Finite Group Databasebeta

← #25086154

Subgroups — #25086154

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#19147525C211yesmaximal, characteristic, complemented
2Q8×#10621883C211yesmaximal, characteristic
3#19147237C211yesmaximal, characteristic
4#19134226C211yesmaximal, characteristic, complemented
5#19134086C211yesmaximal, characteristic, complemented
6#19134027C211yesmaximal, characteristic, complemented
7#19133975C211yesmaximal, characteristic, complemented
8#19133970C211yesmaximal, characteristic
9#19133870C211yesmaximal, characteristic, complemented
10#19133858C211yesmaximal, characteristic
11#19133814C211yesmaximal, characteristic, complemented
12#19133760C211yesmaximal, characteristic
13#19133431C211yesmaximal, characteristic
14#19133414C211yesmaximal, characteristic, complemented
15D4×Q16×C43C211yesmaximal, characteristic
16Q8×Dic86C2211yescharacteristic
17#14583628C2211yescharacteristic
18#14583307—21no—
19#14583297C2211yescharacteristic
20#14583295—21no—
21#14583295C2211yescharacteristic
22#14583295C2211yescharacteristic
23#14583251—21no—
24#14583251C2211yescharacteristic
25#14583251C2211yescharacteristic
26#14583204—21no—
27#14583194C2211yescharacteristic
28#14583189—21no—
29#14583189C2211yescharacteristic
30#14583184C2211yescharacteristic
31#14583182—21no—
32#14583181—21no—
33#14583181—21no—
34#14583181C2211yescharacteristic
35#14583181C2211yescharacteristic
36#14583104—21no—
37#14583104C2211yescharacteristic
38#14583029C2211yescharacteristic
39Q8×C43⋊C8C2211yescharacteristic
40Q8×C43⋊C8C2211yescharacteristic
41#14583006—21no—
42C4×#10621883—21no—
43C4×#10621883—21no—
44C4×#10621883C2211yescharacteristic
45#14582650C2211yescharacteristic
46#14582649C2211yescharacteristic
47#14582647C2211yescharacteristic
48#14582645C2211yescharacteristic
49#14582645C2211yescharacteristic
50#14582644C2211yescharacteristic
51Q8⋊(C43⋊C8)C2211yescharacteristic
52Q8⋊(C43⋊C8)C2211yescharacteristic
53#14582629C2211yescharacteristic
54D4⋊(C43⋊C8)C2211yescharacteristic
55#14582620C2211yescharacteristic
56C22×Q16×C43—21no—
57D4×Q8×C43C2211yescharacteristic
58D4×Q8×C43C2211yescharacteristic
59C43×#494C2211yescharacteristic
60C43×#467—21no—
61C43×#462C2211yescharacteristic
62C43×#462C2211yescharacteristic
63C43×#451—21no—
64C43×#451—21no—
65C4×Q16×C43C2211yescharacteristic
66C8×D4×C43C2211yescharacteristic
67C2×#10621883—21no—
68C2×#10621883D411yescharacteristic
69#13063325—21no—
70#13063325—21no—
71#13063325C2311yescharacteristic
72#13063325C2311yescharacteristic
73#13063324—21no—
74#13063324—41no—
75#13063316—21no—
76#13063316—41no—
77Q8×Dic43—21no—
78Q8×Dic43—21no—
79C4×Dic86—21no—
80C4×Dic86—21no—
81C4×Dic86C2311yescharacteristic
82#13063258—21no—
83#13063257—21no—
84#13063257—21no—
85#13063256—21no—
86#13063256—21no—
87#13063256—21no—
88#13063256—21no—
89#13063256—21no—
90#13063256—21no—
91#13063256—41no—
92#13063256D411yescharacteristic
93C4⋊(C43⋊C8)—21no—
94C4⋊(C43⋊C8)—21no—
95C4⋊(C43⋊C8)C2311yescharacteristic
96C4⋊(C43⋊C8)C2311yescharacteristic
97C4⋊(C43⋊C8)C2311yescharacteristic
98Dic43⋊C8—21no—
99#13063249—21no—
100#13063249—21no—
