Finite Group Databasebeta

← #25086818

Subgroups — #25086818

Order 11008.

580 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147509C211yesmaximal, characteristic
2#19147476C211yesmaximal, characteristic, complemented
3Q8×#10621883C211yesmaximal, characteristic
4#19146471C211yesmaximal, characteristic
5#19134274C211yesmaximal, characteristic, complemented
6#19134100C211yesmaximal, characteristic, complemented
7#19134052C211yesmaximal, characteristic, complemented
8#19133997C211yesmaximal, characteristic, complemented
9#19133992C211yesmaximal, characteristic
10#19133813C211yesmaximal, characteristic, complemented
11#19133737C211yesmaximal, characteristic, complemented
12#19133731C211yesmaximal, characteristic
13#19133442C211yesmaximal, characteristic, complemented
14#19133404C211yesmaximal, characteristic
15C43×#1637C211yesmaximal, characteristic
16Q8×Dic86C2211yescharacteristic
17#14583629C2211yescharacteristic
18#14583606C2211yescharacteristic
19#14583306—21no—
20#14583303—21no—
21#14583302C2211yescharacteristic
22#14583302C2211yescharacteristic
23#14583295—21no—
24#14583295C2211yescharacteristic
25#14583279C2211yescharacteristic
26#14583249C2211yescharacteristic
27#14583232—21no—
28#14583209—21no—
29#14583195C2211yescharacteristic
30#14583195C2211yescharacteristic
31#14583194C2211yescharacteristic
32#14583193—21no—
33#14583192—21no—
34#14583189—21no—
35#14583189C2211yescharacteristic
36#14583181—21no—
37#14583181C2211yescharacteristic
38#14583177—21no—
39#14583138—21no—
40#14583106C2211yescharacteristic
41#14583055—21no—
42#14583026C2211yescharacteristic
43Q8×C43⋊C8C2211yescharacteristic
44#14583023C2211yescharacteristic
45#14582996—21no—
46C4×#10621885C2211yescharacteristic
47#14582991—21no—
48C4×#10621883—21no—
49C4×#10621883C2211yescharacteristic
50D4×C43⋊C8C2211yescharacteristic
51C2×#13063249—21no—
52#14582651C2211yescharacteristic
53#14582651C2211yescharacteristic
54#14582648C2211yescharacteristic
55#14582647C2211yescharacteristic
56#14582647C2211yescharacteristic
57#14582631C2211yescharacteristic
58#14582631C2211yescharacteristic
59#14582621C2211yescharacteristic
60D4×Q8×C43C2211yescharacteristic
61C43×#487C2211yescharacteristic
62C43×#477C2211yescharacteristic
63C43×#476C2211yescharacteristic
64C43×#332C2211yescharacteristic
65C43×Q8⋊C8C2211yescharacteristic
66C43×D4⋊C8C2211yescharacteristic
67C2×Dic172—41no—
68C2×#10621885—21no—
69C2×#10621885D411yescharacteristic
70C2×#10621883D411yescharacteristic
71C22×C43⋊C8—21no—
72#13063325—21no—
73#13063325C2311yescharacteristic
74#13063325C2311yescharacteristic
75#13063324—21no—
76#13063324—21no—
77#13063324—21no—
78#13063324—41no—
79#13063323—41no—
80#13063316—21no—
81#13063304—21no—
82#13063302—41no—
83Q8×Dic43—21no—
84C4×Dic86—21no—
85C4×Dic86C2311yescharacteristic
86C4×Dic86C2311yescharacteristic
87#13063290—21no—
88#13063290—21no—
89D4×Dic43—21no—
90C2×C4⋊Dic43—21no—
91#13063259—21no—
92#13063259—21no—
93#13063258—21no—
94#13063258—21no—
95#13063258—21no—
96#13063258—21no—
97#13063258—21no—
98#13063258D411yescharacteristic
