacteristic
10#16861198C411yescharacteristic
11#16861194—21no—
12#16861185C411yescharacteristic
13#16861184C411yescharacteristic
14C4×D508—21no—
15C4×D508C2211yescharacteristic
16#16861166—21no—
17C42×D127C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19Dic127⋊C8—21no—
20(C2×D127)⋊C8—21no—
21(C2×D127)⋊C8—21no—
22C4×C127⋊C8C2211yescharacteristic
23C8×Dic127—21no—
24C4×Q8×C127C2211yescharacteristic
25C4⋊C8×C127C2211yescharacteristic
26C4×C8×C127C2211yescharacteristic
27C2×D508—21no—
28C2×D508C2×C411yescharacteristic
29C2×C4×D127—21no—
30C2×C4×D127C2×C411yescharacteristic
31C2×C4×D127D411yescharacteristic
32C2×C4×D127D411yescharacteristic
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127C2×C411yescharacteristic
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44C4×Dic127C2×C411yescharacteristic
45C4×Dic127D411yescharacteristic
46C4×Dic127D411yescharacteristic
47C2×Q8×C127C2×C411yescharacteristic
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127C2×C411yescharacteristic
52C42×C127—21no—
53C42×C127C2311yescharacteristic
54C22×D127—21no—
55C22×D127C22⋊C411yescharacteristic
56D508—21no—
57D508—21no—
58D508—42no—
59C2×Dic127—21no—
60C2×Dic127C22⋊C411yescharacteristic
61C2×Dic127C22⋊C411yescharacteristic
62C2×Dic127C22⋊C411yescharacteristic
63C4×D127—21no—
64C4×D127—21no—
65C4×D127—21no—
66C4×D127—21no—
67C4×D127—41no—
68C4×D127—41no—
69C4×D127—41no—
70C127⋊C8—41no—
71C127⋊C8—41no—
72Q8×C127—21no—
73Q8×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127C22×C411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C8×C127—41no—
80C8×C127—41no—
81C22×C127C2×C22⋊C411yescharacteristic
82C2×D127—21no—
83C2×D127—21no—
84C2×D127—21no—
85C2×D127—41no—
86C2×D127—41no—
87C2×D127—41no—
88C4×C127—21no—
89C4×C127—41no—
90C4×C127—41no—
91C4×C127#13111yescharacteristic
92C4×C127#13111yescharacteristic
93C4×C127#13511yescharacteristic
94C4×C127#13511yescharacteristic
95Dic127—21no—
96Dic127—21no—
97Dic127—21no—
98Dic127—41no—
99C2×C127#40811yescharacteristic
100C2×C127#41611yescharacteristic
101C2×C127#42111yescharacteristic
102D127—41no—
103D127—81no—
104#1482—1271nomaximal, Sylow
105C127#148211yescharacteristic, complemented, Sylow
106#529—1271no—
107#431—1271no—
108#431—1271no—
109Q8⋊C8—1271no—
110Q8⋊C8D12711yescharacteristic, complemented
111D4⋊C8—1271no—
112D4⋊C8—1271no—
113#126—2541no—
114#125—1271no—
115#124—1271no—
116#123—2541no—
117C22⋊Q8—1271no—
118#121—1271no—
119C4×Q8C2×D12711yescharacteristic
120C4×D4—1271no—
121C4×D4—2541no—
122#117—2541no—
123C2×C42—1271no—
124C4⋊C8—1271no—
125C4⋊C8—2541no—
126C4⋊C8C2×D12711yescharacteristic
127C22⋊C8—2541no—
128C22⋊C8—2541no—
129C4×C8—1271no—
130C4×C8—2541no—
131C4×C8C2×D12711yescharacteristic
132C2×Q8C4×D12711yescharacteristic
133C2×D4—1271no—
134C2×D4—2541no—
135C22×C4—1271no—
136C22×C4—1271no—
137C22×C4—1271no—
138C22×C4—2541no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—2541no—
142C2×C8—2541no—
143C4⋊C4—21no—
144C4⋊C4—1271no—
145C4⋊C4—2541no—
146C4⋊C4—2541no—
147C4⋊C4—2541no—
148C4⋊C4C4×D12711yescharacteristic
149C22⋊C4—2541no—
150C22⋊C4—2541no—
151C22⋊C4—2541no—
152C22⋊C4—2541no—
153C22⋊C4—2541no—
154C42—21no—
155C42—1271no—
156C42—1271no—
157C42—1271no—
158C42C22×D12711yescharacteristic
159C23—1271no—
160C23—2541no—
161Q8—21no—
162Q8—21no—
163D4—5082no—
164D4—2541no—
165D4—2541no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—1271no—
169C2×C4—1271no—
170C2×C4—1271no—
171C2×C4—5081no—
172C2×C4—5081no—
173C2×C4—5081no—
174C2×C4—2541no—
175C2×C4—2541no—
176C2×C4—2541no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4C2×C4×D12711yescharacteristic
180C2×C4D4×D12711yescharacteristic
181C2×C4D4×D12711yescharacteristic
182C8—41no—
183C8—41no—
184C8—5081no—
185C8—5081no—
186C22—5081no—
187C22—5081no—
188C22—5081no—
189C22—2541no—
190C22—2541no—
191C22—2541no—
192C22C22⋊C4×D12711yescharacteristic
193C4—21no—
194C4—41no—
195C4—41no—
196C4—5081no—
197C4—2541no—
198C4—2541no—
199C4—2541no—
200C4#1686121511yescharacteristic
201C4#1686121711yescharacteristic
202C4#1686123011yescharacteristic
203C4#1686123211yescharacteristic
204C2—10161no—
205C2—5081no—
206C2#2254984111yescharacteristic
207C2#2254987111yescharacteristic
208C2#2254989111yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861184C4
cosocleC42×C127C23
hypercenterQ8⋊C8D127
hypocenterC127#1482
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #22549928C2semidirect
3. #22549927C2semidirect
4. Q8⋊(C127⋊C8)C2semidirect
5. #22549605C2semidirect
6. #22549603C2semidirect
7. Q8⋊C8×C127C2semidirect
105. C127#1482semidirect
110. Q8⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225504132
#225499282
#225499272
Q8⋊(C127⋊C8)2
#225496052
#225496032
Q8⋊C8×C1272
#1482127