Finite Group Databasebeta

← #30648007

Subgroups — #30648007

Order 16256.

752 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#16861239C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3#22550447C211yesmaximal, characteristic, complemented
4#22550038C214yesmaximal, complemented
5#22550027C214yesmaximal, complemented
6#22549881C211yesmaximal, characteristic, complemented
7#22549872C211yesmaximal, characteristic, complemented
8#22549843C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic, complemented
11#16861239—24no—
12C2×#14523238C2212yescomplemented
13C2×#14523236—24no—
14C2×#14523236C2212yescomplemented
15C2×#14523236C2214yescomplemented
16C2×#14523232C2211yescharacteristic, complemented
17C22×C127⋊C8C2211yescharacteristic, complemented
18#16861207—21no—
19#16861196—21no—
20#16861190—22no—
21#16861188C2214yescomplemented
22#16861168C2211yescharacteristic, complemented
23#16861133C2212yescomplemented
24#16861130C2212yescomplemented
25#16861130C2214yescomplemented
26C22⋊(C127⋊C8)C2214yescomplemented
27H(5,𝔽2)×C127—24no—
28H(5,𝔽2)×C127—24no—
29C2×#41×C127—21no—
30C2×#41×C127C2212yes—
31C2×#41×C127C2211yescharacteristic
32C2×#41×C127C2211yescharacteristic
33C22×D4×C127—22no—
34C22×D4×C127—21no—
35C22×D4×C127C2212yes—
36C22×D4×C127C2211yescharacteristic
37C2×Dic254—21no—
38C2×Dic254—22no—
39C2×Dic254—22no—
40C2×Dic254C2312yes—
41C22×Dic127—21no—
42#14523238—24no—
43#14523236—28no—
44#14523236—24no—
45#14523232—22no—
46C2×C127⋊C8—22no—
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2312yes—
49C4⋊Dic127C2312yes—
50Dic127⋊C4—24no—
51C22⋊Dic127—24no—
52C22⋊Dic127—21no—
53C4×Dic127—21no—
54C24×C127—22no—
55C24×C127—21no—
56#41×C127—28no—
57#41×C127—24no—
58#41×C127—24no—
59#41×C127—22no—
60#41×C127—22no—
61#41×C127—22no—
62#41×C127D414yes—
63C2×Q8×C127D411yescharacteristic
64C2×Q8×C127C2311yescharacteristic
65C2×D4×C127—216no—
66C2×D4×C127—28no—
67C2×D4×C127—22no—
68C2×D4×C127—21no—
69C2×D4×C127—24no—
70C2×D4×C127—24no—
71C2×D4×C127—22no—
72C2×D4×C127—21no—
73C2×D4×C127—22no—
74C2×D4×C127D412yes—
75C2×D4×C127D411yescharacteristic
76C2×D4×C127D414yes—
77C2×D4×C127C2311yescharacteristic
78C2×D4×C127C2312yes—
79C22×C4×C127—22no—
80C22×C4×C127—21no—
81C22×C4×C127C2312yes—
82C22×C4×C127C2311yescharacteristic
83Dic254—22no—
84Dic254—22no—
85Dic254—42no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—42no—
89C127⋊C8—22no—
90C127⋊C8—22no—
91C23×C127—28no—
92C23×C127—22no—
93C23×C127—21no—
94C23×C127—24no—
95C23×C127—82no—
96C23×C127—84no—
97C23×C127C2×D412yes—
98C23×C127C2×D411yescharacteristic
99Q8×C127—22no—
100Q8×C127—21no—
101Q8×C127C2×D412yes—
102D4×C127—28no—
103D4×C127—28no—
104D4×C127—24no—
105D4×C127—24no—
106D4×C127—22no—
107D4×C127—24no—
108D4×C127—22no—
109D4×C127—21no—
110D4×C127C2×D412yes—
111D4×C127C2×D414yes—
112C2×C4×C127—28no—
113C2×C4×C127—21no—
114C2×C4×C127—24no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2411yescharacteristic
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—24no—
125C22×C127—22no—
126C22×C127—24no—
127C22×C127—44no—
128C22×C127—42no—
129C22×C127—84no—
130C22×C127—88no—
131C22×C127#12012yes—
132C22×C127C22×D411yescharacteristic
133C4×C127—22no—
134C4×C127—22no—
135C4×C127#12012yes—
136C4×C127C22×D411yescharacteristic
137C4×C127C22×D411yescharacteristic
138Dic127—42no—
139Dic127—42no—
140C2×C127—22no—
141C2×C127—42no—
142C2×C127—44no—
143C2×C127—42no—
144C2×C127C2×#12011yescharacteristic
145C2×C127#57712yes—
146#3009—1271nomaximal, Sylow
