Finite Group Databasebeta

← #25098241

Subgroups — #25098241

Order 11008.

1742 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583475C211yesmaximal, characteristic
2#19147796C211yesmaximal, characteristic
3C2×#14582872C211yesmaximal, characteristic
4#19137179C211yesmaximal, characteristic, complemented
5#19137179C211yesmaximal, characteristic, complemented
6#19135007C218yesmaximal, complemented
7#19134327C211yesmaximal, characteristic, complemented
8C43×#1740C211yesmaximal, characteristic, complemented
9C22×#10621898C2211yescharacteristic
10C23×Dic86C2211yescharacteristic
11#14583475—24no—
12#14583475C418yescomplemented
13C2×#13063325—21no—
14C2×#13063313—21no—
15C2×#13063309C2214yes—
16C2×#13063305—22no—
17#14583380—22no—
18#14583379—22no—
19#14583376—21no—
20#14583373—21no—
21#14583372C2214yes—
22C4×#10621898—22no—
23#14583367—22no—
24C2×#13063284C2211yescharacteristic
25C2×#13063284C2211yescharacteristic
26#117×D43—22no—
27#14582872—24no—
28C2×#13063239C2214yes—
29#14582805—22no—
30#14582805—22no—
31#14582804—22no—
32#14582804—22no—
33#14582760C2214yes—
34#14582760C2214yes—
35#14582665C2214yes—
36C2×#117×C43C2211yescharacteristic
37C2×#117×C43C2211yescharacteristic
38C2×#117×C43C2211yescharacteristic
39C43×#342C2214yes—
40C2×#10621898—21no—
41C2×#10621898—21no—
42C2×#10621898—21no—
43C2×#10621898—21no—
44C2×#10621898—22no—
45C2×#10621898—22no—
46C2×#10621898C2×C414yes—
47C2×#10621898D414yes—
48C22×Dic86—21no—
49C22×Dic86—21no—
50C22×Dic86—21no—
51C22×Dic86—21no—
52C22×Dic86—21no—
53C22×Dic86—21no—
54C22×Dic86—22no—
55C22×Dic86—22no—
56C22×Dic86—22no—
57C22×Dic86C2×C414yescomplemented
58C22×Dic86D411yescharacteristic
59C22×D172D411yescharacteristic
60C22×C22⋊D43C2312yes—
61C23×Dic43C2312yes—
62C22×C4×D43D412yes—
63#13063325—28no—
64#13063313—28no—
65#13063309—28no—
66#13063309C2×C4116yescomplemented
67#13063305—216no—
68C4×Dic86—24no—
69#13063290—24no—
70C4×D172—24no—
71#13063287—24no—
72C4×C22⋊D43—28no—
73(C22⋊D43)⋊C4—28no—
74#13063284—21no—
75#13063284—21no—
76#13063284—21no—
77#13063284—21no—
78#13063284—22no—
79#13063284—22no—
80#13063283—24no—
81#13063282—24no—
82#13063281—24no—
83C2×C4⋊Dic43—24no—
84C2×C4⋊Dic43—21no—
85C2×C4⋊Dic43D412yes—
86C4⋊C4×D43—24no—
87C2×C22⋊Dic43C2312yes—
88C2×C22⋊Dic43C2312yes—
89C2×(C2×D43)⋊C4—24no—
90C22⋊C4×D43—24no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43D412yes—
93C42×D43—24no—
94#13063239—28no—
95C43⋊#95—28no—
96C43⋊#95—28no—
97C23×C4×C43C2311yescharacteristic
98#117×C43—21no—
99#117×C43—21no—
100#117×C43—21no—
101#117×C43—21no—
102#117×C43—22no—
103#117×C43—22no—
104#117×C43—22no—
105#117×C43C2×C414yescomplemented
106C2×C4⋊C4×C43—21no—
107C2×C4⋊C4×C43—21no—
108C2×C4⋊C4×C43D412yes—