99#13063256—21no—
100#13063256—21no—
101#13063256—21no—
102#13063256—21no—
103#13063256D411yescharacteristic
104C4⋊(C43⋊C8)—21no—
105C4⋊(C43⋊C8)C2311yescharacteristic
106C4⋊(C43⋊C8)C2311yescharacteristic
107Dic43⋊C8—21no—
108Dic43⋊C8—21no—
109#13063250—21no—
110#13063250—21no—
111#13063249—21no—
112#13063249—21no—
113#13063249—21no—
114#13063249—21no—
115#13063249D411yescharacteristic
116#13063248—21no—
117#13063248D411yescharacteristic
118#13063248D411yescharacteristic
119#13063246—21no—
120#13063246D411yescharacteristic
121C22⋊(C43⋊C8)—21no—
122#13063235—21no—
123C4×C43⋊C8—21no—
124C4×C43⋊C8C2311yescharacteristic
125C4×C43⋊C8C2311yescharacteristic
126C8×Dic43—21no—
127C22×Q8×C43—21no—
128C4⋊Q8×C43—21no—
129C4⋊Q8×C43C2311yescharacteristic
130C4⋊Q8×C43C2311yescharacteristic
131C22⋊Q8×C43—21no—
132C22⋊Q8×C43—41no—
133C4×Q8×C43C2311yescharacteristic
134C4×D4×C43—21no—
135C4×D4×C43C2311yescharacteristic
136#107×C43—21no—
137#106×C43—21no—
138C4⋊C8×C43C2311yescharacteristic
139C4⋊C8×C43C2311yescharacteristic
140#103×C43—21no—
141#103×C43—41no—
142#102×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
151C22×C4×C43—21no—
152C22×C4×C43—41no—
153C22×Dic43—41no—
154C2×Dic86—21no—
155C2×Dic86—21no—
156C2×Dic86—41no—
157C2×Dic86—41no—
158C2×Dic86C2×D411yescharacteristic
159C2×Dic86C2×D411yescharacteristic
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43—21no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—41no—
169C4⋊C4×C43C2×D411yescharacteristic
170C4⋊C4×C43C2×D411yescharacteristic
171C22⋊C4×C43—21no—
172C22⋊C4×C43—41no—
173C42×C43—21no—
174C42×C43C2411yescharacteristic
175C22⋊Dic43—41no—
176#10621885—22no—
177#10621885—81no—
178#10621883Q1614yescomplemented
179C4⋊Dic43—21no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43—41no—
184C4⋊Dic43—41no—
185C4⋊Dic43C2×D411yescharacteristic
186C4⋊Dic43C2×D411yescharacteristic
187Dic43⋊C4—21no—
188Dic43⋊C4—21no—
189Dic43⋊C4—41no—
190Dic43⋊C4—41no—
191Dic43⋊C4—41no—
192Dic43⋊C4—41no—
193C4×Dic43—21no—
194C4×Dic43—21no—
195C4×Dic43—41no—
196C4×Dic43—41no—
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—81no—
200C2×C43⋊C8C2×D411yescharacteristic
201C2×C43⋊C8C2×D411yescharacteristic
202C2×C43⋊C8C2×D411yescharacteristic
203C2×C43⋊C8C2×D411yescharacteristic
204Dic172—82no—
205C23×C43—21no—
206Q8×C43—21no—
207Q8×C43—41no—
208Q8×C43—41no—
209Q8×C43—82no—
210Q8×C43—81no—
211Q8×C43#13712yes—
212D4×C43—21no—
213D4×C43C2×Q1612yes—
214C2×C4×C43—21no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—82no—
222C2×C4×C43C22×D411yescharacteristic
223C2×C4×C43C22×D411yescharacteristic
224C2×C4×C43H(5,𝔽2)11yescharacteristic
225C2×Dic43—21no—
226C2×Dic43—21no—
227C2×Dic43—21no—
228C2×Dic43—21no—
229C2×Dic43—41no—
230C2×Dic43—41no—
231C2×Dic43—161no—
232Dic86—21no—
233Dic86—21no—
234Dic86—21no—
235Dic86—81no—
236Dic86—81no—
237Dic86—161no—
238Dic86—161no—
239Dic86C2×Q1612yes—
240C8×C43—41no—
241C8×C43—41no—
242C8×C43—41no—
243C43⋊C8—21no—
244C43⋊C8—21no—
245C43⋊C8—21no—
246C43⋊C8—41no—
247C43⋊C8—41no—
248C43⋊C8C2×Q1612yes—
249#19813—431nomaximal, Sylow
