Finite Group Databasebeta

← #25086823

Subgroups — #25086823

Order 11008.

590 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147460C211yesmaximal, characteristic
2#19146983C211yesmaximal, characteristic, complemented
3#19146471C211yesmaximal, characteristic, complemented
4#19134269C211yesmaximal, characteristic, complemented
5#19134123C211yesmaximal, characteristic, complemented
6#19134044C211yesmaximal, characteristic, complemented
7#19133992C211yesmaximal, characteristic, complemented
8#19133974C211yesmaximal, characteristic, complemented
9#19133808C211yesmaximal, characteristic, complemented
10#19133760C211yesmaximal, characteristic, complemented
11#19133729C211yesmaximal, characteristic, complemented
12#19133727C211yesmaximal, characteristic, complemented
13#19133443C211yesmaximal, characteristic, complemented
14#19133419C211yesmaximal, characteristic
15C43×#3384C211yesmaximal, characteristic
16#14583606C2211yescharacteristic
17#14583606C2211yescharacteristic
18#14583304C2211yescharacteristic
19#14583302C2211yescharacteristic
20#14583300—21no—
21#14583278C2211yescharacteristic
22#14583251C2211yescharacteristic
23#14583249C2211yescharacteristic
24#14583234—21no—
25#14583232—21no—
26#14583232—21no—
27#14583197—21no—
28#14583197—21no—
29#14583194C2211yescharacteristic
30#14583194C2211yescharacteristic
31#14583191C2211yescharacteristic
32#14583189C2211yescharacteristic
33#14583189C2211yescharacteristic
34#14583138—21no—
35#14583138—21no—
36#14583106C2211yescharacteristic
37#14583055—21no—
38C8×Dic86C2211yescharacteristic
39C4×#10621885C2211yescharacteristic
40D8×Dic43—21no—
41D4×C43⋊C8C2211yescharacteristic
42D4×C43⋊C8C2211yescharacteristic
43C2×#13063258—21no—
44C2×#13063249—21no—
45#14582651C2211yescharacteristic
46#14582650C2211yescharacteristic
47#14582648C2211yescharacteristic
48#14582647C2211yescharacteristic
49#14582647C2211yescharacteristic
50#14582645C2211yescharacteristic
51Q8⋊(C43⋊C8)C2211yescharacteristic
52#14582629C2211yescharacteristic
53D4⋊(C43⋊C8)C2211yescharacteristic
54D4⋊(C43⋊C8)C2211yescharacteristic
55#14582620C2211yescharacteristic
56C43×#554C2211yescharacteristic
57D4×Q8×C43C2211yescharacteristic
58C43×#501—21no—
59C43×#498C2211yescharacteristic
60C43×#478—21no—
61C43×#476—21no—
62C43×#476C2211yescharacteristic
63C43×#474C2211yescharacteristic
64C43×#446C2211yescharacteristic
65C43×#442—21no—
66C4×D8×C43C2211yescharacteristic
67C2×#10621885—21no—
68C2×#10621885—41no—
69C2×#10621885D411yescharacteristic
70C22×C43⋊C8—21no—
71C22×C43⋊C8—21no—
72#13063325C2311yescharacteristic
73#13063325C2311yescharacteristic
74#13063316—21no—
75#13063316—21no—
76#13063304—21no—
77#13063304—21no—
78#13063302—41no—
79#13063302—41no—
80C4×Dic86C2311yescharacteristic
81D4×Dic43—21no—
82D4×Dic43—21no—
83C2×C4⋊Dic43—21no—
84C2×C4⋊Dic43—21no—
85#13063259—21no—
86#13063259—21no—
87#13063258—21no—
88#13063258—21no—
89#13063258—21no—
90#13063258—21no—
91#13063258—21no—
92#13063258D411yescharacteristic
93#13063256—21no—
94#13063256—21no—
95C4⋊(C43⋊C8)C2311yescharacteristic
96C4⋊(C43⋊C8)C2311yescharacteristic
97C4⋊(C43⋊C8)C2311yescharacteristic
98Dic43⋊C8—21no—
99#13063249—21no—
100#13063249—21no—
101#13063249—21no—