109C2×C22⋊C4×C43C2312yes—
110C2×C22⋊C4×C43C2312yes—
111C2×C22⋊C4×C43C2312yes—
112C2×C42×C43—21no—
113C2×C42×C43—21no—
114C2×C42×C43D412yes—
115C24×C43C2411yescharacteristic
116C23×D43C2×D412yes—
117C22×C4×C43—22no—
118C22×C4×C43—22no—
119C22×C4×C43C22×C412yes—
120C22×C4×C43C2×D414yes—
121C22×C4×C43C2×D411yescharacteristic
122C22×C4×C43C2×D411yescharacteristic
123C22×C4×C43C2×D411yescharacteristic
124C22×C4×C43C2×D411yescharacteristic
125C22×C4×C43C2×D412yes—
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43C2×D411yescharacteristic
128C22×C4×C43#4112yes—
129C22×C4×C43#4112yes—
130C2×C22⋊D43—22no—
131C2×C22⋊D43—22no—
132C2×C22⋊D43—22no—
133C2×C22⋊D43—22no—
134C2×C22⋊D43C22×C414yes—
135C2×C22⋊D43C2×D414yes—
136C22×Dic43—22no—
137C22×Dic43—22no—
138C22×Dic43—22no—
139C22×Dic43—22no—
140C22×Dic43—22no—
141C22×Dic43—22no—
142C22×Dic43—22no—
143C22×Dic43C22×C414yes—
144C22×Dic43C2×D412yes—
145C22×Dic43C2×D414yes—
146#10621898—24no—
147#10621898—24no—
148#10621898—24no—
149#10621898—24no—
150#10621898—24no—
151#10621898—24no—
152#10621898—24no—
153#10621898—24no—
154C2×D172—21no—
155C2×D172—21no—
156C2×D172—22no—
157C2×D172C22⋊C414yes—
158C2×C4×D43—22no—
159C2×C4×D43—22no—
160C2×C4×D43—24no—
161C2×C4×D43—28no—
162C2×C4×D43C22⋊C418yes—
163C2×Dic86—21no—
164C2×Dic86—21no—
165C2×Dic86—22no—
166C2×Dic86—24no—
167C2×Dic86—24no—
168C2×Dic86—24no—
169C2×Dic86—24no—
170C2×Dic86—24no—
171C2×Dic86—24no—
172C2×Dic86—41no—
173C2×Dic86—41no—
174C2×Dic86—41no—
175C2×Dic86—41no—
176C2×Dic86—42no—
177C2×Dic86—42no—
178C2×Dic86—42no—
179C2×Dic86—42no—
180C2×Dic86—42no—
181C2×Dic86—42no—
182C2×Dic86C22⋊C414yes—
183C4⋊C4×C43—24no—
184C4⋊C4×C43—22no—
185C4⋊C4×C43—22no—
186C4⋊C4×C43—42no—
187C22⋊C4×C43—22no—
188C22⋊C4×C43—22no—
189C22⋊C4×C43—22no—
190C22⋊C4×C43—22no—
191C22⋊C4×C43—22no—
192C22⋊C4×C43C22×C414yes—
193C42×C43—24no—
194C42×C43—22no—
195C42×C43—22no—
196C42×C43—42no—
197C22⋊Dic43—22no—
198C22⋊Dic43—22no—
199C22⋊Dic43—22no—
200C22⋊Dic43—22no—
201(C2×D43)⋊C4—28no—
202(C2×D43)⋊C4—216no—
203C4⋊Dic43—22no—
204C4⋊Dic43—22no—
205C4⋊Dic43—216no—
206C4⋊Dic43—42no—
207Dic43⋊C4—28no—
208C4×Dic43—28no—
209C4×Dic43—22no—
210C4×Dic43—22no—
211C4×Dic43—42no—
212C23×C43—42no—
213C23×C43C23×C411yescharacteristic
214C23×C43C22×D411yescharacteristic
215C23×C43C22×D411yescharacteristic
216C23×C43C22×D411yescharacteristic
217C23×C43C22×D411yescharacteristic
218C23×C43C2×#4111yescharacteristic
219C23×C43C2×#4111yescharacteristic
220C22×D43—22no—
221C22×D43—22no—
222C22×D43—44no—
223C22×D43C2×C22⋊C414yes—
224C2×C4×C43—24no—
225C2×C4×C43—24no—
226C2×C4×C43—44no—
227C2×C4×C43—44no—
228C2×C4×C43—44no—
229C2×C4×C43—42no—
230C2×C4×C43C2×C22⋊C412yes—
231C2×C4×C43C2×C22⋊C412yes—
