Finite Group Databasebeta

← #30647985

Subgroups — #30647985

Order 16256.

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

normal subgroups only

This group has 754 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
2C2×#16861238C212yesmaximal, complemented
3C22⋊Q8×D127C211yesmaximal, characteristic, complemented
4#22550039C212yesmaximal, complemented
5#22550035C212yesmaximal, complemented
6#22550034C211yesmaximal, characteristic, complemented
7#22550031C211yesmaximal, characteristic, complemented
8#22550027C211yesmaximal, characteristic, complemented
9#22549874C212yesmaximal, complemented
10#22549838C211yesmaximal, characteristic, complemented
11#450×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—28no—
13C2×#14523247—21no—
14C2×#14523247—21no—
15C2×#14523247C2212yes—
16C2×#14523247C2212yes—
17C22×Dic254C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic, complemented
19C22×C22⋊D127—21no—
20C23×Dic127—21no—
21C22×C4×D127C2211yescharacteristic
22#16861238—28no—
23C2×Dic508C2212yescomplemented
24C2×#14523238C2212yescomplemented
25C2×#14523236C2212yescomplemented
26C2×#14523231C2212yescomplemented
27#16861189C2211yescharacteristic, complemented
28#16861188C2211yescharacteristic
29#16861187—21no—
30#16861186—21no—
31#16861185C2211yescharacteristic, complemented
32C4⋊C4×D127—21no—
33C4⋊C4×D127C2211yescharacteristic, complemented
34C22⋊C4×D127—21no—
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#14523247—28no—
47#14523247—28no—
48#14523247—24no—
49#14523247—24no—
50#14523247—24no—
51#14523247D418yes—
52C2×Dic254—24no—
53C2×Dic254—21no—
54C2×Dic254—21no—
55C2×Dic254C2311yescharacteristic
56C2×Dic254C2312yes—
57Q8×D127—24no—
58C2×C22⋊D127—28no—
59C2×C22⋊D127—22no—
60C2×C22⋊D127—22no—
61C22×Dic127—28no—
62C22×Dic127—22no—
63C22×Dic127—22no—
64C22×Dic127—21no—
65C22×Dic127D411yescharacteristic
66C2×C4×D127—24no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127D411yescharacteristic
70C2×C4×D127D411yescharacteristic
71C2×C4×D127C2311yescharacteristic
72C2×C4×D127C2311yescharacteristic
73Dic508—24no—
74#14523238—24no—
75#14523236—24no—
76#14523231—24no—
77C2×C127⋊C8C2312yes—
78C4⋊Dic127—21no—
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83C22⋊Dic127—21no—
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—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—41no—
100C22×D127C2×D411yescharacteristic
101C22×D127C2×D411yescharacteristic
102Dic254—24no—
103Dic254—22no—
104Dic254—22no—
105Dic254—42no—
106Dic254C2×D414yes—
107C22⋊D127—28no—
108C22⋊D127—28no—
109C2×Dic127—24no—
110C2×Dic127—28no—
111C2×Dic127—22no—
112C2×Dic127—22no—
113C2×Dic127—28no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127C2×D411yescharacteristic
117C2×Dic127C2×D411yescharacteristic
118C4×D127—22no—
119C4×D127—22no—
120C4×D127—22no—
121C4×D127—42no—
122C4×D127—42no—
123C4×D127C2×D414yes—
124C127⋊C8—22no—
125C23×C127—24no—
126C23×C127—22no—
127C23×C127—42no—
128C23×C127—42no—
129C23×C127C2×D411yescharacteristic
130Q8×C127—22no—
131D4×C127—24no—
132D4×C127—22no—
133D4×C127C2×D414yes—
134C2×C4×C127—22no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127C2×D411yescharacteristic
138C2×C4×C127C2411yescharacteristic
139C8×C127—22no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—24no—
143C22×C127—48no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127#13712yescomplemented
147C22×C127C22×D411yescharacteristic
148C2×D127—21no—
149C2×D127—21no—
150C2×D127—21no—
151C2×D127—21no—
152C2×D127—21no—
153C2×D127—44no—
154C2×D127#12012yes—
155C4×C127—21no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127C22×D412yes—
159Dic127—21no—
160Dic127—24no—
161Dic127—41no—
162Dic127—41no—
163Dic127#12012yes—
164C2×C127—22no—
165C2×C127—44no—
166C2×C127C2×#12011yescharacteristic
167C2×C127C2×#13711yescharacteristic
168C2×C127#57511yescharacteristic
169D127—22no—
170D127—41no—
171#3007—1271nomaximal, Sylow
172C127#300711yescharacteristic, complemented, Sylow
173C22×#41—1271no—
174C2×#137—1272no—
175C2×C22⋊Q8—1271no—
176#452—1272no—
177#452—1272no—
178#451—1271no—
179#451—1271no—
