Finite Group Databasebeta

← #30638035

Subgroups — #30638035

Order 16256.

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

normal subgroups only

This group has 228 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#22550618C211yesmaximal, characteristic
2#22550268C211yesmaximal, characteristic, complemented
3#22550238C211yesmaximal, characteristic, complemented
4#22549616C211yesmaximal, characteristic, complemented
5#22549615C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic, complemented
7#489×C127C211yesmaximal, characteristic, complemented
8C2×#14523238—21no—
9C2×#14523235—21no—
10#16861198—21no—
11#16861197C2211yescharacteristic
12#16861197C2211yescharacteristic
13#16861195—21no—
14#16861195—21no—
15#16861179—21no—
16#16861179—21no—
17#16861177—21no—
18#16861132—21no—
19#16861129—21no—
20#16861120C2211yescharacteristic, complemented
21#16861120C2211yescharacteristic, complemented
22C4⋊Q8×C127C2211yescharacteristic, complemented
23#124×C127C2211yescharacteristic
24#103×C127—21no—
25#102×C127—21no—
26#97×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C2×Dic254D411yescharacteristic
29C2×Dic254D411yescharacteristic
30C2×D508—21no—
31C2×D508D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35#14523238—42no—
36#14523235—42no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C22⋊Dic127—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46C4×Dic127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127D411yescharacteristic
49C2×D4×C127D411yescharacteristic
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—
57C22×D127—21no—
58C22×D127—81no—
59C22×D127—81no—
60C22×D127—81no—
61Dic254—41no—
62Dic254—41no—
63D508—41no—
64D508—41no—
65D508—81no—
66C22⋊D127—81no—
67C22⋊D127—81no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C127⋊C8—22no—
71C127⋊C8—41no—
72C23×C127—21no—
73Q8×C127—41no—
74Q8×C127—41no—
75Q8×C127—41no—
76D4×C127—41no—
77C2×C4×C127—21no—
78C2×C4×C127—41no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C8×C127—41no—
83C22×C127—41no—
84C22×C127—41no—
85C22×C127—41no—
86C22×C127#12011yescharacteristic
87C2×D127—41no—
88C2×D127—41no—
89C2×D127—41no—
90C2×D127—41no—
91C2×D127—41no—
92C2×D127—41no—
93C2×D127—84no—
94C4×C127—21no—
95C4×C127—21no—
96C4×C127—21no—
97C4×C127—41no—
98C4×C127—81no—
99Dic127—41no—
100Dic127—41no—
101C2×C127—81no—
102C2×C127#45311yescharacteristic
103C2×C127#45511yescharacteristic
104C2×C127#45511yescharacteristic
105D127—81no—
106D127—81no—
107#1659—1271nomaximal, Sylow
108C127#165911yescharacteristic, complemented, Sylow
109H(3,𝔽4)—1271no—
110#497—1271no—
111#489—1271no—
112#489D12711yescharacteristic, complemented
113#329—1271no—
114#329—1271no—
115#329—1271no—
116C2×Q16—2541no—
117C2×SD16—2541no—
118C4⋊Q8C2×D12711yescharacteristic, complemented
119#124—1271no—
120#124—1271no—
121#124—2541no—
122#124—2541no—
123#124—2541no—
124#124C2×D12711yescharacteristic
125#120—2541no—
126#120—2541no—
127#120—2541no—
128#103—21no—
129#103—2541no—
130#102—21no—
131#102—2541no—
132#97—1271no—
133#97—1271no—
134#97C2×D12711yescharacteristic, complemented
135C24—2541no—
136C2×Q8—21no—
137C2×Q8—1271no—
138C2×Q8—1271no—
139C2×Q8C22⋊D12711yescharacteristic
140C2×D4—1271no—
141C2×D4—1271no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4C22⋊D12711yescharacteristic
146Q16—5082no—
147SD16—5082no—
148C2×C8—21no—
149C2×C8—2541no—
150C2×C8—2541no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—2541no—
156C22⋊C4—2541no—
157C22⋊C4—2541no—
158C22⋊C4—2541no—
159C22⋊C4—2541no—
160C22⋊C4—2541no—
161C22⋊C4—2541no—
162C42—2541no—
163C42C22×D12711yescharacteristic
164C23—21no—
165C23—10161no—
166C23—10161no—
167C23—10161no—
168C23—2541no—
169C23—2541no—
170Q8—41no—
171Q8—41no—
172Q8—41no—
173Q8—5081no—
174Q8—5081no—
175D4—41no—
176D4—10161no—
177D4—10161no—
178D4—10161no—
179D4—5081no—
180D4—5081no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—2541no—
184C2×C4—2541no—
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C8—41no—
189C8—5081no—
190C8—2542no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—10164no—
195C22—5081no—
196C22—5081no—
197C22—5081no—
198C22—5081no—
199C22—5081no—
200C22—5081no—
201C22#1686117711yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—81no—
207C4—5081no—
208C4—5081no—
209C2—81no—
210C2—10161no—
211C2—10161no—
212C2#2255004211yescharacteristic
213C2#2255005211yescharacteristic
214C2#2255005311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#489×C127C2
socleC22×C127#120
prime radical#22550618C2
hypercenter#489D127
hypocenterC127#1659
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #22550268C2semidirect
3. #22550238C2semidirect
4. #22549616C2semidirect
5. #22549615C2semidirect
6. #22549615C2semidirect
7. #489×C127C2semidirect
20. #16861120C22semidirect
21. #16861120C22semidirect
22. C4⋊Q8×C127C22semidirect
26. #97×C127C22semidirect
108. C127#1659semidirect
112. #489D127semidirect
118. C4⋊Q8C2×D127semidirect
134. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506182
#225502682
#225502382
#225496162
#225496152
#489×C1272
#1659127