Finite Group Databasebeta

← #30637918

Subgroups — #30637918

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#22550223C211yesmaximal, characteristic
3#22550162C211yesmaximal, characteristic, complemented
4#22550086C211yesmaximal, characteristic, complemented
5#22549619C211yesmaximal, characteristic
6#22549608C211yesmaximal, characteristic, complemented
7Q8⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523238—21no—
9#16861209C2211yescharacteristic
10#16861206C2211yescharacteristic
11#16861198—21no—
12#16861197—21no—
13#16861194—22no—
14C4×Dic254C2211yescharacteristic
15C4×D508—21no—
16#16861171—21no—
17#16861141—21no—
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861134—21no—
20#16861132—21no—
21#16861131—22no—
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#14523238—42no—
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—41no—
52Dic254—41no—
53D508—42no—
54D508—41no—
55D508—41no—
56D508—41no—
57D508—41no—
58C2×Dic127—21no—
59C2×Dic127—21no—
60C2×Dic127—41no—
61C4×D127—82no—
62C127⋊C8—41no—
63Q8×C127—21no—
64Q8×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C8×C127—41no—
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#13511yescharacteristic
80C4×C127#13511yescharacteristic
81C4×C127AGL(1,ℤ/8)11yescharacteristic
82C4×C127#13711yescharacteristic
83Dic127—41no—
84Dic127—41no—
85Dic127—81no—
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#475—1271no—
95#462—1271no—
96#331—1271no—
97Q8⋊C8—1271no—
98Q8⋊C8D12711yescharacteristic, complemented
99C2×Q16—2541no—
100C4⋊Q8—1271no—
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—2541no—
110C4⋊C8—1271no—
111C4⋊C8C2×D12711yescharacteristic
112#103—2541no—
113#103—2541no—
114#102—2542no—
115C4×C8C2×D12711yescharacteristic
116C2×Q8—1271no—
117C2×Q8—2541no—
118C2×Q8D50811yescharacteristic
119C2×D4—1272no—
120C2×D4—2541no—
121C2×D4—2541no—
122C22×C4—2542no—
123Q16—5082no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—2541no—
127C4⋊C4—21no—
128C4⋊C4—1271no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4—2541no—
132C4⋊C4D50811yescharacteristic
133C22⋊C4—2542no—
134C22⋊C4—2542no—
135C22⋊C4—2542no—
136C42—21no—
137C42—2541no—
138C42C22×D12711yescharacteristic
139C23—2542no—
140Q8—21no—
141Q8—21no—
142Q8—5081no—
143Q8—5081no—
144Q8—2541no—
145Q8—2541no—
146D4—5082no—
147D4—5081no—
148D4—5081no—
149D4—5081no—
150D4—5081no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—10162no—
154C2×C4—5081no—
155C2×C4—2541no—
156C2×C4—2541no—
157C2×C4D4×D12711yescharacteristic
158C2×C4D4×D12711yescharacteristic
159C2×C4C2×D50811yescharacteristic
160C8—41no—
161C8—41no—
162C8—5081no—
163C22—5082no—
164C22—5082no—
165C22—5082no—
166C22#1686117911yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—10161no—
171C4—5081no—
172C4—5081no—
173C4#1686123011yescharacteristic
174C4#1686123111yescharacteristic
175C4#1686123711yescharacteristic
176C4#1686124111yescharacteristic
177C2—10162no—
178C2#2255002311yescharacteristic
179C2#2255004111yescharacteristic
180C2#2255005511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socleC22×C127#120
prime radical#16861206C22
hypercenterQ8⋊C8D127
hypocenterC127#1640
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. #22550162C2semidirect
4. #22550086C2semidirect
6. #22549608C2semidirect
7. Q8⋊C8×C127C2semidirect
91. C127#1640semidirect
98. Q8⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225502232
#225501622
#225500862
#225496192
#225496082
Q8⋊C8×C1272
#1640127