Finite Group Databasebeta

← #30637719

Subgroups — #30637719

Order 16256.

228 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550501C211yesmaximal, characteristic
2#22550206C211yesmaximal, characteristic, complemented
3#22549802C211yesmaximal, characteristic
4#22549614C211yesmaximal, characteristic, complemented
5#22549613C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic
7#483×C127C211yesmaximal, characteristic, complemented
8#16861189—21no—
9#16861188C2211yescharacteristic
10#16861187—21no—
11#16861187—21no—
12#16861186—21no—
13#16861185—21no—
14C2×(C2×D127)⋊C4—21no—
15C2×(C2×D127)⋊C4C2211yescharacteristic
16C22⋊C4×D127—21no—
17#16861140—21no—
18#16861132—21no—
19C22⋊(C127⋊C8)C2211yescharacteristic
20C127⋊#95—21no—
21C127⋊#95C2211yescharacteristic
22C22⋊Q8×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#106×C127—21no—
25#103×C127—21no—
26C22⋊C8×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254D411yescharacteristic
30C22×Dic127—21no—
31C22×Dic127D411yescharacteristic
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40C22⋊Dic127—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—42no—
46C2×Q8×C127D411yescharacteristic
47C22×C4×C127—21no—
48C22×C4×C127C2311yescharacteristic
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127D411yescharacteristic
54C22⋊C4×C127—21no—
55C22×D127—21no—
56C22×D127—21no—
57C22×D127—41no—
58C22×D127—41no—
59C22×D127SD1612yes—
60Dic254—41no—
61Dic254—81no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C4×D127—81no—
70C4×D127—81no—
71C127⋊C8—41no—
72C23×C127C2×D411yescharacteristic
73Q8×C127—41no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C8×C127—41no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127#12011yescharacteristic
86C22×C127C2×SD1611yescharacteristic
87C22×C127#13711yescharacteristic
88C2×D127—22no—
89C2×D127—41no—
90C2×D127—41no—
91C2×D127—42no—
92C2×D127—81no—
93C4×C127—21no—
94C4×C127—21no—
95C4×C127—41no—
96C4×C127—41no—
97C4×C127—81no—
98Dic127—41no—
99Dic127—41no—
100Dic127—41no—
101Dic127—81no—
102C2×C127—21no—
103C2×C127—21no—
104C2×C127#44811yescharacteristic
105C2×C127#45511yescharacteristic
106C2×C127#45811yescharacteristic
107D127—42no—
108#1608—1271nomaximal, Sylow
109C127#160811yescharacteristic, complemented, Sylow
110#543—1271no—
111#483—1271no—
112#483D12711yescharacteristic, complemented
113#397—1271no—
114#328—1271no—
115#328—1271no—
116#323—1271no—
117C22⋊Q8—1271no—
118C22⋊Q8—2541no—
119C22⋊Q8—2541no—
120C22⋊Q8—2541no—
121C22⋊Q8—2541no—
122C22⋊Q8—2541no—
123C22⋊Q8C2×D12711yescharacteristic
124C2×C4⋊C4C2×D12711yescharacteristic
125C2×C22⋊C4—1271no—
126C2×C22⋊C4—2541no—
127C2×C22⋊C4—2541no—
128#106—21no—
129#106—2541no—
130#103—21no—
131#103—2541no—
132C22⋊C8—1271no—
133C22⋊C8C2×D12711yescharacteristic
134#95—1271no—
135#95—2541no—
136C24—1271no—
137C2×Q8—1271no—
138C2×Q8—2541no—
139C2×Q8C22⋊D12711yescharacteristic
140C22×C4—21no—
141C22×C4—1271no—
142C22×C4—2541no—
143C22×C4—2541no—
144C22×C4—2541no—
145C22×C4C22×D12711yescharacteristic
146C2×C8—21no—
147C2×C8—2541no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—1271no—
152C4⋊C4—2541no—
153C4⋊C4—2541no—
154C4⋊C4—2541no—
155C4⋊C4—2541no—
156C4⋊C4C22⋊D12711yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—5082no—
159C22⋊C4—2541no—
160C22⋊C4—2541no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C23—1272no—
165C23—5081no—
166C23—5081no—
167C23—2541no—
168C23—2541no—
169C23D4×D12711yescharacteristic
170Q8—41no—
171Q8—10161no—
172Q8—5081no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—10161no—
180C2×C4—10161no—
181C2×C4—5081no—
182C2×C4—5081no—
183C2×C4—5081no—
184C2×C4—5081no—
185C2×C4—2541no—
186C2×C4—2541no—
187C2×C4—2541no—
188C2×C4D4×D12711yescharacteristic
189C2×C4C2×C22⋊D12711yescharacteristic
190C8—41no—
191C8—5081no—
192C22—21no—
193C22—21no—
194C22—10161no—
195C22—5082no—
196C22—5081no—
197C22—5081no—
198C22—2542no—
199C22#1686117711yescharacteristic
200C22SD16×D12711yescharacteristic
201C22#1686123811yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—10161no—
208C4—5081no—
209C4—5081no—
210C4—5081no—
211C2—21no—
212C2—21no—
213C2—5082no—
214C2#2255001311yescharacteristic
215C2#2255005211yescharacteristic
216C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#483×C127C2
socleC22×C127#120
prime radicalC23×D127D4
hypercenter#483D127
hypocenterC127#1608
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. #22550206C2semidirect
4. #22549614C2semidirect
5. #22549613C2semidirect
7. #483×C127C2semidirect
109. C127#1608semidirect
112. #483D127semidirect

Maximal subgroups

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

subgroupindex
#225505012
#225502062
#225498022
#225496142
#225496132
#225495992
#483×C1272
#1608127