Finite Group Databasebeta

← #25128597

Subgroups — #25128597

Order 11008.

1222 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583522C211yesmaximal, characteristic
2C2×#14583038C211yesmaximal, characteristic
3#19137181C211yesmaximal, characteristic, complemented
4#19135734C212yesmaximal, complemented
5#19135686C212yesmaximal, complemented
6#19134938C212yesmaximal, complemented
7#19134926C212yesmaximal, complemented
8#19134915C211yesmaximal, characteristic
9#19134908C211yesmaximal, characteristic, complemented
10#19134664C211yesmaximal, characteristic, complemented
11C2×C43×#451C211yesmaximal, characteristic, complemented
12C22×#10621884C2212yes—
13#14583522—24no—
14#14583522—24no—
15C2×#13063324C2211yescharacteristic
16C2×#13063320—21no—
17C2×#13063314C2211yescharacteristic
18C2×#13063311—21no—
19C2×#13063307—21no—
20C2×#13063305—21no—
21C2×#13063299C2211yescharacteristic
22C2×Q8×Dic43C2211yescharacteristic
23C2×#13063287C2211yescharacteristic
24C2×#13063284C2211yescharacteristic
25C2×#13063284C2211yescharacteristic
26#14583038—24no—
27C2×#13063249C2212yes—
28C2×#13063249C2212yes—
29C2×#13063247C2212yes—
30C2×C22⋊(C43⋊C8)C2211yescharacteristic
31#14582806—21no—
32#14582806—21no—
33#14582806C2212yes—
34#14582806C2212yes—
35#14582762—21no—
36#14582762—21no—
37#14582662C2212yes—
38#14582661C2212yes—
39#14582653C2212yes—
40#14582653C2212yes—
41C23×Q8×C43C2211yescharacteristic
42C22×Q16×C43C2212yes—
43C2×C22⋊Q8×C43C2211yescharacteristic
44C43×#451—24no—
45C2×#103×C43C2212yes—
46C2×C22⋊C8×C43C2211yescharacteristic
47C22×Dic86—21no—
48C22×D172C2311yescharacteristic
49C22×C22⋊D43—21no—
50C22×C4×D43—21no—
51C2×#10621884—24no—
52C2×#10621884—24no—
53C2×#10621884—42no—
54C22×C43⋊C8C2312yes—
55#13063324—22no—
56#13063324—22no—
57#13063320—28no—
58#13063314—22no—
59#13063314—22no—
60#13063311—28no—
61#13063307—28no—
62#13063305—28no—
63#13063299—22no—
64#13063299D414yes—
65Q8×Dic43—22no—
66Q8×Dic43—22no—
67#13063287—22no—
68#13063287—22no—
69#13063284—22no—
70#13063284—22no—
71#13063284—42no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43D411yescharacteristic
74C2×C4⋊Dic43D411yescharacteristic
75C2×C4⋊Dic43C2312yes—
76C2×C4⋊Dic43C2311yescharacteristic
77C2×Dic43⋊C4—21no—
78C2×C22⋊Dic43—21no—
79C2×C22⋊Dic43D411yescharacteristic
80C2×C22⋊Dic43D411yescharacteristic
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×C4×Dic43—21no—
86C2×C4×Dic43—21no—
87C2×C4×Dic43C2311yescharacteristic
88C2×C4×Dic43C2311yescharacteristic
89#13063249—24no—
90#13063249—42no—
91#13063247—42no—
92C22⋊(C43⋊C8)—22no—
93C43⋊#95—22no—
94C43⋊#95—22no—
95C43⋊#95—22no—
96C43⋊#95—22no—
97C22×Q8×C43—24no—
98C22×Q8×C43—21no—
99C22×Q8×C43—22no—
100C22×Q8×C43—22no—
101C22×Q8×C43—42no—
102C22×Q8×C43C2311yescharacteristic
103C22×Q8×C43C2312yes—
104C23×C4×C43—21no—
105C23×C4×C43C2311yescharacteristic
106C2×Q16×C43—24no—
107C2×Q16×C43—24no—
108C2×Q16×C43—42no—
109C22×C8×C43C2312yes—
110C22⋊Q8×C43—22no—
111C22⋊Q8×C43D414yes—
112C2×C4⋊C4×C43—21no—
113C2×C4⋊C4×C43C2311yescharacteristic
