Finite Group Databasebeta

← #448×D127 #30647915

Subgroups — #448×D127 #30647915

Order 16256.

792 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 155 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×Q8×D127C211yesmaximal, characteristic, complemented
2C2×SD16×D127C212yesmaximal, complemented
3#121×D127C211yesmaximal, characteristic
4#22550017C211yesmaximal, characteristic, complemented
5#22550016C212yesmaximal, complemented
6#22550015C211yesmaximal, characteristic, complemented
7#22550014C211yesmaximal, characteristic, complemented
8#22550013C212yesmaximal, complemented
9#103×D127C212yesmaximal, complemented
10C22⋊C8×D127C211yesmaximal, characteristic, complemented
11#448×C127C211yesmaximal, characteristic, complemented
12C22×Dic254—21no—
13C22×Dic254C2211yescharacteristic, complemented
14C2×Q8×D127—28no—
15C2×Q8×D127—21no—
16C2×Q8×D127—21no—
17C2×Q8×D127C2212yes—
18C2×Q8×D127C2212yescomplemented
19C2×D4×D127—21no—
20C2×D4×D127C2211yescharacteristic
21C22×C4×D127—21no—
22C22×C4×D127C2211yescharacteristic
23C2×#14523237C2212yescomplemented
24C2×#14523236C2212yescomplemented
25C2×#14523235C2212yescomplemented
26SD16×D127—28no—
27C2×C8×D127C2212yes—
28#16861184C2211yescharacteristic
29#16861183C2211yescharacteristic
30#16861182—21no—
31#16861181—21no—
32#16861180C2211yescharacteristic
33C4⋊C4×D127C2211yescharacteristic
34C22⋊C4×D127—21no—
35#16861134C2212yescomplemented
36#16861133C2212yescomplemented
37#16861132C2212yescomplemented
38C22⋊(C127⋊C8)C2211yescharacteristic, complemented
39(C2×D127)⋊C8C2212yescomplemented
40C22×Q8×C127C2211yescharacteristic, complemented
41C2×SD16×C127C2212yescomplemented
42#121×C127C2211yescharacteristic
43#103×C127C2212yescomplemented
44C22⋊C8×C127C2211yescharacteristic, complemented
45C23×D127—21no—
46C23×D127D411yescharacteristic
47C2×Dic254—28no—
48C2×Dic254—24no—
49C2×Dic254—22no—
50C2×Dic254—22no—
51C2×Dic254—21no—
52C2×Dic254C2312yes—
53Q8×D127—28no—
54Q8×D127—28no—
55Q8×D127—24no—
56Q8×D127—24no—
57Q8×D127—24no—
58Q8×D127D418yescomplemented
59C2×D508—21no—
60C2×D508C2311yescharacteristic
61C2×C22⋊D127—21no—
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64D4×D127—24no—
65D4×D127—44no—
66C22×Dic127—21no—
67C22×Dic127D411yescharacteristic
68C2×C4×D127—28no—
69C2×C4×D127—22no—
70C2×C4×D127—22no—
71C2×C4×D127—24no—
72C2×C4×D127—21no—
73C2×C4×D127D411yescharacteristic
74C2×C4×D127D411yescharacteristic
75C2×C4×D127C2311yescharacteristic
76C2×C4×D127C2311yescharacteristic
77#14523237—24no—
78#14523236—24no—
79#14523235—24no—
80C2×C127⋊C8C2312yes—
81C8×D127—24no—
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84C22⋊Dic127—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87C2×Q8×C127—24no—
88C2×Q8×C127—21no—
89C2×Q8×C127C2312yes—
90C2×D4×C127—21no—
91C2×D4×C127C2311yescharacteristic
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94SD16×C127—24no—
95C2×C8×C127C2312yes—
96C4⋊C4×C127C2311yescharacteristic
97C22⋊C4×C127—21no—
98C22×D127—22no—
99C22×D127—22no—
100C22×D127—21no—
101C22×D127—42no—
102C22×D127—42no—
103C22×D127—42no—
104C22×D127SD1614yescomplemented
105C22×D127C2×D411yescharacteristic
