Finite Group Databasebeta

← #30638044

Subgroups — #30638044

Order 16256.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550620C211yesmaximal, characteristic
2#22550282C211yesmaximal, characteristic, complemented
3#22550238C211yesmaximal, characteristic, complemented
4#22549617C211yesmaximal, characteristic, complemented
5#22549616C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic, complemented
7#497×C127C211yesmaximal, characteristic, complemented
8#16861201C2211yescharacteristic
9#16861197C2211yescharacteristic
10#16861193—21no—
11#16861191—21no—
12#16861186—21no—
13#16861185—21no—
14#16861182—21no—
15#16861181—21no—
16#16861142—21no—
17#16861140—21no—
18#16861132—21no—
19#16861129—21no—
20#16861120C2211yescharacteristic
21#16861120C2211yescharacteristic
22C2×Q16×C127—21no—
23C2×SD16×C127—21no—
24C4⋊Q8×C127C2211yescharacteristic
25#124×C127C2211yescharacteristic
26#97×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C22×Dic127—21no—
32C2×C4×D127—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D411yescharacteristic
37C4⋊Dic127D411yescharacteristic
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127D411yescharacteristic
47C2×D4×C127D411yescharacteristic
48Q16×C127—42no—
49SD16×C127—42no—
50C2×C8×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C22⋊C4×C127—21no—
54C22⋊C4×C127—21no—
55C42×C127C2311yescharacteristic
56C22×D127—21no—
57Dic254—41no—
58D508—41no—
59C22⋊D127—81no—
60C22⋊D127—81no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—81no—
65C4×D127—81no—
66C127⋊C8—22no—
67C127⋊C8—41no—
68C23×C127—21no—
69Q8×C127—41no—
70Q8×C127—41no—
71Q8×C127—41no—
72D4×C127—41no—
73C2×C4×C127—21no—
74C2×C4×C127—41no—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C8×C127—22no—
79C22×C127—41no—
80C22×C127—41no—
81C22×C127—41no—
82C22×C127#12011yescharacteristic
83C2×D127—41no—
84C2×D127—41no—
85C2×D127—41no—
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127—41no—
90C4×C127—81no—
91Dic127—41no—
92Dic127—41no—
93Dic127—41no—
94C2×C127—81no—
95C2×C127#45311yescharacteristic
96C2×C127#45511yescharacteristic
97C2×C127#45611yescharacteristic
98D127—81no—
99#1661—1271nomaximal, Sylow
100C127#166111yescharacteristic, complemented, Sylow
101#562—1271no—
102#501—1271no—
103#497D12711yescharacteristic, complemented
104#489—1271no—
105#330—1271no—
106#329—1271no—
107#329—1271no—
108C2×Q16—21no—
109C2×SD16—21no—
110C4⋊Q8C2×D12711yescharacteristic
111#125—1271no—
112#124—1271no—
113#124C2×D12711yescharacteristic
114#123—2541no—
115#123—2541no—
116C22⋊Q8—2541no—
117C22⋊Q8—2541no—
118#121—2541no—
119#121—2541no—
120#107—2541no—
121#106—2541no—
122#103—2541no—
123#102—2541no—
124#97—1271no—
125#97—1271no—
126#97C2×D12711yescharacteristic
127C2×Q8—21no—
128C2×Q8—1271no—
129C2×Q8C22⋊D12711yescharacteristic
130C2×D4—1271no—
131C2×D4—2541no—
132C2×D4—2541no—
133C2×D4C22⋊D12711yescharacteristic
134C22×C4—2541no—
135C22×C4—2541no—
136Q16—42no—
137SD16—42no—
138C2×C8—21no—
139C2×C8—2541no—
140C2×C8—2541no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1271no—
144C4⋊C4—1271no—
145C4⋊C4—2541no—
146C4⋊C4—2541no—
147C4⋊C4—2541no—
148C4⋊C4—2541no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—2541no—
152C22⋊C4—2541no—
153C22⋊C4—2541no—
154C22⋊C4—2541no—
155C42C22×D12711yescharacteristic
156C23—21no—
157C23—2541no—
158Q8—41no—
159Q8—41no—
160Q8—41no—
161Q8—5081no—
162D4—41no—
163D4—10161no—
164D4—10161no—
165D4—5081no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—10161no—
169C2×C4—10161no—
170C2×C4—2541no—
171C2×C4—2541no—
172C2×C4—2541no—
173C2×C4D4×D12711yescharacteristic
174C2×C4D4×D12711yescharacteristic
175C2×C4C2×C22⋊D12711yescharacteristic
176C8—22no—
177C8—5081no—
178C8—2542no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—5081no—
183C22—5081no—
184C22—5081no—
185C22#1686117711yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—81no—
191C4—5081no—
192C4—5081no—
193C4—5081no—
194C2—81no—
195C2—10161no—
196C2#2255004311yescharacteristic
197C2#2255005211yescharacteristic
198C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#497×C127C2
socleC22×C127#120
prime radical#22550620C2
hypercenter#497D127
hypocenterC127#1661
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #22550282C2semidirect
3. #22550238C2semidirect
4. #22549617C2semidirect
5. #22549616C2semidirect
6. #22549615C2semidirect
7. #497×C127C2semidirect
100. C127#1661semidirect
103. #497D127semidirect

Maximal subgroups

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

subgroupindex
#225506202
#225502822
#225502382
#225496172
#225496162
#225496152
#497×C1272
#1661127