Finite Group Databasebeta

← #25146369

Subgroups — #25146369

Order 11008.

764 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144388C211yesmaximal, characteristic, complemented
2C2×#14583091C211yesmaximal, characteristic
3#19143544C211yesmaximal, characteristic, complemented
4#19135955C212yesmaximal, complemented
5#19135850C212yesmaximal, complemented
6#19135504C212yesmaximal, complemented
7#19135436C211yesmaximal, characteristic, complemented
8#19135370C211yesmaximal, characteristic, complemented
9#19135260C212yesmaximal, complemented
10C4×#13063241C211yesmaximal, characteristic, complemented
11C43×#3138C211yesmaximal, characteristic, complemented
12C22×#10621882C2212yes—
13C2×C4×D172C2211yescharacteristic
14#14583196—22no—
15#14583186—21no—
16#14583176—21no—
17#14583091—21no—
18#14583091—21no—
19#14583091—22no—
20#14583030—21no—
21#14583030—22no—
22Q8×C43⋊C8—21no—
23C2×#13063260C2212yes—
24#14582942C2212yes—
25#14582926C2212yes—
26#14582923C2211yescharacteristic
27C2×C4⋊(C43⋊C8)C2211yescharacteristic
28#14582903C2212yes—
29C2×#13063246C2212yes—
30C2×#13063242C2212yes—
31C2×#13063242C2212yes—
32C2×#13063241C2211yescharacteristic
33C2×#13063241C2211yescharacteristic
34#14582855C2212yes—
35C4×#10621877C2211yescharacteristic
36#14582667C2211yescharacteristic
37#14582667C2211yescharacteristic
38#14582663C2212yes—
39C43×#533C2211yescharacteristic
40C2×#127×C43C2211yescharacteristic
41C43×#490—22no—
42C43×#486—21no—
43C43×#486—21no—
44C43×#418C2212yes—
45C43×#418C2212yes—
46C4×M16×C43C2211yescharacteristic
47C22×D172C2311yescharacteristic
48C22×C4×D43—21no—
49C2×#10621882—24no—
50C2×#10621882—22no—
51C2×#10621882—22no—
52C2×#10621882—42no—
53C2×#10621877C2311yescharacteristic
54C2×#10621877C2312yes—
55C2×#10621877C2311yescharacteristic
56C22×C43⋊C8C2312yes—
57C4×D172—21no—
58C4×D172—21no—
59C4×D172—21no—
60C4×D172—21no—
61C2×C4⋊Dic43C2311yescharacteristic
62C2×(C2×D43)⋊C4—21no—
63#13063260—42no—
64#13063258—22no—
65#13063256—22no—
66C4⋊(C43⋊C8)—21no—
67C4⋊(C43⋊C8)—21no—
68C4⋊(C43⋊C8)—22no—
69C4⋊(C43⋊C8)—22no—
70C4⋊(C43⋊C8)D411yescharacteristic
71C4⋊(C43⋊C8)D412yes—
72C4⋊(C43⋊C8)D411yescharacteristic
73#13063246—22no—
74#13063246—22no—
75#13063245—22no—
76#13063245—22no—
77#13063242—24no—
78#13063242—42no—
79#13063241—22no—
80#13063241—22no—
81#13063236—21no—
82#13063236—22no—
83C4×C43⋊C8—21no—
84C4×C43⋊C8—22no—
85C22×D4×C43—21no—
86C22×D4×C43—21no—
87C22×D4×C43C2312yes—
88C2×M16×C43C2311yescharacteristic
89C2×M16×C43C2311yescharacteristic
90C4⋊Q8×C43D411yescharacteristic
91C4⋊Q8×C43D411yescharacteristic
92#127×C43—21no—
93#127×C43—21no—
94#127×C43—21no—
95#127×C43—21no—
96#125×C43D412yes—
97C22⋊Q8×C43—21no—
98C22⋊Q8×C43—21no—
99C4×Q8×C43—21no—
100C4×Q8×C43—21no—
101#117×C43C2312yes—
102C2×C42×C43C2311yescharacteristic
103#102×C43—22no—
104#102×C43—22no—
105#102×C43—22no—
106#102×C43—22no—
