Finite Group Databasebeta

← #25124172

Subgroups — #25124172

Order 11008.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583477C211yesmaximal, characteristic
2C22×#13063257C211yesmaximal, characteristic, complemented
3C2×#14582900C211yesmaximal, characteristic, complemented
4C2×#14582865C211yesmaximal, characteristic, complemented
5#19137183C211yesmaximal, characteristic, complemented
6#19135790C214yesmaximal, complemented
7#19135783C214yesmaximal, complemented
8#19134316C211yesmaximal, characteristic, complemented
9C2×C43×#416C211yesmaximal, characteristic, complemented
10C22×#10621898C2211yescharacteristic
11C22×#10621877C2211yescharacteristic
12#14583477—24no—
13#14583477—24no—
14C2×#13063304—21no—
15C2×#13063301C2212yes—
16C2×#13063299—21no—
17C2×#13063297—22no—
18C2×#13063296C2212yes—
19C22×C4⋊Dic43C2211yescharacteristic
20C22×C4⋊Dic43C2211yescharacteristic
21C2×#13063257—24no—
22C2×#13063257C2214yes—
23#14582900—24no—
24C2×#13063249C2212yes—
25C2×#13063246C2212yes—
26#14582865—24no—
27C2×(C2×D43)⋊C8C2214yes—
28#14582805—21no—
29#14582801—21no—
30#14582776—22no—
31#14582756C2214yes—
32#14582661C2214yes—
33C22×#41×C43C2211yescharacteristic
34C23×C8×C43C2211yescharacteristic
35C2×#117×C43C2211yescharacteristic
36C43×#416—24no—
37C2×#103×C43C2212yes—
38C2×#102×C43C2212yes—
39C2×#10621898—22no—
40C2×#10621898—22no—
41C2×#10621898—22no—
42C2×#10621898—22no—
43C2×#10621898—22no—
44C2×#10621898—22no—
45C22×Dic86D411yescharacteristic
46C22×D172D411yescharacteristic
47C22×C22⋊D43—22no—
48C22×C22⋊D43D412yes—
49C23×Dic43—21no—
50C23×Dic43—21no—
51C22×C4×D43C2312yes—
52C2×#10621877—22no—
53C2×#10621877—22no—
54C2×#10621877—22no—
55C22×C43⋊C8C2312yes—
56#13063304—28no—
57#13063301—24no—
58#13063301—24no—
59#13063299—28no—
60#13063297—216no—
61#13063296—24no—
62#13063296—24no—
63C2×C4⋊Dic43—21no—
64C2×C4⋊Dic43—22no—
65C2×C4⋊Dic43—22no—
66C2×C4⋊Dic43—22no—
67C2×C4⋊Dic43—21no—
68C2×C4⋊Dic43D414yes—
69C2×C4⋊Dic43C2312yes—
70C2×C4⋊Dic43C2312yes—
71C2×Dic43⋊C4—22no—
72C2×C22⋊Dic43—22no—
73C2×(C2×D43)⋊C4—22no—
74C2×(C2×D43)⋊C4—24no—
75#13063257—28no—
76#13063249—24no—
77#13063246—24no—
78(C2×D43)⋊C8—28no—
79C43⋊#95—24no—
80C2×#41×C43—22no—
81C2×#41×C43—22no—
82C2×#41×C43—22no—
83C2×#41×C43—22no—
84C2×#41×C43—22no—
85C2×#41×C43D414yes—
86C22×Q8×C43C2311yescharacteristic
87C22×D4×C43—21no—
88C22×D4×C43C2311yescharacteristic
89C23×C4×C43—21no—
90C23×C4×C43C2311yescharacteristic
91C22×C8×C43—22no—
92C22×C8×C43—22no—
93C22×C8×C43D414yescomplemented
94C22×C8×C43C2312yes—
95#117×C43—22no—
96#117×C43—22no—
97C2×C4⋊C4×C43C2312yes—
98C2×C22⋊C4×C43—21no—
99C2×C42×C43—21no—
100#103×C43—24no—
101#102×C43—24no—
102C24×C43—21no—
103C24×C43C2×D411yescharacteristic
104C23×D43C2×D412yes—
105#41×C43—28no—
106#41×C43—28no—
107#41×C43—44no—
108#41×C43—44no—
109C2×Q8×C43—21no—
110C2×Q8×C43—21no—
111C2×Q8×C43—21no—
112C2×Q8×C43—21no—
113C2×Q8×C43—21no—
114C2×Q8×C43C2×D412yes—
115C2×D4×C43—21no—
116C2×D4×C43—24no—
117C2×D4×C43—21no—
118C2×D4×C43—21no—
119C2×D4×C43—21no—
120C2×D4×C43—21no—
121C2×D4×C43—42no—
122C2×D4×C43—42no—
123C2×D4×C43C2×D412yes—
