Finite Group Databasebeta

← #30647949

Subgroups — #30647949

Order 16256.

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

normal subgroups only

This group has 764 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
1C22×#14523247C211yesmaximal, characteristic, complemented
2C2×#16861230C212yesmaximal, complemented
3#22550389C211yesmaximal, characteristic, complemented
4#22550035C212yesmaximal, complemented
5#22550031C211yesmaximal, characteristic, complemented
6#22550022C212yesmaximal, complemented
7#22550017C211yesmaximal, characteristic, complemented
8#22550010C211yesmaximal, characteristic, complemented
9#22549869C212yesmaximal, complemented
10C22⋊C8×D127C211yesmaximal, characteristic, complemented
11#450×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2211yescharacteristic, complemented
13C2×#14523247—28no—
14C2×#14523247—21no—
15C2×#14523247—21no—
16C2×#14523247C2212yes—
17C2×#14523247C2212yes—
18C22×Dic254C2211yescharacteristic
19C22×C22⋊D127—21no—
20C23×Dic127—21no—
21C22×C4×D127C2211yescharacteristic
22#16861230—28no—
23C2×#14523238C2212yescomplemented
24C2×#14523237C2212yescomplemented
25C2×#14523233C2212yescomplemented
26C2×C8×D127C2212yescomplemented
27#16861194—21no—
28#16861185C2211yescharacteristic, complemented
29#16861184C2211yescharacteristic, complemented
30#16861183C2211yescharacteristic
31#16861179—21no—
32#16861171—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167C2211yescharacteristic, complemented
35#16861134C2212yescomplemented
36#16861132C2212yescomplemented
37#16861130C2212yescomplemented
38C22⋊(C127⋊C8)C2211yescharacteristic
39(C2×D127)⋊C8C2212yescomplemented
40C22×D4×C127C2211yescharacteristic
41C2×SD16×C127C2212yescomplemented
42C22⋊Q8×C127C2211yescharacteristic
43#102×C127C2212yescomplemented
44C22⋊C8×C127C2211yescharacteristic
45C23×D127D411yescharacteristic
46#14523249—24no—
47#14523247—28no—
48#14523247—28no—
49#14523247—24no—
50#14523247—24no—
51#14523247—24no—
52#14523247D418yescomplemented
53C2×Dic254—24no—
54C2×Dic254—21no—
55C2×Dic254C2312yes—
56C2×D508—21no—
57C2×D508—21no—
58C2×D508C2311yescharacteristic
59C2×C22⋊D127—28no—
60C2×C22⋊D127—22no—
61C2×C22⋊D127—22no—
62C2×C22⋊D127—21no—
63C22×Dic127—28no—
64C22×Dic127—22no—
65C22×Dic127—22no—
66C22×Dic127—21no—
67C22×Dic127D411yescharacteristic
68C2×C4×D127—24no—
69C2×C4×D127—21no—
70C2×C4×D127D411yescharacteristic
71C2×C4×D127D411yescharacteristic
72C2×C4×D127C2311yescharacteristic
73C2×C4×D127C2311yescharacteristic
74#14523238—24no—
75#14523237—24no—
76#14523233—24no—
77C2×C127⋊C8C2312yes—
78C8×D127—24no—
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83(C2×D127)⋊C4—21no—
84(C2×D127)⋊C4—21no—
85C4×Dic127—21no—
86C24×C127—21no—
87C2×Q8×C127C2311yescharacteristic
88C2×D4×C127—24no—
89C2×D4×C127—21no—
90C2×D4×C127C2312yes—
91C22×C4×C127C2311yescharacteristic
92SD16×C127—24no—
93C2×C8×C127C2312yes—
94C4⋊C4×C127—21no—
95C4⋊C4×C127C2311yescharacteristic
96C22⋊C4×C127—21no—