101#13063248—21no—
102#13063248—21no—
103#13063248—21no—
104#13063248—21no—
105#13063248—21no—
106#13063248—41no—
107#13063248D411yescharacteristic
108#13063246—21no—
109#13063246D411yescharacteristic
110#13063235—21no—
111C4×C43⋊C8—21no—
112C4×C43⋊C8—21no—
113C4×C43⋊C8C2311yescharacteristic
114C4×C43⋊C8C2311yescharacteristic
115C22×Q8×C43—21no—
116C22×Q8×C43—21no—
117C2×Q16×C43—21no—
118C2×Q16×C43—21no—
119C2×Q16×C43—21no—
120C2×Q16×C43—21no—
121C2×Q16×C43—44no—
122C2×Q16×C43D411yescharacteristic
123C22×C8×C43—21no—
124C4⋊Q8×C43—21no—
125C4⋊Q8×C43—21no—
126C4⋊Q8×C43C2311yescharacteristic
127C4⋊Q8×C43C2311yescharacteristic
128C22⋊Q8×C43—21no—
129C22⋊Q8×C43—21no—
130C22⋊Q8×C43—41no—
131C22⋊Q8×C43—41no—
132C4×Q8×C43C2311yescharacteristic
133C4×Q8×C43C2311yescharacteristic
134C4×D4×C43—21no—
135C4×D4×C43—21no—
136C4×D4×C43C2311yescharacteristic
137#107×C43D411yescharacteristic
138C4⋊C8×C43C2311yescharacteristic
139#103×C43—21no—
140#103×C43—21no—
141#103×C43D411yescharacteristic
142#103×C43D411yescharacteristic
143C22⋊C8×C43—21no—
144C4×C8×C43C2311yescharacteristic
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—41no—
148C2×Q8×C43—41no—
149C2×Q8×C43—42no—
150C2×Q8×C43—42no—
151C2×Q8×C43—81no—
152C2×Q8×C43—81no—
153C2×Q8×C43C2×D411yescharacteristic
154C2×Q8×C43C2×D411yescharacteristic
155C2×D4×C43C2×D411yescharacteristic
156C22×C4×C43—21no—
157C22×C4×C43—41no—
158C22×C4×C43—41no—
159C2×Dic86—21no—
160C2×Dic86—21no—
161C2×Dic86—41no—
162C2×Dic86C2×D411yescharacteristic
163Q16×C43—42no—
164Q16×C43—42no—
165Q16×C43—42no—
166Q16×C43—81no—
167Q16×C43SD1614yescomplemented
168C2×C8×C43—21no—
169C2×C8×C43—42no—
170C2×C8×C43C2×D411yescharacteristic
171C2×C8×C43C2×D411yescharacteristic
172C4⋊C4×C43—21no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—41no—
179C4⋊C4×C43—41no—
180C4⋊C4×C43C2×D411yescharacteristic
181C4⋊C4×C43C2×D411yescharacteristic
182C4⋊C4×C43C2×D411yescharacteristic
183C22⋊C4×C43—21no—
184C22⋊C4×C43—41no—
185C22⋊C4×C43—41no—
186C42×C43—21no—
187C42×C43—21no—
188C42×C43C2411yescharacteristic
189#10621883—22no—
190#10621883—42no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—41no—
196C4⋊Dic43—41no—
197C4⋊Dic43—41no—
198C4⋊Dic43C2×D411yescharacteristic
199Dic43⋊C4—21no—
200Dic43⋊C4—41no—
201Dic43⋊C4—41no—
202C4×Dic43—21no—
203C4×Dic43—41no—
204C4×Dic43—41no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8C2×D411yescharacteristic
210C2×C43⋊C8C2×D411yescharacteristic
211C23×C43—21no—
212Q8×C43—21no—
213Q8×C43—21no—
214Q8×C43—41no—
215Q8×C43—41no—
216Q8×C43—41no—
217Q8×C43—41no—
218Q8×C43—82no—
219Q8×C43—82no—
220Q8×C43C2×SD1612yes—
221Q8×C43C2×SD1612yes—
222D4×C43—21no—
223D4×C43#13712yes—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—41no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43—82no—
234C2×C4×C43—82no—
235C2×C4×C43C22×D411yescharacteristic
236C2×C4×C43C22×D411yescharacteristic
237C2×C4×C43H(5,𝔽2)11yescharacteristic
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242Dic86—21no—
243Dic86—21no—
244Dic86—41no—
245Dic86—41no—
246Dic86—81no—
247Dic86—161no—
248C8×C43—21no—
249C8×C43—41no—
