Finite Group Databasebeta

← #30637855

Subgroups — #30637855

Order 16256.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550548C211yesmaximal, characteristic
2#22550250C211yesmaximal, characteristic, complemented
3#22550163C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#477×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9C2×#14523235—22no—
10#16861206—21no—
11#16861206C2211yescharacteristic
12#16861184—22no—
13#16861184—21no—
14C4×D508—21no—
15C4×D508C2211yescharacteristic
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861129—21no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×Q8×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25#14523249—42no—
26#14523249—42no—
27C2×D508—22no—
28C2×D508—22no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D411yescharacteristic
32C2×D508D412yes—
33C2×C4×D127—22no—
34C2×C4×D127—21no—
35#14523235—44no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C4⋊Dic127D411yescharacteristic
39(C2×D127)⋊C4—22no—
40(C2×D127)⋊C4—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127D411yescharacteristic
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C42×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51C22×D127—22no—
52D508—21no—
53D508—21no—
54D508—42no—
55D508—44no—
56D508—41no—
57D508—41no—
58D508—41no—
59D508—41no—
60D508—82no—
61C2×Dic127—21no—
62C4×D127—44no—
63C4×D127—41no—
64C4×D127—41no—
65C127⋊C8—22no—
66C127⋊C8—41no—
67Q8×C127—21no—
68Q8×C127—42no—
69Q8×C127SD1612yescomplemented
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—41no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C8×C127—41no—
77C22×C127#12011yescharacteristic
78C2×D127—42no—
79C2×D127—41no—
80C2×D127—41no—
81C2×D127—42no—
82C2×D127—42no—
83C2×D127—41no—
84C4×C127—22no—
85C4×C127—21no—
86C4×C127—41no—
87C4×C127—81no—
88C4×C127C2×SD1611yescharacteristic
89C4×C127#13511yescharacteristic
90C4×C127AGL(1,ℤ/8)11yescharacteristic
91C4×C127#13711yescharacteristic
92Dic127—41no—
93C2×C127#45011yescharacteristic
94C2×C127#45211yescharacteristic
95C2×C127#45311yescharacteristic
96D127—81no—
97D127—82no—
98#1631—1271nomaximal, Sylow
99C127#163111yescharacteristic, complemented, Sylow
100#550—1271no—
101#492—1271no—
102#477D12711yescharacteristic, complemented
103#476—1271no—
104#331—1271no—
105Q8⋊C8—1271no—
106D4⋊C8—1271no—
107C2×#41—2541no—
108C2×SD16—2542no—
109C4⋊Q8C2×D12711yescharacteristic
110#127—1271no—
111#127—2541no—
112#121—2542no—
113#121—2541no—
114C4×Q8C2×D12711yescharacteristic
115C4×D4—1271no—
116C4×D4—2541no—
117#107—21no—
118#106—2541no—
119C4⋊C8—1271no—
120C4⋊C8C2×D12711yescharacteristic
121#103—21no—
122#102—2541no—
123C4×C8—1271no—
124#41—5082no—
125#41—5082no—
126C2×Q8—21no—
127C2×Q8C22⋊D12711yescharacteristic
128C2×D4—1272no—
129C2×D4—1271no—
130C2×D4—2542no—
131C2×D4—2542no—
132C2×D4—2541no—
133C2×D4—2541no—
134C22×C4—2542no—
135C22×C4—2541no—
136SD16—5084no—
137C2×C8—21no—
138C2×C8—2541no—
139C2×C8—2541no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1271no—
144C4⋊C4C22⋊D12711yescharacteristic
145C22⋊C4—2542no—
146C22⋊C4—2541no—
147C42—21no—
148C42C22×D12711yescharacteristic
149C23—2542no—
150C23—2541no—
151Q8—21no—
152Q8—42no—
153Q8#1452323512yescomplemented
154D4—10162no—
155D4—5082no—
156D4—5084no—
157D4—5081no—
158D4—5081no—
159D4—5081no—
160D4—5081no—
161D4—2541no—
162D4—2541no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—5084no—
167C2×C4—5081no—
168C2×C4—5081no—
169C2×C4—2541no—
170C2×C4D4×D12711yescharacteristic
171C2×C4D4×D12711yescharacteristic
172C2×C4C2×C22⋊D12711yescharacteristic
173C8—41no—
174C8—5081no—
175C8—2542no—
176C22—5082no—
177C22—5081no—
178C22—5082no—
179C22—5081no—
180C22—5081no—
181C22—5082no—
182C22#1686117711yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—81no—
187C4—5081no—
188C4C2×#1452323511yescharacteristic
189C4#1686123211yescharacteristic
190C4#1686123411yescharacteristic
191C4#1686124111yescharacteristic
192C2—10162no—
193C2—10161no—
194C2#2255002611yescharacteristic
195C2#2255003711yescharacteristic
196C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#477×C127C2
socleC22×C127#120
prime radical#22550548C2
hypercenter#477D127
hypocenterC127#1631
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #22550250C2semidirect
3. #22550163C2semidirect
4. #22549619C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #477×C127C2semidirect
69. Q8×C127SD16semidirect
99. C127#1631semidirect
102. #477D127semidirect
153. Q8#14523235semidirect

Maximal subgroups

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

subgroupindex
#225505482
#225502502
#225501632
#225496192
Q8⋊(C127⋊C8)2
#225496032
#477×C1272
#1631127