97C22×D127—22no—
98C22×D127—22no—
99C22×D127—21no—
100C22×D127—41no—
101C22×D127C2×D411yescharacteristic
102C22×D127C2×D411yescharacteristic
103Dic254—24no—
104Dic254—22no—
105Dic254C2×D414yes—
106D508—22no—
107D508—42no—
108D508—42no—
109C22⋊D127—28no—
110C22⋊D127—28no—
111C22⋊D127—42no—
112C2×Dic127—24no—
113C2×Dic127—28no—
114C2×Dic127—22no—
115C2×Dic127—22no—
116C2×Dic127—28no—
117C2×Dic127—21no—
118C2×Dic127C2×D411yescharacteristic
119C2×Dic127C2×D411yescharacteristic
120C4×D127—22no—
121C4×D127—22no—
122C4×D127—22no—
123C4×D127—42no—
124C4×D127C2×D414yes—
125C127⋊C8—22no—
126C23×C127—24no—
127C23×C127—22no—
128C23×C127—42no—
129C23×C127—42no—
130C23×C127C2×D411yescharacteristic
131Q8×C127—22no—
132D4×C127—24no—
133D4×C127—22no—
134D4×C127C2×D414yes—
135C2×C4×C127—22no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2411yescharacteristic
140C8×C127—22no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—24no—
144C22×C127—48no—
145C22×C127—42no—
146C22×C127—42no—
147C22×C127#13512yescomplemented
148C22×C127C22×D411yescharacteristic
149C2×D127—21no—
150C2×D127—21no—
151C2×D127—21no—
152C2×D127—21no—
153C2×D127—21no—
154C2×D127—44no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127#12012yes—
159C4×C127—21no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127C22×D412yes—
163Dic127—21no—
164Dic127—24no—
165Dic127—41no—
166Dic127#12012yes—
167C2×C127—22no—
168C2×C127—44no—
169C2×C127C2×#12011yescharacteristic
170C2×C127C2×#13511yescharacteristic
171C2×C127#57511yescharacteristic
172D127—22no—
173D127—41no—
174D127—81no—
175#3002—1271nomaximal, Sylow
176C127#300211yescharacteristic, complemented, Sylow
177C22×#41—1271no—
178C2×#135—1272no—
179#525—1271no—
180#452—1272no—
181#451—1271no—
182#450D12711yescharacteristic, complemented
183#449—1272no—
184#448—1271no—
185#447—1271no—
186#416—1272no—
187C2×C22⋊C8—1271no—
188C2×#41—1271no—
189C2×#41—1272no—
190C2×#41—1272no—
191C2×#41—2541no—
192C2×#41—2541no—
193C2×#41—2548no—
194C22×Q8—1271no—
195C22×D4—2541no—
196C22×D4C2×D12711yescharacteristic
197C23×C4—1271no—
198C23×C4—2541no—
199#135—2548no—
200C2×Q16—1272no—
201C2×SD16—1272no—
202C2×SD16C2×D12712yescomplemented
203C2×D8—1272no—
204C22×C8—1272no—
205#123—2541no—
206C22⋊Q8—1271no—
207C22⋊Q8C2×D12711yescharacteristic
208#121—1271no—
209#121—1271no—
210#120—2541no—
211C4×D4—2541no—
212C4×D4—2541no—
213#117—1271no—
214#103—1272no—
215#103—1272no—
216#102—1272no—
217#102C2×D12712yescomplemented
218C22⋊C8—1271no—
219C22⋊C8—1272no—
220C22⋊C8C2×D12711yescharacteristic
221C24—21no—
222C24—1271no—
223#41—1278no—
224#41—2544no—
225#41—2544no—
226#41—2544no—
227#41—2548no—
228#41—2548no—
229#41—2544no—
230C2×Q8—1272no—
231C2×Q8—2544no—
232C2×Q8—2541no—
233C2×Q8C22×D12711yescharacteristic
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—1271no—
237C2×D4—2541no—
238C2×D4—2541no—
239C2×D4—2541no—
240C2×D4—2542no—
241C2×D4—2542no—
242C2×D4—2548no—
243C2×D4C22×D12712yes—
244C22×C4—1271no—