102#13063249—21no—
103#13063249—21no—
104#13063249D411yescharacteristic
105#13063248—21no—
106#13063248—41no—
107#13063248D411yescharacteristic
108#13063246—21no—
109#13063246—21no—
110C22⋊(C43⋊C8)—21no—
111C22⋊(C43⋊C8)—21no—
112C4×C43⋊C8C2311yescharacteristic
113C4×C43⋊C8C2311yescharacteristic
114C8×Dic43—21no—
115C22×Q8×C43—21no—
116C2×D8×C43D411yescharacteristic
117C4⋊Q8×C43—21no—
118C4⋊Q8×C43C2311yescharacteristic
119C4⋊Q8×C43C2311yescharacteristic
120#125×C43—21no—
121C22⋊Q8×C43—21no—
122C22⋊Q8×C43—21no—
123C22⋊Q8×C43—41no—
124C22⋊Q8×C43—41no—
125C4×Q8×C43C2311yescharacteristic
126C4×D4×C43—21no—
127C4×D4×C43—21no—
128C4×D4×C43C2311yescharacteristic
129C4×D4×C43C2311yescharacteristic
130C2×C4⋊C4×C43—21no—
131#107×C43—21no—
132#107×C43D411yescharacteristic
133#106×C43—21no—
134#106×C43—21no—
135#106×C43—21no—
136C4⋊C8×C43—21no—
137C4⋊C8×C43C2311yescharacteristic
138#102×C43—21no—
139#102×C43—21no—
140#102×C43D411yescharacteristic
141#102×C43D411yescharacteristic
142#97×C43—21no—
143C4×C8×C43C2311yescharacteristic
144C2×Q8×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—41no—
147C2×Q8×C43—42no—
148C2×Q8×C43—81no—
149C2×Q8×C43C2×D411yescharacteristic
150C2×D4×C43C2×D411yescharacteristic
151C2×D4×C43C2×D411yescharacteristic
152C22×C4×C43—21no—
153C22×C4×C43—21no—
154C22×C4×C43—41no—
155C22×C4×C43—41no—
156C22×Dic43—41no—
157C22×Dic43—41no—
158C2×Dic86—21no—
159C2×Dic86—21no—
160C2×Dic86C2×D411yescharacteristic
161D8×C43Q1614yescomplemented
162C2×C8×C43—21no—
163C2×C8×C43C2×D411yescharacteristic
164C2×C8×C43C2×D411yescharacteristic
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—41no—
171C4⋊C4×C43—41no—
172C4⋊C4×C43—41no—
173C4⋊C4×C43C2×D411yescharacteristic
174C4⋊C4×C43C2×D411yescharacteristic
175C4⋊C4×C43C2×D411yescharacteristic
176C22⋊C4×C43—21no—
177C22⋊C4×C43—21no—
178C22⋊C4×C43—41no—
179C22⋊C4×C43—41no—
180C42×C43—21no—
181C42×C43C2411yescharacteristic
182C22⋊Dic43—41no—
183C22⋊Dic43—41no—
184#10621885—41no—
185#10621885—81no—
186#10621885—82no—
187C4⋊Dic43—21no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—41no—
192C4⋊Dic43—41no—
193C4⋊Dic43C2×D411yescharacteristic
194Dic43⋊C4—21no—
195Dic43⋊C4—41no—
196Dic43⋊C4—41no—
197C4×Dic43—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202C2×C43⋊C8—42no—
203C2×C43⋊C8—81no—
204C2×C43⋊C8C2×D411yescharacteristic
205C2×C43⋊C8C2×D411yescharacteristic
206C23×C43—21no—
207C23×C43—21no—
208Q8×C43—21no—
209Q8×C43—21no—
210Q8×C43—41no—
211Q8×C43—41no—
212Q8×C43—81no—
213Q8×C43—82no—
214D4×C43—21no—
215D4×C43—21no—
216D4×C43C2×Q1612yes—
217D4×C43C2×Q1612yes—
218C2×C4×C43—21no—
219C2×C4×C43—21no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43—82no—
229C2×C4×C43—161no—
230C2×C4×C43C22×D411yescharacteristic
231C2×C4×C43C22×D411yescharacteristic
232C2×C4×C43H(5,𝔽2)11yescharacteristic
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43—41no—
236C2×Dic43—41no—
237C2×Dic43—161no—
238C2×Dic43—161no—
239Dic86—21no—
240Dic86—21no—
241Dic86—81no—
242Dic86—81no—
243C8×C43—21no—
244C8×C43—41no—
245C8×C43C2×Q1612yes—
246C43⋊C8—21no—
247C43⋊C8—21no—
248C43⋊C8—41no—
249C43⋊C8—41no—
250C43⋊C8—41no—
251C43⋊C8—41no—
252#19814—431nomaximal, Sylow