232C2×C4×C43C4×D412yes—
233C2×C4×C43C4×D412yes—
234C2×C4×C43C4×D412yes—
235C2×C4×C43C4×D412yes—
236C2×C4×C43#12011yescharacteristic
237C2×C4×C43#12011yescharacteristic
238C2×C4×C43#12011yescharacteristic
239C2×C4×C43#12011yescharacteristic
240C2×C4×C43#12112yes—
241C2×C4×C43#12112yes—
242C2×C4×C43#12112yes—
243C2×C4×C43#12112yes—
244C2×C4×C43#12312yes—
245C2×C4×C43#12312yes—
246C22⋊D43—24no—
247C22⋊D43—24no—
248C22⋊D43—24no—
249C22⋊D43—24no—
250C2×Dic43—22no—
251C2×Dic43—22no—
252C2×Dic43—24no—
253C2×Dic43—24no—
254C2×Dic43—24no—
255C2×Dic43—24no—
256C2×Dic43—42no—
257C2×Dic43—42no—
258C2×Dic43—42no—
259C2×Dic43—42no—
260C2×Dic43—44no—
261C2×Dic43—44no—
262C2×Dic43—42no—
263C2×Dic43C2×C22⋊C414yes—
264D172—24no—
265D172—24no—
266C4×D43—28no—
267C4×D43—28no—
268C4×D43—88no—
269Dic86—24no—
270Dic86—24no—
271Dic86—44no—
272Dic86—44no—
273Dic86—44no—
274Dic86—44no—
275#20962—431nomaximal, Sylow
276C22×C43—22no—
277C22×C43—22no—
278C22×C43—22no—
279C22×C43—41no—
280C22×C43—41no—
281C22×C43—41no—
282C22×C43—41no—
283C22×C43C22×C22⋊C411yescharacteristic
284C22×C43C2×C4×D411yescharacteristic
285C22×C43C2×C4×D411yescharacteristic
286C22×C43C2×#12011yescharacteristic
287C22×C43C2×#12111yescharacteristic
288C22×C43C2×#12111yescharacteristic
289C22×C43C2×#12311yescharacteristic
290C2×D43—24no—
291C2×D43—44no—
292C2×D43—44no—
293C2×D43—82no—
294C4×C43—44no—
295C4×C43—44no—
296C4×C43—82no—
297C4×C43#38612yes—
298C4×C43#38612yes—
299C4×C43#38612yes—
300C4×C43#38612yes—
301Dic43—24no—
302Dic43—44no—
303Dic43—44no—
304Dic43—82no—
305C2×#536—431no—
306#3428—431no—
307C2×#410—431no—
308#2310—431no—
309#2310—431no—
310#1890—438no—
311#1740—431no—
312#1740D4311yescharacteristic, complemented
313C2×C43—21no—
314C2×C43—21no—
315C2×C43—21no—
316C2×C43—21no—
317C2×C43—21no—
318C2×C43—21no—
319C2×C43C2×#38611yescharacteristic
320C2×C43#302812yes—
321D43—44no—
322C22×#41—431no—
323C23×Q8—431no—
324#536—438no—
325#536—864no—
326C2×C4⋊Q8—861no—
327C2×#124—861no—
328C2×#123—434no—
329C2×C22⋊Q8—862no—
330#520—862no—
331#520—862no—
332#519—861no—
333#518—434no—
334#518—861no—
335C4×#41—862no—
336C4×#41—862no—
337C2×#117—431no—
338C2×#117—431no—
339C2×#117—862no—
340C2×#117C2×D4311yescharacteristic
341C2×#117C2×D4311yescharacteristic
342C2×#117C2×D4311yescharacteristic
343#410—864no—
344C2×#99—434no—
345#391—862no—
346#391—862no—
347#391—862no—
348#391—862no—
349#381—434no—
350#381—434no—
351#342—434no—
352#342C2×D4314yes—
353C43#2096211yescharacteristic, complemented, Sylow
354C2×#41—434no—
355C2×#41—434no—
356C2×#41—861no—
357C2×#41—861no—
358C2×#41—862no—
359C2×#41—861no—
360C2×#41—861no—
361C2×#41—862no—
362C22×Q8—434no—
363C22×Q8—431no—
364C22×Q8—861no—
365C22×Q8—861no—
366C22×Q8—862no—
