Finite Group Databasebeta

← #25130526

Subgroups — #25130526

Order 11008.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583623C211yesmaximal, characteristic, complemented
2C2×#14583176C211yesmaximal, characteristic, complemented
3C2×C4×#10621873C211yesmaximal, characteristic, complemented
4#19139511C211yesmaximal, characteristic
5#19135801C212yesmaximal, complemented
6#19135601C212yesmaximal, complemented
7#19135389C211yesmaximal, characteristic, complemented
8#19134916C212yesmaximal, complemented
9#19134795C212yesmaximal, complemented
10#19134597C211yesmaximal, characteristic, complemented
11C43×#1990C211yesmaximal, characteristic, complemented
12C22×#10621873C2211yescharacteristic
13#14583623—24no—
14#14583623—24no—
15C2×#13063320—22no—
16C2×#13063314—21no—
17C2×#13063305C2212yes—
18C2×#13063300—21no—
19C2×#13063300C2212yes—
20C2×Q8×Dic43—21no—
21C2×C4×Dic86C2211yescharacteristic
22C2×C4×D172C2211yescharacteristic
23C2×C4×D172C2211yescharacteristic
24C2×#13063283C2212yes—
25#14583176—24no—
26C4×#10621873—24no—
27C2×#13063258C2212yes—
28C2×#13063257C2211yescharacteristic
29C2×C4⋊(C43⋊C8)C2211yescharacteristic
30C2×#13063250C2211yescharacteristic
31C2×#13063248C2212yes—
32C2×#13063247C2211yescharacteristic
33C2×#13063245C2212yes—
34C2×(C2×D43)⋊C8C2212yes—
35C2×(C2×D43)⋊C8C2212yes—
36#14582816—22no—
37#14582756—21no—
38#14582756C2212yes—
39#14582661C2212yes—
40C2×C4⋊Q8×C43C2211yescharacteristic
41C2×C4×C8×C43C2211yescharacteristic
42C43×#384C2211yescharacteristic
43C43×#340C2212yes—
44C43×#340C2212yes—
45C22×Dic86C2311yescharacteristic
46C22×D172—22no—
47C22×D172D411yescharacteristic
48C22×D172C2311yescharacteristic
49C22×C4×D43—21no—
50C22×C4×D43C2312yes—
51C2×#10621873—22no—
52C2×#10621873—22no—
53C2×#10621873—22no—
54C22×C43⋊C8C2312yes—
55#13063320—216no—
56#13063314—28no—
57#13063305—24no—
58#13063305—24no—
59#13063300—24no—
60#13063300—42no—
61#13063300—44no—
62Q8×Dic43—28no—
63C4×Dic86—22no—
64C4×Dic86D414yescomplemented
65C4×D172—22no—
66C4×D172—22no—
67C4×D172—22no—
68C4×D172—22no—
69#13063283—24no—
70#13063283—24no—
71C2×C4⋊Dic43—22no—
72C2×C4⋊Dic43D411yescharacteristic
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×Dic43⋊C4—21no—
76C2×(C2×D43)⋊C4—22no—
77C2×(C2×D43)⋊C4—22no—
78C2×(C2×D43)⋊C4—22no—
79C2×(C2×D43)⋊C4—21no—
80C2×(C2×D43)⋊C4—22no—
81C2×(C2×D43)⋊C4D412yes—
82C2×C4×Dic43—21no—
83C2×C4×Dic43—22no—
84#13063258—24no—
85#13063257—22no—
86C4⋊(C43⋊C8)—22no—
87#13063250—22no—
88#13063248—42no—
89#13063247—22no—
90#13063245—24no—
91(C2×D43)⋊C8—24no—
92(C2×D43)⋊C8—42no—
93C43⋊#95—22no—
94C22×Q8×C43—21no—
95C22×C8×C43C2311yescharacteristic
96C22×C8×C43C2311yescharacteristic
97C4⋊Q8×C43—22no—
98C4⋊Q8×C43D414yescomplemented
99C2×C4⋊C4×C43—21no—
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43C2312yes—
103C2×C4⋊C4×C43C2312yes—
104C2×C42×C43C2311yescharacteristic
105C4×C8×C43—22no—
106C23×D43—21no—
107C23×D43C2×D412yes—
108C2×Q8×C43—24no—
109C2×Q8×C43—42no—
110C2×Q8×C43—42no—
111C22×C4×C43—22no—
112C22×C4×C43—22no—
113C22×C4×C43C2×D411yescharacteristic
114C22×C4×C43C2×D411yescharacteristic
115C22×C4×C43C2411yescharacteristic
116C22×Dic43—21no—
117C22×Dic43—21no—
118C22×Dic43—21no—
119C22×Dic43C2×D412yes—
