Finite Group Databasebeta

← #30637915

Subgroups — #30637915

Order 16256.

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

normal subgroups only

This group has 212 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#22550224C211yesmaximal, characteristic
3#22550083C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549606C211yesmaximal, characteristic, complemented
7#475×C127C211yesmaximal, characteristic, complemented
8C2×#14523238—21no—
9#16861206C2211yescharacteristic
10#16861198—21no—
11#16861197—21no—
12#16861194—22no—
13C4×Dic254C2211yescharacteristic
14C4×D508—21no—
15#16861171—21no—
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861133—21no—
18#16861129—22no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×Q8×C127C2211yescharacteristic
22#106×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25C2×Dic254D411yescharacteristic
26C2×D508—21no—
27C2×D508—21no—
28C2×D508D412yes—
29C2×C4×D127—22no—
30#14523238—42no—
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—41no—
60C127⋊C8—41no—
61Q8×C127—21no—
62Q8×C127—21no—
63Q8×C127—41no—
64Q8×C127—41no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—41no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—41no—
72C22×C127#12011yescharacteristic
73C2×D127—42no—
74C2×D127—42no—
75C2×D127—42no—
76C4×C127—21no—
77C4×C127—41no—
78C4×C127—41no—
79C4×C127—81no—
80C4×C127#13511yescharacteristic
81C4×C127#13511yescharacteristic
82C4×C127AGL(1,ℤ/8)11yescharacteristic
83C4×C127#13711yescharacteristic
84Dic127—41no—
85Dic127—41no—
86C2×C127#44911yescharacteristic
87C2×C127#45211yescharacteristic
88C2×C127#45511yescharacteristic
89D127—82no—
90#1640—1271nomaximal, Sylow
91C127#164011yescharacteristic, complemented, Sylow
92#559—1271no—
93#486—1271no—
94#475D12711yescharacteristic, complemented
95#462—1271no—
96#331—1271no—
97Q8⋊C8—1271no—
98Q8⋊C8—1271no—
99C2×Q16—2541no—
100C4⋊Q8C2×D12711yescharacteristic
101#127—1271no—
102#124—2541no—
103#124—2541no—
104#123—2542no—
105C4×Q8—1271no—
106C4×Q8C2×D12711yescharacteristic
107C4×D4—2541no—
108C4×D4—2541no—
109#106—21no—
110C4⋊C8—1271no—
111C4⋊C8C2×D12711yescharacteristic
112#103—21no—
113#103—2541no—
114#102—2542no—
115C4×C8—1271no—
116C2×Q8—21no—
117C2×Q8—1271no—
118C2×Q8C22⋊D12711yescharacteristic
119C2×D4—1272no—
120C2×D4—2541no—
121C2×D4—2541no—
122C22×C4—2542no—
123Q16—5082no—
124C2×C8—21no—
125C2×C8—2541no—
126C2×C8—2541no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—1271no—
131C4⋊C4—2541no—
132C4⋊C4C22⋊D12711yescharacteristic
133C22⋊C4—2542no—
134C22⋊C4—2542no—
135C22⋊C4—2542no—
136C42—21no—
137C42—2541no—
138C42C22×D12711yescharacteristic
139C23—2542no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—2541no—
145Q8—2541no—
146D4—5082no—
147D4—5081no—
148D4—5081no—
149D4—5081no—
150D4—5081no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—10162no—
155C2×C4—2541no—
156C2×C4—2541no—
157C2×C4D4×D12711yescharacteristic
158C2×C4D4×D12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C8—41no—
161C8—5081no—
162C8—5081no—
163C22—5082no—
164C22—5082no—
165C22—5082no—
166C22#1686117711yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—5081no—
172C4—5081no—
173C4#1686122911yescharacteristic
174C4#1686123011yescharacteristic
175C4#1686123611yescharacteristic
176C4#1686124011yescharacteristic
177C2—10162no—
178C2#2255002011yescharacteristic
179C2#2255003611yescharacteristic
180C2#2255005211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#475×C127C2
socleC22×C127#120
prime radical#16861206C22
hypercenter#475D127
hypocenterC127#1640
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #22550083C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. #22549606C2semidirect
7. #475×C127C2semidirect
91. C127#1640semidirect
94. #475D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225502242
#225500832
#225496192
Q8⋊(C127⋊C8)2
#225496062
#475×C1272
#1640127