124C22×C4×C43—24no—
125C22×C4×C43—21no—
126C22×C4×C43—22no—
127C22×C4×C43—42no—
128C22×C4×C43—42no—
129C22×C4×C43C2×D412yes—
130C22×C4×C43C2×D412yes—
131C22×C4×C43C2×D412yes—
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43#4112yes—
134C22×C4×C43#4112yes—
135C22×C4×C43#4112yes—
136C22×C4×C43C2411yescharacteristic
137C2×C22⋊D43—28no—
138C2×C22⋊D43—24no—
139C2×C22⋊D43—24no—
140C2×C22⋊D43—24no—
141C2×C22⋊D43—84no—
142C22×Dic43—24no—
143C22×Dic43—22no—
144C22×Dic43—22no—
145C22×Dic43—42no—
146C22×Dic43—82no—
147C22×Dic43—82no—
148C22×Dic43C2×D412yes—
149#10621898—44no—
150#10621898—44no—
151#10621898—44no—
152#10621898—44no—
153C2×D172—22no—
154C2×D172—22no—
155C2×D172—22no—
156C2×C4×D43—22no—
157C2×C4×D43—22no—
158C2×C4×D43—22no—
159C2×C4×D43—22no—
160C2×C4×D43—22no—
161C2×C4×D43—22no—
162C2×Dic86—22no—
163C2×Dic86—22no—
164C2×Dic86—22no—
165C2×C8×C43—28no—
166C2×C8×C43—22no—
167C2×C8×C43—22no—
168C2×C8×C43C2×D414yes—
169C4⋊C4×C43—22no—
170C4⋊C4×C43—22no—
171C22⋊C4×C43—42no—
172C42×C43—42no—
173C22⋊Dic43—28no—
174(C2×D43)⋊C4—28no—
175(C2×D43)⋊C4—84no—
176C4⋊Dic43—24no—
177C4⋊Dic43—22no—
178C4⋊Dic43—22no—
179C4⋊Dic43—22no—
180C4⋊Dic43—42no—
181C4⋊Dic43C2×D414yes—
182Dic43⋊C4—28no—
183#10621877—44no—
184C2×C43⋊C8—22no—
185C2×C43⋊C8—22no—
186C2×C43⋊C8—22no—
187C23×C43—22no—
188C23×C43—22no—
189C23×C43—22no—
190C23×C43—41no—
191C23×C43—41no—
192C23×C43—42no—
193C23×C43—42no—
194C23×C43#12012yes—
195C23×C43#12112yes—
196C23×C43#12312yes—
197C23×C43C22×D411yescharacteristic
198C22×D43—22no—
199C22×D43—22no—
200C22×D43—22no—
201C22×D43—24no—
202C22×D43—24no—
203Q8×C43—22no—
204Q8×C43—22no—
205Q8×C43—41no—
206Q8×C43—41no—
207D4×C43—22no—
208D4×C43—22no—
209D4×C43—48no—
210D4×C43—41no—
211D4×C43—41no—
212C2×C4×C43—22no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—48no—
216C2×C4×C43—42no—
217C2×C4×C43—42no—
218C2×C4×C43—42no—
219C2×C4×C43#12012yes—
220C2×C4×C43#12112yes—
221C2×C4×C43#12114yes—
222C2×C4×C43#12114yes—
223C2×C4×C43#12312yes—
224C2×C4×C43#12314yes—
225C2×C4×C43#12414yes—
226C2×C4×C43C22×D411yescharacteristic
227C2×C4×C43C22×D411yescharacteristic
228C2×C4×C43C22×D411yescharacteristic
229C2×C4×C43C2×#4111yescharacteristic
230C2×C4×C43C2×#4111yescharacteristic
231C2×C4×C43C2×#4111yescharacteristic
232C22⋊D43—48no—
233C22⋊D43—88no—
234C2×Dic43—22no—
235C2×Dic43—24no—
236C2×Dic43—22no—
237C2×Dic43—22no—
238C2×Dic43—44no—
239C2×Dic43—42no—
240C2×Dic43—44no—
241C2×Dic43—84no—
242C2×Dic43—84no—
243D172—44no—
244C4×D43—42no—
245C4×D43—42no—
246C4×D43—42no—
247C4×D43—42no—
248Dic86—44no—
249C8×C43—24no—
250C43⋊C8—42no—
251#23489—431nomaximal, Sylow
252C22×C43—24no—
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—24no—
256C22×C43—42no—
257C22×C43—42no—
258C22×C43—42no—
259C22×C43—82no—
260C22×C43—82no—
261C22×C43#39414yes—
262C22×C43C2×#12011yescharacteristic
263C22×C43C2×#12111yescharacteristic
264C22×C43C2×#12311yescharacteristic
265C22×C43#57712yes—
266C22×C43#57712yes—
267C2×D43—44no—
268C2×D43—42no—
269C2×D43—42no—
270C2×D43—42no—
271C2×D43—44no—
272C4×C43—42no—
273C4×C43—82no—
274C4×C43#39414yes—
