Finite Group Databasebeta

← #25130439

Subgroups — #25130439

Order 11008.

1088 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583589C211yesmaximal, characteristic
2C2×#14583235C211yesmaximal, characteristic, complemented
3C2×#14582941C211yesmaximal, characteristic, complemented
4#19138903C211yesmaximal, characteristic
5#19135934C212yesmaximal, complemented
6#19135792C212yesmaximal, complemented
7#19135028C211yesmaximal, characteristic, complemented
8#19134852C212yesmaximal, complemented
9#19134785C212yesmaximal, complemented
10#19134691C211yesmaximal, characteristic, complemented
11C43×#1870C211yesmaximal, characteristic
12#14583589—24no—
13#14583589—24no—
14C2×#13063318—21no—
15C2×#13063317C2211yescharacteristic
16C2×#13063315—21no—
17C2×#13063306—21no—
18C2×#13063306—21no—
19C2×#13063304C2211yescharacteristic
20C2×#13063304C2211yescharacteristic
21C2×C4×Dic86C2211yescharacteristic
22C2×#13063284C2211yescharacteristic
23C2×#13063281—21no—
24C22×C4⋊Dic43C2211yescharacteristic
25C22×C4⋊Dic43C2211yescharacteristic
26#14583235—24no—
27#14582941—24no—
28C2×#13063259C2212yes—
29C2×#13063259C2212yes—
30C2×#13063257C2212yes—
31C2×#13063250C2212yes—
32C2×#13063250C2212yes—
33#14582829—21no—
34#14582824—21no—
35#14582797C2212yes—
36#14582760—21no—
37#14582759C2212yes—
38#14582759C2211yescharacteristic
39#14582662C2212yes—
40#14582662C2212yes—
41#14582662C2212yes—
42C22×M16×C43C2211yescharacteristic
43C2×C4×D4×C43C2211yescharacteristic
44C2×C22⋊C8×C43C2212yes—
45C2×C22⋊C8×C43C2211yescharacteristic
46C43×#336C2212yes—
47C22×Dic86C2311yescharacteristic
48C23×Dic43—21no—
49C23×Dic43—21no—
50#13063318—28no—
51#13063317—22no—
52#13063317—22no—
53#13063315—28no—
54#13063306—28no—
55#13063306—44no—
56#13063304—22no—
57#13063304—22no—
58#13063304—22no—
59#13063304D414yes—
60C4×Dic86—22no—
61C4×Dic86—22no—
62#13063284—22no—
63#13063284D414yes—
64#13063281—28no—
65C2×C4⋊Dic43—21no—
66C2×C4⋊Dic43—22no—
67C2×C4⋊Dic43—22no—
68C2×C4⋊Dic43—22no—
69C2×C4⋊Dic43—22no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43C2312yes—
72C2×C4⋊Dic43C2312yes—
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×Dic43⋊C4—21no—
77C2×Dic43⋊C4—21no—
78C2×Dic43⋊C4—21no—
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—22no—
82C2×C22⋊Dic43—21no—
83C2×C22⋊Dic43—21no—
84C2×C22⋊Dic43—21no—
85C2×C22⋊Dic43—22no—
86C2×C22⋊Dic43—21no—
87C2×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89#13063259—24no—
90#13063259—24no—
91#13063257—24no—
92#13063250—24no—
93#13063250—24no—
94C43⋊#95—22no—
95C43⋊#95—21no—
96C43⋊#95—22no—
97C22×D4×C43D411yescharacteristic
98C23×C4×C43C2311yescharacteristic
99C23×C4×C43C2311yescharacteristic
100C2×M16×C43—22no—
101C2×M16×C43—24no—
102C22×C8×C43C2312yes—
103C22×C8×C43C2312yes—
104C4×D4×C43—22no—
105C4×D4×C43—22no—
106C2×C4⋊C4×C43D411yescharacteristic
107C2×C22⋊C4×C43D411yescharacteristic
108C2×C22⋊C4×C43D411yescharacteristic
109C2×C42×C43C2311yescharacteristic
