Finite Group Databasebeta

← #25086800

Subgroups — #25086800

Order 11008.

616 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147512C211yesmaximal, characteristic
2Q8×#10621884C211yesmaximal, characteristic
3Q16×D172C211yesmaximal, characteristic
4#19134274C211yesmaximal, characteristic, complemented
5#19134103C211yesmaximal, characteristic, complemented
6#19134043C211yesmaximal, characteristic
7#19133987C211yesmaximal, characteristic, complemented
8#19133965C211yesmaximal, characteristic, complemented
9#19133842C211yesmaximal, characteristic
10#19133837C211yesmaximal, characteristic, complemented
11#19133816C211yesmaximal, characteristic, complemented
12#19133758C211yesmaximal, characteristic
13#19133446C211yesmaximal, characteristic, complemented
14#19133405C211yesmaximal, characteristic
15C43×#3363C211yesmaximal, characteristic, complemented
16C2×Q16×D43—21no—
17Q8×D172C2211yescharacteristic
18Q8×D172C2211yescharacteristic
19#14583309—21no—
20#14583302C2211yescharacteristic
21#14583302C2211yescharacteristic
22#14583295—21no—
23#14583295C2211yescharacteristic
24#14583279C2211yescharacteristic
25#14583249C2211yescharacteristic
26#14583196—21no—
27#14583194—21no—
28#14583186—21no—
29#14583186C2211yescharacteristic
30#14583184—21no—
31#14583184C2211yescharacteristic
32#14583184C2211yescharacteristic
33#14583176C2211yescharacteristic
34#14583141—21no—
35#14583109C2211yescharacteristic
36#14583109C2211yescharacteristic
37#14583056—21no—
38#14583056—21no—
39#14583030C2211yescharacteristic
40Q8×C43⋊C8C2211yescharacteristic
41#14582997—21no—
42C4×#10621884—21no—
43C4×#10621884C2211yescharacteristic
44C8×D172C2211yescharacteristic
45#14582652C2211yescharacteristic
46#14582651C2211yescharacteristic
47#14582648C2211yescharacteristic
48#14582646C2211yescharacteristic
49#14582646C2211yescharacteristic
50#14582645C2211yescharacteristic
51Q8⋊(C43⋊C8)C2211yescharacteristic
52Q8⋊(C43⋊C8)C2211yescharacteristic
53Q8⋊(C43⋊C8)C2211yescharacteristic
54#14582626C2211yescharacteristic
55#14582621C2211yescharacteristic
56Q82×C43C2211yescharacteristic
57C43×#557C2211yescharacteristic
58C43×#487—21no—
59C43×#487C2211yescharacteristic
60C43×#477—21no—
61C43×#477C2211yescharacteristic
62C43×#462—21no—
63C43×#462C2211yescharacteristic
64C8×Q8×C43C2211yescharacteristic
65C4×Q16×C43—21no—
66C4×Q16×C43C2211yescharacteristic
67C2×Q8×D43—21no—
68C2×Q8×D43—21no—
69C2×Dic172—21no—
70C2×Dic172—21no—
71C2×#10621885—21no—
72C2×#10621885—21no—
73Q16×D43—44no—
74C2×#10621884D411yescharacteristic
75C2×C8×D43—21no—
76#13063325—21no—
77#13063325—21no—
78#13063325C2311yescharacteristic
79#13063325C2311yescharacteristic
80#13063305—21no—
81#13063305—21no—
82#13063305—41no—
83#13063305—41no—
84C4×D172—21no—
85C4×D172—21no—
86C4×D172C2311yescharacteristic
87#13063258—21no—
88#13063258—21no—
89#13063258—21no—
90#13063258—41no—
91#13063256—21no—
92#13063256—21no—
93#13063256—21no—
94#13063256—21no—
95#13063256—21no—
96#13063256D411yescharacteristic
97C4⋊(C43⋊C8)—21no—
98C4⋊(C43⋊C8)—21no—
99C4⋊(C43⋊C8)C2311yescharacteristic
100C4⋊(C43⋊C8)C2311yescharacteristic
101C4⋊(C43⋊C8)C2311yescharacteristic
102#13063249—21no—
103#13063249—21no—