367C22×Q8—861no—
368C22×Q8—861no—
369C22×Q8—862no—
370C22×Q8—861no—
371C22×Q8—861no—
372C22×Q8—862no—
373C22×D4—432no—
374C22×D4—431no—
375C23×C4—432no—
376C23×C4—432no—
377C23×C4C22×D4311yescharacteristic
378C4⋊Q8—868no—
379#124—868no—
380#123—4316no—
381#123—868no—
382C22⋊Q8—8616no—
383C4×Q8—864no—
384C4×Q8—864no—
385C4×D4—864no—
386C4×D4—868no—
387C4×D4—864no—
388C4×D4—868no—
389#117—21no—
390#117—21no—
391#117—21no—
392#117—21no—
393#117—22no—
394#117—22no—
395#117—22no—
396#117—864no—
397#117—864no—
398#117—864no—
399#117—861no—
400#117—861no—
401#117—861no—
402#117—861no—
403#117—862no—
404#117—862no—
405#117C4×D4314yescomplemented
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—432no—
409C2×C4⋊C4—864no—
410C2×C4⋊C4—864no—
411C2×C4⋊C4—861no—
412C2×C4⋊C4D17212yes—
413C2×C22⋊C4—432no—
414C2×C22⋊C4—432no—
415C2×C22⋊C4—864no—
416C2×C22⋊C4—864no—
417C2×C22⋊C4C22×D4312yes—
418C2×C22⋊C4C22×D4312yes—
419C2×C22⋊C4C22×D4312yes—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—432no—
423C2×C42—864no—
424C2×C42—861no—
425C2×C42D17212yes—
426#99—868no—
427#95—868no—
428#95—868no—
429C24—432no—
430C24C23×D4311yescharacteristic
431#41—864no—
432#41—864no—
433#41—864no—
434#41—864no—
435#41—864no—
436#41—864no—
437#41—864no—
438#41—864no—
439C2×Q8—434no—
440C2×Q8—864no—
441C2×Q8—864no—
442C2×Q8—864no—
443C2×Q8—864no—
444C2×Q8—864no—
445C2×Q8—864no—
446C2×Q8—861no—
447C2×Q8—861no—
448C2×Q8—862no—
449C2×Q8—1721no—
450C2×Q8—1721no—
451C2×Q8—1721no—
452C2×Q8—1721no—
453C2×Q8—1722no—
454C2×Q8—1722no—
455C2×Q8—1722no—
456C2×Q8—1722no—
457C2×Q8—1722no—
458C2×Q8—1722no—
459C2×D4—434no—
460C2×D4—434no—
461C2×D4—434no—
462C2×D4—862no—
463C2×D4—862no—
464C2×D4—862no—
465C2×D4—862no—
466C2×D4—861no—
467C2×D4—861no—
468C2×D4—862no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—432no—
472C22×C4—438no—
473C22×C4—434no—
474C22×C4—434no—
475C22×C4—868no—
476C22×C4—862no—
477C22×C4—862no—
478C22×C4—864no—
479C22×C4—862no—
480C22×C4—862no—
481C22×C4—862no—
482C22×C4—862no—
483C22×C4—862no—
484C22×C4—862no—
485C22×C4—862no—
486C22×C4C2×C4×D4312yes—
487C22×C4C2×D17214yes—
488C22×C4C2×D17211yescharacteristic
489C22×C4C2×D17211yescharacteristic
490C22×C4#1062189812yes—
491C22×C4D4×D4311yescharacteristic
492C22×C4D4×D4311yescharacteristic
493C22×C4D4×D4312yes—
494C22×C4D4×D4311yescharacteristic
495C22×C4D4×D4311yescharacteristic
496C22×C4#1062190212yes—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—8616no—
502C4⋊C4—868no—
503C4⋊C4—862no—
504C4⋊C4—862no—
505C4⋊C4—1722no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—8616no—
512C22⋊C4—868no—
513C22⋊C4—862no—
514C22⋊C4—862no—
515C22⋊C4—862no—
516C22⋊C4—862no—
517C22⋊C4C2×C4×D4314yes—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—42no—