110C22⋊C8×C43—22no—
111C22⋊C8×C43—24no—
112C24×C43C2×D411yescharacteristic
113C24×C43C2×D411yescharacteristic
114C2×D4×C43—22no—
115C2×D4×C43—22no—
116C2×D4×C43—41no—
117C22×C4×C43—22no—
118C22×C4×C43—22no—
119C22×C4×C43—21no—
120C22×C4×C43—21no—
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43C2×D412yes—
124C22×C4×C43C2×D412yes—
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43C2×D411yescharacteristic
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2411yescharacteristic
130C22×Dic43—24no—
131C22×Dic43—21no—
132C22×Dic43—21no—
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43—22no—
136C22×Dic43—22no—
137C22×Dic43—42no—
138C22×Dic43—42no—
139C22×Dic43—42no—
140C22×Dic43—42no—
141C22×Dic43—42no—
142C2×Dic86—22no—
143C2×Dic86—22no—
144C2×Dic86—21no—
145C2×Dic86C2×D412yes—
146M16×C43—28no—
147C2×C8×C43—22no—
148C2×C8×C43—22no—
149C2×C8×C43—22no—
150C2×C8×C43—22no—
151C2×C8×C43—22no—
152C2×C8×C43#4114yes—
153C4⋊C4×C43—22no—
154C22⋊C4×C43—22no—
155C22⋊C4×C43—22no—
156C42×C43—21no—
157C42×C43—21no—
158C22⋊Dic43—24no—
159C22⋊Dic43—24no—
160C22⋊Dic43—24no—
161C22⋊Dic43—42no—
162C22⋊Dic43—44no—
163C4⋊Dic43—24no—
164C4⋊Dic43—21no—
165C4⋊Dic43—21no—
166C4⋊Dic43—21no—
167C4⋊Dic43—22no—
168C4⋊Dic43—22no—
169C4⋊Dic43—22no—
170C4⋊Dic43—22no—
171C4⋊Dic43—42no—
172C4⋊Dic43C2×D412yes—
173C4⋊Dic43C2×D412yes—
174C4⋊Dic43C2×D412yes—
175Dic43⋊C4—24no—
176Dic43⋊C4—24no—
177Dic43⋊C4—24no—
178Dic43⋊C4—24no—
179Dic43⋊C4—24no—
180Dic43⋊C4—44no—
181C4×Dic43—24no—
182C4×Dic43—24no—
183C23×C43—21no—
184C23×C43—22no—
185C23×C43—22no—
186C23×C43—21no—
187C23×C43—21no—
188C23×C43—21no—
189C23×C43—22no—
190C23×C43—22no—
191C23×C43#12012yes—
192C23×C43AGL(1,ℤ/8)12yes—
193C23×C43C22×D411yescharacteristic
194D4×C43—44no—
195C2×C4×C43—22no—
196C2×C4×C43—22no—
197C2×C4×C43—22no—
198C2×C4×C43—24no—
199C2×C4×C43—21no—
200C2×C4×C43—21no—
201C2×C4×C43—21no—
202C2×C4×C43—22no—
203C2×C4×C43—21no—
204C2×C4×C43—21no—
205C2×C4×C43—22no—
206C2×C4×C43—41no—
207C2×C4×C43—41no—
208C2×C4×C43—42no—
209C2×C4×C43#12012yes—
210C2×C4×C43C22×D411yescharacteristic
211C2×C4×C43C22×D411yescharacteristic
212C2×C4×C43C2×#4112yes—
213C2×C4×C432−1+412yes—
214C2×Dic43—22no—
215C2×Dic43—22no—
216C2×Dic43—22no—
217C2×Dic43—22no—
218C2×Dic43—48no—
219C2×Dic43—42no—
220C2×Dic43—41no—
221C2×Dic43—41no—
222C2×Dic43—41no—
223C2×Dic43—41no—
224C2×Dic43—42no—
225C2×Dic43—42no—
226C2×Dic43—42no—
227C2×Dic43—42no—
228C2×Dic43—42no—
229C2×Dic43—42no—
230C2×Dic43—42no—
231C2×Dic43—84no—
232Dic86—24no—
233Dic86—41no—
234Dic86—41no—
235C8×C43—24no—
236C8×C43—42no—
237#24110—431nomaximal, Sylow
238C22×C43—24no—
239C22×C43—22no—
240C22×C43—22no—
241C22×C43—22no—
242C22×C43—22no—
243C22×C43—42no—
244C22×C43—42no—
245C22×C43—42no—
246C22×C43—41no—
247C22×C43—41no—
248C22×C43C2×#12011yescharacteristic
249C22×C43#54712yes—
250C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C43#57511yescharacteristic