104#13063249—41no—
105#13063249D411yescharacteristic
106#13063248—21no—
107#13063248—21no—
108#13063245—21no—
109#13063245—21no—
110#13063245D411yescharacteristic
111(C2×D43)⋊C8—21no—
112#13063236—21no—
113C4×C43⋊C8—21no—
114C4×C43⋊C8C2311yescharacteristic
115C4×C43⋊C8C2311yescharacteristic
116C2×Q16×C43—21no—
117C2×Q16×C43D411yescharacteristic
118C4⋊Q8×C43—21no—
119C4⋊Q8×C43—21no—
120C4⋊Q8×C43—21no—
121C4⋊Q8×C43—41no—
122C4⋊Q8×C43C2311yescharacteristic
123C4⋊Q8×C43C2311yescharacteristic
124#125×C43—21no—
125#125×C43—41no—
126C4×Q8×C43—21no—
127C4×Q8×C43—21no—
128C4×Q8×C43—21no—
129C4×Q8×C43—21no—
130C4×Q8×C43C2311yescharacteristic
131C4×Q8×C43C2311yescharacteristic
132C4×Q8×C43C2311yescharacteristic
133#107×C43—21no—
134#107×C43D411yescharacteristic
135C4⋊C8×C43—21no—
136C4⋊C8×C43C2311yescharacteristic
137#103×C43—21no—
138#103×C43—21no—
139#103×C43—21no—
140#103×C43—21no—
141#103×C43D411yescharacteristic
142#103×C43D411yescharacteristic
143C4×C8×C43—21no—
144C4×C8×C43C2311yescharacteristic
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—41no—
148C2×Q8×C43C2×D411yescharacteristic
149C2×Q8×C43C2×D411yescharacteristic
150C2×Q8×C43C2×D411yescharacteristic
151Q8×D43—42no—
152Q8×D43—42no—
153Q8×D43—81no—
154Q8×D43—81no—
155C2×D172C2×D411yescharacteristic
156C2×C4×D43—21no—
157C2×C4×D43—41no—
158C2×C4×D43—41no—
159C2×Dic86—21no—
160C2×Dic86—21no—
161C2×Dic86—41no—
162C2×Dic86—41no—
163Q16×C43—22no—
164Q16×C43—42no—
165C2×C8×C43—21no—
166C2×C8×C43C2×D411yescharacteristic
167C2×C8×C43C2×D411yescharacteristic
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—21no—
171C4⋊C4×C43—21no—
172C4⋊C4×C43—21no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—41no—
176C4⋊C4×C43—41no—
177C4⋊C4×C43—41no—
178C4⋊C4×C43—41no—
179C4⋊C4×C43—41no—
180C4⋊C4×C43C2×D411yescharacteristic
181C4⋊C4×C43C2×D411yescharacteristic
182C4⋊C4×C43C2×D411yescharacteristic
183C42×C43—21no—
184C42×C43—21no—
185C42×C43—21no—
186C42×C43—41no—
187C42×C43—41no—
188C42×C43C2411yescharacteristic
189#10621885—42no—
190#10621885—42no—
191#10621884Q1614yescomplemented
192(C2×D43)⋊C4—21no—
193(C2×D43)⋊C4—41no—
194(C2×D43)⋊C4—41no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—41no—
200C4⋊Dic43—41no—
201C4⋊Dic43C2×D411yescharacteristic
202C2×C43⋊C8—21no—
203C2×C43⋊C8—21no—
204C2×C43⋊C8—21no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8C2×D411yescharacteristic
207C2×C43⋊C8C2×D411yescharacteristic
208Dic172—42no—
209Dic172—81no—
210C8×D43—42no—
211C22×D43—21no—
212Q8×C43—21no—
213Q8×C43—21no—
214Q8×C43—21no—
215Q8×C43—21no—
216Q8×C43—41no—
217Q8×C43—41no—
218Q8×C43—81no—
219Q8×C43—161no—
220Q8×C43C2×Q1612yescomplemented
221Q8×C43AGL(1,ℤ/8)12yes—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43C22×D411yescharacteristic
231C2×C4×C43C22×D411yescharacteristic
232C2×C4×C43H(5,𝔽2)11yescharacteristic
233C2×Dic43—21no—
234C2×Dic43—41no—
235C2×Dic43—41no—
236D172—21no—
237D172C2×Q1612yes—
238C4×D43—41no—
239C4×D43—41no—
240C4×D43—82no—
241C4×D43—82no—
242Dic86—41no—
243Dic86—41no—
244Dic86—41no—
245Dic86—41no—
246Dic86—82no—
247Dic86—82no—
248C8×C43—21no—
249C8×C43—21no—
250C8×C43—41no—
251C43⋊C8—21no—
252C43⋊C8—41no—
253C43⋊C8—41no—