245C22×C4—1271no—
246C22×C4—1271no—
247C22×C4—1271no—
248C22×C4—1271no—
249C22×C4—2541no—
250C22×C4—2544no—
251C22×C4—2542no—
252C22×C4—2542no—
253C22×C4—2548no—
254C22×C4—2541no—
255C22×C4C22×D12711yescharacteristic
256Q16—2544no—
257SD16—24no—
258SD16—2544no—
259D8—2544no—
260C2×C8—1272no—
261C2×C8—2544no—
262C2×C8C22×D12712yes—
263C4⋊C4—21no—
264C4⋊C4—1271no—
265C4⋊C4—2541no—
266C4⋊C4C22×D12711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—2541no—
269C22⋊C4—2541no—
270C22⋊C4—2541no—
271C22⋊C4—2541no—
272C42—2541no—
273C23—24no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—1271no—
278C23—1271no—
279C23—5081no—
280C23—2542no—
281C23—2541no—
282C23—2542no—
283C23D4×D12711yescharacteristic
284Q8—22no—
285Q8—1274no—
286Q8—2544no—
287Q8—2542no—
288D4—24no—
289D4—22no—
290D4—5082no—
291D4—5082no—
292D4—5082no—
293D4—2548no—
294D4—2548no—
295D4—2542no—
296D4D4×D12714yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—1271no—
301C2×C4—1274no—
302C2×C4—1271no—
303C2×C4—5082no—
304C2×C4—2542no—
305C2×C4—2542no—
306C2×C4—2542no—
307C2×C4—2548no—
308C2×C4—2541no—
309C2×C4—2544no—
310C2×C4—2548no—
311C2×C4—2542no—
312C2×C4—2542no—
313C2×C4D4×D12711yescharacteristic
314C2×C4C23×D12711yescharacteristic
315C8—22no—
316C8—2542no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—1272no—
324C22—5084no—
325C22—5081no—
326C22—5081no—
327C22—5081no—
328C22—2541no—
329C22—2541no—
330C22—2541no—
331C22—2541no—
332C22—2541no—
333C22#1686123012yescomplemented
334C22C2×D4×D12711yescharacteristic
335C4—21no—
336C4—41no—
337C4—41no—
338C4—1272no—
339C4—5081no—
340C4—2541no—
341C4—2544no—
342C4C2×D4×D12712yes—
343C2—22no—
344C2—44no—
345C2—10161no—
346C2—5081no—
347C2—2542no—
348C2#120×D12711yescharacteristic
349C2C2×#1686123011yescharacteristic
350C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
hypercenter#450D127
hypocenterC127#3002
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14523247C2semidirect
2. C2×#16861230C2semidirect
3. #22550389C2semidirect
4. #22550035C2semidirect
5. #22550031C2semidirect
6. #22550022C2semidirect
7. #22550017C2semidirect
8. #22550010C2semidirect
9. #22549869C2semidirect
10. C22⋊C8×D127C2semidirect
11. #450×C127C2semidirect
12. C2×#14523249C22semidirect
23. C2×#14523238C22semidirect
24. C2×#14523237C22semidirect
25. C2×#14523233C22semidirect
26. C2×C8×D127C22semidirect
28. #16861185C22semidirect
29. #16861184C22semidirect
34. #16861167C22semidirect
35. #16861134C22semidirect
36. #16861132C22semidirect
37. #16861130C22semidirect
39. (C2×D127)⋊C8C22semidirect
41. C2×SD16×C127C22semidirect
43. #102×C127C22semidirect
52. #14523247D4semidirect
147. C22×C127#135semidirect
176. C127#3002semidirect
182. #450D127semidirect
202. C2×SD16C2×D127semidirect
217. #102C2×D127semidirect
333. C22#16861230semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168612302
#225503892
#225500352
#225500312
#225500222
#225500172
#225500102
#225498692
C22⋊C8×D1272
#450×C1272
#3002127