114C2×C22⋊C4×C43—21no—
115#103×C43—24no—
116C22⋊C8×C43—22no—
117C24×C43C2×D411yescharacteristic
118C23×D43—21no—
119C2×Q8×C43—24no—
120C2×Q8×C43—21no—
121C2×Q8×C43—22no—
122C2×Q8×C43—22no—
123C2×Q8×C43—22no—
124C2×Q8×C43—22no—
125C2×Q8×C43—22no—
126C2×Q8×C43—22no—
127C2×Q8×C43—22no—
128C2×Q8×C43—48no—
129C2×Q8×C43—48no—
130C2×Q8×C43—42no—
131C2×Q8×C43—42no—
132C2×Q8×C43C2×D412yes—
133C2×Q8×C43#4114yes—
134C22×C4×C43—22no—
135C22×C4×C43—22no—
136C22×C4×C43—24no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—22no—
140C22×C4×C43—42no—
141C22×C4×C43—42no—
142C22×C4×C43C2×D412yes—
143C22×C4×C43C2×D412yes—
144C22×C4×C43C2×D411yescharacteristic
145C22×C4×C43C2411yescharacteristic
146C2×C22⋊D43—24no—
147C2×C22⋊D43—82no—
148C22×Dic43—21no—
149C22×Dic43—22no—
150C22×Dic43C2×D411yescharacteristic
151C22×Dic43C2×D411yescharacteristic
152C22×Dic43C2×D411yescharacteristic
153C22×Dic43C2×D411yescharacteristic
154C2×D172—21no—
155C2×D172—22no—
156C2×D172—22no—
157C2×D172C2×D412yes—
158C2×C4×D43—24no—
159C2×C4×D43—82no—
160C2×Dic86—24no—
161C2×Dic86—82no—
162Q16×C43—48no—
163C2×C8×C43—22no—
164C2×C8×C43—22no—
165C2×C8×C43—22no—
166C4⋊C4×C43—24no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43C2×D412yes—
169C22⋊C4×C43—24no—
170C22⋊Dic43—22no—
171C22⋊Dic43—22no—
172C22⋊Dic43—42no—
173#10621884—48no—
174(C2×D43)⋊C4—24no—
175(C2×D43)⋊C4—24no—
176(C2×D43)⋊C4—24no—
177(C2×D43)⋊C4—24no—
178C4⋊Dic43—24no—
179C4⋊Dic43—22no—
180C4⋊Dic43—22no—
181C4⋊Dic43—21no—
182C4⋊Dic43—22no—
183C4⋊Dic43—22no—
184C4⋊Dic43C2×D412yes—
185Dic43⋊C4—24no—
186C4×Dic43—24no—
187C4×Dic43—21no—
188C4×Dic43—21no—
189C4×Dic43—21no—
190C4×Dic43—21no—
191C4×Dic43—42no—
192C2×C43⋊C8—22no—
193C2×C43⋊C8—22no—
194C2×C43⋊C8—22no—
195C23×C43—21no—
196C23×C43—21no—
197C23×C43—22no—
198C23×C43—22no—
199C23×C43#12012yes—
200C23×C43C22×D411yescharacteristic
201C22×D43—22no—
202C22×D43—41no—
203C22×D43—41no—
204C22×D43—42no—
205C22×D43—42no—
206Q8×C43—24no—
207Q8×C43—24no—
208Q8×C43—24no—
209Q8×C43—48no—
210Q8×C43—42no—
211Q8×C43—42no—
212Q8×C43—42no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—24no—
216C2×C4×C43—24no—
217C2×C4×C43—22no—
218C2×C4×C43—22no—
219C2×C4×C43—24no—
220C2×C4×C43—48no—
221C2×C4×C43—42no—
222C2×C4×C43—41no—
223C2×C4×C43—42no—
224C2×C4×C43—41no—
225C2×C4×C43—42no—
226C2×C4×C43#12012yes—
227C2×C4×C43C22×D411yescharacteristic
228C2×C4×C43C22×D411yescharacteristic
229C2×C4×C43C2×#4112yes—
230C2×C4×C43H(5,𝔽2)12yes—
231C22⋊D43—84no—
232C2×Dic43—21no—
233C2×Dic43—21no—
234C2×Dic43—22no—
235C2×Dic43—21no—
236C2×Dic43—21no—
237C2×Dic43—22no—
238C2×Dic43—22no—
239C2×Dic43—22no—
240C2×Dic43—22no—
241C2×Dic43—42no—
242C2×Dic43—42no—
243C2×Dic43—42no—
244C2×Dic43—42no—
245D172—24no—
246D172—42no—
247C4×D43—84no—
248Dic86—84no—
249C8×C43—42no—
250C43⋊C8—42no—
251#23946—431nomaximal, Sylow
252C22×C43—22no—
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—42no—
256C22×C43—42no—
257C22×C43C2×#12011yescharacteristic
258C22×C43#52511yescharacteristic
259C22×C43#53511yescharacteristic
