Finite Group Databasebeta

← C2×C127×#1654 #44582169

Subgroups — C2×C127×#1654 #44582169

Order 32512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547×C127C211yesmaximal, characteristic
2C2×#498×C127C211yesmaximal, characteristic, complemented
3C2×#474×C127C211yesmaximal, characteristic, complemented
4C2×#461×C127C211yesmaximal, characteristic, complemented
5C127×#1654C218yesmaximal, direct product factor
6C2×#332×C127C211yesmaximal, characteristic, complemented
7C2×D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×D4⋊C8×C127C211yesmaximal, characteristic, complemented
9C22×#41×C127—21no—
10C22×SD16×C127—21no—
11#547×C127—22no—
12#547×C127—22no—
13#547×C127—22no—
14#547×C127C2214yes—
15C2×C4⋊Q8×C127C2211yescharacteristic
16C2×C4⋊Q8×C127C2211yescharacteristic
17C2×#123×C127—22no—
18C2×C22⋊Q8×C127—21no—
19C2×#121×C127—21no—
20C2×C4×D4×C127C2211yescharacteristic
21C2×C4×D4×C127C2211yescharacteristic
22C22×C4⋊C4×C127—21no—
23#498×C127—22no—
24#498×C127C2214yescomplemented
25#474×C127—22no—
26#474×C127C2214yescomplemented
27#461×C127—22no—
28#461×C127C2214yescomplemented
29C2×#107×C127—21no—
30C2×#106×C127—22no—
31C2×C4⋊C8×C127C2211yescharacteristic
32C2×C4⋊C8×C127C2211yescharacteristic
33C2×#103×C127—21no—
34C2×#102×C127—21no—
35C2×C4×C8×C127C2211yescharacteristic
36#332×C127C2214yescomplemented
37D4⋊C8×C127C2214yescomplemented
38D4⋊C8×C127C2214yescomplemented
39C2×#41×C127—24no—
40C2×#41×C127—24no—
41C2×#41×C127—44no—
42C2×#41×C127—44no—
43C22×Q8×C127—21no—
44C22×Q8×C127—21no—
45C22×D4×C127—22no—
46C22×D4×C127D411yescharacteristic
47C22×D4×C127D411yescharacteristic
48C23×C4×C127—22no—
49C23×C4×C127—21no—
50C23×C4×C127—21no—
51C2×SD16×C127—24no—
52C2×SD16×C127—44no—
53C22×C8×C127—21no—
54C22×C8×C127—21no—
55C22×C8×C127—21no—
56C4⋊Q8×C127—21no—
57C4⋊Q8×C127—21no—
58C4⋊Q8×C127—21no—
59C4⋊Q8×C127—21no—
60C4⋊Q8×C127—21no—
61C4⋊Q8×C127D412yes—
62C4⋊Q8×C127C2312yes—
63C4⋊Q8×C127C2312yes—
64#123×C127—28no—
65#123×C127—28no—
66C22⋊Q8×C127—24no—
67C22⋊Q8×C127—24no—
68#121×C127—24no—
69#121×C127—24no—
70C4×D4×C127—21no—
71C4×D4×C127—21no—
72C4×D4×C127—21no—
73C4×D4×C127—21no—
74C4×D4×C127D412yes—
75C4×D4×C127D412yes—
76C4×D4×C127C2312yes—
77C4×D4×C127C2312yes—
78C2×C4⋊C4×C127—22no—
79C2×C4⋊C4×C127—24no—
80C2×C4⋊C4×C127—21no—
81C2×C4⋊C4×C127—24no—
82C2×C4⋊C4×C127—21no—
83C2×C4⋊C4×C127—21no—
84C2×C4⋊C4×C127D411yescharacteristic
85C2×C4⋊C4×C127D411yescharacteristic
86C2×C4⋊C4×C127D412yes—
87C2×C22⋊C4×C127—22no—
88C2×C22⋊C4×C127—21no—
89C2×C22⋊C4×C127—21no—
90C2×C42×C127C2311yescharacteristic
91#107×C127—24no—
92#106×C127—28no—
93C4⋊C8×C127—21no—
94C4⋊C8×C127—21no—
95C4⋊C8×C127C2312yes—
96C4⋊C8×C127C2312yes—
97#103×C127—24no—
98#102×C127—24no—
99C4×C8×C127—21no—
100C4×C8×C127C2312yes—
101C24×C127—21no—
102C24×C127—21no—
103#41×C127—48no—
104#41×C127—416no—
105#41×C127—48no—
106C2×Q8×C127—22no—
107C2×Q8×C127—22no—
108C2×Q8×C127—22no—
109C2×Q8×C127—41no—
110C2×Q8×C127—42no—
111C2×Q8×C127—42no—
112C2×Q8×C127—42no—
113C2×Q8×C127—42no—
114C2×D4×C127—28no—
115C2×D4×C127—21no—
