Finite Group Databasebeta

← #25067515

Subgroups — #25067515

Order 11008.

670 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144624C211yesmaximal, characteristic
2Q8×#10621872C211yesmaximal, characteristic, complemented
3#19143334C211yesmaximal, characteristic
4#19133488C211yesmaximal, characteristic, complemented
5#19133487C211yesmaximal, characteristic, complemented
6#19133439C211yesmaximal, characteristic, complemented
7#19133438C211yesmaximal, characteristic, complemented
8#19133431C211yesmaximal, characteristic, complemented
9#19133430C211yesmaximal, characteristic, complemented
10#19133392C211yesmaximal, characteristic, complemented
11#19133364C211yesmaximal, characteristic, complemented
12#19133044C211yesmaximal, characteristic
13Q8⋊C8×D43C211yesmaximal, characteristic, complemented
14C8×#10621872C211yesmaximal, characteristic, complemented
15C8×SD16×C43C211yesmaximal, characteristic, complemented
16C2×#13063331—21no—
17#14583446C2211yescharacteristic
18C4×Q8×D43C2211yescharacteristic
19#14583219C411yescharacteristic
20#14583216C411yescharacteristic
21#14583183C411yescharacteristic
22#14583182C411yescharacteristic
23#14583098C411yescharacteristic
24#14583097C411yescharacteristic
25#14583039C411yescharacteristic
26#14583036C411yescharacteristic
27#14583029—21no—
28#14583029C2211yescharacteristic
29#14583026—21no—
30#14583026C2211yescharacteristic
31Q8×C43⋊C8C2211yescharacteristic
32C4×#10621873C2211yescharacteristic
33C4×#10621884C2211yescharacteristic
34C4×#10621883C2211yescharacteristic
35#14582978C2211yescharacteristic
36#14582977C2211yescharacteristic
37#14582973—21no—
38#14582970—21no—
39#14582964C2211yescharacteristic
40#14582950C2211yescharacteristic
41#14582921—21no—
42#14582921C2211yescharacteristic
43C4⋊C8×D43C2211yescharacteristic
44#14582864—21no—
45C4×#10621872—21no—
46C4×#10621872C2211yescharacteristic
47C4×C8×D43C2211yescharacteristic
48#14582650C2211yescharacteristic
49#14582649C2211yescharacteristic
50#14582649C2211yescharacteristic
51#14582631C2211yescharacteristic
52Q8⋊(C43⋊C8)C2211yescharacteristic
53#14582629C2211yescharacteristic
54#14582628C2211yescharacteristic
55D4⋊(C43⋊C8)C2211yescharacteristic
56#14582626C2211yescharacteristic
57#14582621C2211yescharacteristic
58#14582620C2211yescharacteristic
59C8×C43⋊C8C2211yescharacteristic
60C8×Q8×C43C2211yescharacteristic
61C4×SD16×C43C2211yescharacteristic
62C8×D4×C43C2211yescharacteristic
63C43×#334C2211yescharacteristic
64C43×Q8⋊C8C2211yescharacteristic
65C43×D4⋊C8C2211yescharacteristic
66C82×C43C2211yescharacteristic
67C2×#10621898—21no—
68C2×Q8×D43C2×C411yescharacteristic
69C2×#10621873C2×C411yescharacteristic
70C2×#10621883C2×C411yescharacteristic
71C2×#10621884C2×C411yescharacteristic
72#13063331—44no—
73C2×#10621877—21no—
74C2×#10621872—21no—
75C2×#10621872—21no—
76C2×#10621872D411yescharacteristic
77C2×#10621872D411yescharacteristic
78C2×C8×D43—21no—
79C2×C8×D43—21no—
80C2×C8×D43—21no—
81#13063324C2×C411yescharacteristic
82#13063321C2×C411yescharacteristic
83#13063311—21no—
84#13063300C2×C411yescharacteristic
85#13063298—21no—
86#13063296C2×C411yescharacteristic
87Q8×Dic43C2×C411yescharacteristic
88C4×Dic86—21no—
89C4×Dic86C2311yescharacteristic
90#13063290—21no—
91#13063290C2×C411yescharacteristic
92C4×D172C2311yescharacteristic
93C4×C22⋊D43—21no—
94C4⋊C4×D43—21no—
