Finite Group Databasebeta

← #30637843

Subgroups — #30637843

Order 16256.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550599C211yesmaximal, characteristic, complemented
2#22550223C211yesmaximal, characteristic, complemented
3#22550076C211yesmaximal, characteristic, complemented
4#22550072C211yesmaximal, characteristic
5#22549623C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7Q8⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523235—21no—
9C2×D1016—21no—
10#16861209C2211yescharacteristic
11#16861206C2211yescharacteristic
12#16861204—22no—
13#16861197—21no—
14#16861185—21no—
15C4×Dic254—21no—
16C4×D508C2211yescharacteristic
17#16861167—21no—
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861131—21no—
20#16861131—22no—
21#16861129—21no—
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508D411yescharacteristic
30C2×C4×D127—21no—
31#14523235—42no—
32D1016—42no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D412yes—
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—22no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—22no—
40(C2×D127)⋊C4—21no—
41C4×Dic127—22no—
42C2×Q8×C127D411yescharacteristic
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C42×C127—21no—
48C42×C127C2311yescharacteristic
49C22×D127—21no—
50C22×D127—41no—
51Dic254—41no—
52Dic254—41no—
53D508—21no—
54D508—21no—
55D508—42no—
56D508—41no—
57D508—41no—
58D508—41no—
59D508—41no—
60C2×Dic127—21no—
61C2×Dic127—22no—
62C4×D127—41no—
63C4×D127—41no—
64C127⋊C8—41no—
65Q8×C127—21no—
66Q8×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127—41no—
74C22×C127#12011yescharacteristic
75C2×D127—41no—
76C2×D127—41no—
77C2×D127—41no—
78C2×D127—81no—
79C2×D127—81no—
80C2×D127—81no—
81C4×C127—21no—
82C4×C127—41no—
83C4×C127—41no—
84C4×C127#13511yescharacteristic
85C4×C127#13511yescharacteristic
86C4×C127AGL(1,ℤ/8)11yescharacteristic
87C4×C127AGL(1,ℤ/8)11yescharacteristic
88Dic127—41no—
89Dic127—42no—
90C2×C127#44911yescharacteristic
91C2×C127#44911yescharacteristic
92C2×C127#45411yescharacteristic
93D127—81no—
94D127—161no—
95#1628—1271nomaximal, Sylow
96C127#162811yescharacteristic, complemented, Sylow
97#556—1271no—
98#486—1271no—
99#460—1271no—
100#459—1271no—
101#332—1271no—
102Q8⋊C8D12711yescharacteristic, complemented
103D4⋊C8—1271no—
104C2×SD16—2541no—
105C2×D8—2541no—
106C4⋊Q8—1271no—
107#127—1271no—
108#126—2542no—
109#124—2541no—
110C22⋊Q8—2541no—
111C4×Q8—2541no—
112C4×Q8C2×D12711yescharacteristic
113C4×D4—1271no—
114#117—2541no—
115C4⋊C8—1271no—
116C4⋊C8C2×D12711yescharacteristic
117#102—2541no—
118#102—2542no—
119#102—2541no—
120C4×C8C2×D12711yescharacteristic
121C2×Q8—2541no—
122C2×Q8D50811yescharacteristic
123C2×D4—1271no—
124C2×D4—2541no—
125C2×D4—2541no—
126C2×D4—2541no—
127C22×C4—2541no—
128SD16—5082no—
129D8—5082no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—2541no—
133C4⋊C4—21no—
134C4⋊C4—1271no—
135C4⋊C4—1272no—
136C4⋊C4—2542no—
137C4⋊C4—2541no—
138C4⋊C4D50811yescharacteristic
139C22⋊C4—2542no—
140C22⋊C4—2542no—
141C22⋊C4—2541no—
142C42—21no—
143C42—2542no—
144C42C22×D12711yescharacteristic
145C23—5081no—
146C23—2541no—
147Q8—21no—
148Q8—21no—
149Q8—5081no—
150Q8—5081no—
151D4—5082no—
152D4—5081no—
153D4—5081no—
154D4—5081no—
155D4—5081no—
156D4—2541no—
157D4—2541no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—5081no—
161C2×C4—5081no—
162C2×C4—2541no—
163C2×C4—2542no—
164C2×C4D4×D12711yescharacteristic
165C2×C4D4×D12711yescharacteristic
166C2×C4C2×D50811yescharacteristic
167C8—41no—
168C8—41no—
169C8—5081no—
170C22—10161no—
171C22—10161no—
172C22—10161no—
173C22—5081no—
174C22—5081no—
175C22—5081no—
176C22#1686117911yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—5081no—
181C4—5082no—
182C4#1686123111yescharacteristic
183C4#1686123211yescharacteristic
184C4#1686123611yescharacteristic
185C4#1686123711yescharacteristic
186C2—20321no—
187C2—10161no—
188C2#2255002311yescharacteristic
189C2#2255002411yescharacteristic
190C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socleC22×C127#120
prime radical#22550072C2
hypercenterQ8⋊C8D127
hypocenterC127#1628
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #22550599C2semidirect
2. #22550223C2semidirect
3. #22550076C2semidirect
5. #22549623C2semidirect
6. #22549605C2semidirect
7. Q8⋊C8×C127C2semidirect
96. C127#1628semidirect
102. Q8⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225505992
#225502232
#225500762
#225500722
#225496232
#225496052
Q8⋊C8×C1272
#1628127