Finite Group Databasebeta

← #30638021

Subgroups — #30638021

Order 16256.

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

normal subgroups only

This group has 230 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, complemented
2#22550263C211yesmaximal, characteristic, complemented
3#22550240C211yesmaximal, characteristic, complemented
4#22549617C211yesmaximal, characteristic, complemented
5#22549616C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic, complemented
7#496×C127C211yesmaximal, characteristic, complemented
8C2×#14523236—21no—
9C2×#14523233—21no—
10#16861201C2211yescharacteristic
11#16861197C2211yescharacteristic
12#16861193—21no—
13#16861191—21no—
14#16861186—21no—
15#16861185—21no—
16#16861182—21no—
17#16861181—21no—
18#16861130—21no—
19#16861130—21no—
20#16861120C2211yescharacteristic
21#16861120C2211yescharacteristic
22C2×SD16×C127—21no—
23C2×D8×C127—21no—
24#127×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—
33#14523236—42no—
34#14523233—42no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D411yescharacteristic
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47C2×Q8×C127D411yescharacteristic
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127D411yescharacteristic
52SD16×C127—42no—
53D8×C127—42no—
54C2×C8×C127—21no—
55C22⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C42×C127C2311yescharacteristic
58C22×D127—21no—
59Dic254—41no—
60D508—41no—
61C22⋊D127—81no—
62C22⋊D127—81no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—81no—
67C4×D127—81no—
68C127⋊C8—22no—
69C127⋊C8—41no—
70C23×C127—21no—
71C23×C127—41no—
72Q8×C127—41no—
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—41no—
76D4×C127—42no—
77D4×C127—41no—
78D4×C127—41no—
79C2×C4×C127—21no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C8×C127—22no—
84C22×C127—41no—
85C22×C127—41no—
86C22×C127—41no—
87C22×C127—81no—
88C22×C127—81no—
89C22×C127—81no—
90C22×C127#12011yescharacteristic
91C2×D127—41no—
92C2×D127—41no—
93C2×D127—41no—
94C4×C127—21no—
95C4×C127—21no—
96C4×C127—21no—
97C4×C127—41no—
98Dic127—41no—
99Dic127—41no—
100Dic127—41no—
101C2×C127—81no—
102C2×C127—161no—
103C2×C127#45311yescharacteristic
104C2×C127#45311yescharacteristic
105C2×C127#45411yescharacteristic
106D127—81no—
107#1656—1271nomaximal, Sylow
108C127#165611yescharacteristic, complemented, Sylow
109#562—1271no—
110#496—1271no—
111#496D12711yescharacteristic, complemented
112#490—1271no—
113#330—1271no—
114#329—1271no—
115#329—1271no—
116C2×SD16—21no—
117C2×SD16—2541no—
118C2×D8—21no—
119C2×D8—2541no—
120#127C2×D12711yescharacteristic
121#125—1271no—
122#124—1271no—
123#124C2×D12711yescharacteristic
124#123—2541no—
125#123—2541no—
126C22⋊Q8—2541no—
127C22⋊Q8—2541no—
128#121—2541no—
129#121—2541no—
130#102—2541no—
131#102—2541no—
132#97—1271no—
133#97—1271no—
134#97C2×D12711yescharacteristic
135C2×Q8—1271no—
136C2×Q8C22⋊D12711yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—1271no—
141C2×D4—2541no—
142C2×D4—2541no—
143C2×D4C22⋊D12711yescharacteristic
144C22×C4—2541no—
145C22×C4—2541no—
146SD16—42no—
147SD16—5082no—
148D8—42no—
149D8—5082no—
150C2×C8—21no—
151C2×C8—2541no—
152C2×C8—2541no—
153C4⋊C4—1271no—
154C4⋊C4—1271no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C22⋊C4—2541no—
165C42C22×D12711yescharacteristic
166C23—21no—
167C23—41no—
168C23—2541no—
169Q8—41no—
170Q8—5081no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—42no—
177D4—10161no—
178D4—10161no—
179D4—5081no—
180C2×C4—21no—
181C2×C4—10161no—
182C2×C4—10161no—
183C2×C4—2541no—
184C2×C4—2541no—
185C2×C4—2541no—
186C2×C4D4×D12711yescharacteristic
187C2×C4D4×D12711yescharacteristic
188C2×C4C2×C22⋊D12711yescharacteristic
189C8—22no—
190C8—5081no—
191C8—2542no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—5081no—
199C22—5081no—
200C22—5081no—
201C22#1686117711yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—5081no—
207C4—5081no—
208C4—5081no—
209C2—81no—
210C2—161no—
211C2—10161no—
212C2#2255004211yescharacteristic
213C2#2255004311yescharacteristic
214C2#2255004711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#496×C127C2
socleC22×C127#120
hypercenter#496D127
hypocenterC127#1656
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #22550620C2semidirect
2. #22550263C2semidirect
3. #22550240C2semidirect
4. #22549617C2semidirect
5. #22549616C2semidirect
6. #22549615C2semidirect
7. #496×C127C2semidirect
108. C127#1656semidirect
111. #496D127semidirect

Maximal subgroups

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

subgroupindex
#225506202
#225502632
#225502402
#225496172
#225496162
#225496152
#496×C1272
#1656127