250C22×C43—22no—
251C22×C43—41no—
252C22×C43—41no—
253C22×C43D4211yescharacteristic
254C4×C43—22no—
255C4×C43—21no—
256C4×C43—41no—
257C4×C43—41no—
258C4×C43—81no—
259C4×C43—81no—
260C4×C43C22×Q1611yescharacteristic
261C4×C43C2×#13711yescharacteristic
262C4×C43#57711yescharacteristic
263C4×C43#57811yescharacteristic
264Dic43—22no—
265Dic43—41no—
266Dic43—41no—
267Dic43—41no—
268Dic43—81no—
269Dic43—81no—
270#3385—431no—
271Q8×Q16—431no—
272Q8×SD16—431no—
273#3299—431no—
274#1723—431no—
275#1692—431no—
276#1684—431no—
277#1676—431no—
278#1675—431no—
279#1645—431no—
280#1637D4311yescharacteristic
281#1633—431no—
282#1632—431no—
283#1584—431no—
284SD16⋊C8—431no—
285C2×C43—42no—
286C2×C43#326611yescharacteristic
287C2×C43D4×Q1611yescharacteristic
288C2×C43#328811yescharacteristic
289Q82—431no—
290Q82—431no—
291#554—431no—
292D4×Q8C2×D4311yescharacteristic
293#501—861no—
294#500—431no—
295#500—431no—
296#500—861no—
297#498—431no—
298#498—861no—
299#494—431no—
300#487—431no—
301#487C2×D4311yescharacteristic
302#484—861no—
303#479—861no—
304#477—431no—
305#477—431no—
306#477—431no—
307#477—861no—
308#477—861no—
309#477C2×D4311yescharacteristic
310#476—431no—
311#476—861no—
312#476C2×D4311yescharacteristic
313#475—431no—
314#475—861no—
315#474—861no—
316#467—861no—
317#462—431no—
318#451—861no—
319#446—431no—
320C8×Q8—431no—
321C8×Q8—431no—
322#440—861no—
323C4×Q16—431no—
324C4×Q16—861no—
325C4×SD16—431no—
326C4×SD16—861no—
327C8×D4—431no—
328C2×#103—861no—
329#335—431no—
330#335—431no—
331#333—431no—
332#333—431no—
333#333—431no—
334#332C2×D4311yescharacteristic
335Q8⋊C8—431no—
336Q8⋊C8—431no—
337Q8⋊C8C2×D4311yescharacteristic
338D4⋊C8C2×D4311yescharacteristic
339#322—431no—
340C43#1981311yescharacteristic, complemented, Sylow
341C22×Q8—21no—
342C2×Q16—431no—
343C2×Q16—861no—
344C2×Q16—1721no—
345C2×SD16—431no—
346C22×C8—861no—
347C4⋊Q8—21no—
348C4⋊Q8—431no—
349C4⋊Q8—431no—
350C4⋊Q8—861no—
351C4⋊Q8—861no—
352C4⋊Q8—861no—
353C4⋊Q8—861no—
354C4⋊Q8—1721no—
355C4⋊Q8—1721no—
356C4⋊Q8C22×D4311yescharacteristic
357C4⋊Q8C22×D4311yescharacteristic
358#125—861no—
359C22⋊Q8—21no—
360C22⋊Q8—41no—
361C22⋊Q8—861no—
362C22⋊Q8—1721no—
363C4×Q8—431no—
364C4×Q8—431no—
365C4×Q8—861no—
366C4×Q8—861no—
367C4×Q8—861no—
368C4×Q8—861no—
369C4×Q8C22×D4311yescharacteristic
370C4×D4—21no—
371C4×D4—861no—
372C4×D4C22×D4311yescharacteristic
373C2×C4⋊C4—861no—
374#107—21no—
375#107—431no—
376#107—861no—
377#107—861no—
378#107—861no—
379#107—861no—
380#107—861no—
381#107—861no—
382#107—861no—
383#106—21no—
384#106—431no—
385#106—861no—
386#106—861no—
387#106—861no—
388#106—861no—
389C4⋊C8—431no—
390C4⋊C8—431no—
391C4⋊C8—861no—
392C4⋊C8—861no—
393C4⋊C8—861no—
394C4⋊C8C22×D4311yescharacteristic
395C4⋊C8C22×D4311yescharacteristic
396#103—21no—
397#103—41no—
398#103—431no—
399#103—431no—
400#103—431no—
401#103—861no—
402#103—861no—
403#103—861no—
404#103—861no—
405#103—861no—
406#103—861no—
407#103—861no—
408#102—21no—
409#102—431no—
410#102—861no—
411C22⋊C8—861no—
412#97—861no—
413C4×C8—431no—
414C4×C8—431no—
415C4×C8—861no—
