Finite Group Databasebeta

← #30638038

Subgroups — #30638038

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#22550270C211yesmaximal, characteristic, complemented
3#22550236C211yesmaximal, characteristic, complemented
4#22550235C211yesmaximal, characteristic, complemented
5#22549615C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic, complemented
7#329×C127C211yesmaximal, characteristic, complemented
8C2×Dic508—21no—
9C2×#14523237—21no—
10#16861209C2211yescharacteristic, complemented
11#16861198—21no—
12#16861197C2211yescharacteristic
13#16861197C2211yescharacteristic
14#16861195—21no—
15#16861195—21no—
16#16861179—21no—
17#16861179—21no—
18#16861177—21no—
19#16861134—21no—
20#16861133—21no—
21#16861131—21no—
22#16861130—21no—
23#16861120C2211yescharacteristic, complemented
24#124×C127C2211yescharacteristic
25#97×C127C2211yescharacteristic, complemented
26#97×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C2×Dic254—21no—
29C2×Dic254D411yescharacteristic
30C2×Dic254D411yescharacteristic
31C2×D508—21no—
32C2×D508D411yescharacteristic
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36Dic508—42no—
37#14523237—42no—
38C2×C127⋊C8—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48C4×Dic127—21no—
49C2×Q8×C127D411yescharacteristic
50C2×D4×C127D411yescharacteristic
51C2×C8×C127—21no—
52C2×C8×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—
63Dic254—41no—
64Dic254—41no—
65D508—41no—
66D508—41no—
67D508—81no—
68C22⋊D127—81no—
69C22⋊D127—81no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—41no—
73C127⋊C8—41no—
74C23×C127—21no—
75Q8×C127—41no—
76D4×C127—41no—
77C2×C4×C127—21no—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C8×C127—22no—
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—
98Dic127—41no—
99Dic127—41no—
100Dic127—81no—
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#489—1271no—
113#329—1271no—
114#329—1271no—
115#329D12711yescharacteristic, complemented
116C2×Q16—2541no—
117C2×SD16—2541no—
118C4⋊Q8—1271no—
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—2541no—
129#103—2541no—
130#102—2541no—
131#102—2541no—
132#97—1271no—
133#97C2×D12711yescharacteristic, complemented
134#97C2×D12711yescharacteristic, complemented
135C24—2541no—
136C2×Q8—1271no—
137C2×Q8—1271no—
138C2×Q8—2541no—
139C2×Q8D50811yescharacteristic
140C2×D4—1271no—
141C2×D4—1271no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4D50811yescharacteristic
146Q16—5082no—
147SD16—5082no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—2541no—
151C4⋊C4—2541no—
152C4⋊C4—2541no—
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—5081no—
172Q8—5081no—
173Q8—5081no—
174Q8—5081no—
175D4—41no—
176D4—10161no—
177D4—10161no—
178D4—10161no—
179D4—5081no—
180D4—5081no—
181C2×C4—21no—
182C2×C4—5081no—
183C2×C4—2541no—
184C2×C4—2541no—
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×D50811yescharacteristic
188C8—22no—
189C8—41no—
190C8—5081no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—10164no—
195C22—5081no—
196C22—5081no—
197C22—5081no—
198C22—5081no—
199C22—5081no—
200C22—5081no—
201C22#1686117911yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—10161no—
207C4—5081no—
208C4—5081no—
209C2—81no—
210C2—10161no—
211C2—10161no—
212C2#2255004511yescharacteristic
213C2#2255005511yescharacteristic
214C2#2255005611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127#120
prime radical#22550618C2
hypercenter#329D127
hypocenterC127#1659
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22550270C2semidirect
3. #22550236C2semidirect
4. #22550235C2semidirect
5. #22549615C2semidirect
6. #22549615C2semidirect
7. #329×C127C2semidirect
10. #16861209C22semidirect
23. #16861120C22semidirect
25. #97×C127C22semidirect
26. #97×C127C22semidirect
108. C127#1659semidirect
115. #329D127semidirect
133. #97C2×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
#225502702
#225502362
#225502352
#225496152
#329×C1272
#1659127