120C2×D172—22no—
121C2×D172—22no—
122C2×D172—22no—
123C2×D172—21no—
124C2×D172—21no—
125C2×D172—21no—
126C2×D172—21no—
127C2×D172—21no—
128C2×D172—44no—
129C2×D172—44no—
130C2×D172—82no—
131C2×D172#4112yes—
132C2×C4×D43—22no—
133C2×C4×D43—22no—
134C2×C4×D43—22no—
135C2×C4×D43—22no—
136C2×C4×D43—22no—
137C2×C4×D43—22no—
138C2×C4×D43—42no—
139C2×C4×D43—42no—
140C2×C4×D43—42no—
141C2×Dic86—21no—
142C2×Dic86—21no—
143C2×Dic86—21no—
144C2×Dic86—21no—
145C2×Dic86—21no—
146C2×Dic86C2×D412yes—
147C2×C8×C43—21no—
148C2×C8×C43—21no—
149C2×C8×C43—21no—
150C2×C8×C43—21no—
151C2×C8×C43—21no—
152C2×C8×C43#4112yes—
153C4⋊C4×C43—24no—
154C4⋊C4×C43—22no—
155C4⋊C4×C43—22no—
156C4⋊C4×C43—22no—
157C4⋊C4×C43—42no—
158C4⋊C4×C43—81no—
159C4⋊C4×C43C2×D414yes—
160C42×C43C2×D412yes—
161C42×C43C2×D412yes—
162(C2×D43)⋊C4—28no—
163(C2×D43)⋊C4—28no—
164(C2×D43)⋊C4—24no—
165(C2×D43)⋊C4—44no—
166(C2×D43)⋊C4—44no—
167(C2×D43)⋊C4—42no—
168C4⋊Dic43—28no—
169C4⋊Dic43—22no—
170C4⋊Dic43—21no—
171C4⋊Dic43—21no—
172C4⋊Dic43—21no—
173C4⋊Dic43C2×D412yes—
174Dic43⋊C4—24no—
175C4×Dic43—28no—
176C4×Dic43—24no—
177C2×C43⋊C8—22no—
178C2×C43⋊C8—22no—
179C2×C43⋊C8—22no—
180#10621873—28no—
181C23×C43C22×D411yescharacteristic
182C22×D43—22no—
183C22×D43—22no—
184C22×D43—22no—
185C22×D43—22no—
186C22×D43—22no—
187C22×D43—22no—
188C22×D43—22no—
189C22×D43—22no—
190C22×D43—41no—
191C22×D43—41no—
192C22×D43—41no—
193C22×D43—41no—
194C22×D43—41no—
195C22×D43—41no—
196Q8×C43—48no—
197C2×C4×C43—21no—
198C2×C4×C43—21no—
199C2×C4×C43—24no—
200C2×C4×C43—42no—
201C2×C4×C43—42no—
202C2×C4×C43—42no—
203C2×C4×C43—42no—
204C2×C4×C43—42no—
205C2×C4×C43#12012yes—
206C2×C4×C43#12012yes—
207C2×C4×C43#13512yes—
208C2×C4×C43AGL(1,ℤ/8)12yes—
209C2×C4×C43C22×D411yescharacteristic
210C2×C4×C43C22×D411yescharacteristic
211C2×C4×C43C2×#4111yescharacteristic
212C2×C4×C43C2×#4111yescharacteristic
213C2×C4×C43H(5,𝔽2)11yescharacteristic
214C2×C4×C432−1+411yescharacteristic
215C2×Dic43—22no—
216C2×Dic43—22no—
217C2×Dic43—22no—
218C2×Dic43—22no—
219C2×Dic43—22no—
220C2×Dic43—41no—
221C2×Dic43—41no—
222C2×Dic43—41no—
223C2×Dic43—41no—
224C2×Dic43—41no—
225C2×Dic43—41no—
226C2×Dic43—41no—
227D172—22no—
228D172—22no—
229D172—41no—
230D172—41no—
231D172—44no—
232D172—88no—
233C4×D43—42no—
234C4×D43—42no—
235C4×D43—42no—
236C4×D43—42no—
237C4×D43—84no—
238Dic86—22no—
239Dic86—22no—
240Dic86—41no—
241Dic86—41no—
242C8×C43—22no—
243C8×C43—41no—
244C43⋊C8—42no—
245#24119—431nomaximal, Sylow
246C22×C43C2×#12011yescharacteristic
247C22×C43#54611yescharacteristic
248C22×C43#54711yescharacteristic
249C22×C43C2×#13511yescharacteristic
250C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C43#57711yescharacteristic
252C22×C43#57811yescharacteristic
253C2×D43—42no—
254C2×D43—42no—
255C2×D43—42no—
256C2×D43—42no—
257C2×D43—42no—
258C2×D43—42no—
259C2×D43—42no—
260C2×D43—42no—
261C2×D43—42no—
262C2×D43—42no—
263C2×D43—81no—
264C2×D43—81no—
265C2×D43—81no—
266C2×D43—81no—
267C4×C43—22no—
268C4×C43—44no—
269C4×C43—82no—
270C4×C43#44914yes—
271C4×C43#52511yescharacteristic
272C4×C43#53711yescharacteristic
273C4×C43#54811yescharacteristic
274C4×C43#55011yescharacteristic
275Dic43—42no—
276Dic43—42no—
277Dic43—42no—
278Dic43—81no—