252C22×C43#57812yes—
253C4×C43—22no—
254C4×C43—41no—
255C4×C43—41no—
256C4×C43—41no—
257C4×C43#52511yescharacteristic
258C4×C43#53611yescharacteristic
259C4×C43#54712yes—
260Dic43—42no—
261Dic43—42no—
262Dic43—42no—
263Dic43—42no—
264Dic43—82no—
265Dic43—82no—
266C2×#552—431no—
267C2×#484—431no—
268C2×#428—431no—
269#2486—431no—
270#2059—432no—
271#2035—432no—
272#1894—431no—
273#1870D4311yescharacteristic
274#1863—432no—
275#1852—432no—
276#1825—431no—
277C2×C43—42no—
278C2×C43—42no—
279C2×C43#240611yescharacteristic
280C2×C43#300611yescharacteristic
281C2×C43#301111yescharacteristic
282C2×C43#334312yes—
283C2×C43#335212yes—
284C22×M16C2×D4311yescharacteristic
285#552—864no—
286#552—864no—
287C2×#126—861no—
288C2×#125—431no—
289C2×#125—861no—
290C2×#123—861no—
291C2×#123—861no—
292C2×C22⋊Q8—431no—
293C2×C22⋊Q8—431no—
294C2×C4×Q8—431no—
295C2×C4×D4C2×D4311yescharacteristic
296C2×#117—431no—
297C2×#117—861no—
298C22×C4⋊C4—431no—
299C22×C4⋊C4—431no—
300#484—864no—
301#428—864no—
302C2×#107—432no—
303C2×#107—432no—
304C2×#106—432no—
305C2×#103—432no—
306C2×#103—432no—
307C2×C22⋊C8C2×D4311yescharacteristic
308C2×C22⋊C8C2×D4312yes—
309#398—861no—
310#397—861no—
311#389—432no—
312#381—861no—
313#380—431no—
314#380—432no—
315#340—432no—
316#340—432no—
317#340—432no—
318#336C2×D4312yes—
319C43#2411011yescharacteristic, complemented, Sylow
320C22×Q8—431no—
321C22×D4D17211yescharacteristic
322C23×C4—861no—
323C23×C4—861no—
324C23×C4C22×D4311yescharacteristic
325C23×C4C22×D4311yescharacteristic
326C2×M16—22no—
327C2×M16—24no—
328C22×C8C22×D4312yes—
329C22×C8C22×D4312yes—
330#126—868no—
331#125—862no—
332#125—862no—
333#125—868no—
334#123—868no—
335#123—1724no—
336C22⋊Q8—434no—
337C22⋊Q8—862no—
338C22⋊Q8—862no—
339C22⋊Q8—862no—
340C4×Q8—862no—
341C4×Q8—862no—
342C4×D4—22no—
343C4×D4—22no—
344#117—434no—
345#117—868no—
346#117—862no—
347C2×C4⋊C4—431no—
348C2×C4⋊C4—431no—
349C2×C4⋊C4—431no—
350C2×C4⋊C4—432no—
351C2×C4⋊C4—432no—
352C2×C4⋊C4—862no—
353C2×C4⋊C4—862no—
354C2×C4⋊C4—861no—
355C2×C4⋊C4—861no—
356C2×C4⋊C4—861no—
357C2×C4⋊C4—861no—
358C2×C4⋊C4—861no—
359C2×C4⋊C4—861no—
360C2×C4⋊C4—862no—
361C2×C4⋊C4—862no—
362C2×C4⋊C4—861no—
363C2×C4⋊C4—862no—
364C2×C4⋊C4D17211yescharacteristic
365C2×C22⋊C4—861no—
366C2×C22⋊C4—861no—
367C2×C22⋊C4—861no—
368C2×C22⋊C4—861no—
369C2×C22⋊C4—862no—
370C2×C22⋊C4D17211yescharacteristic
371C2×C22⋊C4D17211yescharacteristic
372C2×C42—861no—
373C2×C42—861no—
374C2×C42C22×D4311yescharacteristic
375#107—864no—
376#107—864no—
377#106—864no—
378#103—864no—
379#103—864no—
380C22⋊C8—22no—
381C22⋊C8—24no—
382#95—862no—
383#95—861no—
384#95—862no—
385C24C2×D17211yescharacteristic
386C24C2×D17211yescharacteristic
387C2×Q8—432no—
388C2×Q8—861no—
389C2×Q8—862no—
390C2×Q8—862no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—864no—
401C22×C4—861no—
402C22×C4—861no—
403C22×C4—861no—
404C22×C4—861no—
405C22×C4—862no—
406C22×C4—862no—
407C22×C4—1722no—
