Finite Group Databasebeta

← #30637777

Subgroups — #30637777

Order 16256.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550466C211yesmaximal, characteristic
2#22550194C211yesmaximal, characteristic, complemented
3#22549811C211yesmaximal, characteristic, complemented
4#22549613C211yesmaximal, characteristic, complemented
5#22549611C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#484×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861198—21no—
10#16861194—21no—
11#16861184—21no—
12#16861183C2211yescharacteristic
13#16861179—21no—
14(C22⋊D127)⋊C4—21no—
15C2×Dic127⋊C4—21no—
16C2×(C2×D127)⋊C4—21no—
17C2×(C2×D127)⋊C4C2211yescharacteristic
18#16861140—21no—
19#16861129—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C127⋊#95C2211yescharacteristic
22C22⋊Q8×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#107×C127—21no—
25#103×C127—21no—
26C22⋊C8×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×D508—21no—
29C2×D508—21no—
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C2×C4×D127—21no—
35C2×C127⋊C8—21no—
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—21no—
38Dic127⋊C4—42no—
39(C2×D127)⋊C4—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—42no—
46C4×Dic127—21no—
47C2×Q8×C127D411yescharacteristic
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50C2×C8×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127D411yescharacteristic
55C22⋊C4×C127—21no—
56C22×D127—21no—
57C22×D127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—41no—
61C22×D127—41no—
62D508—41no—
63D508—81no—
64D508—81no—
65C22⋊D127—41no—
66C22⋊D127—41no—
67C2×Dic127—22no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C4×D127—81no—
74C127⋊C8—41no—
75C23×C127C2×D411yescharacteristic
76Q8×C127—41no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C8×C127—41no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127#12011yescharacteristic
89C22×C127#13511yescharacteristic
90C22×C127#13711yescharacteristic
91C2×D127—42no—
92C2×D127—41no—
93C2×D127—41no—
94C2×D127—41no—
95C2×D127—41no—
96C2×D127—41no—
97C2×D127—41no—
98C2×D127—81no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—41no—
102C4×C127—41no—
103C4×C127—81no—
104Dic127—42no—
105Dic127—41no—
106Dic127—41no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127#45211yescharacteristic
110C2×C127#45311yescharacteristic
111C2×C127#45811yescharacteristic
112D127—81no—
113D127—81no—
114#1618—1271nomaximal, Sylow
115C127#161811yescharacteristic, complemented, Sylow
116#538—1271no—
117#484D12711yescharacteristic, complemented
118#481—1271no—
119#399—1271no—
120#328—1271no—
121#327—1271no—
122#323—1271no—
123#126—2541no—
124#124—2541no—
125#123—2541no—
126C22⋊Q8C2×D12711yescharacteristic
127#121—1271no—
128#121—2541no—
129#120—2541no—
130C4×D4—2541no—
131C2×C4⋊C4—2541no—
132C2×C4⋊C4C2×D12711yescharacteristic
133C2×C22⋊C4—1271no—
134C2×C22⋊C4—2541no—
135#107—21no—
136#106—2541no—
137#103—21no—
138#102—2541no—
139C22⋊C8—1271no—
140C22⋊C8C2×D12711yescharacteristic
141#95—1271no—
142C24—1271no—
143C2×Q8C22⋊D12711yescharacteristic
144C2×D4—1271no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4—2541no—
148C22×C4—21no—
149C22×C4—1271no—
150C22×C4—2541no—
151C22×C4—2541no—
152C22×C4C22×D12711yescharacteristic
153C2×C8—21no—
154C2×C8—2541no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—1271no—
159C4⋊C4—5082no—
160C4⋊C4—2541no—
161C4⋊C4C22⋊D12711yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—5082no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C22⋊C4—2541no—
169C22⋊C4—2541no—
170C42—2541no—
171C23—5081no—
172C23—5081no—
173C23—2541no—
174C23—2541no—
175C23—2541no—
176C23—2541no—
177C23D4×D12711yescharacteristic
178Q8—41no—
179D4—10161no—
180D4—10161no—
181D4—5081no—
182D4—5081no—
183D4—5081no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—10161no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2542no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4D4×D12711yescharacteristic
198C2×C4C2×C22⋊D12711yescharacteristic
199C8—41no—
200C8—5081no—
201C22—21no—
202C22—21no—
203C22—10161no—
204C22—5082no—
205C22—5081no—
206C22—5081no—
207C22—5081no—
208C22—5081no—
209C22—5081no—
210C22—5081no—
211C22#1686117711yescharacteristic
212C22#1686123211yescharacteristic
213C22#1686124111yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—5082no—
220C4—5081no—
221C4—5081no—
222C2—21no—
223C2—21no—
224C2—10161no—
225C2—10161no—
226C2#2255003711yescharacteristic
227C2#2255004211yescharacteristic
228C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#484×C127C2
socleC22×C127#120
prime radical#22550466C2
hypercenter#484D127
hypocenterC127#1618
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #22550194C2semidirect
3. #22549811C2semidirect
4. #22549613C2semidirect
5. #22549611C2semidirect
6. #22549599C2semidirect
7. #484×C127C2semidirect
115. C127#1618semidirect
117. #484D127semidirect

Maximal subgroups

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

subgroupindex
#225504662
#225501942
#225498112
#225496132
#225496112
#225495992
#484×C1272
#1618127