260C22×C43#57712yes—
261C22×C43#57812yes—
262C2×D43—42no—
263C2×D43—42no—
264C2×D43—42no—
265C2×D43—82no—
266C2×D43—82no—
267C4×C43—22no—
268C4×C43—44no—
269C4×C43—42no—
270C4×C43—42no—
271C4×C43#54612yes—
272C4×C43#54612yes—
273Dic43—41no—
274Dic43—42no—
275Dic43—41no—
276Dic43—41no—
277Dic43—41no—
278Dic43—82no—
279C2×#542—431no—
280C2×#451D4311yescharacteristic, complemented
281C2×#448—431no—
282#2310—431no—
283#2024—432no—
284#2015—432no—
285#1878—432no—
286#1877—432no—
287#1875—431no—
288#1873—431no—
289#1818—431no—
290C2×C43—42no—
291C2×C43#240411yescharacteristic
292C2×C43#300911yescharacteristic
293C2×C43#301111yescharacteristic
294C2×C43#312112yes—
295C2×C43#322512yes—
296D43—82no—
297C23×Q8C2×D4311yescharacteristic
298C22×Q16C2×D4312yes—
299C22×SD16—432no—
300#542—864no—
301#542—864no—
302C2×C4⋊Q8—431no—
303C2×#126—861no—
304C2×#124—431no—
305C2×#124—861no—
306C2×#123—861no—
307C2×C22⋊Q8—861no—
308C2×C22⋊Q8C2×D4311yescharacteristic
309C2×#121—431no—
310C2×C4×Q8—431no—
311C2×C4×D4—431no—
312C2×#117—431no—
313C2×#117—431no—
314#451—24no—
315#448—864no—
316C2×#103—432no—
317C2×#103—432no—
318C2×#103C2×D4312yes—
319C2×#102—432no—
320C2×C22⋊C8—431no—
321C2×C22⋊C8C2×D4311yescharacteristic
322#391—432no—
323#391—432no—
324#391—861no—
325#391—861no—
326#381—861no—
327#381—861no—
328#340—432no—
329#340—432no—
330#336—432no—
331#336—432no—
332C43#2394611yescharacteristic, complemented, Sylow
333C22×Q8—24no—
334C22×Q8—21no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—42no—
338C22×Q8—861no—
339C22×Q8C22×D4312yes—
340C22×Q8C22×D4311yescharacteristic
341C22×D4—431no—
342C22×D4—861no—
343C23×C4—21no—
344C23×C4—861no—
345C23×C4C22×D4311yescharacteristic
346C2×Q16—24no—
347C2×Q16—24no—
348C2×Q16—42no—
349C2×SD16—864no—
350C2×SD16—864no—
351C2×SD16—1722no—
352C22×C8—432no—
353C22×C8C22×D4312yes—
354C4⋊Q8—862no—
355C4⋊Q8—862no—
356#126—868no—
357#124—868no—
358#124—862no—
359#124—862no—
360#123—868no—
361C22⋊Q8—22no—
362C22⋊Q8—868no—
363C22⋊Q8C22⋊D4314yes—
364#121—434no—
365#121—862no—
366C4×Q8—862no—
367C4×Q8—862no—
368C4×D4—862no—
369C4×D4—862no—
370#117—862no—
371#117—862no—
372#117—1722no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—431no—
375C2×C4⋊C4—431no—
376C2×C4⋊C4—431no—
377C2×C4⋊C4—432no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4C22×D4311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—431no—
383C2×C22⋊C4—431no—
384C2×C22⋊C4—861no—
385C2×C22⋊C4—861no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—861no—
388C2×C22⋊C4—861no—
389C2×C42—431no—
390C2×C42—431no—
391C2×C42—861no—
392C2×C42—861no—
393#103—24no—
394#103—864no—
395#103—1722no—
396#102—1722no—
397C22⋊C8—22no—
398C22⋊C8—862no—
399#95—862no—
400#95—862no—
401#95—862no—
402#95—862no—
403C24—861no—
404C24D4×D4311yescharacteristic
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—48no—
415C2×Q8—48no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—864no—
419C2×Q8—3442no—
420C2×Q8#1062190014yes—
421C2×Q8C2×C22⋊D4312yes—
422C2×D4—432no—
423C2×D4—864no—