250C8×C43C2×SD1612yes—
251C43⋊C8—21no—
252C43⋊C8—21no—
253C43⋊C8—41no—
254C43⋊C8—41no—
255C43⋊C8—41no—
256C43⋊C8—41no—
257#19769—431nomaximal, Sylow
258C22×C43—22no—
259C22×C43—41no—
260C22×C43—41no—
261C22×C43D4211yescharacteristic
262C4×C43—22no—
263C4×C43—22no—
264C4×C43—21no—
265C4×C43—41no—
266C4×C43—41no—
267C4×C43—41no—
268C4×C43—81no—
269C4×C43—81no—
270C4×C43C22×SD1611yescharacteristic
271C4×C43C2×#13711yescharacteristic
272C4×C43#57711yescharacteristic
273C4×C43#57811yescharacteristic
274Dic43—41no—
275Dic43—41no—
276Dic43—81no—
277Dic43—81no—
278#3387—431no—
279Q8×SD16—431no—
280#3359—431no—
281D4×Q16D4311yescharacteristic
282#1716—431no—
283#1690—431no—
284#1680—431no—
285#1672—431no—
286#1671—431no—
287#1653—431no—
288#1651—431no—
289#1645—431no—
290#1637—431no—
291#1582—431no—
292#1579—431no—
293C2×C43—42no—
294C2×C43#327011yescharacteristic
295C2×C43D4×SD1611yescharacteristic
296C2×C43#328911yescharacteristic
297C22×Q16—21no—
298Q82—431no—
299#557—431no—
300D4×Q8C2×D4311yescharacteristic
301D4×Q8C2×D4311yescharacteristic
302#501—861no—
303#498—431no—
304#498—431no—
305#498—431no—
306#498—861no—
307#494C2×D4311yescharacteristic
308#487—431no—
309#487—431no—
310#487—861no—
311#479—861no—
312#477—431no—
313#476—431no—
314#476—861no—
315#475—431no—
316#475—431no—
317#475—431no—
318#475—861no—
319#475—861no—
320#475—861no—
321#467—21no—
322#462C2×D4311yescharacteristic
323#462C2×D4311yescharacteristic
324#461—431no—
325#461—861no—
326#451—21no—
327#451—21no—
328#446—431no—
329C8×Q8—431no—
330C8×Q8—431no—
331#441—861no—
332C4×Q16C2×D4311yescharacteristic
333C4×SD16—431no—
334C4×SD16—861no—
335C4×SD16—861no—
336C8×D4C2×D4311yescharacteristic
337#334—431no—
338#334—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#1976911yescharacteristic, complemented, Sylow
349C22×Q8—21no—
350C22×Q8—21no—
351C2×Q16—21no—
352C2×Q16—21no—
353C2×Q16—21no—
354C2×Q16—21no—
355C2×Q16—44no—
356C2×Q16C22⋊D4311yescharacteristic
357C2×SD16—431no—
358C2×SD16—861no—
359C22×C8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—431no—
363C4⋊Q8—431no—
364C4⋊Q8—861no—
365C4⋊Q8—861no—
366C4⋊Q8—861no—
367C4⋊Q8—1721no—
368C4⋊Q8C22×D4311yescharacteristic
369C4⋊Q8C22×D4311yescharacteristic
370#125—861no—
371#125—1721no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—41no—
375C22⋊Q8—41no—
376C4×Q8—431no—
377C4×Q8—861no—
378C4×Q8—861no—
379C4×Q8—861no—
380C4×Q8—861no—
381C4×Q8C22×D4311yescharacteristic
382C4×Q8C22×D4311yescharacteristic
383C4×D4—21no—
384C4×D4—21no—
385C4×D4C22×D4311yescharacteristic
386#107—861no—
387#107C22⋊D4311yescharacteristic
388#106—431no—
389#106—861no—
390#106—861no—
391#106—861no—
392#106—861no—
393#106—861no—
394#106—861no—
395#106—861no—
396#106—861no—
397#106—1721no—
398C4⋊C8—431no—
399C4⋊C8—431no—
400C4⋊C8—431no—
401C4⋊C8—861no—
402C4⋊C8—861no—
403C4⋊C8—861no—
404C4⋊C8C22×D4311yescharacteristic
405#103—21no—
406#103—21no—
407#103—431no—
408#103—861no—
409#103—861no—
410#103—861no—
411#103—861no—
412#103—861no—
413#103—861no—
414#103—861no—
415#103—1721no—
416#103C22⋊D4311yescharacteristic
417#103C22⋊D4311yescharacteristic