107#97×C43—21no—
108C4×C8×C43—21no—
109C24×C43—22no—
110C23×D43—21no—
111C2×Q8×C43—21no—
112C2×Q8×C43—21no—
113C2×D4×C43—22no—
114C2×D4×C43—22no—
115C2×D4×C43—22no—
116C2×D4×C43—22no—
117C2×D4×C43—22no—
118C2×D4×C43—22no—
119C2×D4×C43—41no—
120C2×D4×C43—41no—
121C2×D4×C43—41no—
122C2×D4×C43—41no—
123C2×D4×C43—82no—
124C2×D4×C43—82no—
125C22×C4×C43#4111yescharacteristic
126C22×C4×C43#4111yescharacteristic
127C22×C4×C43C2411yescharacteristic
128C22×Dic43—21no—
129C2×D172—21no—
130C2×D172—21no—
131C2×D172—21no—
132C2×D172—21no—
133C2×D172—21no—
134C2×D172#4112yes—
135C2×C4×D43—41no—
136C2×C4×D43—41no—
137C2×C4×D43—82no—
138M16×C43—21no—
139M16×C43—21no—
140M16×C43—21no—
141M16×C43—21no—
142C2×C8×C43—21no—
143C2×C8×C43—21no—
144C4⋊C4×C43—22no—
145C4⋊C4×C43—22no—
146C4⋊C4×C43—21no—
147C4⋊C4×C43—21no—
148C4⋊C4×C43C2×D412yes—
149C4⋊C4×C43C2×D412yes—
150C22⋊C4×C43—22no—
151C42×C43—22no—
152C42×C43C2×D411yescharacteristic
153C42×C43C2×D411yescharacteristic
154C42×C43C2×D411yescharacteristic
155C42×C43C2×D411yescharacteristic
156#10621882—44no—
157#10621882—44no—
158(C2×D43)⋊C4—41no—
159(C2×D43)⋊C4—41no—
160C4⋊Dic43—21no—
161C4⋊Dic43—21no—
162#10621877—21no—
163#10621877—21no—
164#10621877—22no—
165#10621877—21no—
166#10621877—22no—
167#10621877#4112yes—
168C2×C43⋊C8—21no—
169C2×C43⋊C8—22no—
170C2×C43⋊C8—22no—
171C2×C43⋊C8—21no—
172C2×C43⋊C8—22no—
173C2×C43⋊C8C2×D412yes—
174C2×C43⋊C8C2×D412yes—
175C23×C43—42no—
176C23×C43—42no—
177C23×C43—42no—
178C23×C43—82no—
179C23×C43—82no—
180C23×C43C2×#4111yescharacteristic
181C22×D43—22no—
182C22×D43—41no—
183C22×D43—41no—
184C22×D43—81no—
185C22×D43—81no—
186Q8×C43—42no—
187Q8×C43—81no—
188D4×C43—42no—
189D4×C43—42no—
190D4×C43—42no—
191D4×C43—42no—
192D4×C43—82no—
193D4×C43—82no—
194D4×C43—82no—
195D4×C43—82no—
196C2×C4×C43—22no—
197C2×C4×C43—22no—
198C2×C4×C43—22no—
199C2×C4×C43—41no—
200C2×C4×C43#12111yescharacteristic
201C2×C4×C43#12111yescharacteristic
202C2×C4×C43#12111yescharacteristic
203C2×C4×C43#12111yescharacteristic
204C2×C4×C43C22×D411yescharacteristic
205C2×C4×C43C22×D411yescharacteristic
206C2×C4×C43C2×#4111yescharacteristic
207C2×C4×C43C2×#4111yescharacteristic
208C2×C4×C43H(5,𝔽2)11yescharacteristic
209C2×C4×C432−1+411yescharacteristic
210C2×Dic43—41no—
211C2×Dic43—81no—
212D172—22no—
213D172—22no—
214D172—41no—
215D172—41no—
216C4×D43—82no—
217C4×D43—82no—
218C8×C43—41no—
219C8×C43—41no—
220C43⋊C8—22no—
221C43⋊C8—22no—
222C43⋊C8—22no—
223C43⋊C8—41no—
224C43⋊C8—42no—
225#25312—431nomaximal, Sylow
226C22×C43—21no—
227C22×C43—21no—
228C22×C43—82no—
229C22×C43—82no—
230C22×C43—82no—
231C22×C43—162no—
232C22×C43—162no—
233C22×C43C2×#12111yescharacteristic
234C22×C43#52911yescharacteristic
235C22×C43#54211yescharacteristic
236C2×D43—42no—
237C2×D43—82no—
238C2×D43—81no—
239C2×D43—81no—
240C2×D43—82no—
241C4×C43—21no—
242C4×C43—21no—