95C4⋊C4×D43C2×C411yescharacteristic
96C42×D43—21no—
97C42×D43C2311yescharacteristic
98#13063257C2×C411yescharacteristic
99#13063256C2×C411yescharacteristic
100#13063256C2×C411yescharacteristic
101C4⋊(C43⋊C8)—21no—
102C4⋊(C43⋊C8)C2311yescharacteristic
103C4⋊(C43⋊C8)C2311yescharacteristic
104Dic43⋊C8—21no—
105Dic43⋊C8—21no—
106Dic43⋊C8—21no—
107Dic43⋊C8—21no—
108Dic43⋊C8C2×C411yescharacteristic
109Dic43⋊C8C2×C411yescharacteristic
110#13063250C2×C411yescharacteristic
111#13063249C2×C411yescharacteristic
112#13063248C2×C411yescharacteristic
113#13063247C2×C411yescharacteristic
114#13063246C2×C411yescharacteristic
115#13063245C2×C411yescharacteristic
116C22⋊(C43⋊C8)—21no—
117(C2×D43)⋊C8—21no—
118(C2×D43)⋊C8—21no—
119(C2×D43)⋊C8C2×C411yescharacteristic
120(C2×D43)⋊C8C2×C411yescharacteristic
121#13063235—21no—
122#13063235—21no—
123#13063235D411yescharacteristic
124#13063235D411yescharacteristic
125C4×C43⋊C8—21no—
126C4×C43⋊C8C2311yescharacteristic
127C4×C43⋊C8C2311yescharacteristic
128C4×C43⋊C8C2311yescharacteristic
129C8×Dic43—21no—
130C2×SD16×C43C2×C411yescharacteristic
131C22×C8×C43—21no—
132C4×Q8×C43C2311yescharacteristic
133C4×D4×C43C2311yescharacteristic
134#106×C43C2×C411yescharacteristic
135C4⋊C8×C43—21no—
136C4⋊C8×C43C2311yescharacteristic
137C4⋊C8×C43C2311yescharacteristic
138#103×C43C2×C411yescharacteristic
139#102×C43C2×C411yescharacteristic
140C22⋊C8×C43—21no—
141C4×C8×C43—21no—
142C4×C8×C43C2311yescharacteristic
143C4×C8×C43C2311yescharacteristic
144C4×C8×C43C2311yescharacteristic
145C2×Q8×C43C22×C411yescharacteristic
146C2×D4×C43C22×C411yescharacteristic
147C22×C4×C43—21no—
148C2×C22⋊D43—21no—
149C2×C22⋊D43—21no—
150C2×C22⋊D43—21no—
151Q8×D43—22no—
152Q8×D43—41no—
153#10621898—42no—
154#10621898—42no—
155C2×D172—21no—
156C2×D172C22×C411yescharacteristic
157C2×C4×D43—21no—
158C2×C4×D43—21no—
159C2×C4×D43—21no—
160C2×C4×D43C22×C411yescharacteristic
161C2×C4×D43C2×D411yescharacteristic
162C2×C4×D43#4111yescharacteristic
163C2×Dic86—21no—
164C2×Dic86—21no—
165C2×Dic86C22×C411yescharacteristic
166SD16×C43—22no—
167C2×C8×C43—21no—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—42no—
171C2×C8×C43C22×C411yescharacteristic
172C2×C8×C43C22×C411yescharacteristic
173C2×C8×C43C2×D411yescharacteristic
174C2×C8×C43C2×D411yescharacteristic
175C4⋊C4×C43—21no—
176C4⋊C4×C43C22×C411yescharacteristic
177C4⋊C4×C43C22×C411yescharacteristic
178C22⋊C4×C43—21no—
179C42×C43—21no—
180C42×C43C2411yescharacteristic
181C22⋊Dic43—21no—
182#10621884—22no—
183#10621883—41no—
184(C2×D43)⋊C4—21no—
185(C2×D43)⋊C4—21no—
186C4⋊Dic43—21no—
187C4⋊Dic43C22×C411yescharacteristic
188C4⋊Dic43C22×C411yescharacteristic
189Dic43⋊C4—21no—
190Dic43⋊C4—21no—
191Dic43⋊C4—21no—
192Dic43⋊C4—21no—
193C4×Dic43—21no—
194C4×Dic43—21no—
195C4×Dic43—21no—
196C4×Dic43C22×C411yescharacteristic
197C4×Dic43C2×D411yescharacteristic
198C4×Dic43#4111yescharacteristic
199#10621877—42no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202C2×C43⋊C8—21no—
203C2×C43⋊C8C22×C411yescharacteristic
204C2×C43⋊C8C22×C411yescharacteristic
205C2×C43⋊C8C2×D411yescharacteristic
206C2×C43⋊C8C2×D411yescharacteristic
207#10621873—41no—
208#10621872—22no—
209#10621872—42no—
210#10621872—42no—
211#10621872SD1614yescomplemented
212C8×D43—42no—
213C8×D43—42no—
214C8×D43—42no—
215C23×C43—21no—
216C22×D43—21no—
217C22×D43C4×D411yescharacteristic
