Finite Group Databasebeta

← #25082009

Subgroups — #25082009

Order 11008.

758 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145510C211yesmaximal, characteristic, complemented
2#19144075C211yesmaximal, characteristic, complemented
3#19143706C211yesmaximal, characteristic, complemented
4#19134041C212yesmaximal, complemented
5#19134034C212yesmaximal, complemented
6#19133953C212yesmaximal, complemented
7#19133948C212yesmaximal, complemented
8#19133369C211yesmaximal, characteristic, complemented
9#19133272C211yesmaximal, characteristic, complemented
10C2×#14582642C211yesmaximal, characteristic, complemented
11C43×#3035C211yesmaximal, characteristic, complemented
12C2×#13063350—21no—
13C2×#13063322C2211yescharacteristic
14#14583275C2212yes—
15#14583247C2212yes—
16#14583099—21no—
17#14583099C2212yes—
18#14583094—21no—
19#14583094—21no—
20#14583094—21no—
21#14583094C2212yes—
22#14583032—22no—
23Q8×C43⋊C8C2212yescomplemented
24D4×C43⋊C8—22no—
25D4×C43⋊C8C2212yescomplemented
26#14582952—21no—
27#14582952C2211yescharacteristic
28#14582923C2211yescharacteristic
29C2×C4⋊(C43⋊C8)—21no—
30C2×C4⋊(C43⋊C8)C2211yescharacteristic
31#14582903—21no—
32#14582651C2212yescomplemented
33#14582651C2212yescomplemented
34#14582645C2212yescomplemented
35#14582643C2212yescomplemented
36#14582642C2212yes—
37#14582642C2212yes—
38C2×#135×C43—21no—
39C43×#532C2211yescharacteristic
40C4×#41×C43C2211yescharacteristic
41C43×#464—21no—
42C43×#463—21no—
43C43×#462C2211yescharacteristic, complemented
44C43×#461C2211yescharacteristic, complemented
45C43×#460C2211yescharacteristic, complemented
46C43×#459C2211yescharacteristic, complemented
47C2×C4⋊C8×C43C2211yescharacteristic
48C43×#416—21no—
49C22×D172—21no—
50C22×D172—21no—
51C22×D172—21no—
52#13063350—44no—
53C2×#10621884—22no—
54C2×#10621882—22no—
55C2×#10621882—22no—
56C2×#10621882—22no—
57C2×#10621877—21no—
58C22×C43⋊C8—21no—
59C22×C43⋊C8—22no—
60#13063322—21no—
61#13063322—21no—
62#13063322—21no—
63#13063322C2312yes—
64C4⋊(C43⋊C8)—21no—
65C4⋊(C43⋊C8)—21no—
66C4⋊(C43⋊C8)—22no—
67C4⋊(C43⋊C8)D411yescharacteristic
68C4⋊(C43⋊C8)D411yescharacteristic
69C4⋊(C43⋊C8)C2312yes—
70C4⋊(C43⋊C8)C2312yes—
71#13063245—22no—
72#13063245—22no—
73#13063245—22no—
74#13063245—22no—
75C22⋊(C43⋊C8)—22no—
76C22⋊(C43⋊C8)D412yes—
77C4×C43⋊C8—22no—
78C4×C43⋊C8C2312yes—
79C2×#41×C43—21no—
80C2×#41×C43D411yescharacteristic
81#135×C43—44no—
82C2×Q16×C43—21no—
83C2×SD16×C43—21no—
84C2×SD16×C43—21no—
85C2×D8×C43—21no—
86C22×C8×C43—21no—
87C4⋊Q8×C43C2311yescharacteristic
88#127×C43C2311yescharacteristic
89#124×C43—21no—
90#121×C43—21no—
91#121×C43—21no—
92C4×Q8×C43D411yescharacteristic
93C4×Q8×C43C2311yescharacteristic
94C4×D4×C43—21no—
95C4×D4×C43—21no—
96C4×D4×C43—21no—
97C4×D4×C43—21no—
98C4×D4×C43D411yescharacteristic
99C4×D4×C43C2311yescharacteristic
100#117×C43—21no—
101#117×C43D411yescharacteristic
102C2×C42×C43—21no—
103C2×C42×C43C2311yescharacteristic
104C4⋊C8×C43—21no—
105C4⋊C8×C43C2311yescharacteristic
106C4⋊C8×C43C2311yescharacteristic
107#103×C43—21no—
