Finite Group Databasebeta

← #30648377

Subgroups — #30648377

Order 16256.

524 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#14523233C211yesmaximal, characteristic, complemented
2C2×#16861217C211yesmaximal, characteristic, complemented
3#22550445C211yesmaximal, characteristic, complemented
4#22550443C211yesmaximal, characteristic, complemented
5#22550128C214yesmaximal, complemented
6#22550111C214yesmaximal, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549882C211yesmaximal, characteristic, complemented
9#418×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×D508C2211yescharacteristic
12C2×#14523233—24no—
13C2×#14523233C2212yes—
14C2×#14523233C2212yes—
15#16861217—24no—
16C2×#14523231C2212yescomplemented
17C22×C127⋊C8C2211yescharacteristic
18C2×C8×D127C2212yescomplemented
19#16861206—21no—
20#16861205—21no—
21#16861197—21no—
22#16861196—21no—
23#16861184C2214yescomplemented
24#16861180C2214yescomplemented
25#16861179—22no—
26#16861178—22no—
27#16861168C2211yescharacteristic
28#16861142C2212yescomplemented
29#16861140C2212yescomplemented
30#16861131C2214yescomplemented
31C22×D4×C127C2211yescharacteristic
32C2×M16×C127C2211yescharacteristic
33#117×C127C2211yescharacteristic
34#102×C127C2214yescomplemented
35C23×D127—21no—
36#14523248—22no—
37#14523248—22no—
38C2×Dic254D411yescharacteristic
39C2×D508—24no—
40C2×D508—22no—
41C2×D508—21no—
42C2×D508—22no—
43C2×D508D411yescharacteristic
44C2×D508C2312yes—
45C2×C22⋊D127—24no—
46C2×C22⋊D127D412yes—
47C2×C4×D127C2312yes—
48#14523233—24no—
49#14523233—24no—
50#14523231—24no—
51C2×C127⋊C8D414yescomplemented
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C8×D127—24no—
55C4⋊Dic127C2312yes—
56C22⋊Dic127—24no—
57C22⋊Dic127—21no—
58(C2×D127)⋊C4—24no—
59C4×Dic127—21no—
60C24×C127—21no—
61C2×D4×C127—22no—
62C2×D4×C127—21no—
63C2×D4×C127—22no—
64C2×D4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66M16×C127—22no—
67C2×C8×C127C2312yes—
68C4⋊C4×C127C2312yes—
69C22⋊C4×C127—21no—
70C42×C127—21no—
71C22×D127—22no—
72C22×D127—42no—
73C22×D127—82no—
74C22×D127C2×D412yes—
75Dic254—22no—
76D508—22no—
77D508—22no—
78D508—22no—
79D508—42no—
80D508—84no—
81C22⋊D127—24no—
82C22⋊D127—84no—
83C2×Dic127—22no—
84C2×Dic127C2×D412yes—
85C4×D127—22no—
86C4×D127—22no—
87C127⋊C8C2×D412yes—
88C127⋊C8C2×D412yes—
89C23×C127—22no—
90C23×C127—42no—
91C23×C127—82no—
92C23×C127C2×D411yescharacteristic
93D4×C127—22no—
94D4×C127—22no—
95D4×C127—42no—
96C2×C4×C127—22no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D412yes—
99C2×C4×C127#4112yes—
100C2×C4×C127C2411yescharacteristic
101C8×C127—22no—
102C22×C127—22no—
103C22×C127—42no—
104C22×C127—44no—
105C22×C127—84no—
106C22×C127#12112yes—
107C22×C127C22×D411yescharacteristic
108C2×D127—22no—
109C2×D127—24no—
110C2×D127—42no—
111C2×D127—44no—
112C2×D127—84no—
113C4×C127—42no—
114C4×C127#12112yes—
115C4×C127C22×D411yescharacteristic
116C4×C127C2×#4111yescharacteristic
117Dic127—22no—
118Dic127—42no—
119C2×C127—22no—
120C2×C127—82no—
121C2×C127C2×#12111yescharacteristic
122C2×C127#57612yes—
123D127—42no—
124D127—82no—
125#3058—1271nomaximal, Sylow