275C4×C43C2×#12111yescharacteristic
276C4×C43C2×#12111yescharacteristic
277C4×C43C2×#12311yescharacteristic
278C4×C43C2×#12411yescharacteristic
279Dic43—42no—
280Dic43—44no—
281Dic43—82no—
282C2×#537—431no—
283C22×#106—431no—
284C2×#417—431no—
285C2×#416D4311yescharacteristic, complemented
286C2×#408—431no—
287#2311—431no—
288#2035—434no—
289#2034—434no—
290#1736—431no—
291C2×C43—24no—
292C2×C43—82no—
293C2×C43C2×#39411yescharacteristic
294C2×C43#300911yescharacteristic
295C2×C43#301011yescharacteristic
296C2×C43#306012yes—
297C2×C43#309912yes—
298D43—82no—
299C22×#41—431no—
300C22×#41C2×D4311yescharacteristic
301C22×M16—431no—
302C23×C8C2×D4311yescharacteristic
303#537—864no—
304#537—864no—
305C2×C22⋊Q8—432no—
306C2×C22⋊Q8—861no—
307C2×#121—432no—
308C2×#121—861no—
309C2×#121—862no—
310C2×#117C2×D4311yescharacteristic
311C22×C4⋊C4—431no—
312C22×C4⋊C4—431no—
313C2×#106—434no—
314C2×#106—864no—
315#417—864no—
316#416—24no—
317C2×#103—432no—
318C2×#103C2×D4312yes—
319C2×#102—432no—
320C2×#102C2×D4312yes—
321#408—864no—
322C2×C22⋊C8—434no—
323#391—861no—
324#390—861no—
325#386—862no—
326#380—434no—
327#340—434no—
328C43#2348911yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—862no—
335C2×#41—862no—
336C2×#41—862no—
337C2×#41—862no—
338C2×#41—862no—
339C2×#41—862no—
340C2×#41D17214yes—
341C22×Q8—431no—
342C22×Q8C22×D4311yescharacteristic
343C22×D4—21no—
344C22×D4—432no—
345C22×D4—431no—
346C22×D4—862no—
347C22×D4C22×D4311yescharacteristic
348C23×C4—21no—
349C23×C4—432no—
350C23×C4—861no—
351C23×C4—861no—
352C23×C4C22×D4311yescharacteristic
353C2×M16—862no—
354C2×M16—862no—
355C2×M16—862no—
356C22×C8—22no—
357C22×C8—22no—
358C22×C8—432no—
359C22×C8C22⋊D4314yescomplemented
360C22×C8C22×D4312yes—
361C22⋊Q8—864no—
362C22⋊Q8—868no—
363C22⋊Q8—864no—
364#121—864no—
365#121—868no—
366#121—8616no—
367#121—864no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—434no—
371C2×C4⋊C4—432no—
372C2×C4⋊C4—432no—
373C2×C4⋊C4—862no—
374C2×C4⋊C4—862no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4—862no—
377C2×C4⋊C4—862no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4C22×D4312yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—862no—
382C2×C22⋊C4—862no—
383C2×C22⋊C4—864no—
384C2×C42—21no—
385#106—868no—
386#103—24no—
387#103—864no—
388#102—24no—
389#102—864no—
390C22⋊C8—868no—
391#95—864no—
392C24—21no—
393C24—432no—
394C24D4×D4311yescharacteristic
395#41—28no—
396#41—28no—
397#41—44no—
398#41—44no—
399#41—1724no—
400#41—1724no—
401#41—1724no—
402#41—1724no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—862no—
409C2×Q8—862no—
410C2×Q8—862no—
411C2×Q8C2×D17212yes—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—868no—
421C2×D4—864no—
422C2×D4—864no—
423C2×D4—864no—
424C2×D4—862no—
425C2×D4—862no—
426C2×D4—862no—
427C2×D4—3444no—
428C2×D4C2×D17212yes—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—432no—
435C22×C4—864no—
436C22×C4—862no—
437C22×C4—862no—
438C22×C4—862no—
439C22×C4—862no—
440C22×C4—862no—
441C22×C4—862no—
442C22×C4—862no—
443C22×C4—862no—
444C22×C4—3442no—
445C22×C4—3442no—