108#103×C43—21no—
109#102×C43—21no—
110#102×C43—21no—
111C23×D43—41no—
112#41×C43—22no—
113#41×C43—42no—
114#41×C43—42no—
115#41×C43D814yescomplemented
116C2×Q8×C43—21no—
117C2×Q8×C43C2×D411yescharacteristic
118C2×D4×C43—21no—
119C2×D4×C43—21no—
120C2×D4×C43—21no—
121C2×D4×C43—21no—
122C2×D4×C43—41no—
123C2×D4×C43C2×D411yescharacteristic
124C22×C4×C43—21no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—41no—
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43#4111yescharacteristic
131C2×D172—22no—
132C2×D172—22no—
133C2×D172—22no—
134C2×D172—41no—
135C2×D172—41no—
136C2×D172—41no—
137C2×D172—42no—
138C2×D172—42no—
139C2×D172—81no—
140C2×D172—81no—
141C2×D172—82no—
142Q16×C43—42no—
143SD16×C43—42no—
144SD16×C43—42no—
145D8×C43—42no—
146C2×C8×C43—21no—
147C2×C8×C43—21no—
148C2×C8×C43—42no—
149C4⋊C4×C43—21no—
150C4⋊C4×C43—21no—
151C4⋊C4×C43—21no—
152C4⋊C4×C43—21no—
153C4⋊C4×C43C2×D411yescharacteristic
154C4⋊C4×C43C2×D411yescharacteristic
155C22⋊C4×C43—21no—
156C22⋊C4×C43—21no—
157C22⋊C4×C43—21no—
158C22⋊C4×C43—41no—
159C22⋊C4×C43—41no—
160C42×C43—21no—
161C42×C43—21no—
162C42×C43—21no—
163C42×C43C2×D411yescharacteristic
164C42×C43C2×D411yescharacteristic
165C42×C43#4111yescharacteristic
166C42×C43C2411yescharacteristic
167#10621884—44no—
168#10621882—44no—
169#10621882—82no—
170#10621882—82no—
171#10621877—42no—
172C2×C43⋊C8—22no—
173C2×C43⋊C8—22no—
174C2×C43⋊C8—82no—
175C2×C43⋊C8—81no—
176C2×C43⋊C8C2×D412yes—
177C2×C43⋊C8C2×D412yes—
178C23×C43—21no—
179C23×C43—41no—
180C23×C43#12111yescharacteristic
181C22×D43—42no—
182C22×D43—82no—
183C22×D43—82no—
184C22×D43—82no—
185Q8×C43—21no—
186Q8×C43—41no—
187Q8×C43—41no—
188Q8×C43C2×D812yes—
189D4×C43—24no—
190D4×C43—21no—
191D4×C43—41no—
192D4×C43—41no—
193D4×C43—41no—
194D4×C43—41no—
195D4×C43—81no—
196D4×C43—82no—
197D4×C43C2×D812yes—
198C2×C4×C43—21no—
199C2×C4×C43—21no—
200C2×C4×C43—21no—
201C2×C4×C43—24no—
202C2×C4×C43—21no—
203C2×C4×C43—21no—
204C2×C4×C43—41no—
205C2×C4×C43—41no—
206C2×C4×C43—41no—
207C2×C4×C43—41no—
208C2×C4×C43—81no—
209C2×C4×C43—82no—
210C2×C4×C43#12111yescharacteristic
211C2×C4×C43#12111yescharacteristic
212C2×C4×C43#12111yescharacteristic
213C2×C4×C43C2×D812yes—
214C2×C4×C43#13512yes—
215C2×C4×C43C22×D411yescharacteristic
216C2×C4×C43C22×D411yescharacteristic
217C2×C4×C43C2×#4111yescharacteristic
218D172—42no—
219D172—42no—
220D172—42no—
221D172—42no—
222D172—82no—
223D172—82no—
224D172—82no—
225D172—82no—
226C8×C43—41no—
227C8×C43—41no—
228C43⋊C8—22no—
229C43⋊C8—22no—
230C43⋊C8—42no—
231C43⋊C8—42no—
232#19410—431nomaximal, Sylow
233C22×C43—21no—
234C22×C43—21no—
235C22×C43—22no—
236C22×C43—41no—
237C22×C43—41no—
238C22×C43—81no—
239C22×C43—81no—
240C22×C43—81no—
241C22×C43#46612yes—
242C22×C43C2×#12111yescharacteristic
243C2×D43—82no—
244C2×D43—82no—
245C2×D43—82no—
246C2×D43—164no—
247C4×C43—21no—
248C4×C43—22no—
249C4×C43—21no—
250C4×C43—21no—
251C4×C43—41no—
252C4×C43—41no—
253C4×C43—81no—
254C4×C43#46612yes—
255C4×C43C22×D811yescharacteristic
256C4×C43C2×#13511yescharacteristic