218Q8×C43—21no—
219Q8×C43#13112yescomplemented
220D4×C43—21no—
221D4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—21no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43C23×C411yescharacteristic
228C2×C4×C43C22×D411yescharacteristic
229C2×C4×C43C2×#4111yescharacteristic
230C22⋊D43—42no—
231C22⋊D43—42no—
232C22⋊D43—81no—
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43C4×D411yescharacteristic
237C2×Dic43C4×D411yescharacteristic
238C2×Dic43C4×D411yescharacteristic
239D172—21no—
240D172—21no—
241D172—42no—
242C4×D43—21no—
243C4×D43—21no—
244C4×D43—41no—
245C4×D43—42no—
246C4×D43—41no—
247C4×D43—41no—
248C4×D43—81no—
249C4×D43C2×SD1612yes—
250C4×D43#13512yes—
251Dic86—21no—
252Dic86—21no—
253Dic86—42no—
254Dic86—42no—
255C8×C43—22no—
256C8×C43—21no—
257C8×C43—21no—
258C8×C43—21no—
259C8×C43—41no—
260C8×C43—41no—
261C8×C43C2×SD1612yes—
262C43⋊C8—21no—
263C43⋊C8—21no—
264C43⋊C8—21no—
265C43⋊C8—41no—
266C43⋊C8—41no—
267C43⋊C8—41no—
268C43⋊C8C2×SD1612yes—
269#17997—431nomaximal, Sylow
270C22×C43—41no—
271C22×C43—41no—
272C22×C43—41no—
273C22×C43C2×C4×D411yescharacteristic
274C2×D43—21no—
275C2×D43—21no—
276C2×D43—41no—
277C2×D43—41no—
278C2×D43—41no—
279C2×D43C4×SD1612yes—
280C4×C43—22no—
281C4×C43—21no—
282C4×C43—41no—
283C4×C43—41no—
284C4×C43C2×#13111yescharacteristic
285C4×C43#56811yescharacteristic
286C4×C43C22×SD1611yescharacteristic
287C4×C43C2×#13511yescharacteristic
288Dic43—21no—
289Dic43—21no—
290Dic43—41no—
291Dic43—41no—
292Dic43—41no—
293Dic43C4×SD1612yes—
294#3103—431no—
295Q8×M16—431no—
296#2935—431no—
297#1590—431no—
298#1590—431no—
299#1583—431no—
300#1583—431no—
301#1582—431no—
302#1582—431no—
303C8×SD16—431no—
304C8×SD16D4311yescharacteristic, complemented
305#1570—431no—
306#1492—431no—
307C2×Q8⋊C8—431no—
308C8×M16—431no—
309C2×C43—81no—
310C2×C43#293311yescharacteristic
311C2×C43C2×C4×SD1611yescharacteristic
312C2×C43#295611yescharacteristic
313D43—22no—
314D43—81no—
315C2×#131—861no—
316#532—431no—
317C2×C4×Q8—431no—
318#481—431no—
319#481—431no—
320#475—431no—
321#475—431no—
322#460—431no—
323#460—431no—
324#448—431no—
325#448—431no—
326#446—431no—
327#446—431no—
328#446—861no—
329#446—861no—
330C8×Q8—431no—
331C8×Q8C2×D4311yescharacteristic
332C4×SD16—431no—
333C4×SD16—431no—
334C4×SD16—431no—
335C4×SD16C2×D4311yescharacteristic
336#436—431no—
337#436—431no—
338#435—861no—
339#435—861no—
340C8×D4—431no—
341C8×D4C2×D4311yescharacteristic
342#431—431no—
343#423—431no—
344#423—861no—
345C2×C4⋊C8—431no—
346#408—861no—
347C4×M16—431no—
348C4×M16—861no—
349C2×C4×C8—431no—
350#334—431no—
351#334—431no—
352#334—431no—
353#334C2×D4311yescharacteristic
354Q8⋊C8—431no—
355Q8⋊C8—431no—
356Q8⋊C8—431no—
357Q8⋊C8C2×D4311yescharacteristic
358D4⋊C8—431no—
359D4⋊C8—431no—
360D4⋊C8—431no—
361D4⋊C8C2×D4311yescharacteristic
362#322—431no—
363#322—431no—
364C82—431no—
365C82C2×D4311yescharacteristic
366C43#1799711yescharacteristic, complemented, Sylow
367C2×#41—861no—
368C22×Q8—431no—
369C2×SD16—431no—
370C2×SD16—431no—
371C2×SD16—431no—
372C2×SD16C4×D4311yescharacteristic
373#131—1724no—
374C2×M16—431no—
375C2×M16—431no—
376C2×M16—861no—
377C2×M16—861no—
378C2×M16—861no—
379C22×C8—21no—
380C22×C8—861no—
381C22×C8—861no—
382C22×C8—861no—
383C4⋊Q8—431no—
384#127—431no—
385#124—861no—
386#121—431no—
387#121—431no—
388#121—861no—
389C4×Q8—431no—
390C4×Q8—431no—