279C2×#556—431no—
280C2×#474—431no—
281C2×C4×SD16—431no—
282#2537—431no—
283#2036—432no—
284#2000—432no—
285#1990D4311yescharacteristic, complemented
286#1954—431no—
287#1875—432no—
288#1854—432no—
289#1803—431no—
290C2×C43#240711yescharacteristic
291C2×C43C2×#44911yescharacteristic
292C2×C43#301311yescharacteristic
293C2×C43#330211yescharacteristic
294C2×C43#332411yescharacteristic
295C2×C43#334011yescharacteristic
296C2×C43#335311yescharacteristic
297D43—82no—
298D43—82no—
299C22×SD16—431no—
300#556—864no—
301#556—864no—
302C2×C4⋊Q8C2×D4311yescharacteristic
303C2×#126—862no—
304C2×#124—861no—
305C2×C22⋊Q8—432no—
306C2×#121—432no—
307C2×#121—861no—
308C2×C4×Q8—431no—
309C2×C4×Q8—861no—
310C2×C4×D4—431no—
311C2×C4×D4—431no—
312C2×#117—432no—
313#474—864no—
314C4×SD16—864no—
315C2×#107—432no—
316C2×#106—431no—
317C2×C4⋊C8—431no—
318C2×#103—431no—
319C2×#103—432no—
320C2×#102—431no—
321C2×#102—432no—
322C2×C22⋊C8—432no—
323C2×C22⋊C8—432no—
324C2×C4×C8C2×D4311yescharacteristic
325#394—862no—
326#384C2×D4311yescharacteristic
327#380—432no—
328#380—861no—
329#340—432no—
330#340C2×D4312yes—
331#340C2×D4312yes—
332C43#2411911yescharacteristic, complemented, Sylow
333C22×Q8—21no—
334C22×Q8—431no—
335C22×D4—431no—
336C22×D4—431no—
337C22×D4—862no—
338C23×C4—432no—
339C23×C4—861no—
340C2×SD16—862no—
341C2×SD16—862no—
342C2×SD16—862no—
343C22×C8—432no—
344C22×C8C22×D4311yescharacteristic
345C22×C8C22×D4311yescharacteristic
346C4⋊Q8—22no—
347C4⋊Q8D17214yescomplemented
348#126—8616no—
349#124—868no—
350C22⋊Q8—864no—
351C22⋊Q8—864no—
352#121—864no—
353#121—1722no—
354#121—1724no—
355C4×Q8—434no—
356C4×Q8—868no—
357C4×Q8—862no—
358C4×D4—862no—
359C4×D4—862no—
360C4×D4—862no—
361C4×D4—862no—
362#117—864no—
363#117—864no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—431no—
368C2×C4⋊C4—431no—
369C2×C4⋊C4—431no—
370C2×C4⋊C4—862no—
371C2×C4⋊C4—861no—
372C2×C4⋊C4C22×D4312yes—
373C2×C4⋊C4C22×D4312yes—
374C2×C22⋊C4—432no—
375C2×C22⋊C4—862no—
376C2×C22⋊C4—862no—
377C2×C22⋊C4—862no—
378C2×C22⋊C4—862no—
379C2×C22⋊C4—861no—
380C2×C42—862no—
381C2×C42—861no—
382C2×C42C22×D4311yescharacteristic
383#107—864no—
384#106—862no—
385C4⋊C8—862no—
386#103—862no—
387#103—1722no—
388#102—862no—
389#102—864no—
390C22⋊C8—864no—
391C22⋊C8—1722no—
392C4×C8—22no—
393#95—862no—
394C24—432no—
395C24—861no—
396C2×Q8—24no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—432no—
400C2×Q8—861no—
401C2×Q8—861no—
402C2×Q8—861no—
403C2×Q8—861no—
404C2×Q8—861no—
405C2×D4—432no—
406C2×D4—862no—
407C2×D4—862no—
408C2×D4—862no—
409C2×D4—861no—
410C2×D4—861no—
411C2×D4—861no—
412C2×D4—861no—
413C2×D4—861no—
414C2×D4—3442no—
415C2×D4—1724no—
416C2×D4—1724no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—432no—
420C22×C4—861no—
421C22×C4—861no—
422C22×C4—862no—
423C22×C4—862no—
424C22×C4—862no—
425C22×C4—862no—
426C22×C4—862no—
427C22×C4—862no—
428C22×C4—861no—
429C22×C4—1722no—
430C22×C4—1722no—
431C22×C4—1722no—
432C22×C4D4×D4311yescharacteristic
433C22×C4D4×D4311yescharacteristic
434C22×C4C23×D4311yescharacteristic
435SD16—868no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—862no—
442C2×C8—862no—
443C2×C8—862no—
444C2×C8#1062190212yes—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—81no—