257C4×C43#57611yescharacteristic
258C4×C43#57711yescharacteristic
259#3205—431no—
260#3038—431no—
261#3035D4311yescharacteristic, complemented
262#2988—431no—
263#1682—432no—
264#1681—432no—
265#1668—432no—
266#1667—432no—
267#1571—431no—
268#1547—431no—
269C2×#331—431no—
270C2×C43—22no—
271C2×C43—42no—
272C2×C43—161no—
273C2×C43#304511yescharacteristic
274C2×C43C2×#46611yescharacteristic
275C2×C43#306811yescharacteristic
276D43—162no—
277C2×AGL(1,ℤ/8)—861no—
278C2×#135—21no—
279#532C2×D4311yescharacteristic
280C2×#127—431no—
281C4×#41C2×D4311yescharacteristic
282#493—432no—
283#486—432no—
284#464—21no—
285#463—21no—
286#462C2×D4311yescharacteristic, complemented
287#461C2×D4311yescharacteristic, complemented
288#460—432no—
289#460—861no—
290#460C2×D4311yescharacteristic, complemented
291#459—432no—
292#459—861no—
293#459—861no—
294#459—861no—
295#459C2×D4311yescharacteristic, complemented
296#447—862no—
297C8×Q8—432no—
298C8×D4—432no—
299C8×D4—862no—
300#431—431no—
301#431—861no—
302#423—431no—
303C2×C4⋊C8—431no—
304C2×C4⋊C8—861no—
305C2×C4⋊C8C2×D4311yescharacteristic
306#418—861no—
307#416—21no—
308#335—432no—
309#335—432no—
310#332—432no—
311#331—432no—
312#331—432no—
313#331—432no—
314C43#1941011yescharacteristic, complemented, Sylow
315C2×#41—21no—
316C2×#41C22⋊D4311yescharacteristic
317C22×D4—861no—
318C22×D4—861no—
319C22×D4—861no—
320AGL(1,ℤ/8)—1724no—
321#135—44no—
322C2×Q16—21no—
323C2×SD16—21no—
324C2×SD16—21no—
325C2×SD16—862no—
326C2×D8—21no—
327C2×D8—862no—
328C2×D8—862no—
329C2×D8—862no—
330C2×M16—861no—
331C22×C8—21no—
332C22×C8—862no—
333C22×C8—861no—
334C4⋊Q8C22×D4311yescharacteristic
335#127—432no—
336#127—861no—
337#127—861no—
338#127—861no—
339#127C22×D4311yescharacteristic
340#124—21no—
341#121—21no—
342#121—21no—
343C4×Q8C22⋊D4311yescharacteristic
344C4×Q8C22×D4311yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4C22⋊D4311yescharacteristic
350C4×D4C22×D4311yescharacteristic
351#117—21no—
352#117C22⋊D4311yescharacteristic
353C2×C42—21no—
354C2×C42C22×D4311yescharacteristic
355C4⋊C8—21no—
356C4⋊C8—431no—
357C4⋊C8—431no—
358C4⋊C8—432no—
359C4⋊C8—432no—
360C4⋊C8—862no—
361C4⋊C8—861no—
362C4⋊C8—861no—
363C4⋊C8C22×D4311yescharacteristic
364C4⋊C8C22×D4311yescharacteristic
365#103—21no—
366#103—21no—
367#102—21no—
368#102—21no—
369#102—862no—
370#102—862no—
371#102—862no—
372#102—862no—
373C22⋊C8—432no—
374C22⋊C8—862no—
375C4×C8—432no—
376C4×C8—862no—
377C24—1721no—
378#41—22no—
379#41—42no—
380#41—42no—
381#41#1062188214yescomplemented
382C2×Q8—21no—
383C2×Q8C2×C22⋊D4311yescharacteristic
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—862no—
390C2×D4—862no—
391C2×D4—862no—
392C2×D4—3441no—
393C2×D4—3441no—
394C2×D4—3442no—
395C2×D4—1721no—
396C2×D4—1721no—
397C2×D4—1721no—
398C2×D4—1722no—
399C2×D4—1722no—
400C2×D4C2×C22⋊D4311yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4D4×D4311yescharacteristic
406C22×C4#1062190011yescharacteristic
407C22×C4C2×C22⋊D4311yescharacteristic
408Q16—42no—
409SD16—42no—
410SD16—42no—
411SD16—1724no—
412D8—42no—