416C4×C8—861no—
417C4×C8C22×D4311yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—41no—
421C2×Q8—42no—
422C2×Q8—81no—
423C2×Q8—431no—
424C2×Q8—431no—
425C2×Q8—861no—
426C2×Q8—861no—
427C2×Q8—1721no—
428C2×Q8—1721no—
429C2×Q8D4×D4311yescharacteristic
430C2×D4D4×D4311yescharacteristic
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—1721no—
434Q16—862no—
435Q16—3442no—
436Q16—3441no—
437SD16—434no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—431no—
442C2×C8—431no—
443C2×C8—431no—
444C2×C8—431no—
445C2×C8—861no—
446C2×C8—861no—
447C2×C8—3441no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—431no—
455C4⋊C4—431no—
456C4⋊C4—861no—
457C4⋊C4—861no—
458C4⋊C4—861no—
459C4⋊C4—861no—
460C4⋊C4—861no—
461C4⋊C4—861no—
462C4⋊C4—1721no—
463C4⋊C4—1721no—
464C4⋊C4—1721no—
465C4⋊C4—1721no—
466C4⋊C4—1721no—
467C4⋊C4—1721no—
468C4⋊C4D4×D4311yescharacteristic
469C4⋊C4D4×D4311yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—41no—
472C22⋊C4—1721no—
473C42—21no—
474C42—861no—
475C42—861no—
476C42—1721no—
477C42—1721no—
478C42C23×D4311yescharacteristic
479C23—21no—
480Q8—21no—
481Q8—41no—
482Q8—41no—
483Q8—82no—
484Q8—81no—
485Q8—432no—
486Q8—861no—
487Q8—861no—
488Q8—861no—
489Q8—3441no—
490Q8—3441no—
491Q8—6881no—
492Q8—6881no—
493Q8#1306335412yes—
494D4—21no—
495D4Q16×D4312yes—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—82no—
504C2×C4—861no—
505C2×C4—861no—
506C2×C4—861no—
507C2×C4—861no—
508C2×C4—1721no—
509C2×C4—1721no—
510C2×C4—6881no—
511C2×C4C2×D4×D4311yescharacteristic
512C2×C4C2×D4×D4311yescharacteristic
513C2×C4#1306337011yescharacteristic
514C8—41no—
515C8—41no—
516C8—41no—
517C8—432no—
518C8—861no—
519C8—861no—
520C8—861no—
521C8—1721no—
522C8—1721no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22#1458353111yescharacteristic
527C4—21no—
528C4—22no—
529C4—41no—
530C4—41no—
531C4—81no—
532C4—81no—
533C4—862no—
534C4—3441no—
535C4—3441no—
536C4—1721no—
537C4—1721no—
538C4—1721no—
539C4C2×Q16×D4311yescharacteristic
540C4C2×#1306335411yescharacteristic
541C4#1458370911yescharacteristic
542C4#1458371511yescharacteristic
543C2—42no—
544C2#1914610911yescharacteristic
545C2#1914634011yescharacteristic
546C2#1914636311yescharacteristic
subgroupG/H
centerC22#14583531
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#1637C2
socleC22×C43D42
prime radicalC2×D4×C43C2×D4
hypercenter#1637D43
hypocenterC43#19813
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #1637

Complements

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

NH ≅ G/Ntype
2. #19147476C2semidirect
5. #19134274C2semidirect
6. #19134100C2semidirect
7. #19134052C2semidirect
8. #19133997C2semidirect
10. #19133813C2semidirect
11. #19133737C2semidirect
13. #19133442C2semidirect
178. #10621883Q16semidirect
340. C43#19813semidirect

Maximal subgroups

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

subgroupindex
#191475092
#191474762
Q8×#106218832
#191464712
#191342742
#191341002
#191340522
#191339972
#191339922
#191338132
#191337372
#191337312
#191334422
#191334042
C43×#16372
#1981343