Finite Group Databasebeta

← #30637850

Subgroups — #30637850

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#22550250C211yesmaximal, characteristic
3#22550168C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549606C211yesmaximal, characteristic, complemented
7#477×C127C211yesmaximal, characteristic, complemented
8C2×#14523235—22no—
9#16861206C2211yescharacteristic
10#16861198—21no—
11#16861197—21no—
12#16861194—22no—
13C4×Dic254C2211yescharacteristic
14C4×D508—21no—
15#16861171—21no—
16#16861142—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861132—21no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×Q8×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25C2×Dic254D411yescharacteristic
26C2×D508—21no—
27C2×D508—21no—
28C2×D508D412yes—
29C2×C4×D127—22no—
30#14523235—44no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C4⋊Dic127D411yescharacteristic
34Dic127⋊C4—21no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—22no—
37(C2×D127)⋊C4—22no—
38C4×Dic127—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127D411yescharacteristic
41C2×C8×C127—21no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127—22no—
49Dic254—21no—
50Dic254—21no—
51D508—42no—
52D508—41no—
53D508—41no—
54D508—41no—
55D508—41no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C4×D127—82no—
59C127⋊C8—22no—
60C127⋊C8—41no—
61Q8×C127—21no—
62Q8×C127—21no—
63Q8×C127—42no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—41no—
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
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—81no—
79C4×C127#13511yescharacteristic
80C4×C127#13511yescharacteristic
81C4×C127#13711yescharacteristic
82C4×C127#13711yescharacteristic
83Dic127—41no—
84Dic127—41no—
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#477D12711yescharacteristic, complemented
95#331—1271no—
96Q8⋊C8—1271no—
97Q8⋊C8—1271no—
98C2×SD16—2542no—
99C4⋊Q8C2×D12711yescharacteristic
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—21no—
109#107—2541no—
110C4⋊C8—1271no—
111C4⋊C8C2×D12711yescharacteristic
112#103—21no—
113#103—2541no—
114C4×C8—1271no—
115C2×Q8—21no—
116C2×Q8—1271no—
117C2×Q8C22⋊D12711yescharacteristic
118C2×D4—1272no—
119C2×D4—2541no—
120C2×D4—2541no—
121C22×C4—2542no—
122SD16—5084no—
123C2×C8—21no—
124C2×C8—2541no—
125C2×C8—2541no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1271no—
130C4⋊C4—2541no—
131C4⋊C4C22⋊D12711yescharacteristic
132C22⋊C4—2542no—
133C22⋊C4—2542no—
134C22⋊C4—2542no—
135C42—21no—
136C42—2541no—
137C42C22×D12711yescharacteristic
138C23—2542no—
139Q8—21no—
140Q8—21no—
141Q8—42no—
142Q8—2541no—
143Q8—2541no—
144D4—5082no—
145D4—5081no—
146D4—5081no—
147D4—5081no—
148D4—5081no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—10162no—
153C2×C4—2541no—
154C2×C4—2541no—
155C2×C4D4×D12711yescharacteristic
156C2×C4D4×D12711yescharacteristic
157C2×C4C2×C22⋊D12711yescharacteristic
158C8—41no—
159C8—5081no—
160C8—2542no—
161C22—5082no—
162C22—5082no—
163C22—5082no—
164C22#1686117711yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—81no—
169C4—5081no—
170C4—5081no—
171C4#1686122911yescharacteristic
172C4#1686123211yescharacteristic
173C4#1686124011yescharacteristic
174C4#1686124111yescharacteristic
175C2—10162no—
176C2#2255003611yescharacteristic
177C2#2255003711yescharacteristic
178C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#477×C127C2
socleC22×C127#120
prime radical#16861206C22
hypercenter#477D127
hypocenterC127#1630
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #22550168C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. #22549606C2semidirect
7. #477×C127C2semidirect
90. C127#1630semidirect
94. #477D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225502502
#225501682
#225496192
Q8⋊(C127⋊C8)2
#225496062
#477×C1272
#1630127