413D8—3442no—
414D8—3442no—
415D8—1724no—
416M16—1722no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—42no—
420C2×C8—432no—
421C2×C8—432no—
422C2×C8—862no—
423C2×C8—862no—
424C2×C8—3442no—
425C2×C8—3441no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4C2×C22⋊D4311yescharacteristic
431C4⋊C4C2×C22⋊D4311yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42D4×D4311yescharacteristic
441C42#1062190011yescharacteristic
442C42C2×C22⋊D4311yescharacteristic
443C42C23×D4311yescharacteristic
444C23—21no—
445C23—41no—
446C23—3442no—
447C23—3442no—
448C23—3442no—
449C23—1722no—
450C23#1306329711yescharacteristic
451Q8—21no—
452Q8—41no—
453Q8—41no—
454Q8C2×#1062188212yes—
455D4—24no—
456D4—21no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—81no—
462D4—82no—
463D4—3442no—
464D4—3442no—
465D4—3442no—
466D4—3442no—
467D4—1722no—
468D4—1722no—
469D4—1722no—
470D4—1722no—
471D4C2×#1062188212yes—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4#1306329711yescharacteristic
485C2×C4#1306329711yescharacteristic
486C2×C4#1306329711yescharacteristic
487C2×C4C2×#1062188212yes—
488C2×C4#1306334512yes—
489C2×C4C2×D4×D4311yescharacteristic
490C2×C4C22×C22⋊D4311yescharacteristic
491C2×C4C2×#1062190011yescharacteristic
492C8—41no—
493C8—41no—
494C8—862no—
495C8—862no—
496C8—1722no—
497C8—1722no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—3442no—
507C22—3442no—
508C22—3442no—
509C22—6884no—
510C22#1458313312yes—
511C22C2×#1306329711yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—81no—
519C4#1458313312yes—
520C4C22×#1062188211yescharacteristic
521C4C2×#1306334511yescharacteristic
522C4#1458370111yescharacteristic
523C4#1458370611yescharacteristic
524C2—22no—
525C2—42no—
526C2—161no—
527C2—6882no—
528C2#1914413211yescharacteristic
529C2C2×#1458313311yescharacteristic
530C2#1914432511yescharacteristic
subgroupG/H
centerC22C2×#13063297
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3035C2
socleC22×C43C2×#121
hypercenter#3035D43
hypocenterC43#19410
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #19145510C2semidirect
2. #19144075C2semidirect
3. #19143706C2semidirect
4. #19134041C2semidirect
5. #19134034C2semidirect
6. #19133953C2semidirect
7. #19133948C2semidirect
8. #19133369C2semidirect
9. #19133272C2semidirect
10. C2×#14582642C2semidirect
11. C43×#3035C2semidirect
23. Q8×C43⋊C8C22semidirect
25. D4×C43⋊C8C22semidirect
32. #14582651C22semidirect
33. #14582651C22semidirect
34. #14582645C22semidirect
35. #14582643C22semidirect
43. C43×#462C22semidirect
44. C43×#461C22semidirect
45. C43×#460C22semidirect
46. C43×#459C22semidirect
115. #41×C43D8semidirect
261. #3035D43semidirect
286. #462C2×D43semidirect
287. #461C2×D43semidirect
290. #460C2×D43semidirect
295. #459C2×D43semidirect
314. C43#19410semidirect
381. #41#10621882semidirect

Maximal subgroups

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

subgroupindex
#191455102
#191440752
#191437062
#191340412
#191340342
#191339532
#191339482
#191333692
#191332722
C2×#145826422
C43×#30352
#1941043