116C2×D4×C127—21no—
117C2×D4×C127—22no—
118C2×D4×C127—22no—
119C2×D4×C127—22no—
120C2×D4×C127—48no—
121C2×D4×C127SD1614yescomplemented
122C2×D4×C127C2×D412yes—
123C2×D4×C127C2×D412yes—
124C22×C4×C127—28no—
125C22×C4×C127—21no—
126C22×C4×C127—21no—
127C22×C4×C127—22no—
128C22×C4×C127—22no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—22no—
132C22×C4×C127—48no—
133C22×C4×C127—42no—
134C22×C4×C127—42no—
135C22×C4×C127—42no—
136C22×C4×C127—42no—
137C22×C4×C127—41no—
138C22×C4×C127C2×D411yescharacteristic
139C22×C4×C127C2×D411yescharacteristic
140C22×C4×C127C2×D411yescharacteristic
141SD16×C127—48no—
142C2×C8×C127—24no—
143C2×C8×C127—22no—
144C2×C8×C127—22no—
145C2×C8×C127—22no—
146C2×C8×C127—42no—
147C2×C8×C127—42no—
148C4⋊C4×C127—28no—
149C4⋊C4×C127—21no—
150C4⋊C4×C127—22no—
151C4⋊C4×C127—21no—
152C4⋊C4×C127—22no—
153C4⋊C4×C127—22no—
154C4⋊C4×C127—22no—
155C4⋊C4×C127—22no—
156C4⋊C4×C127—41no—
157C4⋊C4×C127—41no—
158C4⋊C4×C127C2×D412yes—
159C4⋊C4×C127C2×D414yes—
160C4⋊C4×C127C2×D412yes—
161C22⋊C4×C127—28no—
162C22⋊C4×C127—22no—
163C22⋊C4×C127—22no—
164C22⋊C4×C127—22no—
165C22⋊C4×C127—22no—
166C42×C127C2×D411yescharacteristic
167C42×C127C2×D411yescharacteristic
168C42×C127C2×D411yescharacteristic
169C42×C127C2411yescharacteristic
170C23×C127—22no—
171C23×C127—24no—
172C23×C127—22no—
173C23×C127—42no—
174C23×C127—42no—
175C23×C127—42no—
176C23×C127—42no—
177C23×C127—42no—
178C23×C127#12011yescharacteristic
179Q8×C127—42no—
180Q8×C127—42no—
181Q8×C127—44no—
182Q8×C127—42no—
183Q8×C127—42no—
184Q8×C127—82no—
185D4×C127—22no—
186D4×C127—22no—
187D4×C127—22no—
188D4×C127—24no—
189D4×C127—416no—
190D4×C127—42no—
191D4×C127C2×SD1614yescomplemented
192C2×C4×C127—22no—
193C2×C4×C127—22no—
194C2×C4×C127—22no—
195C2×C4×C127—24no—
196C2×C4×C127—416no—
197C2×C4×C127—42no—
198C2×C4×C127—42no—
199C2×C4×C127—44no—
200C2×C4×C127—42no—
201C2×C4×C127—42no—
202C2×C4×C127—42no—
203C2×C4×C127—42no—
204C2×C4×C127—44no—
205C2×C4×C127—44no—
206C2×C4×C127—42no—
207C2×C4×C127—82no—
208C2×C4×C127#12011yescharacteristic
209C2×C4×C127#12011yescharacteristic
210C2×C4×C127#12011yescharacteristic
211C2×C4×C127C2×SD1612yes—
212C2×C4×C127#13512yes—
213C2×C4×C127AGL(1,ℤ/8)12yes—
214C2×C4×C127AGL(1,ℤ/8)12yes—
215C2×C4×C127C22×D411yescharacteristic
216C2×C4×C127C22×D411yescharacteristic
217C2×C4×C127C22×D411yescharacteristic
218C8×C127—24no—
219C8×C127—42no—
220C8×C127—42no—
221C22×C127—24no—
222C22×C127—48no—
223C22×C127—42no—
224C22×C127—42no—
225C22×C127—42no—
226C22×C127—42no—
227C22×C127—42no—
228C22×C127—84no—
229C22×C127#44912yes—
230C22×C127#45012yes—
231C22×C127#45612yes—
232C22×C127C2×#12011yescharacteristic
233C4×C127—21no—
234C4×C127—21no—
235C4×C127—42no—
236C4×C127—44no—
237C4×C127—42no—
238C4×C127—82no—
239C4×C127#44912yes—
240C4×C127#45012yes—
241C4×C127C22×SD1611yescharacteristic
242C4×C127C2×#13511yescharacteristic
243C4×C127C2×AGL(1,ℤ/8)11yescharacteristic
244C4×C127C2×AGL(1,ℤ/8)11yescharacteristic
245C2×#1654C12711yesmaximal, characteristic, direct product factor, Sylow