418#102—431no—
419#102—861no—
420C22⋊C8—21no—
421#97—861no—
422C4×C8—431no—
423C4×C8—431no—
424C4×C8—861no—
425C4×C8—861no—
426C4×C8C22×D4311yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—41no—
430C2×Q8—41no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—81no—
434C2×Q8—81no—
435C2×Q8—431no—
436C2×Q8—861no—
437C2×Q8—861no—
438C2×Q8—1721no—
439C2×Q8C2×C22⋊D4311yescharacteristic
440C2×Q8C2×C22⋊D4311yescharacteristic
441C2×D4D4×D4311yescharacteristic
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—41no—
445Q16—42no—
446Q16—42no—
447Q16—42no—
448Q16—81no—
449Q16#1062188314yescomplemented
450SD16—862no—
451SD16—1722no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—431no—
455C2×C8—431no—
456C2×C8—861no—
457C2×C8—861no—
458C2×C8—861no—
459C2×C8—861no—
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—41no—
469C4⋊C4—41no—
470C4⋊C4—431no—
471C4⋊C4—861no—
472C4⋊C4—861no—
473C4⋊C4—861no—
474C4⋊C4—861no—
475C4⋊C4—861no—
476C4⋊C4—1721no—
477C4⋊C4—1721no—
478C4⋊C4—1721no—
479C4⋊C4—1721no—
480C4⋊C4—1721no—
481C4⋊C4D4×D4311yescharacteristic
482C4⋊C4C2×C22⋊D4311yescharacteristic
483C4⋊C4C2×C22⋊D4311yescharacteristic
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C42—21no—
488C42—21no—
489C42—861no—
490C42—1721no—
491C42—1721no—
492C42C23×D4311yescharacteristic
493C23—21no—
494Q8—21no—
495Q8—21no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—82no—
501Q8—82no—
502Q8—861no—
503Q8—861no—
504Q8—3441no—
505Q8—1721no—
506Q8—1721no—
507Q8—6881no—
508Q8C2×#1062188312yes—
509Q8C2×#1062188312yes—
510D4—21no—
511D4#1306335712yes—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—861no—
524C2×C4—861no—
525C2×C4—1721no—
526C2×C4—1721no—
527C2×C4C2×D4×D4311yescharacteristic
528C2×C4C22×C22⋊D4311yescharacteristic
529C2×C4#1306337011yescharacteristic
530C8—21no—
531C8—41no—
532C8—861no—
533C8—861no—
534C8—1721no—
535C8—1721no—
536C8—1721no—
537C8—1721no—
538C8C2×#1062188312yes—
539C22—22no—
540C22—41no—
541C22—41no—
542C22D4×C22⋊D4311yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—3441no—
552C4—3441no—
553C4—1721no—
554C4—1721no—
555C4C22×#1062188311yescharacteristic
556C4C2×#1306335711yescharacteristic
557C4#1458371011yescharacteristic
558C4#1458371211yescharacteristic
559C2—42no—
560C2#1914616511yescharacteristic
561C2D4×#1062188311yescharacteristic
562C2#1914636611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingD4×Q16×C43C2
socleC22×C43D42
prime radicalC2×D4×C43C2×D4
hypercenterD4×Q16D43
hypocenterC43#19769
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. #19147525C2semidirect
4. #19134226C2semidirect
5. #19134086C2semidirect
6. #19134027C2semidirect
7. #19133975C2semidirect
9. #19133870C2semidirect
11. #19133814C2semidirect
14. #19133414C2semidirect
167. Q16×C43SD16semidirect
348. C43#19769semidirect
449. Q16#10621883semidirect

Maximal subgroups

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

subgroupindex
#191475252
Q8×#106218832
#191472372
#191342262
#191340862
#191340272
#191339752
#191339702
#191338702
#191338582
#191338142
#191337602
#191334312
#191334142
D4×Q16×C432
#1976943