Finite Group Databasebeta

← C2×#20242477 #26191975

Subgroups — C2×#20242477 #26191975

Order 12032.

1322 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689471C211yesmaximal, characteristic, complemented
2C2×#15689318C211yesmaximal, characteristic, complemented
3C2×#15689318C211yesmaximal, characteristic, complemented
4#20242477C218yesmaximal, direct product factor
5C2×#15688869C211yesmaximal, characteristic, complemented
6C2×D4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
7C2×D4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
8C2×C47×#547C211yesmaximal, characteristic, complemented
9C22×#10625334—21no—
10C22×#10625334—21no—
11C2×#13094021C2211yescharacteristic
12#15689471—22no—
13#15689471C2214yescomplemented
14#15689318—22no—
15#15689318—22no—
16#15689318C2214yescomplemented
17#15689318C2214yescomplemented
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093944—21no—
21C2×#13093944—21no—
22C2×#13093944—22no—
23C2×C4×C47⋊C8C2211yescharacteristic
24#15688869C2214yescomplemented
25D4⋊(C47⋊C8)C2214yescomplemented
26D4⋊(C47⋊C8)C2214yescomplemented
27C22×#41×C47—21no—
28C47×#547—22no—
29C47×#547—22no—
30C47×#547—22no—
31C47×#547C2214yescomplemented
32C2×C4⋊Q8×C47C2211yescharacteristic
33C2×#123×C47—22no—
34C2×C22⋊Q8×C47—21no—
35C2×#121×C47—21no—
36C2×C4×D4×C47C2211yescharacteristic
37C2×C4×D4×C47C2211yescharacteristic
38C22×C4⋊C4×C47—21no—
39C22×D188—21no—
40C22×D188—21no—
41C22×D188—21no—
42C2×#10625334—24no—
43C2×#10625334—24no—
44C2×#10625334—44no—
45C2×#10625334—44no—
46C22×C47⋊C8—21no—
47C22×C47⋊C8—21no—
48C22×C47⋊C8—21no—
49#13094021—21no—
50#13094021—21no—
51#13094021—21no—
52#13094021C2312yes—
53C4⋊(C47⋊C8)—21no—
54C4⋊(C47⋊C8)—21no—
55C4⋊(C47⋊C8)C2312yes—
56C4⋊(C47⋊C8)C2312yes—
57#13093944—24no—
58#13093944—24no—
59#13093944—28no—
60C4×C47⋊C8—21no—
61C4×C47⋊C8C2312yes—
62C2×#41×C47—24no—
63C2×#41×C47—24no—
64C2×#41×C47—44no—
65C2×#41×C47—44no—
66C22×Q8×C47—21no—
67C22×D4×C47—22no—
68C22×D4×C47D411yescharacteristic
69C22×D4×C47D411yescharacteristic
70C23×C4×C47—22no—
71C23×C4×C47—21no—
72C23×C4×C47—21no—
73C4⋊Q8×C47—21no—
74C4⋊Q8×C47—21no—
75C4⋊Q8×C47D412yes—
76C4⋊Q8×C47C2312yes—
77#123×C47—28no—
78#123×C47—28no—
79C22⋊Q8×C47—24no—
80C22⋊Q8×C47—24no—
81#121×C47—24no—
82#121×C47—24no—
83C4×D4×C47—21no—
84C4×D4×C47—21no—
85C4×D4×C47—21no—
86C4×D4×C47—21no—
87C4×D4×C47D412yes—
88C4×D4×C47D412yes—
89C4×D4×C47C2312yes—
90C4×D4×C47C2312yes—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47—24no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—24no—
95C2×C4⋊C4×C47D411yescharacteristic
96C2×C4⋊C4×C47D411yescharacteristic
97C2×C4⋊C4×C47D412yes—
98C2×C22⋊C4×C47—22no—
99C2×C22⋊C4×C47—21no—
100C2×C22⋊C4×C47—21no—
101C2×C42×C47C2311yescharacteristic
102C24×C47—21no—
103C24×C47—21no—
104C23×D47—41no—
105#41×C47—48no—
106#41×C47—416no—
107#41×C47—48no—
108C2×Q8×C47—22no—
109C2×Q8×C47—22no—
110C2×Q8×C47—42no—
111C2×Q8×C47—42no—
112C2×D4×C47—28no—
113C2×D4×C47—21no—
114C2×D4×C47—21no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47—48no—
119C2×D4×C47D814yescomplemented
120C2×D4×C47C2×D412yes—
121C2×D4×C47C2×D412yes—
122C22×C4×C47—28no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—48no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C2×D188—22no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188—44no—
142C2×D188—42no—
143C2×D188—42no—
144C2×D188—42no—
145C2×D188—42no—
146C2×D188—41no—
147C2×D188—41no—
148C2×D188—41no—
149C4⋊C4×C47—28no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47C2×D412yes—
156C4⋊C4×C47C2×D412yes—
157C4⋊C4×C47C2×D414yes—
158C22⋊C4×C47—28no—
159C22⋊C4×C47—22no—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—22no—
163C42×C47C2×D411yescharacteristic
164C42×C47C2×D411yescharacteristic
165C42×C47C2×D411yescharacteristic
166C42×C47C2411yescharacteristic