254C43⋊C8—41no—
255C43⋊C8—41no—
256C43⋊C8C2×Q1612yescomplemented
257#19812—431nomaximal, Sylow
258C22×C43D4211yescharacteristic
259C2×D43—22no—
260C2×D43—41no—
261C2×D43—41no—
262C4×C43—22no—
263C4×C43—22no—
264C4×C43—21no—
265C4×C43—41no—
266C4×C43—41no—
267C4×C43—41no—
268C4×C43—41no—
269C4×C43—81no—
270C4×C43—81no—
271C4×C43C22×Q1611yescharacteristic
272C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
273C4×C43#57711yescharacteristic
274C4×C43#57811yescharacteristic
275Dic43—41no—
276Dic43—81no—
277Dic43—81no—
278#3386—431no—
279Q8×SD16—431no—
280#3363D4311yescharacteristic, complemented
281D4×Q16—431no—
282#1723—431no—
283#1692—431no—
284#1683—431no—
285#1674—431no—
286#1670—431no—
287#1649—431no—
288#1648—431no—
289#1645—431no—
290#1637—431no—
291#1584—431no—
292SD16⋊C8—431no—
293C2×C43#326511yescharacteristic
294C2×C43D4×Q1611yescharacteristic
295C2×C43#328911yescharacteristic
296D43—42no—
297C22×Q16—861no—
298Q82C2×D4311yescharacteristic
299#557C2×D4311yescharacteristic
300D4×Q8—431no—
301D4×Q8—431no—
302#501—861no—
303#500—431no—
304#500—431no—
305#498—431no—
306#498—861no—
307#494—431no—
308#487—21no—
309#487—431no—
310#487C2×D4311yescharacteristic
311#478—861no—
312#477—21no—
313#477—861no—
314#477C2×D4311yescharacteristic
315#476—431no—
316#476—861no—
317#475—431no—
318#475—431no—
319#475—861no—
320#474—431no—
321#467—861no—
322#462—21no—
323#462—431no—
324#462—431no—
325#462C2×D4311yescharacteristic
326#451—861no—
327#451—861no—
328#446—431no—
329C8×Q8—431no—
330C8×Q8C2×D4311yescharacteristic
331#440—861no—
332C4×Q16—21no—
333C4×Q16C2×D4311yescharacteristic
334C4×SD16—431no—
335C4×SD16—861no—
336C8×D4—431no—
337#335—431no—
338#335—431no—
339#333—431no—
340#332—431no—
341#332—431no—
342#332—431no—
343Q8⋊C8—431no—
344Q8⋊C8—431no—
345Q8⋊C8—431no—
346D4⋊C8—431no—
347#322—431no—
348C43#1981211yescharacteristic, complemented, Sylow
349C22×Q8—861no—
350C22×Q8—861no—
351C2×Q16—21no—
352C2×Q16—861no—
353C2×Q16—861no—
354C2×Q16—861no—
355C2×Q16—861no—
356C2×Q16—1724no—
357C2×Q16C22⋊D4311yescharacteristic
358C2×SD16—431no—
359C22×C8—861no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—41no—
364C4⋊Q8—431no—
365C4⋊Q8—431no—
366C4⋊Q8—861no—
367C4⋊Q8—861no—
368C4⋊Q8C22×D4311yescharacteristic
369C4⋊Q8C22×D4311yescharacteristic
370#125—21no—
371#125—41no—
372C22⋊Q8—861no—
373C22⋊Q8—861no—
374C22⋊Q8—1721no—
375C22⋊Q8—1721no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8C22×D4311yescharacteristic
381C4×Q8C22×D4311yescharacteristic
382C4×Q8C22×D4311yescharacteristic
383C4×D4—431no—
384C4×D4—861no—
385C4×D4—861no—
386#107—21no—
387#107—861no—
388#107—861no—
389#107—861no—
390#107—1721no—
391#107C22⋊D4311yescharacteristic
392#106—431no—
393#106—861no—
394#106—861no—
395#106—861no—
396#106—861no—
397#106—861no—
398C4⋊C8—21no—
399C4⋊C8—431no—
400C4⋊C8—431no—
401C4⋊C8—431no—
402C4⋊C8—861no—
403C4⋊C8—861no—
404C4⋊C8C22×D4311yescharacteristic
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—431no—
410#103—861no—
411#103—861no—
412#103—861no—
413#103—861no—
414#103—1721no—
415#103C22⋊D4311yescharacteristic
416#103C22⋊D4311yescharacteristic
417#102—431no—
418#102—861no—
419#102—861no—