424C2×D4—861no—
425C2×D4—862no—
426C2×D4—862no—
427C2×D4—3442no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—431no—
437C22×C4—431no—
438C22×C4—431no—
439C22×C4—431no—
440C22×C4—864no—
441C22×C4—861no—
442C22×C4—862no—
443C22×C4—3442no—
444C22×C4D4×D4311yescharacteristic
445C22×C4D4×D4312yes—
446C22×C4C2×C22⋊D4312yes—
447C22×C4C23×D4311yescharacteristic
448Q16—48no—
449SD16—1728no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—862no—
454C2×C8—862no—
455C2×C8—862no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—432no—
459C4⋊C4—864no—
460C4⋊C4—864no—
461C4⋊C4—862no—
462C4⋊C4—862no—
463C4⋊C4—861no—
464C4⋊C4—862no—
465C4⋊C4—862no—
466C4⋊C4C2×C22⋊D4312yes—
467C22⋊C4—24no—
468C22⋊C4—864no—
469C22⋊C4—864no—
470C22⋊C4—864no—
471C22⋊C4—862no—
472C22⋊C4—862no—
473C22⋊C4—864no—
474C22⋊C4—1722no—
475C42—864no—
476C42—861no—
477C42—861no—
478C42—861no—
479C42—861no—
480C42—1722no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—21no—
485C23—862no—
486C23—1721no—
487C23—1722no—
488C23—1722no—
489C23—1721no—
490C23#1306329312yes—
491C23C2×D4×D4311yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—24no—
495Q8—48no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—3444no—
500D4—864no—
501D4—3444no—
502D4—1722no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—861no—
517C2×C4—861no—
518C2×C4—861no—
519C2×C4—862no—
520C2×C4—861no—
521C2×C4—862no—
522C2×C4—862no—
523C2×C4—862no—
524C2×C4—862no—
525C2×C4—3444no—
526C2×C4—1722no—
527C2×C4—1722no—
528C2×C4—1722no—
529C2×C4—1722no—
530C2×C4#1306329312yes—
531C2×C4C2×D4×D4311yescharacteristic
532C2×C4C22×C22⋊D4311yescharacteristic
533C2×C4C2×#1062190012yes—
534C2×C4#1306337012yes—
535C8—42no—
536C8—1722no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—3442no—
543C22—3442no—
544C22—1722no—
545C22—1722no—
546C22—1722no—
547C22C2×#1306329311yescharacteristic
548C22#1458340511yescharacteristic
549C22#1458346411yescharacteristic
550C22#1458370812yes—
551C22#1458371212yes—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—3442no—
557C4—1721no—
558C4—1721no—
559C4—1722no—
560C4—1721no—
561C4—1721no—
562C4#1458353612yes—
563C4#1458354312yes—
564C2—42no—
565C2—3442no—
566C2#1913791111yescharacteristic
567C2#1914387111yescharacteristic
568C2#1914388611yescharacteristic
569C2#1914482512yes—
570C2#1914571012yes—
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#451C2
socleC23×C43C22×D4
prime radicalC2×#13063299C22
hypercenterC2×#451D43
hypocenterC43#23946
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
3. #19137181C2semidirect
4. #19135734C2semidirect
5. #19135686C2semidirect
6. #19134938C2semidirect
7. #19134926C2semidirect
9. #19134908C2semidirect
10. #19134664C2semidirect
11. C2×C43×#451C2semidirect
280. C2×#451D43semidirect
332. C43#23946semidirect

Maximal subgroups

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

subgroupindex
C2×#145835222
C2×#145830382
#191371812
#191357342
#191356862
#191349382
#191349262
#191349152
#191349082
#191346642
C2×C43×#4512
#2394643