Finite Group Databasebeta

← #30648043

Subgroups — #30648043

Order 16256.

712 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861256C211yesmaximal, characteristic, complemented
2C2×#16861238C211yesmaximal, characteristic, complemented
3C2×#16861228C211yesmaximal, characteristic, complemented
4#22550455C211yesmaximal, characteristic, complemented
5#22550039C211yesmaximal, characteristic, complemented
6#22550038C211yesmaximal, characteristic, complemented
7#22550034C211yesmaximal, characteristic, complemented
8#22550033C211yesmaximal, characteristic, complemented
9#22550018C211yesmaximal, characteristic, complemented
10#22550014C211yesmaximal, characteristic, complemented
11#22550013C211yesmaximal, characteristic, complemented
12#22549874C211yesmaximal, characteristic, complemented
13#103×D127C211yesmaximal, characteristic, complemented
14#22549841C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16#16861256—24no—
17#16861256—24no—
18C2×#14523248—21no—
19C2×#14523248C2211yescharacteristic
20C2×#14523247—21no—
21C2×#14523247—21no—
22C2×#14523247C2211yescharacteristic
23C2×#14523247C2211yescharacteristic
24C2×#14523247C2211yescharacteristic
25C22×Dic254—21no—
26C22×Dic254C2211yescharacteristic
27C2×Q8×D127C2211yescharacteristic
28C2×Q8×D127C2211yescharacteristic
29#16861238—24no—
30#16861228—24no—
31C2×Dic508C2211yescharacteristic
32C2×Dic508C2211yescharacteristic
33C2×#14523238C2211yescharacteristic
34C2×#14523236C2211yescharacteristic
35C2×#14523236C2211yescharacteristic
36C2×#14523236C2211yescharacteristic
37C2×#14523231C2211yescharacteristic
38C2×C8×D127C2211yescharacteristic
39#16861189C2211yescharacteristic
40#16861188C2211yescharacteristic
41#16861186—21no—
42#16861182—21no—
43#16861181—21no—
44#16861180—21no—
45#16861180C2211yescharacteristic
46C4⋊C4×D127C2211yescharacteristic
47#16861134C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861133C2211yescharacteristic
50#16861132C2211yescharacteristic
51#16861132C2211yescharacteristic
52#16861130C2211yescharacteristic
53C22⋊(C127⋊C8)C2211yescharacteristic
54(C2×D127)⋊C8C2211yescharacteristic
55(C2×D127)⋊C8C2211yescharacteristic
56C2×#41×C127C2211yescharacteristic
57C2×SD16×C127C2211yescharacteristic
58C2×D8×C127C2211yescharacteristic
59#121×C127C2211yescharacteristic
60#103×C127C2211yescharacteristic
61#102×C127C2211yescharacteristic
62C22⋊C8×C127C2211yescharacteristic
63#14523248—28no—
64#14523248—22no—
65#14523248—22no—
66#14523247—28no—
67#14523247—28no—
68#14523247—22no—
69#14523247—22no—
70#14523247—22no—
71#14523247—22no—
72#14523247—22no—
73#14523247D414yescomplemented
74C2×Dic254—28no—
75C2×Dic254—22no—
76C2×Dic254—22no—
77C2×Dic254—21no—
78C2×Dic254—21no—
79C2×Dic254—21no—
80C2×Dic254D411yescharacteristic
81C2×Dic254C2311yescharacteristic
82C2×Dic254C2311yescharacteristic
83C2×Dic254C2311yescharacteristic
84Q8×D127—22no—
85Q8×D127—22no—
86Q8×D127—22no—
87Q8×D127D414yescomplemented
88C2×D508D411yescharacteristic
89C2×C22⋊D127—21no—
90C2×C22⋊D127—21no—
91C2×C22⋊D127—21no—
92C2×C22⋊D127—21no—
93C2×C22⋊D127D411yescharacteristic
94C2×C22⋊D127D411yescharacteristic
95C22×Dic127—21no—
96C22×Dic127—21no—
97C22×Dic127—21no—
98C22×Dic127—21no—
99C2×C4×D127—21no—
100C2×C4×D127—21no—
101C2×C4×D127—21no—
102C2×C4×D127C2311yescharacteristic
103C2×C4×D127C2311yescharacteristic
104Dic508—22no—
105Dic508—22no—
106#14523238—22no—
107#14523236—22no—
108#14523236—22no—
109#14523236—22no—
110#14523231—22no—
111C2×C127⋊C8C2311yescharacteristic
112C2×C127⋊C8C2311yescharacteristic
113C8×D127—22no—
114C4⋊Dic127—21no—
115C4⋊Dic127C2311yescharacteristic
116Dic127⋊C4—21no—
117Dic127⋊C4—21no—