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—41no—
256C22×C43—41no—
257C22×C43—41no—
258C22×C43—41no—
259C22×C43D4211yescharacteristic
260C4×C43—21no—
261C4×C43—41no—
262C4×C43—41no—
263C4×C43—41no—
264C4×C43—41no—
265C4×C43—81no—
266C4×C43—81no—
267C4×C43C22×Q1611yescharacteristic
268C4×C43#57511yescharacteristic
269C4×C43#57711yescharacteristic
270C4×C43#57811yescharacteristic
271Dic43—41no—
272Dic43—41no—
273Dic43—81no—
274Dic43—81no—
275#3384D4311yescharacteristic
276#3380—431no—
277#3338—431no—
278#3299—431no—
279#1723—431no—
280#1696—431no—
281#1683—431no—
282#1675—431no—
283#1672—431no—
284#1644—431no—
285#1637—431no—
286#1632—431no—
287#1632—431no—
288#1584—431no—
289#1580—431no—
290C2×C43—42no—
291C2×C43—42no—
292C2×C43#326711yescharacteristic
293C2×C43D4×Q1611yescharacteristic
294C2×C43#328811yescharacteristic
295#554—431no—
296#554—431no—
297#554C2×D4311yescharacteristic
298D4×Q8C2×D4311yescharacteristic
299#501—21no—
300#500—431no—
301#500—431no—
302#499—861no—
303#498C2×D4311yescharacteristic
304#494—431no—
305#487—431no—
306#487—431no—
307#484—861no—
308#484—861no—
309#484—861no—
310#478—21no—
311#478—861no—
312#478—861no—
313#477—431no—
314#477—431no—
315#477—431no—
316#476—21no—
317#476—431no—
318#476—431no—
319#476C2×D4311yescharacteristic
320#474C2×D4311yescharacteristic
321#467—861no—
322#467—861no—
323#462—431no—
324#451—861no—
325#446C2×D4311yescharacteristic
326C8×Q8—431no—
327#442—21no—
328C4×Q16—431no—
329C4×D8—861no—
330C4×D8C2×D4311yescharacteristic
331C8×D4—431no—
332C8×D4—431no—
333C2×#107—861no—
334C2×#103—861no—
335#335—431no—
336#334—431no—
337#333—431no—
338#333—431no—
339#333—431no—
340#332—431no—
341Q8⋊C8—431no—
342Q8⋊C8—431no—
343D4⋊C8—431no—
344D4⋊C8—431no—
345#322—431no—
346C43#1981411yescharacteristic, complemented, Sylow
347C22×Q8—21no—
348C2×Q16—431no—
349C2×Q16—861no—
350C2×Q16—1721no—
351C2×D8C22⋊D4311yescharacteristic
352C22×C8—861no—
353C22×C8—861no—
354C4⋊Q8—21no—
355C4⋊Q8—431no—
356C4⋊Q8—431no—
357C4⋊Q8C22×D4311yescharacteristic
358C4⋊Q8C22×D4311yescharacteristic
359#125—21no—
360#125—861no—
361#125—861no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—41no—
365C22⋊Q8—41no—
366C22⋊Q8—861no—
367C22⋊Q8—861no—
368C22⋊Q8—1721no—
369C22⋊Q8—1721no—
370C4×Q8—431no—
371C4×Q8C22×D4311yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4—861no—
375C4×D4—861no—
376C4×D4C22×D4311yescharacteristic
377C4×D4C22×D4311yescharacteristic
378C2×C4⋊C4—21no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4—861no—
381#107—21no—
382#107—431no—
383#107—861no—
384#107—861no—
385#107—861no—
386#107—861no—
387#107—861no—
388#107—861no—
389#107—861no—
390#107C22⋊D4311yescharacteristic
391#106—21no—
392#106—21no—
393#106—21no—
394#106—861no—
395#106—861no—
396C4⋊C8—21no—
397C4⋊C8—431no—
398C4⋊C8—431no—
399C4⋊C8—431no—
400C4⋊C8—861no—
401C4⋊C8C22×D4311yescharacteristic
402#103—431no—
403#103—431no—
404#103—861no—
405#103—861no—
406#103—861no—
407#103—861no—
408#103—861no—
409#103—861no—
410#103—1721no—
411#102—21no—
412#102—21no—
413#102—861no—
414#102—861no—
415#102C22⋊D4311yescharacteristic
416#102C22⋊D4311yescharacteristic
417C22⋊C8—861no—
418C22⋊C8—861no—
419#97—21no—
420C4×C8—431no—
421C4×C8—431no—