451C4⋊C4—432no—
452C4⋊C4—868no—
453C4⋊C4—862no—
454C4⋊C4—864no—
455C4⋊C4—861no—
456C4⋊C4—861no—
457C4⋊C4—861no—
458C4⋊C4C2×D17214yes—
459C22⋊C4—868no—
460C22⋊C4—868no—
461C22⋊C4—864no—
462C22⋊C4—1724no—
463C22⋊C4—1724no—
464C22⋊C4—1722no—
465C42—868no—
466C42—864no—
467C42C2×D17212yes—
468C42D4×D4312yes—
469C23—862no—
470C23—862no—
471C23—862no—
472C23—862no—
473C23—862no—
474C23—862no—
475C23—862no—
476C23—862no—
477C23—1721no—
478C23—1721no—
479C23—1721no—
480C23—1721no—
481C23—1721no—
482C23—1721no—
483C23C2×D4×D4311yescharacteristic
484Q8—48no—
485Q8—862no—
486Q8—862no—
487Q8—1721no—
488Q8—1721no—
489D4—862no—
490D4—862no—
491D4—3448no—
492D4—1724no—
493D4—1721no—
494D4—1721no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—862no—
504C2×C4—862no—
505C2×C4—862no—
506C2×C4—862no—
507C2×C4—862no—
508C2×C4—3444no—
509C2×C4—1721no—
510C2×C4—1721no—
511C2×C4—1721no—
512C2×C4—1721no—
513C2×C4—1721no—
514C2×C4—1721no—
515C2×C4—1721no—
516C2×C4—1722no—
517C2×C4—1722no—
518C2×C4—1722no—
519C2×C4—1722no—
520C2×C4#1306329512yes—
521C2×C4#1306329512yes—
522C2×C4#1306334612yes—
523C2×C4#1306334912yes—
524C2×C4C2×D4×D4311yescharacteristic
525C2×C4C22×D17211yescharacteristic
526C2×C4#41×D4311yescharacteristic
527C2×C4C2×#1062190211yescharacteristic
528C2×C4#1306337111yescharacteristic
529C2×C4#1306337311yescharacteristic
530C8—22no—
531C8—41no—
532C8—1722no—
533C22—3441no—
534C22—3441no—
535C22—3441no—
536C22—3441no—
537C22—1722no—
538C22—1722no—
539C22—1722no—
540C22—1722no—
541C22—1722no—
542C22—1722no—
543C22—1722no—
544C22—1722no—
545C22—1722no—
546C22—1722no—
547C22C2×#1306329511yescharacteristic
548C22#1458354211yescharacteristic
549C22#1458355211yescharacteristic
550C22C2×#1306334611yescharacteristic
551C22C2×#1306334911yescharacteristic
552C22#1458370511yescharacteristic
553C22#1458371211yescharacteristic
554C4—22no—
555C4—44no—
556C4—82no—
557C4—3441no—
558C4—1722no—
559C4—1722no—
560C4—1722no—
561C4#1458304514yes—
562C4#1458341211yescharacteristic
563C4#1458347911yescharacteristic
564C4#1458356111yescharacteristic
565C4#1458357111yescharacteristic
566C2—3442no—
567C2—3442no—
568C2#1913794311yescharacteristic
569C2C2×#1458304511yescharacteristic
570C2#1914391511yescharacteristic
571C2#1914651111yescharacteristic
572C2#1914682511yescharacteristic
573C2#1914700711yescharacteristic
574C2#1914717811yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1990C2
socleC23×C43C22×D4
prime radical#19139511C2
hypercenter#1990D43
hypocenterC43#24119
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. C2×#14583623C2semidirect
2. C2×#14583176C2semidirect
3. C2×C4×#10621873C2semidirect
5. #19135801C2semidirect
6. #19135601C2semidirect
7. #19135389C2semidirect
8. #19134916C2semidirect
9. #19134795C2semidirect
10. #19134597C2semidirect
11. C43×#1990C2semidirect
64. C4×Dic86D4semidirect
98. C4⋊Q8×C43D4semidirect
285. #1990D43semidirect
332. C43#24119semidirect
347. C4⋊Q8D172semidirect

Maximal subgroups

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

subgroupindex
C2×#145836232
C2×#145831762
C2×C4×#106218732
#191395112
#191358012
#191356012
#191353892
#191349162
#191347952
#191345972
C43×#19902
#2411943