118C22⋊Dic127—21no—
119C22⋊Dic127—21no—
120(C2×D127)⋊C4—21no—
121#41×C127—22no—
122#41×C127—22no—
123C2×Q8×C127C2311yescharacteristic
124C2×D4×C127—21no—
125C2×D4×C127—21no—
126C2×D4×C127C2311yescharacteristic
127C2×D4×C127C2311yescharacteristic
128C22×C4×C127—21no—
129C22×C4×C127C2311yescharacteristic
130SD16×C127—22no—
131D8×C127—22no—
132C2×C8×C127C2311yescharacteristic
133C2×C8×C127C2311yescharacteristic
134C4⋊C4×C127C2311yescharacteristic
135C22⋊C4×C127—21no—
136C22×D127C2×D411yescharacteristic
137C22×D127C2×D411yescharacteristic
138Dic254—24no—
139Dic254—24no—
140Dic254—21no—
141Dic254—21no—
142Dic254—24no—
143Dic254—22no—
144Dic254—21no—
145Dic254—21no—
146Dic254C2×D412yes—
147Dic254C2×D412yes—
148D508—22no—
149C22⋊D127—24no—
150C22⋊D127—24no—
151C22⋊D127—22no—
152C22⋊D127—22no—
153C22⋊D127—42no—
154C22⋊D127—81no—
155C2×Dic127—24no—
156C2×Dic127—24no—
157C2×Dic127—21no—
158C2×Dic127—21no—
159C2×Dic127—24no—
160C2×Dic127—21no—
161C2×Dic127—21no—
162C2×Dic127—42no—
163C2×Dic127C2×D411yescharacteristic
164C2×Dic127C2×D411yescharacteristic
165C4×D127—24no—
166C4×D127—24no—
167C4×D127—21no—
168C4×D127—21no—
169C4×D127—81no—
170C4×D127C2×D412yes—
171C4×D127C2×D412yes—
172C127⋊C8—21no—
173C127⋊C8—21no—
174C23×C127—21no—
175C23×C127—21no—
176C23×C127C2×D411yescharacteristic
177Q8×C127—21no—
178Q8×C127C2×D412yes—
179D4×C127—24no—
180D4×C127—21no—
181D4×C127—21no—
182D4×C127—21no—
183D4×C127—81no—
184D4×C127C2×D412yes—
185C2×C4×C127—24no—
186C2×C4×C127—21no—
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127C2×D411yescharacteristic
191C2×C4×C127C2411yescharacteristic
192C8×C127—21no—
193C8×C127—21no—
194C22×C127—21no—
195C22×C127—22no—
196C22×C127—21no—
197C22×C127—21no—
198C22×C127—41no—
199C22×C127—41no—
200C22×C127—41no—
201C22×C127—41no—
202C22×C127—41no—
203C22×C127C22×D411yescharacteristic
204C2×D127—21no—
205C2×D127—21no—
206C2×D127—21no—
207C2×D127—21no—
208C2×D127—21no—
209C2×D127#12012yes—
210C4×C127—21no—
211C4×C127—22no—
212C4×C127—41no—
213C4×C127C22×D411yescharacteristic
214C4×C127C22×D411yescharacteristic
215Dic127—22no—
216Dic127—22no—
217Dic127—21no—
218Dic127—41no—
219Dic127—41no—
220Dic127#12012yes—
221C2×C127—41no—
222C2×C127—42no—
223C2×C127—81no—
224C2×C127C2×#12011yescharacteristic
225C2×C127#57711yescharacteristic
226C2×C127#57811yescharacteristic
227D127—22no—
228D127—41no—
229#3013—1271nomaximal, Sylow
230C127#301311yescharacteristic, complemented, Sylow
231C2×2−1+4—1271no—
232C2×#137—1271no—
233C2×#135—1271no—
234#537—1271no—
235#452—1271no—
236#452—1271no—
237#451—1271no—
238#451—1271no—
239#449—1271no—
240#449D12711yescharacteristic, complemented
241#448—1271no—
242#448—1271no—
243#417—1271no—
244C2×#103—1271no—
245#408—1271no—
2462−1+4—2544no—
2472−1+4—2544no—
248C2×#41—1271no—
249C2×#41—1271no—
250C2×#41—1271no—
251C2×#41—1271no—
252C2×#41—2541no—
253C2×#41—2541no—
254C2×#41—2541no—
255C2×#41C2×D12711yescharacteristic
256C22×Q8—1271no—
257C22×Q8—1271no—
258C22×Q8—1271no—
259C22×Q8—2541no—
260#137—2544no—
261#135—2544no—
262C2×Q16—1271no—
263C2×Q16—1271no—
264C2×Q16—1271no—
265C2×SD16—1271no—
266C2×SD16—1271no—
267C2×SD16—1271no—
268C2×SD16C2×D12711yescharacteristic
269C2×D8C2×D12711yescharacteristic
270C2×M16—1271no—
271C22×C8—1271no—
272C22⋊Q8—1271no—
273C22⋊Q8—1271no—
274C22⋊Q8—2541no—
275#121—1271no—
276#121—2541no—
277#121—2541no—
278#121—2541no—
279#121C2×D12711yescharacteristic
280C2×C4⋊C4—1271no—
281#103—1271no—
282#103—1271no—
283#103—1271no—
284#103—1271no—
285#103—1271no—
286#103C2×D12711yescharacteristic
287#102—1271no—
288#102C2×D12711yescharacteristic
289C22⋊C8—1271no—
290C22⋊C8—1271no—