106C22×D127C2×D411yescharacteristic
107Dic254—24no—
108Dic254—28no—
109Dic254—22no—
110Dic254—28no—
111Dic254C2×D414yescomplemented
112D508—22no—
113D508—42no—
114C22⋊D127—42no—
115C22⋊D127—42no—
116C22⋊D127—42no—
117C2×Dic127—24no—
118C2×Dic127—22no—
119C2×Dic127—22no—
120C2×Dic127—21no—
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C4×D127—28no—
124C4×D127—28no—
125C4×D127—22no—
126C4×D127—22no—
127C4×D127—22no—
128C4×D127—42no—
129C4×D127C2×D414yes—
130C127⋊C8—22no—
131C23×C127—21no—
132C23×C127C2×D411yescharacteristic
133Q8×C127—24no—
134Q8×C127—22no—
135Q8×C127C2×D414yescomplemented
136D4×C127—22no—
137D4×C127—42no—
138C2×C4×C127—24no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—21no—
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2411yescharacteristic
144C8×C127—22no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127C2×SD1612yescomplemented
151C22×C127C22×D411yescharacteristic
152C2×D127—21no—
153C2×D127—21no—
154C2×D127—28no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—84no—
159C2×D127#12012yes—
160C2×D127C2×SD1612yes—
161C2×D127C2×SD1612yescomplemented
162C2×D127#13712yes—
163C4×C127—24no—
164C4×C127—21no—
165C4×C127—41no—
166C4×C127C22×D412yes—
167Dic127—24no—
168Dic127—21no—
169Dic127—41no—
170Dic127#12012yes—
171C2×C127—22no—
172C2×C127—81no—
173C2×C127C2×#12011yescharacteristic
174C2×C127C22×SD1611yescharacteristic
175C2×C127C2×#13711yescharacteristic
176D127—22no—
177D127—81no—
178D127#44814yesdirect product factor
179C2×#448—1271nomaximal, Sylow
180C127C2×#44811yescharacteristic, complemented, Sylow
181C23×Q8—1271no—
182C22×SD16—1272no—
183C2×#121—1271no—
184#448—1271no—
185#448—1272no—
186#448—1271no—
187#448—1272no—
188#448—1271no—
189#448D12711yescharacteristic, direct product factor
190C2×#103—1272no—
191C2×C22⋊C8—1271no—
192C22×Q8—1272no—
193C22×Q8—1272no—
194C22×Q8—1271no—
195C22×Q8—2541no—
196C22×Q8—2541no—
197C22×Q8—2541no—
198C22×Q8—2548no—
199C22×Q8C2×D12711yescharacteristic, complemented
200C22×D4—1271no—
201C22×D4—2541no—
202C23×C4—1271no—
203C23×C4—2541no—
204C2×SD16—1272no—
205C2×SD16—1272no—
206C2×SD16—1272no—
207C2×SD16—2548no—
208C2×SD16C2×D12712yescomplemented
209C22×C8—1272no—
210#121—1271no—
211#121—1271no—
212#121—1271no—
213#121—2541no—
214#121—2541no—
215#121C2×D12711yescharacteristic
216C2×C4⋊C4—1271no—
217C2×C22⋊C4—2541no—
218#103—1272no—
219#103—1272no—
220#103—1272no—
221#103C2×D12712yescomplemented
222C22⋊C8—1271no—
223C22⋊C8—1272no—
224C22⋊C8C2×D12711yescharacteristic, complemented
225C24—1271no—
226C24—2541no—
227C2×Q8—24no—
228C2×Q8—21no—
229C2×Q8—1278no—
230C2×Q8—1272no—
231C2×Q8—2542no—
232C2×Q8—2542no—
233C2×Q8—2544no—
234C2×Q8—2548no—
235C2×Q8—2544no—
236C2×Q8—2544no—
237C2×Q8—2548no—
238C2×Q8—2548no—
239C2×Q8—2544no—
240C2×Q8—2541no—
241C2×Q8C22×D12712yes—
242C2×D4—21no—
243C2×D4—1271no—
244C2×D4—5084no—
245C2×D4—2541no—
246C2×D4—2544no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4—2541no—
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—1271no—
253C22×C4—1271no—
254C22×C4—1271no—
255C22×C4—1271no—
256C22×C4—1271no—
257C22×C4—2544no—