420C22⋊C8—861no—
421#97—861no—
422C4×C8—21no—
423C4×C8—431no—
424C4×C8—431no—
425C4×C8—861no—
426C4×C8C22×D4311yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—41no—
430C2×Q8—861no—
431C2×Q8—861no—
432C2×Q8—3441no—
433C2×Q8—3441no—
434C2×Q8—1721no—
435C2×Q8—1721no—
436C2×Q8—1722no—
437C2×Q8—1722no—
438C2×Q8D4×D4311yescharacteristic
439C2×Q8C2×C22⋊D4311yescharacteristic
440C2×Q8C2×C22⋊D4311yescharacteristic
441C2×D4—431no—
442C22×C4—861no—
443C22×C4—1721no—
444C22×C4—1721no—
445Q16—22no—
446Q16—42no—
447Q16—3441no—
448Q16—1722no—
449Q16—1722no—
450Q16—1722no—
451SD16—434no—
452C2×C8—21no—
453C2×C8—431no—
454C2×C8—431no—
455C2×C8—861no—
456C2×C8—861no—
457C2×C8—861no—
458C2×C8—861no—
459C2×C8—1722no—
460C2×C8C2×C22⋊D4311yescharacteristic
461C2×C8C2×C22⋊D4311yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—431no—
475C4⋊C4—861no—
476C4⋊C4—861no—
477C4⋊C4—861no—
478C4⋊C4—861no—
479C4⋊C4—1721no—
480C4⋊C4—1721no—
481C4⋊C4D4×D4311yescharacteristic
482C4⋊C4C2×C22⋊D4311yescharacteristic
483C4⋊C4C2×C22⋊D4311yescharacteristic
484C22⋊C4—861no—
485C22⋊C4—1721no—
486C22⋊C4—1721no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42—41no—
492C42C23×D4311yescharacteristic
493C23—861no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—81no—
501Q8—161no—
502Q8—3442no—
503Q8—3442no—
504Q8—1721no—
505Q8—1721no—
506Q8—1721no—
507Q8—1721no—
508Q8Q16×D4312yescomplemented
509Q8#1306335012yes—
510D4—432no—
511D4—861no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—861no—
521C2×C4—3442no—
522C2×C4—3442no—
523C2×C4—1721no—
524C2×C4—1721no—
525C2×C4—1721no—
526C2×C4—1721no—
527C2×C4C2×D4×D4311yescharacteristic
528C2×C4C22×C22⋊D4311yescharacteristic
529C2×C4#1306337011yescharacteristic
530C8—21no—
531C8—21no—
532C8—41no—
533C8—432no—
534C8—861no—
535C8—1721no—
536C8—1721no—
537C8—1721no—
538C8—1721no—
539C22—862no—
540C22—1721no—
541C22—1721no—
542C22D4×C22⋊D4311yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—81no—
551C4—81no—
552C4—3441no—
553C4—3441no—
554C4—1721no—
555C4C2×Q16×D4311yescharacteristic
556C4C2×#1306335011yescharacteristic
557C4#1458370711yescharacteristic
558C4#1458371211yescharacteristic
559C2—1722no—
560C2#1914609111yescharacteristic
561C2Q16×C22⋊D4311yescharacteristic
562C2#1914636811yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3363C2
socleC22×C43D42
prime radicalC2×D172C2×D4
hypercenter#3363D43
hypocenterC43#19812
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #3363

Complements

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

NH ≅ G/Ntype
4. #19134274C2semidirect
5. #19134103C2semidirect
7. #19133987C2semidirect
8. #19133965C2semidirect
10. #19133837C2semidirect
11. #19133816C2semidirect
13. #19133446C2semidirect
15. C43×#3363C2semidirect
191. #10621884Q16semidirect
220. Q8×C43C2×Q16semidirect
256. C43⋊C8C2×Q16semidirect
280. #3363D43semidirect
348. C43#19812semidirect
508. Q8Q16×D43semidirect

Maximal subgroups

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

subgroupindex
#191475122
Q8×#106218842
Q16×D1722
#191342742
#191341032
#191340432
#191339872
#191339652
#191338422
#191338372
#191338162
#191337582
#191334462
#191334052
C43×#33632
#1981243