246C2×C127—44no—
247C2×C127—82no—
248C2×C127#165414yesdirect product factor
249C2×C127C2×#45011yescharacteristic
250C2×C127C2×#44911yescharacteristic
251C2×C127C2×#45611yescharacteristic
252C2×#547C2×C12711yescharacteristic
253C2×#498C2×C12711yescharacteristic, complemented
254C2×#474C2×C12711yescharacteristic, complemented
255C2×#461C2×C12711yescharacteristic, complemented
256#1654C2×C12718yesdirect product factor
257C2×#332C2×C12711yescharacteristic, complemented
258C2×D4⋊C8C2×C12711yescharacteristic, complemented
259C2×D4⋊C8C2×C12711yescharacteristic, complemented
260C127C2×#165411yescharacteristic, direct product factor, Sylow
261C22×#41—21no—
262C22×SD16—21no—
263#547—22no—
264#547—22no—
265#547—22no—
266#547C22×C12714yes—
267C2×C4⋊Q8C22×C12711yescharacteristic
268C2×C4⋊Q8C22×C12711yescharacteristic
269C2×#123—22no—
270C2×C22⋊Q8—21no—
271C2×#121—21no—
272C2×C4×D4C22×C12711yescharacteristic
273C2×C4×D4C22×C12711yescharacteristic
274C22×C4⋊C4—21no—
275#498—22no—
276#498C22×C12714yescomplemented
277#474—22no—
278#474C22×C12714yescomplemented
279#461—22no—
280#461C22×C12714yescomplemented
281C2×#107—21no—
282C2×#106—22no—
283C2×C4⋊C8C22×C12711yescharacteristic
284C2×C4⋊C8C22×C12711yescharacteristic
285C2×#103—21no—
286C2×#102—21no—
287C2×C4×C8C22×C12711yescharacteristic
288#332C22×C12714yescomplemented
289D4⋊C8C22×C12714yescomplemented
290D4⋊C8C22×C12714yescomplemented
291C2×#41—24no—
292C2×#41—24no—
293C2×#41—44no—
294C2×#41—44no—
295C22×Q8—21no—
296C22×Q8—21no—
297C22×D4—22no—
298C22×D4D4×C12711yescharacteristic
299C22×D4D4×C12711yescharacteristic
300C23×C4—22no—
301C23×C4—21no—
302C23×C4—21no—
303C2×SD16—24no—
304C2×SD16—44no—
305C22×C8—21no—
306C22×C8—21no—
307C22×C8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8D4×C12712yes—
314C4⋊Q8C23×C12712yes—
315C4⋊Q8C23×C12712yes—
316#123—28no—
317#123—28no—
318C22⋊Q8—24no—
319C22⋊Q8—24no—
320#121—24no—
321#121—24no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4D4×C12712yes—
327C4×D4D4×C12712yes—
328C4×D4C23×C12712yes—
329C4×D4C23×C12712yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4D4×C12711yescharacteristic
337C2×C4⋊C4D4×C12711yescharacteristic
338C2×C4⋊C4D4×C12712yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C42C23×C12711yescharacteristic
343#107—24no—
344#106—28no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8C23×C12712yes—
348C4⋊C8C23×C12712yes—
349#103—24no—
350#102—24no—
351C4×C8—21no—
352C4×C8C23×C12712yes—
353C24—21no—
354C24—21no—
355#41—48no—
356#41—416no—
357#41—48no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×D4—28no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—48no—
373C2×D4SD16×C12714yescomplemented
374C2×D4C2×D4×C12712yes—
375C2×D4C2×D4×C12712yes—
376C22×C4—28no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4C2×D4×C12711yescharacteristic
391C22×C4C2×D4×C12711yescharacteristic
392C22×C4C2×D4×C12711yescharacteristic
393SD16—48no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—42no—
399C2×C8—42no—