258C22×C4—2541no—
259C22×C4—2542no—
260C22×C4—2542no—
261C22×C4—2548no—
262C22×C4—2541no—
263C22×C4C22×D12711yescharacteristic
264SD16—24no—
265SD16—2544no—
266SD16—2544no—
267SD16—2544no—
268C2×C8—1272no—
269C2×C8—2544no—
270C2×C8C22×D12712yes—
271C4⋊C4—1271no—
272C4⋊C4—2541no—
273C4⋊C4C22×D12711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C23—21no—
279C23—1271no—
280C23—1274no—
281C23—1271no—
282C23—5082no—
283C23—5082no—
284C23—5082no—
285C23—2542no—
286C23—2542no—
287C23—2541no—
288C23D4×D12711yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8—1274no—
292Q8—2548no—
293Q8—2548no—
294Q8—2544no—
295Q8—2542no—
296Q8D4×D12714yescomplemented
297D4—22no—
298D4—42no—
299D4—5082no—
300D4—5082no—
301D4—5082no—
302D4—5082no—
303D4—2542no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—1271no—
309C2×C4—1274no—
310C2×C4—1271no—
311C2×C4—5082no—
312C2×C4—2542no—
313C2×C4—2542no—
314C2×C4—2548no—
315C2×C4—2548no—
316C2×C4—2542no—
317C2×C4—2541no—
318C2×C4—2544no—
319C2×C4—2542no—
320C2×C4—2542no—
321C2×C4D4×D12711yescharacteristic
322C2×C4C23×D12711yescharacteristic
323C8—22no—
324C8—2542no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—1272no—
331C22—1272no—
332C22—1272no—
333C22—1272no—
334C22—10164no—
335C22—5081no—
336C22—5081no—
337C22—5081no—
338C22—2548no—
339C22—2541no—
340C22—2541no—
341C22SD16×D12712yescomplemented
342C22C2×D4×D12711yescharacteristic
343C4—24no—
344C4—21no—
345C4—41no—
346C4—1272no—
347C4—5081no—
348C4—2541no—
349C4—2544no—
350C4C2×D4×D12712yes—
351C2—22no—
352C2—81no—
353C2—1274no—
354C2—10161no—
355C2—2542no—
356C2#120×D12711yescharacteristic
357C2C2×SD16×D12711yescharacteristic
358C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#448×C127C2
socleC22×C127C22×D4
prime radical#121×D127C2
hypercenter#448D127
hypocenterC127C2×#448
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D127C2semidirect
2. C2×SD16×D127C2semidirect
4. #22550017C2semidirect
5. #22550016C2semidirect
6. #22550015C2semidirect
7. #22550014C2semidirect
8. #22550013C2semidirect
9. #103×D127C2semidirect
10. C22⋊C8×D127C2semidirect
11. #448×C127C2semidirect
13. C22×Dic254C22semidirect
18. C2×Q8×D127C22semidirect
23. C2×#14523237C22semidirect
24. C2×#14523236C22semidirect
25. C2×#14523235C22semidirect
35. #16861134C22semidirect
36. #16861133C22semidirect
37. #16861132C22semidirect
38. C22⋊(C127⋊C8)C22semidirect
39. (C2×D127)⋊C8C22semidirect
40. C22×Q8×C127C22semidirect
41. C2×SD16×C127C22semidirect
43. #103×C127C22semidirect
44. C22⋊C8×C127C22semidirect
58. Q8×D127D4semidirect
104. C22×D127SD16semidirect
111. Dic254C2×D4semidirect
135. Q8×C127C2×D4semidirect
150. C22×C127C2×SD16semidirect
161. C2×D127C2×SD16semidirect
178. D127#448direct (N × H)
180. C127C2×#448semidirect
189. #448D127direct (N × H)
199. C22×Q8C2×D127semidirect
208. C2×SD16C2×D127semidirect
221. #103C2×D127semidirect
224. C22⋊C8C2×D127semidirect
296. Q8D4×D127semidirect
341. C22SD16×D127semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1272
C2×SD16×D1272
#121×D1272
#225500172
#225500162
#225500152
#225500142
#225500132
#103×D1272
C22⋊C8×D1272
#448×C1272
C2×#448127