Finite Group Databasebeta

← #30637928

Subgroups — #30637928

Order 16256.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550547C211yesmaximal, characteristic
2#22550224C211yesmaximal, characteristic, complemented
3#22550158C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7#461×C127C211yesmaximal, characteristic, complemented
8C2×#14523248—21no—
9#16861206C2211yescharacteristic
10#16861205—21no—
11#16861183—21no—
12#16861182—22no—
13C4×Dic254C2211yescharacteristic
14#16861171—21no—
15#16861140—21no—
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861132—21no—
18#16861129—22no—
19C4×C127⋊C8C2211yescharacteristic
20C2×SD16×C127—21no—
21C4⋊Q8×C127C2211yescharacteristic
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25#14523248—42no—
26#14523248—42no—
27C2×Dic254D411yescharacteristic
28C2×D508—21no—
29C2×D508—21no—
30C2×D508D412yes—
31C2×C22⋊D127—22no—
32C2×C22⋊D127—22no—
33C2×C4×D127—22no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—21no—
38(C2×D127)⋊C4—22no—
39C4×Dic127—21no—
40C2×Q8×C127—21no—
41C2×D4×C127D411yescharacteristic
42C22×C4×C127—21no—
43SD16×C127—42no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—22no—
51Dic254—21no—
52Dic254SD1612yescomplemented
53D508—42no—
54D508—41no—
55D508—41no—
56D508—41no—
57D508—41no—
58C22⋊D127—44no—
59C22⋊D127—82no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C4×D127—44no—
63C127⋊C8—41no—
64C127⋊C8—41no—
65C23×C127—21no—
66Q8×C127—41no—
67Q8×C127—41no—
68D4×C127—21no—
69D4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C8×C127—41no—
78C22×C127—41no—
79C22×C127—41no—
80C22×C127—41no—
81C22×C127#12011yescharacteristic
82C2×D127—42no—
83C2×D127—42no—
84C2×D127—42no—
85C4×C127—21no—
86C4×C127—41no—
87C4×C127—81no—
88C4×C127C2×SD1611yescharacteristic
89C4×C127#13511yescharacteristic
90C4×C127AGL(1,ℤ/8)11yescharacteristic
91C4×C127AGL(1,ℤ/8)11yescharacteristic
92Dic127—22no—
93Dic127—41no—
94C2×C127—81no—
95C2×C127#44911yescharacteristic
96C2×C127#45011yescharacteristic
97C2×C127#45511yescharacteristic
98D127—82no—
99#1642—1271nomaximal, Sylow
100C127#164211yescharacteristic, complemented, Sylow
101#550—1271no—
102#486—1271no—
103#475—1271no—
104#461D12711yescharacteristic, complemented
105#331—1271no—
106Q8⋊C8—1271no—
107D4⋊C8—1271no—
108C2×#41—2541no—
109C2×SD16—21no—
110C4⋊Q8C2×D12711yescharacteristic
111#127—1271no—
112#127—2541no—
113#121—2542no—
114#121—2541no—
115C4×Q8—1271no—
116C4×D4—2541no—
117C4×D4C2×D12711yescharacteristic
118#106—2541no—
119C4⋊C8—1271no—
120C4⋊C8C2×D12711yescharacteristic
121#103—21no—
122#103—2541no—
123#102—2542no—
124C4×C8—1271no—
125#41—5082no—
126#41—5082no—
127C2×Q8—21no—
128C2×Q8—1271no—
129C2×D4—1272no—
130C2×D4—2542no—
131C2×D4—2542no—
132C2×D4—2541no—
133C2×D4—2541no—
134C2×D4C22⋊D12711yescharacteristic
135C22×C4—21no—
136C22×C4—2542no—
137SD16—42no—
138C2×C8—21no—
139C2×C8—2541no—
140C2×C8—2541no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1271no—
144C4⋊C4—2541no—
145C4⋊C4C22⋊D12711yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—2542no—
148C42—2541no—
149C42C22×D12711yescharacteristic
150C23—21no—
151C23—2542no—
152Q8—41no—
153Q8—41no—
154Q8—1272no—
155Q8—2541no—
156D4—21no—
157D4—21no—
158D4—10162no—
159D4—5082no—
160D4—5084no—
161D4—5081no—
162D4—5081no—
163D4—5081no—
164D4—5081no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—5084no—
170C2×C4—2541no—
171C2×C4—2541no—
172C2×C4D4×D12711yescharacteristic
173C2×C4D4×D12711yescharacteristic
174C2×C4C2×C22⋊D12711yescharacteristic
175C8—41no—
176C8—5081no—
177C8—5081no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—5082no—
182C22—5082no—
183C22—5082no—
184C22#1686117711yescharacteristic
185C4—21no—
186C4—41no—
187C4—81no—
188C4—5081no—
189C4—2542no—
190C4SD16×D12711yescharacteristic
191C4#1686122911yescharacteristic
192C4#1686123511yescharacteristic
193C4#1686123611yescharacteristic
194C2—81no—
195C2—10162no—
196C2#2255001911yescharacteristic
197C2#2255002511yescharacteristic
198C2#2255005211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#461×C127C2
socleC22×C127#120
prime radical#22550547C2
hypercenter#461D127
hypocenterC127#1642
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #22550224C2semidirect
3. #22550158C2semidirect
4. #22549619C2semidirect
5. #22549606C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
7. #461×C127C2semidirect
52. Dic254SD16semidirect
100. C127#1642semidirect
104. #461D127semidirect

Maximal subgroups

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

subgroupindex
#225505472
#225502242
#225501582
#225496192
#225496062
D4⋊(C127⋊C8)2
#461×C1272
#1642127