400C4⋊C4—28no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4C2×D4×C12712yes—
411C4⋊C4C2×D4×C12714yes—
412C4⋊C4C2×D4×C12712yes—
413C22⋊C4—28no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C42C2×D4×C12711yescharacteristic
419C42C2×D4×C12711yescharacteristic
420C42C2×D4×C12711yescharacteristic
421C42C24×C12711yescharacteristic
422C23—22no—
423C23—24no—
424C23—22no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23#120×C12711yescharacteristic
431Q8—42no—
432Q8—42no—
433Q8—44no—
434Q8—42no—
435Q8—42no—
436Q8—82no—
437D4—22no—
438D4—22no—
439D4—22no—
440D4—24no—
441D4—416no—
442D4—42no—
443D4C2×SD16×C12714yescomplemented
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—416no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4#120×C12711yescharacteristic
461C2×C4#120×C12711yescharacteristic
462C2×C4#120×C12711yescharacteristic
463C2×C4C2×SD16×C12712yes—
464C2×C4#135×C12712yes—
465C2×C4AGL(1,ℤ/8)×C12712yes—
466C2×C4AGL(1,ℤ/8)×C12712yes—
467C2×C4C22×D4×C12711yescharacteristic
468C2×C4C22×D4×C12711yescharacteristic
469C2×C4C22×D4×C12711yescharacteristic
470C8—24no—
471C8—42no—
472C8—42no—
473C22—24no—
474C22—48no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—84no—
481C22#449×C12712yes—
482C22#450×C12712yes—
483C22#456×C12712yes—
484C22C2×#120×C12711yescharacteristic
485C4—21no—
486C4—21no—
487C4—42no—
488C4—44no—
489C4—42no—
490C4—82no—
491C4#449×C12712yes—
492C4#450×C12712yes—
493C4C22×SD16×C12711yescharacteristic
494C4C2×#135×C12711yescharacteristic
495C4C2×AGL(1,ℤ/8)×C12711yescharacteristic
496C4C2×AGL(1,ℤ/8)×C12711yescharacteristic
497C2—44no—
498C2—82no—
499C2C127×#165414yesdirect product factor
500C2C2×#450×C12711yescharacteristic
501C2C2×#449×C12711yescharacteristic
502C2C2×#456×C12711yescharacteristic
subgroupG/H
center, socleC23×C127#120
commutator, Frattini, cosocleC42C24×C127
prime radicalC2×#547×C127C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#498×C127C2semidirect
3. C2×#474×C127C2semidirect
4. C2×#461×C127C2semidirect
5. C127×#1654C2direct (N × H)
6. C2×#332×C127C2semidirect
7. C2×D4⋊C8×C127C2semidirect
8. C2×D4⋊C8×C127C2semidirect
24. #498×C127C22semidirect
26. #474×C127C22semidirect
28. #461×C127C22semidirect
36. #332×C127C22semidirect
37. D4⋊C8×C127C22semidirect
38. D4⋊C8×C127C22semidirect
121. C2×D4×C127SD16semidirect
191. D4×C127C2×SD16semidirect
245. C2×#1654C127direct (N × H)
248. C2×C127#1654direct (N × H)
253. C2×#498C2×C127semidirect
254. C2×#474C2×C127semidirect
255. C2×#461C2×C127semidirect
256. #1654C2×C127direct (N × H)
257. C2×#332C2×C127semidirect
258. C2×D4⋊C8C2×C127semidirect
259. C2×D4⋊C8C2×C127semidirect
260. C127C2×#1654direct (N × H)
276. #498C22×C127semidirect
278. #474C22×C127semidirect
280. #461C22×C127semidirect
288. #332C22×C127semidirect
289. D4⋊C8C22×C127semidirect
290. D4⋊C8C22×C127semidirect
373. C2×D4SD16×C127semidirect
443. D4C2×SD16×C127semidirect
499. C2C127×#1654direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#547×C1272
C2×#498×C1272
C2×#474×C1272
C2×#461×C1272
C127×#16542
C2×#332×C1272
C2×D4⋊C8×C1272
C2×#1654127