291C22⋊C8—1271no—
292C22⋊C8C2×D12711yescharacteristic
293#41—22no—
294#41—22no—
295#41—1274no—
296#41—2542no—
297#41—2542no—
298#41—2548no—
299#41—2542no—
300#41—2542no—
301#41—2542no—
302#41—2542no—
303#41—2548no—
304#41—2548no—
305#41—2542no—
306C2×Q8—1271no—
307C2×Q8—1274no—
308C2×Q8—1271no—
309C2×Q8—1271no—
310C2×Q8—1271no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8—2542no—
314C2×Q8—2542no—
315C2×Q8—2548no—
316C2×Q8—2542no—
317C2×Q8—2542no—
318C2×Q8—2542no—
319C2×Q8—2541no—
320C2×Q8C22×D12711yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—1271no—
324C2×D4—1271no—
325C2×D4—1271no—
326C2×D4—2541no—
327C2×D4—2541no—
328C2×D4—2541no—
329C2×D4—2541no—
330C2×D4C22×D12711yescharacteristic
331C2×D4C22×D12711yescharacteristic
332C22×C4—21no—
333C22×C4—1271no—
334C22×C4—1271no—
335C22×C4—2541no—
336C22×C4—2541no—
337C22×C4—2541no—
338C22×C4—2541no—
339C22×C4—2541no—
340C22×C4—2541no—
341C22×C4—2541no—
342C22×C4C22×D12711yescharacteristic
343Q16—2542no—
344Q16—2542no—
345Q16—2542no—
346SD16—22no—
347SD16—2542no—
348SD16—2542no—
349SD16—2542no—
350D8—22no—
351M16—2542no—
352C2×C8—1271no—
353C2×C8—1271no—
354C2×C8—2542no—
355C2×C8C22×D12711yescharacteristic
356C2×C8C22×D12711yescharacteristic
357C4⋊C4—1271no—
358C4⋊C4—2541no—
359C4⋊C4—2541no—
360C4⋊C4—2541no—
361C4⋊C4C22×D12711yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—2541no—
364C22⋊C4—2541no—
365C22⋊C4—2541no—
366C23—21no—
367C23—21no—
368C23—1271no—
369C23—1271no—
370C23D4×D12711yescharacteristic
371Q8—21no—
372Q8—1272no—
373Q8—1272no—
374Q8—2542no—
375Q8—2544no—
376Q8—2544no—
377Q8—2544no—
378Q8—2541no—
379Q8—2541no—
380Q8—2541no—
381Q8—2541no—
382Q8D4×D12712yes—
383D4—24no—
384D4—21no—
385D4—21no—
386D4—21no—
387D4—81no—
388D4—10161no—
389D4—5082no—
390D4—2542no—
391D4—2542no—
392D4—2542no—
393D4—2544no—
394D4—2544no—
395D4D4×D12712yes—
396C2×C4—24no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—1271no—
402C2×C4—1272no—
403C2×C4—1272no—
404C2×C4—1271no—
405C2×C4—10161no—
406C2×C4—5082no—
407C2×C4—2544no—
408C2×C4—2544no—
409C2×C4—2544no—
410C2×C4—2541no—
411C2×C4—2541no—
412C2×C4—2544no—
413C2×C4—2544no—
414C2×C4—2541no—
415C2×C4—2541no—
416C2×C4—2541no—
417C2×C4—2541no—
418C2×C4D4×D12711yescharacteristic
419C2×C4C23×D12711yescharacteristic
420C8—21no—
421C8—21no—
422C8—2541no—
423C8—2541no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—1272no—
434C22—2541no—
435C22—2541no—
436C22—2541no—
437C22—2541no—
438C22—2541no—
439C22C2×D4×D12711yescharacteristic
440C4—21no—
441C4—22no—
442C4—41no—
443C4—1272no—
444C4—5081no—
445C4—5081no—
446C4—2541no—
447C4—2542no—
448C4—2542no—
449C4C2×D4×D12711yescharacteristic
450C4C2×D4×D12711yescharacteristic
451C2—41no—
452C2—42no—
453C2—81no—
454C2—5081no—
455C2—2542no—
456C2#120×D12711yescharacteristic
457C2#2255068611yescharacteristic
458C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3013
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861256C2semidirect
2. C2×#16861238C2semidirect
3. C2×#16861228C2semidirect
4. #22550455C2semidirect
5. #22550039C2semidirect
6. #22550038C2semidirect
7. #22550034C2semidirect
8. #22550033C2semidirect
9. #22550018C2semidirect
10. #22550014C2semidirect
11. #22550013C2semidirect
12. #22549874C2semidirect
13. #103×D127C2semidirect
14. #22549841C2semidirect
15. #449×C127C2semidirect
73. #14523247D4semidirect
87. Q8×D127D4semidirect
230. C127#3013semidirect
240. #449D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#168612382
C2×#168612282
#225504552
#225500392
#225500382
#225500342
#225500332
#225500182
#225500142
#225500132
#225498742
#103×D1272
#225498412
#449×C1272
#3013127