408C22×C4—1722no—
409C22×C4—1722no—
410C22×C4—1722no—
411C22×C4—1722no—
412C22×C4C2×D17211yescharacteristic
413C22×C4C2×D17211yescharacteristic
414C22×C4C2×D17211yescharacteristic
415C22×C4C2×D17211yescharacteristic
416C22×C4D4×D4312yes—
417C22×C4D4×D4312yes—
418C22×C4C23×D4311yescharacteristic
419M16—28no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8#1062190014yes—
426C4⋊C4—22no—
427C4⋊C4—432no—
428C4⋊C4—432no—
429C4⋊C4—432no—
430C4⋊C4—864no—
431C4⋊C4—864no—
432C4⋊C4—864no—
433C4⋊C4—864no—
434C4⋊C4—864no—
435C4⋊C4—864no—
436C4⋊C4—861no—
437C4⋊C4—861no—
438C4⋊C4—861no—
439C4⋊C4—862no—
440C4⋊C4—862no—
441C4⋊C4—862no—
442C4⋊C4—862no—
443C4⋊C4—1724no—
444C4⋊C4—1722no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—864no—
448C22⋊C4—864no—
449C22⋊C4—864no—
450C22⋊C4—1722no—
451C22⋊C4—1724no—
452C42—21no—
453C42—21no—
454C42—864no—
455C42—864no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23#1306329512yes—
465C23#1306335312yes—
466C23C22×D17211yescharacteristic
467Q8—864no—
468Q8—1721no—
469Q8—1721no—
470D4—44no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—862no—
486C2×C4—862no—
487C2×C4—862no—
488C2×C4—862no—
489C2×C4—3444no—
490C2×C4—1722no—
491C2×C4—1722no—
492C2×C4—1722no—
493C2×C4—1722no—
494C2×C4—1722no—
495C2×C4—1722no—
496C2×C4—1728no—
497C2×C4—1722no—
498C2×C4—1721no—
499C2×C4—1721no—
500C2×C4—1721no—
501C2×C4—1721no—
502C2×C4—1722no—
503C2×C4#1306329512yes—
504C2×C4C2×D4×D4311yescharacteristic
505C2×C4C2×D4×D4311yescharacteristic
506C2×C4C2×#1062190012yes—
507C2×C4#1306337212yes—
508C8—24no—
509C8—42no—
510C22—24no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—42no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22C2×#1306329511yescharacteristic
521C22#1458355312yes—
522C22C2×#1306335311yescharacteristic
523C22#1458369911yescharacteristic
524C22#1458371612yes—
525C4—22no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—3442no—
530C4—3442no—
531C4—1722no—
532C4—1722no—
533C4—1722no—
534C4—1722no—
535C4#1458340511yescharacteristic
536C4#1458347111yescharacteristic
537C4#1458354912yes—
538C2—42no—
539C2—42no—
540C2#1913792911yescharacteristic
541C2#1914384911yescharacteristic
542C2#1914389211yescharacteristic
543C2#1914703712yes—
544C2#1914716912yes—
subgroupG/H
centerC23C22×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1870C2
socleC23×C43C22×D4
prime radicalC22×D4×C43D4
hypercenter#1870D43
hypocenterC43#24110
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. C2×#14583235C2semidirect
3. C2×#14582941C2semidirect
5. #19135934C2semidirect
6. #19135792C2semidirect
7. #19135028C2semidirect
8. #19134852C2semidirect
9. #19134785C2semidirect
10. #19134691C2semidirect
319. C43#24110semidirect

Maximal subgroups

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

subgroupindex
C2×#145835892
C2×#145832352
C2×#145829412
#191389032
#191359342
#191357922
#191350282
#191348522
#191347852
#191346912
C43×#18702
#2411043