Finite Group Databasebeta

← #30648653

Subgroups — #30648653

Order 16256.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#22550387C211yesmaximal, characteristic
2#22550211C212yesmaximal, complemented
3#22550206C212yesmaximal, complemented
4#22550194C212yesmaximal, complemented
5#22550189C212yesmaximal, complemented
6#22549916C212yesmaximal, complemented
7#22549875C212yesmaximal, complemented
8#22549840C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C2×#14523232C2212yes—
12#16861192—21no—
13#16861188C2211yescharacteristic
14#16861188C2211yescharacteristic
15#16861183C2211yescharacteristic
16#16861183C2211yescharacteristic
17#16861178—21no—
18C4×C22⋊D127—21no—
19C4×C22⋊D127—21no—
20#16861168C2211yescharacteristic
21#16861142C2214yescomplemented
22#16861140C2214yescomplemented
23#16861132C2214yescomplemented
24#16861129C2214yescomplemented
25C22⋊(C127⋊C8)C2212yes—
26C22⋊(C127⋊C8)C2212yes—
27C23×C4×C127—21no—
28C23×C4×C127C2211yescharacteristic
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127C2212yes—
33C2×C4⋊C4×C127C2212yes—
34C2×C4⋊C4×C127C2212yes—
35#14523248—24no—
36C2×Dic254C2311yescharacteristic
37C2×D508C2311yescharacteristic
38C2×C22⋊D127—21no—
39C2×C22⋊D127—21no—
40C2×C22⋊D127—21no—
41C2×C4×D127—21no—
42#14523232—24no—
43C2×C127⋊C8C2314yes—
44C4⋊Dic127C2311yescharacteristic
45C4⋊Dic127C2311yescharacteristic
46Dic127⋊C4—21no—
47Dic127⋊C4—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53C4×Dic127—21no—
54C24×C127D411yescharacteristic
55C22×C4×C127—24no—
56C22×C4×C127—22no—
57C22×C4×C127—22no—
58C22×C4×C127—22no—
59C22×C4×C127—44no—
60C22×C4×C127D411yescharacteristic
61C22×C4×C127D411yescharacteristic
62C22×C4×C127D411yescharacteristic
63C22×C4×C127C2311yescharacteristic
64C22×C4×C127C2311yescharacteristic
65C22×C4×C127C2311yescharacteristic
66C4⋊C4×C127—22no—
67C4⋊C4×C127—22no—
68C4⋊C4×C127—22no—
69C4⋊C4×C127C2314yes—
70C22×D127—21no—
71Dic254—22no—
72D508—22no—
73C22⋊D127—42no—
74C22⋊D127—42no—
75C22⋊D127—42no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C4×D127—42no—
80C127⋊C8—24no—
81C23×C127—22no—
82C23×C127—22no—
83C23×C127—22no—
84C23×C127C2×D411yescharacteristic
85C23×C127C2×D411yescharacteristic
86C23×C127C2×D411yescharacteristic
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—24no—
93C2×C4×C127—44no—
94C2×C4×C127—44no—
95C2×C4×C127—44no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127#4114yes—
100C2×C4×C127C2411yescharacteristic
101C22×C127—28no—
102C22×C127—21no—
103C22×C127—21no—
104C22×C127—21no—
105C22×C127—21no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127#12312yes—
109C22×C127AGL(1,ℤ/8)12yes—
110C22×C127#13712yes—
111C22×C127C22×D411yescharacteristic
112C2×D127—41no—
113C2×D127—41no—
114C2×D127—41no—
115C4×C127—21no—
116C4×C127—21no—
117C4×C127—44no—
118C4×C127#12312yes—
119C4×C127C2×#4112yes—
120Dic127—41no—
121Dic127—41no—
122Dic127—41no—
123C2×C127—22no—
124C2×C127—22no—
125C2×C127—22no—
126C2×C127C2×#12311yescharacteristic
127C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C127C2×#13711yescharacteristic
129D127—81no—
130#3092—1271nomaximal, Sylow
131C127#309211yescharacteristic, complemented, Sylow
132#525—1271no—
133C22×C4⋊C4D12711yescharacteristic, complemented
134#484—1272no—
135#483—1272no—
136#481—1272no—
137#480—1272no—
138#428—1272no—
139#417—1272no—
140#407—1271no—
141C2×#41—1271no—
142C23×C4—21no—
143C23×C4C2×D12711yescharacteristic
144C2×M16—1272no—
145#123—2541no—
146C22⋊Q8—1271no—
147C22⋊Q8—1271no—
148#121—1271no—
149#121—1271no—
150#120—2541no—
151C4×D4—2541no—
152C4×D4—2541no—
153#117—1271no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D12712yes—
158C2×C4⋊C4C2×D12712yes—
159C2×C4⋊C4C2×D12712yes—
160#107—1274no—
161#106—1274no—
162#103—1274no—
163#102—1274no—
164C22⋊C8—1272no—
165C22⋊C8—1272no—
166C24C22⋊D12711yescharacteristic
167#41—2544no—
168C2×Q8—1271no—
169C2×D4—1271no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—2541no—
179C22×C4C22⋊D12711yescharacteristic
180C22×C4C22⋊D12711yescharacteristic
181C22×C4C22⋊D12711yescharacteristic
182C22×C4C22×D12711yescharacteristic
183C22×C4C22×D12711yescharacteristic
184C22×C4C22×D12711yescharacteristic
185M16—2544no—
186C2×C8—1274no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—1271no—
191C4⋊C4—1271no—
192C4⋊C4—2541no—
193C4⋊C4—2541no—
194C4⋊C4C22×D12714yes—
195C22⋊C4—2541no—
196C22⋊C4—2541no—
197C22⋊C4—2541no—
198C22⋊C4—2541no—
199C22⋊C4—2541no—
200C42—2541no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—2541no—
205C23C2×C22⋊D12711yescharacteristic
206C23C2×C22⋊D12711yescharacteristic
207C23C2×C22⋊D12711yescharacteristic
208Q8—2542no—
209D4—5082no—
210D4—5082no—
211D4—5082no—
212D4—2542no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—5082no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2541no—
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4C2×C22⋊D12711yescharacteristic
228C2×C4C2×C22⋊D12711yescharacteristic
229C2×C4#1452324814yes—
230C2×C4C23×D12711yescharacteristic
231C8—2544no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—5081no—
240C22—5081no—
241C22—5081no—
242C22#1686119212yes—
243C22#1686123512yes—
244C22#1686124012yes—
245C22C22×C22⋊D12711yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—5081no—
250C4—5081no—
251C4—5081no—
252C4#1686119212yes—
253C4C2×#1452324812yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—10161no—
258C2C2×#1686119211yescharacteristic
259C2C2×#1686123511yescharacteristic
260C2C2×#1686124011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×C4⋊C4×C127C2
socleC22×C127C22×D4
prime radical#22550387C2
hypercenterC22×C4⋊C4D127
hypocenterC127#3092
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22550211C2semidirect
3. #22550206C2semidirect
4. #22550194C2semidirect
5. #22550189C2semidirect
6. #22549916C2semidirect
7. #22549875C2semidirect
8. #22549840C2semidirect
9. C22×C4⋊C4×C127C2semidirect
21. #16861142C22semidirect
22. #16861140C22semidirect
23. #16861132C22semidirect
24. #16861129C22semidirect
131. C127#3092semidirect
133. C22×C4⋊C4D127semidirect

Maximal subgroups

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

subgroupindex
#225503872
#225502112
#225502062
#225501942
#225501892
#225499162
#225498752
#225498402
C22×C4⋊C4×C1272
#3092127