167#10625334—48no—
168#10625334—48no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—42no—
173C2×C47⋊C8—42no—
174C2×C47⋊C8—42no—
175C23×C47—22no—
176C23×C47—24no—
177C23×C47—22no—
178C23×C47—42no—
179C23×C47—42no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47#12011yescharacteristic
184C22×D47—42no—
185C22×D47—82no—
186C22×D47—82no—
187C22×D47—82no—
188Q8×C47—42no—
189Q8×C47—42no—
190Q8×C47—44no—
191D4×C47—22no—
192D4×C47—22no—
193D4×C47—22no—
194D4×C47—24no—
195D4×C47—416no—
196D4×C47—42no—
197D4×C47C2×D814yescomplemented
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—24no—
202C2×C4×C47—416no—
203C2×C4×C47—42no—
204C2×C4×C47—44no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—44no—
211C2×C4×C47—44no—
212C2×C4×C47#12011yescharacteristic
213C2×C4×C47#12011yescharacteristic
214C2×C4×C47#12011yescharacteristic
215C2×C4×C47C2×D812yes—
216C2×C4×C47#13512yes—
217C2×C4×C47AGL(1,ℤ/8)12yes—
218C2×C4×C47AGL(1,ℤ/8)12yes—
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222D188—44no—
223D188—42no—
224D188—42no—
225D188—42no—
226D188—42no—
227D188—82no—
228D188—82no—
229D188—82no—
230C47⋊C8—42no—
231C47⋊C8—42no—
232C47⋊C8—42no—
233C2×#1625—471nomaximal, Sylow
234C22×C47—24no—
235C22×C47—48no—
236C22×C47—42no—
237C22×C47—42no—
238C22×C47—42no—
239C22×C47—42no—
240C22×C47—42no—
241C22×C47—84no—
242C22×C47#44712yes—
243C22×C47#44912yes—
244C22×C47#45412yes—
245C22×C47C2×#12011yescharacteristic
246C2×D47—82no—
247C2×D47—82no—
248C2×D47—82no—
249C2×D47—164no—
250C4×C47—21no—
251C4×C47—21no—
252C4×C47—42no—
253C4×C47—44no—
254C4×C47—42no—
255C4×C47#44712yes—
256C4×C47#44912yes—
257C4×C47C22×D811yescharacteristic
258C4×C47C2×#13511yescharacteristic
259C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
260C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
261C2×#547D4711yescharacteristic, complemented
262C2×#486—471no—
263C2×#459—471no—
264C2×#459—471no—
265#1625—478no—
266C2×#332—471no—
267C2×D4⋊C8—471no—
268C2×D4⋊C8—471no—
269C2×C47—44no—
270C2×C47—82no—
271C2×C47#162514yescomplemented
272C2×C47C2×#44711yescharacteristic
273C2×C47C2×#44911yescharacteristic
274C2×C47C2×#45411yescharacteristic
275D47—162no—
276C22×#41—21no—
277C22×D8—941no—
278C22×D8—941no—
279#547—22no—
280#547—22no—
281#547—22no—
282#547C2×D4714yescomplemented
283C2×C4⋊Q8C2×D4711yescharacteristic
284C2×#127—471no—
285C2×#123—22no—
286C2×C22⋊Q8—21no—
287C2×#121—21no—
288C2×C4×D4C2×D4711yescharacteristic
289C2×C4×D4C2×D4711yescharacteristic
290C22×C4⋊C4—21no—
291#486—474no—
292#486—942no—
293#459—474no—
294#459—474no—
295#459—942no—
296#459—942no—
297C2×C4⋊C8—471no—
298C2×C4⋊C8—471no—
299C2×#102—942no—
300C2×#102—941no—
301C2×#102—941no—
302C2×C4×C8—471no—
303#332—474no—
304D4⋊C8—474no—
305D4⋊C8—474no—
306C47C2×#162511yescharacteristic, complemented, Sylow
307C2×#41—24no—
308C2×#41—24no—
309C2×#41—44no—
310C2×#41—44no—
311C22×Q8—21no—
312C22×D4—22no—
313C22×D4—941no—
314C22×D4—941no—
315C22×D4—941no—
316C22×D4C22⋊D4711yescharacteristic
317C22×D4C22⋊D4711yescharacteristic
318C23×C4—22no—
319C23×C4—21no—
320C23×C4—21no—
321C2×D8—944no—
322C2×D8—944no—
323C2×D8—1884no—
324C2×D8—1884no—
325C22×C8—941no—
326C22×C8—941no—
327C22×C8—941no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8C22⋊D4712yes—
331C4⋊Q8C22×D4712yes—
332#127—472no—
333#127—941no—
334#127—941no—
335#127—941no—
336#123—28no—
337#123—28no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340#121—24no—
341#121—24no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4C22⋊D4712yes—
347C4×D4C22⋊D4712yes—
348C4×D4C22×D4712yes—
349C4×D4C22×D4712yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4C22⋊D4711yescharacteristic
355C2×C4⋊C4C22⋊D4711yescharacteristic
356C2×C4⋊C4C22⋊D4712yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C42C22×D4711yescharacteristic
361C4⋊C8—472no—