243C4×C43—21no—
244C4×C43—42no—
245C4×C43—42no—
246C4×C43#47212yes—
247C4×C43#54611yescharacteristic
248C4×C43#54811yescharacteristic
249C4×C43#54811yescharacteristic
250C4×C43#55011yescharacteristic
251Dic43—81no—
252#3138D4311yescharacteristic, complemented
253#3077—431no—
254C2×#459—431no—
255#2962—431no—
256#2063—432no—
257#2045—432no—
258#1978—432no—
259#1965—431no—
260#1950—431no—
261#1931—432no—
262C4×#100—431no—
263C2×C43—21no—
264C2×C43—21no—
265C2×C43—162no—
266C2×C43#242611yescharacteristic
267C2×C43C2×#47211yescharacteristic
268C2×C43#306811yescharacteristic
269D43—82no—
270C22×D8—432no—
271#533C2×D4311yescharacteristic
272C2×#127C2×D4311yescharacteristic
273C2×C4×D4—431no—
274#490—22no—
275#486—21no—
276#486—21no—
277#478—862no—
278#476—861no—
279#474—861no—
280#459—861no—
281#459—862no—
282#459—861no—
283#446—862no—
284#446—861no—
285C8×Q8—861no—
286C2×#108—432no—
287#428—432no—
288#424—432no—
289#423—431no—
290C2×C4⋊C8—431no—
291#418—432no—
292#418C2×D4312yes—
293#418C2×D4312yes—
294C2×#102—432no—
295C2×#100—432no—
296C2×#100—431no—
297C2×#100—431no—
298C2×#100—432no—
299#405—432no—
300C4×M16—431no—
301C4×M16C2×D4311yescharacteristic
302#343—431no—
303#343—431no—
304#341—432no—
305C43#2531211yescharacteristic, complemented, Sylow
306C22×D4—21no—
307C22×D4—21no—
308C22×D4—431no—
309C22×D4C22×D4312yes—
310C23×C4—861no—
311C2×D8—862no—
312C2×D8—862no—
313C2×D8—864no—
314C2×D8—1722no—
315C2×M16—431no—
316C2×M16—432no—
317C2×M16—431no—
318C2×M16C22×D4311yescharacteristic
319C2×M16C22×D4311yescharacteristic
320C22×C8—432no—
321C4⋊Q8C22⋊D4311yescharacteristic
322C4⋊Q8C22⋊D4311yescharacteristic
323#127—21no—
324#127—21no—
325#127—21no—
326#127—21no—
327#125C22⋊D4312yes—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—861no—
333C4×D4—861no—
334C4×D4—861no—
335C4×D4—861no—
336#117C22×D4312yes—
337C2×C4⋊C4—431no—
338C2×C22⋊C4—861no—
339C2×C42C22×D4311yescharacteristic
340#108—1722no—
341#107—862no—
342#106—862no—
343C4⋊C8—431no—
344C4⋊C8—431no—
345C4⋊C8—432no—
346C4⋊C8—862no—
347C4⋊C8—862no—
348C4⋊C8—861no—
349C4⋊C8—861no—
350#102—22no—
351#102—22no—
352#102—22no—
353#102—22no—
354#102—862no—
355#102—862no—
356#102—862no—
357#102—862no—
358#100—862no—
359#100—862no—
360#100—864no—
361#100—1722no—
362#97—21no—
363#97—862no—
364#97—861no—
365C4×C8—21no—
366C4×C8—862no—
367C4×C8—861no—
368C24—22no—
369C24—861no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—432no—
385C2×D4—861no—
386C2×D4—861no—
387C2×D4—861no—
388C2×D4—861no—
389C2×D4—861no—
390C22×C4—861no—
391C22×C4—3442no—
392C22×C4—1721no—
393C22×C4—1721no—
394C22×C4#1062190011yescharacteristic
395C22×C4#1062190011yescharacteristic
396C22×C4C23×D4311yescharacteristic
397D8—1724no—
398D8—1724no—
399M16—21no—
400M16—21no—
401M16—21no—
402M16—21no—
403M16—432no—
404M16—861no—
405M16—862no—
406M16—861no—
407M16—861no—