147C127#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)D12711yescharacteristic
149C2×#137—1271no—
150C22×SD16—1271no—
151#536—1271no—
152#452—1274no—
153#450—1274no—
154#418—1271no—
155#416—1271no—
156#408—1271no—
157H(5,𝔽2)—24no—
158H(5,𝔽2)—24no—
159C2×#41—21no—
160C2×#41C2×D12712yes—
161C2×#41C2×D12711yescharacteristic
162C2×#41C2×D12711yescharacteristic
163C22×Q8—1271no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4C2×D12712yes—
167C22×D4C2×D12711yescharacteristic
168#137—2544no—
169C2×Q16—1272no—
170C2×SD16—1274no—
171C2×SD16—1272no—
172C2×SD16—2544no—
173C2×M16—1271no—
174C22×C8—1271no—
175C4⋊Q8—2541no—
176#124—2541no—
177#123—2542no—
178C22⋊Q8—1274no—
179#117—1271no—
180#103—1272no—
181#102—1272no—
182#102—1274no—
183C22⋊C8—1274no—
184C24—22no—
185C24—21no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41C22⋊D12714yes—
193C2×Q8—1272no—
194C2×Q8—2542no—
195C2×Q8—2542no—
196C2×Q8—2541no—
197C2×Q8C22⋊D12711yescharacteristic
198C2×Q8C22×D12711yescharacteristic
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4C22⋊D12712yes—
209C2×D4C22⋊D12711yescharacteristic
210C2×D4C22⋊D12714yes—
211C2×D4C22×D12711yescharacteristic
212C2×D4C22×D12712yes—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—2541no—
216C22×C4C22×D12712yes—
217C22×C4C22×D12711yescharacteristic
218Q16—2544no—
219SD16—2548no—
220SD16—2544no—
221M16—2542no—
222C2×C8—1272no—
223C2×C8—1272no—
224C2×C8—2542no—
225C4⋊C4—1272no—
226C4⋊C4—2544no—
227C22⋊C4—2541no—
228C22⋊C4—2544no—
229C42—2541no—
230C23—28no—
231C23—22no—
232C23—21no—
233C23—24no—
234C23—82no—
235C23—84no—
236C23C2×C22⋊D12712yes—
237C23C2×C22⋊D12711yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—5082no—
241Q8—2542no—
242Q8—2542no—
243Q8C2×C22⋊D12712yes—
244D4—28no—
245D4—28no—
246D4—24no—
247D4—24no—
248D4—22no—
249D4—21no—
250D4—22no—
251D4—24no—
252D4C2×C22⋊D12712yes—
253D4C2×C22⋊D12714yes—
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—5082no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4C2×C22⋊D12712yes—
263C2×C4C2×C22⋊D12712yes—
264C2×C4C2×C22⋊D12712yes—
265C2×C4C2×C22⋊D12711yescharacteristic
266C2×C4C23×D12711yescharacteristic
267C8—2542no—
268C8—2542no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—22no—
273C22—24no—
274C22—44no—
275C22—42no—
276C22—84no—
277C22—88no—
278C22#1686117812yes—
279C22C22×C22⋊D12711yescharacteristic
280C4—22no—
281C4—22no—
282C4—5082no—
283C4—5082no—
284C4#1686117812yes—
285C4C22×C22⋊D12711yescharacteristic
286C4C22×C22⋊D12711yescharacteristic
287C2—22no—
288C2—42no—
289C2—44no—
290C2—42no—
291C2C2×#1686117811yescharacteristic
292C2#2255068712yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×H(5,𝔽2)×C127C2
socleC22×C127C22×D4
hypercenterC2×H(5,𝔽2)D127
hypocenterC127#3009
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C22×#14523236C2semidirect
3. #22550447C2semidirect
4. #22550038C2semidirect
5. #22550027C2semidirect
6. #22549881C2semidirect
7. #22549872C2semidirect
8. #22549843C2semidirect
10. C22×Dic254C22semidirect
12. C2×#14523238C22semidirect
14. C2×#14523236C22semidirect
15. C2×#14523236C22semidirect
16. C2×#14523232C22semidirect
17. C22×C127⋊C8C22semidirect
21. #16861188C22semidirect
22. #16861168C22semidirect
23. #16861133C22semidirect
24. #16861130C22semidirect
25. #16861130C22semidirect
26. C22⋊(C127⋊C8)C22semidirect
147. C127#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232362
#225504472
#225500382
#225500272
#225498812
#225498722
#225498432
C2×H(5,𝔽2)×C1272
#3009127