446C22×C4—1722no—
447C22×C4C2×D17212yes—
448C22×C4#1062189812yes—
449C22×C4D4×D4311yescharacteristic
450C22×C4D4×D4312yes—
451C22×C4#1062190012yes—
452C22×C4#1062190012yes—
453C22×C4C2×C22⋊D4312yes—
454C22×C4C23×D4311yescharacteristic
455M16—1724no—
456C2×C8—28no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—862no—
460C2×C8—862no—
461C2×C8—862no—
462C2×C8C2×C22⋊D4314yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—434no—
466C4⋊C4—868no—
467C4⋊C4—864no—
468C4⋊C4—862no—
469C4⋊C4—862no—
470C4⋊C4—862no—
471C4⋊C4—1722no—
472C22⋊C4—42no—
473C22⋊C4—868no—
474C22⋊C4—868no—
475C22⋊C4—3444no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—862no—
485C23—862no—
486C23—862no—
487C23—864no—
488C23—864no—
489C23#1306329512yes—
490C23#1306329612yes—
491C23#1306330712yes—
492C23C2×D4×D4311yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—1724no—
498D4—22no—
499D4—22no—
500D4—48no—
501D4—41no—
502D4—41no—
503D4—3448no—
504D4—1728no—
505D4—1724no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—864no—
514C2×C4—862no—
515C2×C4—862no—
516C2×C4—862no—
517C2×C4—3444no—
518C2×C4—3444no—
519C2×C4—1722no—
520C2×C4—1724no—
521C2×C4—1724no—
522C2×C4—1722no—
523C2×C4—1722no—
524C2×C4—1722no—
525C2×C4—1722no—
526C2×C4#1306329512yes—
527C2×C4#1306329612yes—
528C2×C4#1306329714yes—
529C2×C4#1306329914yes—
530C2×C4#1306330712yes—
531C2×C4#1306330914yes—
532C2×C4#1306331114yes—
533C2×C4C2×D4×D4311yescharacteristic
534C2×C4C22×C22⋊D4311yescharacteristic
535C2×C4C22×D17211yescharacteristic
536C2×C4C2×#1062190011yescharacteristic
537C2×C4C2×#1062190011yescharacteristic
538C2×C4C2×#1062189811yescharacteristic
539C8—24no—
540C8—1722no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—82no—
549C22—82no—
550C22—1722no—
551C22—1724no—
552C22—1724no—
553C22—1722no—
554C22—1722no—
555C22#1458281514yes—
556C22C2×#1306329511yescharacteristic
557C22C2×#1306329611yescharacteristic
558C22C2×#1306330711yescharacteristic
559C22#1458370612yes—
560C22#1458370612yes—
561C4—42no—
562C4—82no—
563C4—3442no—
564C4—1722no—
565C4—1724no—
566C4#1458281514yes—
567C4C2×#1306329711yescharacteristic
568C4C2×#1306329911yescharacteristic
569C4C2×#1306330911yescharacteristic
570C4C2×#1306331111yescharacteristic
571C2—24no—
572C2—82no—
573C2—3442no—
574C2C2×#1458281511yescharacteristic
575C2#1914387511yescharacteristic
576C2#1914388411yescharacteristic
577C2#1914425312yes—
578C2#1914458212yes—
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#416C2
socleC23×C43C22×D4
prime radicalC2×#14583477C2
hypercenterC2×#416D43
hypocenterC43#23489
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C22×#13063257C2semidirect
3. C2×#14582900C2semidirect
4. C2×#14582865C2semidirect
5. #19137183C2semidirect
6. #19135790C2semidirect
7. #19135783C2semidirect
8. #19134316C2semidirect
9. C2×C43×#416C2semidirect
93. C22×C8×C43D4semidirect
285. C2×#416D43semidirect
328. C43#23489semidirect
359. C22×C8C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145834772
C22×#130632572
C2×#145829002
C2×#145828652
#191371832
#191357902
#191357832
#191343162
C2×C43×#4162
#2348943