Finite Group Databasebeta

← #30637920

Subgroups — #30637920

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#22550546C211yesmaximal, characteristic
2#22550153C211yesmaximal, characteristic, complemented
3#22550083C211yesmaximal, characteristic, complemented
4#22549620C211yesmaximal, characteristic, complemented
5#22549608C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7#486×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523238—21no—
10#16861205—21no—
11#16861181—22no—
12#16861180—21no—
13C4×Dic254C2211yescharacteristic
14C4×D508C2211yescharacteristic
15D4×Dic127—21no—
16#16861142—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861133—21no—
20#16861129—21no—
21C4⋊Q8×C127C2211yescharacteristic
22#127×C127C2211yescharacteristic
23#102×C127—22no—
24C4×C8×C127C2211yescharacteristic
25#14523247—42no—
26#14523247—42no—
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—22no—
30C2×C22⋊D127—22no—
31C22×Dic127—22no—
32C2×C4×D127—21no—
33#14523238—42no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C4⋊Dic127D411yescharacteristic
37C4⋊Dic127D411yescharacteristic
38Dic127⋊C4—21no—
39C22⋊Dic127—22no—
40(C2×D127)⋊C4—21no—
41C4×Dic127—21no—
42C2×Q8×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127D412yes—
46C2×C8×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—21no—
52Dic254D812yes—
53D508—21no—
54D508—21no—
55C22⋊D127—44no—
56C22⋊D127—82no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—21no—
60C2×Dic127—44no—
61C4×D127—41no—
62C4×D127—41no—
63C127⋊C8—41no—
64C127⋊C8—41no—
65C23×C127—22no—
66Q8×C127—41no—
67Q8×C127—41no—
68D4×C127—42no—
69D4×C127—41no—
70D4×C127—41no—
71D4×C127—41no—
72D4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C8×C127—41no—
78C22×C127—42no—
79C22×C127—42no—
80C22×C127—42no—
81C22×C127#12011yescharacteristic
82C2×D127—41no—
83C2×D127—41no—
84C2×D127—41no—
85C4×C127—21no—
86C4×C127—81no—
87C4×C127C2×D811yescharacteristic
88C4×C127#13511yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90C4×C127#13711yescharacteristic
91Dic127—22no—
92Dic127—41no—
93Dic127—41no—
94C2×C127—82no—
95C2×C127#44711yescharacteristic
96C2×C127#45211yescharacteristic
97C2×C127#45511yescharacteristic
98D127—81no—
99#1641—1271nomaximal, Sylow
100C127#164111yescharacteristic, complemented, Sylow
101#550—1271no—
102#486D12711yescharacteristic, complemented
103#474—1271no—
104#462—1271no—
105#331—1271no—
106Q8⋊C8—1271no—
107D4⋊C8—1271no—
108C2×#41—2541no—
109C2×Q16—2541no—
110C4⋊Q8C2×D12711yescharacteristic
111#127—2541no—
112#127C2×D12711yescharacteristic
113#121—2541no—
114#121—2542no—
115C4×Q8—1271no—
116C4×D4—1271no—
117C4×D4—2541no—
118#107—2541no—
119C4⋊C8—1271no—
120C4⋊C8—1271no—
121#103—2541no—
122#102—22no—
123#102—2541no—
124C4×C8C2×D12711yescharacteristic
125#41—5082no—
126#41—5082no—
127C2×Q8—21no—
128C2×Q8—1271no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—1271no—
132C2×D4—2542no—
133C2×D4—2542no—
134C2×D4C22⋊D12712yes—
135C22×C4—2542no—
136C22×C4—2541no—
137Q16—5082no—
138C2×C8—21no—
139C2×C8—2541no—
140C2×C8—2541no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1271no—
144C4⋊C4—1271no—
145C4⋊C4—2541no—
146C22⋊C4—2542no—
147C22⋊C4—2541no—
148C42—2541no—
149C42C22×D12711yescharacteristic
150C23—22no—
151C23—2541no—
152Q8—41no—
153Q8—41no—
154Q8—1272no—
155Q8—2541no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—10162no—
162D4—5084no—
163D4—2541no—
164D4—2541no—
165C2×C4—41no—
166C2×C4—5084no—
167C2×C4—5081no—
168C2×C4—5081no—
169C2×C4—2541no—
170C2×C4—2541no—
171C2×C4—2541no—
172C2×C4D4×D12711yescharacteristic
173C2×C4D4×D12711yescharacteristic
174C2×C4C2×C22⋊D12711yescharacteristic
175C8—41no—
176C8—5081no—
177C8—5081no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—5081no—
182C22—5081no—
183C22—5081no—
184C22#1686117711yescharacteristic
185C4—21no—
186C4—81no—
187C4—5081no—
188C4—5081no—
189C4—2542no—
190C4D8×D12711yescharacteristic
191C4#1686123011yescharacteristic
192C4#1686123311yescharacteristic
193C4#1686123811yescharacteristic
194C2—82no—
195C2—10161no—
196C2#2255000811yescharacteristic
197C2#2255003511yescharacteristic
198C2#2255005311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#486×C127C2
socleC22×C127#120
prime radical#22550546C2
hypercenter#486D127
hypocenterC127#1641
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #22550153C2semidirect
3. #22550083C2semidirect
4. #22549620C2semidirect
5. #22549608C2semidirect
6. #22549605C2semidirect
7. #486×C127C2semidirect
100. C127#1641semidirect
102. #486D127semidirect

Maximal subgroups

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

subgroupindex
#225505462
#225501532
#225500832
#225496202
#225496082
#225496052
#486×C1272
#1641127