362C4⋊C8—472no—
363C4⋊C8—941no—
364C4⋊C8—941no—
365#102—944no—
366#102—944no—
367#102—948no—
368C4×C8—472no—
369C4×C8—941no—
370C24—21no—
371C24—21no—
372C24—1881no—
373#41—416no—
374#41—48no—
375#41—48no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×D4—28no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—48no—
387C2×D4—942no—
388C2×D4—942no—
389C2×D4—942no—
390C2×D4—1884no—
391C2×D4—1881no—
392C2×D4—1881no—
393C2×D4—1881no—
394C2×D4—1882no—
395C2×D4—1882no—
396C2×D4—1882no—
397C2×D4—1882no—
398C2×D4#1062533414yescomplemented
399C2×D4C2×C22⋊D4712yes—
400C2×D4C2×C22⋊D4712yes—
401C22×C4—28no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C2×C22⋊D4711yescharacteristic
415C22×C4C2×C22⋊D4711yescharacteristic
416C22×C4C2×C22⋊D4711yescharacteristic
417D8—1888no—
418D8—1888no—
419C2×C8—942no—
420C2×C8—942no—
421C2×C8—942no—
422C2×C8—1882no—
423C2×C8—1882no—
424C2×C8—1882no—
425C4⋊C4—28no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4C2×C22⋊D4712yes—
432C4⋊C4C2×C22⋊D4712yes—
433C4⋊C4C2×C22⋊D4714yes—
434C22⋊C4—28no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C42C2×C22⋊D4711yescharacteristic
440C42C2×C22⋊D4711yescharacteristic
441C42C2×C22⋊D4711yescharacteristic
442C42C23×D4711yescharacteristic
443C23—24no—
444C23—22no—
445C23—22no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—3762no—
452C23—3762no—
453C23—3762no—
454C23—1882no—
455C23#1309399311yescharacteristic
456Q8—44no—
457Q8—42no—
458Q8—42no—
459D4—24no—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—416no—
464D4—42no—
465D4—3762no—
466D4—3762no—
467D4—3762no—
468D4—1884no—
469D4—1882no—
470D4—1882no—
471D4—1882no—
472D4—1882no—
473D4C2×#1062533414yescomplemented
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—416no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4#1309399311yescharacteristic
489C2×C4#1309399311yescharacteristic
490C2×C4#1309399311yescharacteristic
491C2×C4C2×#1062533412yes—
492C2×C4#1309404412yes—
493C2×C4#1309404912yes—
494C2×C4#1309405012yes—
495C2×C4C22×C22⋊D4711yescharacteristic
496C2×C4C22×C22⋊D4711yescharacteristic
497C2×C4C22×C22⋊D4711yescharacteristic
498C8—1882no—
499C8—1882no—
500C8—1882no—
501C22—24no—
502C22—48no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—84no—
509C22—3762no—
510C22—3762no—
511C22—3762no—
512C22—7524no—
513C22#1568925712yes—
514C22#1568926812yes—
515C22#1568929612yes—
516C22C2×#1309399311yescharacteristic
517C4—21no—
518C4—21no—
519C4—42no—
520C4—44no—
521C4—42no—
522C4#1568925712yes—
523C4#1568926812yes—
524C4C22×#1062533411yescharacteristic
525C4C2×#1309404411yescharacteristic
526C4C2×#1309404911yescharacteristic
527C4C2×#1309405011yescharacteristic
528C2—44no—
529C2—82no—
530C2—7522no—
531C2#2024247714yesdirect product factor
532C2C2×#1568925711yescharacteristic
533C2C2×#1568926811yescharacteristic
534C2C2×#1568929611yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#547C2
socleC23×C47#120
hypercenterC2×#547D47
hypocenterC47C2×#1625
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#15689471C2semidirect
2. C2×#15689318C2semidirect
3. C2×#15689318C2semidirect
4. #20242477C2direct (N × H)
5. C2×#15688869C2semidirect
6. C2×D4⋊(C47⋊C8)C2semidirect
7. C2×D4⋊(C47⋊C8)C2semidirect
8. C2×C47×#547C2semidirect
13. #15689471C22semidirect
16. #15689318C22semidirect
17. #15689318C22semidirect
24. #15688869C22semidirect
25. D4⋊(C47⋊C8)C22semidirect
26. D4⋊(C47⋊C8)C22semidirect
31. C47×#547C22semidirect
119. C2×D4×C47D8semidirect
197. D4×C47C2×D8semidirect
261. C2×#547D47semidirect
271. C2×C47#1625semidirect
282. #547C2×D47semidirect
306. C47C2×#1625semidirect
398. C2×D4#10625334semidirect
473. D4C2×#10625334semidirect
531. C2#20242477direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894712
C2×#156893182
#202424772
C2×#156888692
C2×D4⋊(C47⋊C8)2
C2×C47×#5472
C2×#162547