Finite Group Databasebeta

← #30637853

Subgroups — #30637853

Order 16256.

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

normal subgroups only

This group has 218 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#22550611C211yesmaximal, characteristic
2#22550249C211yesmaximal, characteristic
3#22550172C211yesmaximal, characteristic, complemented
4#22550171C211yesmaximal, characteristic, complemented
5#22549619C211yesmaximal, characteristic
6#22549608C211yesmaximal, characteristic, complemented
7Q8⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523237—22no—
9#16861209C2211yescharacteristic
10#16861206C2211yescharacteristic
11#16861198—21no—
12#16861197—21no—
13#16861194—22no—
14C4×Dic254C2211yescharacteristic
15C4×D508—21no—
16#16861171—21no—
17#16861143—21no—
18#16861142—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861134—21no—
21#16861133—21no—
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×Dic254D411yescharacteristic
27C2×D508—21no—
28C2×D508—21no—
29C2×D508D412yes—
30C2×C4×D127—22no—
31#14523237—44no—
32C2×C127⋊C8—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—21no—
37(C2×D127)⋊C4—22no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—22no—
40C4×Dic127—21no—
41C2×Q8×C127D411yescharacteristic
42C2×C8×C127—21no—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127—22no—
49Dic254—21no—
50Dic254—21no—
51Dic254—42no—
52D508—42no—
53D508—41no—
54D508—41no—
55D508—41no—
56D508—41no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—41no—
60C4×D127—82no—
61C127⋊C8—41no—
62Q8×C127—21no—
63Q8×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C8×C127—22no—
70C8×C127—41no—
71C22×C127#12011yescharacteristic
72C2×D127—42no—
73C2×D127—42no—
74C2×D127—42no—
75C4×C127—21no—
76C4×C127—41no—
77C4×C127—41no—
78C4×C127#13511yescharacteristic
79C4×C127#13511yescharacteristic
80C4×C127#13711yescharacteristic
81C4×C127#13711yescharacteristic
82Dic127—41no—
83Dic127—41no—
84Dic127—81no—
85C2×C127#45211yescharacteristic
86C2×C127#45211yescharacteristic
87C2×C127#45311yescharacteristic
88D127—82no—
89#1630—1271nomaximal, Sylow
90C127#163011yescharacteristic, complemented, Sylow
91#559—1271no—
92#492—1271no—
93#477—1271no—
94#477—1271no—
95#331—1271no—
96Q8⋊C8—1271no—
97Q8⋊C8D12711yescharacteristic, complemented
98C2×SD16—2542no—
99C4⋊Q8—1271no—
100#127—1271no—
101#124—2541no—
102#124—2541no—
103#123—2542no—
104C4×Q8—1271no—
105C4×Q8C2×D12711yescharacteristic
106C4×D4—2541no—
107C4×D4—2541no—
108#107—2541no—
109#107—2541no—
110C4⋊C8—1271no—
111C4⋊C8C2×D12711yescharacteristic
112#103—2541no—
113#103—2541no—
114C4×C8C2×D12711yescharacteristic
115C2×Q8—1271no—
116C2×Q8—2541no—
117C2×Q8D50811yescharacteristic
118C2×D4—1272no—
119C2×D4—2541no—
120C2×D4—2541no—
121C22×C4—2542no—
122SD16—5084no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—2541no—
126C4⋊C4—21no—
127C4⋊C4—1271no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4D50811yescharacteristic
132C22⋊C4—2542no—
133C22⋊C4—2542no—
134C22⋊C4—2542no—
135C42—21no—
136C42—2541no—
137C42C22×D12711yescharacteristic
138C23—2542no—
139Q8—21no—
140Q8—21no—
141Q8—5082no—
142Q8—2541no—
143Q8—2541no—
144D4—5082no—
145D4—5081no—
146D4—5081no—
147D4—5081no—
148D4—5081no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—10162no—
152C2×C4—5081no—
153C2×C4—2541no—
154C2×C4—2541no—
155C2×C4D4×D12711yescharacteristic
156C2×C4D4×D12711yescharacteristic
157C2×C4C2×D50811yescharacteristic
158C8—22no—
159C8—41no—
160C8—5081no—
161C22—5082no—
162C22—5082no—
163C22—5082no—
164C22#1686117911yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—10161no—
169C4—5081no—
170C4—5081no—
171C4#1686123011yescharacteristic
172C4#1686123111yescharacteristic
173C4#1686124111yescharacteristic
174C4#1686124211yescharacteristic
175C2—10162no—
176C2#2255004011yescharacteristic
177C2#2255004111yescharacteristic
178C2#2255004511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socleC22×C127#120
prime radical#16861206C22
hypercenterQ8⋊C8D127
hypocenterC127#1630
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
3. #22550172C2semidirect
4. #22550171C2semidirect
6. #22549608C2semidirect
7. Q8⋊C8×C127C2semidirect
90. C127#1630semidirect
97. Q8⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225502492
#225501722
#225501712
#225496192
#225496082
Q8⋊C8×C1272
#1630127