408M16—862no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—432no—
412C2×C8—432no—
413C2×C8—861no—
414C2×C8—862no—
415C2×C8—862no—
416C2×C8—861no—
417C2×C8—862no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—861no—
423C4⋊C4—861no—
424C4⋊C4C2×C22⋊D4312yes—
425C4⋊C4C2×C22⋊D4312yes—
426C22⋊C4—22no—
427C22⋊C4—1721no—
428C22⋊C4—1721no—
429C42—22no—
430C42D4×D4311yescharacteristic
431C42D4×D4311yescharacteristic
432C42C2×C22⋊D4311yescharacteristic
433C42C2×C22⋊D4311yescharacteristic
434C23—42no—
435C23—42no—
436C23—42no—
437C23—82no—
438C23—82no—
439C23—862no—
440C23—3441no—
441C23—3441no—
442C23—1721no—
443C23—1721no—
444C23C2×#1062190011yescharacteristic
445Q8—42no—
446Q8—81no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—82no—
452D4—82no—
453D4—82no—
454D4—82no—
455D4—862no—
456D4—862no—
457D4—1721no—
458D4—1721no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—3442no—
464C2×C4—3442no—
465C2×C4—3441no—
466C2×C4—1721no—
467C2×C4#1306329711yescharacteristic
468C2×C4#1306329711yescharacteristic
469C2×C4#1306329711yescharacteristic
470C2×C4#1306329711yescharacteristic
471C2×C4C2×D4×D4311yescharacteristic
472C2×C4C22×C22⋊D4311yescharacteristic
473C2×C4#41×D4311yescharacteristic
474C2×C4#41×D4311yescharacteristic
475C2×C4#1306337111yescharacteristic
476C2×C4#1306337211yescharacteristic
477C8—41no—
478C8—41no—
479C8—862no—
480C8—862no—
481C8—862no—
482C8—1722no—
483C8—1721no—
484C22—21no—
485C22—21no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—162no—
490C22—162no—
491C22—3442no—
492C22—3442no—
493C22—3441no—
494C22—3441no—
495C22—1722no—
496C22C2×#1306329711yescharacteristic
497C22#1458342811yescharacteristic
498C22#1458351811yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—42no—
503C4—42no—
504C4—3441no—
505C4#1458316712yes—
506C4#1458354211yescharacteristic
507C4#1458355611yescharacteristic
508C4#1458355711yescharacteristic
509C4#1458356811yescharacteristic
510C2—21no—
511C2—21no—
512C2—162no—
513C2—3442no—
514C2#1913816411yescharacteristic
515C2C2×#1458316711yescharacteristic
516C2#1914432011yescharacteristic
subgroupG/H
centerC22C2×#13063297
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#3138C2
socleC22×C43C2×#121
prime radicalC2×#14583091C2
hypercenter#3138D43
hypocenterC43#25312
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #19144388C2semidirect
3. #19143544C2semidirect
4. #19135955C2semidirect
5. #19135850C2semidirect
6. #19135504C2semidirect
7. #19135436C2semidirect
8. #19135370C2semidirect
9. #19135260C2semidirect
10. C4×#13063241C2semidirect
11. C43×#3138C2semidirect
252. #3138D43semidirect
305. C43#25312semidirect

Maximal subgroups

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

subgroupindex
#191443882
C2×#145830912
#191435442
#191359552
#191358502
#191355042
#191354362
#191353702
#191352602
C4×#130632412
C43×#31382
#2531243