126C127#305811yescharacteristic, complemented, Sylow
127C22×D8—1271no—
128C2×#131—1271no—
129#535—1271no—
130#535—1271no—
131#469—1274no—
132#466—1274no—
133#427—1271no—
134#418—1271no—
135#418D12711yescharacteristic, complemented
136C2×#41—1271no—
137C22×D4—1271no—
138C22×D4C2×D12711yescharacteristic
139C2×D8—1272no—
140C2×D8—1272no—
141C2×D8—2544no—
142#131—2544no—
143C2×M16—1272no—
144C2×M16C2×D12711yescharacteristic
145C22×C8—1271no—
146C22×C8—1272no—
147#127—2541no—
148#127—2541no—
149#124—2541no—
150#124—2541no—
151#121—1274no—
152#121—1274no—
153#120—2542no—
154#120—2542no—
155#117—1271no—
156#117C2×D12711yescharacteristic
157#107—1272no—
158#106—1272no—
159#102—1274no—
160#102C2×D12714yescomplemented
161C24—21no—
162C24—2541no—
163#41—2542no—
164#41—2542no—
165C2×Q8—1271no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—22no—
169C2×D4—1272no—
170C2×D4—1271no—
171C2×D4—1272no—
172C2×D4—2544no—
173C2×D4—2544no—
174C2×D4—2541no—
175C2×D4—2542no—
176C2×D4—2542no—
177C2×D4C22×D12712yes—
178C22×C4—1272no—
179C22×C4C22×D12711yescharacteristic
180D8—2544no—
181D8—2544no—
182M16—22no—
183M16—2544no—
184C2×C8—1274no—
185C2×C8—1271no—
186C2×C8—1271no—
187C2×C8—2544no—
188C2×C8C22×D12712yes—
189C4⋊C4—1272no—
190C4⋊C4C22×D12712yes—
191C22⋊C4—21no—
192C22⋊C4—2544no—
193C22⋊C4—2544no—
194C22⋊C4—2541no—
195C42—21no—
196C42—2541no—
197C23—22no—
198C23—42no—
199C23—82no—
200C23—1272no—
201C23—10162no—
202C23—5082no—
203C23—2542no—
204C23D4×D12711yescharacteristic
205Q8—2542no—
206D4—22no—
207D4—22no—
208D4—42no—
209D4—10164no—
210D4—10164no—
211D4—5082no—
212D4—2544no—
213D4—2542no—
214D4—2542no—
215D4—2542no—
216C2×C4—22no—
217C2×C4—1272no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4D4×D12712yes—
222C2×C4D4×D12711yescharacteristic
223C2×C4#1452324812yes—
224C2×C4C23×D12711yescharacteristic
225C8—22no—
226C8—1272no—
227C8—1272no—
228C22—22no—
229C22—42no—
230C22—44no—
231C22—84no—
232C22—10164no—
233C22—5082no—
234C22—5084no—
235C22—2542no—
236C22—2544no—
237C22#1686118212yes—
238C22C2×D4×D12711yescharacteristic
239C4—42no—
240C4—5082no—
241C4—2542no—
242C4#1686118212yes—
243C4C2×D4×D12711yescharacteristic
244C4C2×#1452324811yescharacteristic
245C2—22no—
246C2—82no—
247C2—10162no—
248C2—5082no—
249C2C2×#1686118211yescharacteristic
250C2#2255068012yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
hypercenter#418D127
hypocenterC127#3058
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#14523233C2semidirect
2. C2×#16861217C2semidirect
3. #22550445C2semidirect
4. #22550443C2semidirect
5. #22550128C2semidirect
6. #22550111C2semidirect
7. #22549912C2semidirect
8. #22549882C2semidirect
9. #418×C127C2semidirect
16. C2×#14523231C22semidirect
18. C2×C8×D127C22semidirect
23. #16861184C22semidirect
24. #16861180C22semidirect
28. #16861142C22semidirect
29. #16861140C22semidirect
30. #16861131C22semidirect
34. #102×C127C22semidirect
51. C2×C127⋊C8D4semidirect
126. C127#3058semidirect
135. #418D127semidirect
160. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168612172
#225504452
#225504432
#225501282
#225501112
#225499122
#225498822
#418×C1272
#3058127