180#450—1271no—
181#450D12711yescharacteristic, complemented
182#417—1272no—
183#407—1271no—
184C2×#41—1272no—
185C2×#41—1272no—
186C2×#41—2541no—
187C2×#41—2541no—
188C2×#41—2548no—
189C22×Q8—1271no—
190C22×Q8—1271no—
191C22×D4—2541no—
192C22×D4C2×D12711yescharacteristic
193C23×C4—1271no—
194C23×C4—2541no—
195#137—2548no—
196C2×Q16—1272no—
197C2×Q16—1272no—
198C2×SD16—1272no—
199C2×SD16C2×D12712yescomplemented
200C2×M16—1272no—
201C22⋊Q8—1271no—
202C22⋊Q8—1271no—
203C22⋊Q8—1271no—
204C22⋊Q8—2541no—
205C22⋊Q8—2541no—
206C22⋊Q8C2×D12711yescharacteristic
207C2×C4⋊C4—1271no—
208C2×C4⋊C4—2541no—
209C2×C22⋊C4—2541no—
210#103—1272no—
211#103—1272no—
212#102—1272no—
213#102C2×D12712yescomplemented
214C22⋊C8—1271no—
215C22⋊C8—1272no—
216C22⋊C8C2×D12711yescharacteristic
217C24—21no—
218C24—1271no—
219#41—1278no—
220#41—2544no—
221#41—2544no—
222#41—2548no—
223#41—2548no—
224#41—2544no—
225C2×Q8—1272no—
226C2×Q8—1271no—
227C2×Q8—2541no—
228C2×Q8—2544no—
229C2×Q8—2544no—
230C2×Q8—2541no—
231C2×Q8C22×D12711yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—2542no—
235C2×D4—2542no—
236C2×D4—2548no—
237C2×D4C22×D12712yes—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—1271no—
241C22×C4—1271no—
242C22×C4—1271no—
243C22×C4—2544no—
244C22×C4—2541no—
245C22×C4—2541no—
246C22×C4—2542no—
247C22×C4—2542no—
248C22×C4—2548no—
249C22×C4—2541no—
250C22×C4C22×D12711yescharacteristic
251Q16—2544no—
252Q16—2544no—
253SD16—24no—
254SD16—2544no—
255M16—2544no—
256C2×C8—1272no—
257C2×C8C22×D12712yes—
258C4⋊C4—21no—
259C4⋊C4—1271no—
260C4⋊C4—2541no—
261C4⋊C4—2541no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4C22×D12711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2541no—
267C22⋊C4—2541no—
268C22⋊C4—2541no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—1271no—
274C23—1271no—
275C23—5081no—
276C23—2542no—
277C23—2542no—
278C23D4×D12711yescharacteristic
279Q8—22no—
280Q8—1274no—
281Q8—5082no—
282Q8—2544no—
283Q8—2542no—
284Q8—2542no—
285D4—24no—
286D4—22no—
287D4—2548no—
288D4—2548no—
289D4D4×D12714yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—1271no—
294C2×C4—1274no—
295C2×C4—1271no—
296C2×C4—5082no—
297C2×C4—5082no—
298C2×C4—2542no—
299C2×C4—2542no—
300C2×C4—2542no—
301C2×C4—2548no—
302C2×C4—2541no—
303C2×C4—2541no—
304C2×C4—2544no—
305C2×C4—2548no—
306C2×C4—2542no—
307C2×C4—2542no—
308C2×C4D4×D12711yescharacteristic
309C2×C4C23×D12711yescharacteristic
310C8—22no—
311C8—2542no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—1272no—
319C22—5084no—
320C22—2541no—
321C22—2541no—
322C22—2541no—
323C22—2541no—
324C22—2541no—
325C22#1686123812yescomplemented
326C22C2×D4×D12711yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—1272no—
331C4—5081no—
332C4—5081no—
333C4—2541no—
334C4—2544no—
335C4C2×D4×D12712yes—
336C2—22no—
337C2—44no—
338C2—5081no—
339C2—2542no—
340C2#120×D12711yescharacteristic
341C2C2×#1686123811yescharacteristic
342C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radicalC22×#14523247C2
hypercenter#450D127
hypocenterC127#3007
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
2. C2×#16861238C2semidirect
3. C22⋊Q8×D127C2semidirect
4. #22550039C2semidirect
5. #22550035C2semidirect
6. #22550034C2semidirect
7. #22550031C2semidirect
8. #22550027C2semidirect
9. #22549874C2semidirect
10. #22549838C2semidirect
11. #450×C127C2semidirect
18. C2×Q8×D127C22semidirect
23. C2×Dic508C22semidirect
24. C2×#14523238C22semidirect
25. C2×#14523236C22semidirect
26. C2×#14523231C22semidirect
27. #16861189C22semidirect
31. #16861185C22semidirect
33. C4⋊C4×D127C22semidirect
35. #16861134C22semidirect
36. #16861132C22semidirect
37. #16861130C22semidirect
39. (C2×D127)⋊C8C22semidirect
41. C2×SD16×C127C22semidirect
43. #102×C127C22semidirect
146. C22×C127#137semidirect
172. C127#3007semidirect
181. #450D127semidirect
199. C2×SD16C2×D127semidirect
213. #102C2×D127semidirect
325. C22#16861238semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168612382
C22⋊Q8×D1272
#225500392
#225500352
#225500342
#225500312
#225500272
#225498742
#225498382
#450×C1272
#3007127