522C42—868no—
523C42—862no—
524C42—862no—
525C42—1722no—
526C23—42no—
527C23—434no—
528C23—862no—
529C23—862no—
530C23—1724no—
531C23C22×C4×D4311yescharacteristic
532C23C2×D4×D4311yescharacteristic
533C23C2×D4×D4311yescharacteristic
534C23C2×D4×D4311yescharacteristic
535C23C22×D17211yescharacteristic
536C23C2×#1062189811yescharacteristic
537C23C2×#1062190211yescharacteristic
538Q8—864no—
539Q8—864no—
540Q8—1724no—
541Q8—1724no—
542Q8—1724no—
543Q8—1724no—
544D4—864no—
545D4—864no—
546D4—864no—
547D4—864no—
548D4—864no—
549D4—864no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—434no—
557C2×C4—862no—
558C2×C4—862no—
559C2×C4—868no—
560C2×C4—868no—
561C2×C4—864no—
562C2×C4—864no—
563C2×C4—864no—
564C2×C4—864no—
565C2×C4—3448no—
566C2×C4—1724no—
567C2×C4—1722no—
568C2×C4—1722no—
569C2×C4—1722no—
570C2×C4—1722no—
571C2×C4—1722no—
572C2×C4—1724no—
573C2×C4C22⋊C4×D4312yes—
574C2×C4C22⋊C4×D4312yes—
575C2×C4#1306328712yes—
576C2×C4#1306328712yes—
577C2×C4C4×D17212yes—
578C2×C4C4×D17212yes—
579C2×C4#1306329511yescharacteristic
580C2×C4#1306329511yescharacteristic
581C2×C4#1306329511yescharacteristic
582C2×C4#1306329511yescharacteristic
583C2×C4#1306329812yes—
584C2×C4#1306329812yes—
585C2×C4#1306330012yes—
586C2×C4#1306330012yes—
587C2×C4#1306331012yes—
588C2×C4#1306331012yes—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—864no—
597C22—3442no—
598C22—1724no—
599C22—1724no—
600C22C2×C22⋊C4×D4311yescharacteristic
601C22C2×#1306328711yescharacteristic
602C22C2×C4×D17211yescharacteristic
603C22C2×#1306329511yescharacteristic
604C22C2×#1306329811yescharacteristic
605C22C2×#1306330011yescharacteristic
606C22C2×#1306331011yescharacteristic
607C4—44no—
608C4—44no—
609C4—82no—
610C4—864no—
611C4—3442no—
612C4—1724no—
613C4—1724no—
614C4#1458277712yes—
615C4#1458277712yes—
616C4#1458277712yes—
617C4#1458277712yes—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—1724no—
625C2C2×#1458277711yescharacteristic
626C2#1914400212yes—
subgroupG/H
centerC22C2×#13063295
commutatorC23×C43C23×C4
FrattiniC24C23×D43
FittingC43×#1740C2
socleC22×C43C2×#120
prime radicalC22×#10621898C22
cosocleC24×C43C24
hypercenter#1740D43
hypocenterC43#20962
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. #19137179C2semidirect
5. #19137179C2semidirect
6. #19135007C2semidirect
7. #19134327C2semidirect
8. C43×#1740C2semidirect
12. #14583475C4semidirect
57. C22×Dic86C2×C4semidirect
66. #13063309C2×C4semidirect
105. #117×C43C2×C4semidirect
312. #1740D43semidirect
353. C43#20962semidirect
405. #117C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145834752
#191477962
C2×#145828722
#191371792
#191350072
#191343272
C43×#17402
#2096243