422C4×C8—861no—
423C4×C8C22×D4311yescharacteristic
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—41no—
427C2×Q8—42no—
428C2×Q8—81no—
429C2×Q8—431no—
430C2×Q8—861no—
431C2×Q8—861no—
432C2×Q8D4×D4311yescharacteristic
433C2×D4C2×C22⋊D4311yescharacteristic
434C2×D4C2×C22⋊D4311yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—1721no—
440C22×C4—1721no—
441Q16—3441no—
442Q16—3442no—
443Q16—1721no—
444D8#1062188514yescomplemented
445C2×C8—21no—
446C2×C8—431no—
447C2×C8—431no—
448C2×C8—861no—
449C2×C8—861no—
450C2×C8—861no—
451C2×C8—861no—
452C2×C8—3441no—
453C2×C8—1722no—
454C2×C8C2×C22⋊D4311yescharacteristic
455C2×C8C2×C22⋊D4311yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—431no—
465C4⋊C4—861no—
466C4⋊C4—861no—
467C4⋊C4—861no—
468C4⋊C4—861no—
469C4⋊C4—861no—
470C4⋊C4—1721no—
471C4⋊C4—1721no—
472C4⋊C4—1721no—
473C4⋊C4—1721no—
474C4⋊C4D4×D4311yescharacteristic
475C4⋊C4C2×C22⋊D4311yescharacteristic
476C4⋊C4C2×C22⋊D4311yescharacteristic
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—1721no—
482C22⋊C4—1721no—
483C42—21no—
484C42—861no—
485C42C23×D4311yescharacteristic
486C23—21no—
487C23—21no—
488Q8—21no—
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—82no—
494Q8—861no—
495Q8—861no—
496Q8—3441no—
497Q8—3441no—
498D4—21no—
499D4—21no—
500D4C2×#1062188512yes—
501D4C2×#1062188512yes—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—82no—
513C2×C4—161no—
514C2×C4—861no—
515C2×C4—861no—
516C2×C4—1721no—
517C2×C4—1721no—
518C2×C4—6881no—
519C2×C4—6881no—
520C2×C4C2×D4×D4311yescharacteristic
521C2×C4C22×C22⋊D4311yescharacteristic
522C2×C4#1306337011yescharacteristic
523C8—21no—
524C8—41no—
525C8—861no—
526C8—861no—
527C8—1721no—
528C8—1721no—
529C8—1721no—
530C8—1721no—
531C8C2×#1062188512yes—
532C22—22no—
533C22—22no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22D4×C22⋊D4311yescharacteristic
539C4—21no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—81no—
545C4—81no—
546C4—3441no—
547C4—3441no—
548C4—1721no—
549C4—1721no—
550C4C22×#1062188511yescharacteristic
551C4#1458369511yescharacteristic
552C4#1458370911yescharacteristic
553C4#1458371411yescharacteristic
554C2—42no—
555C2—42no—
556C2#1914611411yescharacteristic
557C2D4×#1062188511yescharacteristic
558C2#1914635711yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3384C2
socleC22×C43D42
prime radicalC2×D8×C43D4
hypercenter#3384D43
hypocenterC43#19814
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #3384

Complements

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

NH ≅ G/Ntype
2. #19146983C2semidirect
3. #19146471C2semidirect
4. #19134269C2semidirect
5. #19134123C2semidirect
6. #19134044C2semidirect
7. #19133992C2semidirect
8. #19133974C2semidirect
9. #19133808C2semidirect
10. #19133760C2semidirect
11. #19133729C2semidirect
12. #19133727C2semidirect
13. #19133443C2semidirect
161. D8×C43Q16semidirect
346. C43#19814semidirect
444. D8#10621885semidirect

Maximal subgroups

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

subgroupindex
#191474602
#191469832
#191464712
#191342692
#191341232
#191340442
#191339922
#191339742
#191338082
#191337602
#191337292
#191337272
#191334432
#191334192
C43×#33842
#1981443