391C4×Q8—431no—
392C4×Q8—861no—
393C4×Q8—861no—
394C4×Q8C22×D4311yescharacteristic
395C4×D4—431no—
396C4×D4—861no—
397C4×D4C22×D4311yescharacteristic
398C2×C4⋊C4—431no—
399C2×C4⋊C4—861no—
400C2×C42—431no—
401C2×C42—861no—
402#106—431no—
403#106—431no—
404#106—431no—
405#106C4×D4311yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—431no—
408C4⋊C8—431no—
409C4⋊C8—431no—
410C4⋊C8—431no—
411C4⋊C8—861no—
412C4⋊C8—861no—
413C4⋊C8—861no—
414C4⋊C8—861no—
415C4⋊C8—861no—
416C4⋊C8C22×D4311yescharacteristic
417C4⋊C8C22×D4311yescharacteristic
418#103—431no—
419#103—431no—
420#103—431no—
421#103C4×D4311yescharacteristic
422#102—431no—
423#102—431no—
424#102—431no—
425#102C4×D4311yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—431no—
428C22⋊C8—431no—
429C22⋊C8—861no—
430C22⋊C8—861no—
431C22⋊C8—861no—
432#97—431no—
433#97—431no—
434#97—861no—
435#97—861no—
436C4×C8—21no—
437C4×C8—431no—
438C4×C8—431no—
439C4×C8—431no—
440C4×C8—861no—
441C4×C8—861no—
442C4×C8C22×D4311yescharacteristic
443C4×C8C22×D4311yescharacteristic
444C4×C8C22×D4311yescharacteristic
445#41—1722no—
446#41—1722no—
447C2×Q8—431no—
448C2×Q8—861no—
449C2×Q8—861no—
450C2×Q8—862no—
451C2×Q8—1721no—
452C2×Q8C2×C4×D4311yescharacteristic
453C2×D4—431no—
454C2×D4—861no—
455C2×D4—861no—
456C2×D4—861no—
457C2×D4—861no—
458C2×D4C2×C4×D4311yescharacteristic
459C22×C4—21no—
460C22×C4—431no—
461C22×C4—431no—
462C22×C4—431no—
463C22×C4—861no—
464C22×C4—861no—
465C22×C4—861no—
466SD16—22no—
467SD16—862no—
468SD16—1721no—
469SD16—1721no—
470M16—434no—
471M16—862no—
472M16—1722no—
473M16—1722no—
474M16—1722no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—42no—
479C2×C8—431no—
480C2×C8—431no—
481C2×C8—431no—
482C2×C8—431no—
483C2×C8—861no—
484C2×C8—861no—
485C2×C8—861no—
486C2×C8—1722no—
487C2×C8—1722no—
488C2×C8—1722no—
489C2×C8C2×C4×D4311yescharacteristic
490C2×C8C2×C4×D4311yescharacteristic
491C2×C8D4×D4311yescharacteristic
492C2×C8D4×D4311yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—431no—
495C4⋊C4—431no—
496C4⋊C4—861no—
497C4⋊C4—861no—
498C4⋊C4—861no—
499C4⋊C4—861no—
500C4⋊C4—861no—
501C4⋊C4C2×C4×D4311yescharacteristic
502C4⋊C4C2×C4×D4311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—861no—
505C22⋊C4—861no—
506C22⋊C4—861no—
507C42—21no—
508C42—431no—
509C42—431no—
510C42—431no—
511C42—861no—
512C42—861no—
513C42—861no—
514C42C23×D4311yescharacteristic
515C23—21no—
516C23—431no—
517C23—861no—
518Q8—21no—
519Q8—861no—
520Q8—861no—
521Q8—1722no—
522Q8—1722no—
523Q8#1306333112yescomplemented
524D4—21no—
525D4—21no—
526D4—861no—
527D4—861no—
528D4—3441no—
529D4—1722no—
530D4—1722no—
531D4—1722no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—431no—
538C2×C4—432no—
539C2×C4—432no—
540C2×C4—431no—
541C2×C4—431no—
542C2×C4—861no—
543C2×C4—861no—
544C2×C4—861no—
545C2×C4—861no—
546C2×C4—861no—
547C2×C4—3441no—
548C2×C4—1721no—
549C2×C4—1722no—
550C2×C4—1721no—
551C2×C4—1721no—
552C2×C4C22×C4×D4311yescharacteristic
553C2×C4C2×D4×D4311yescharacteristic
554C2×C4#41×D4311yescharacteristic
555C8—22no—
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—432no—
562C8—861no—
563C8—861no—
564C8—861no—
565C8—1721no—
566C8—1721no—
567C8—1721no—
568C8SD16×D4312yes—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—432no—
573C22—861no—
574C22—861no—
575C22—1721no—
576C22—1721no—
577C22—1721no—
578C22C4×D4×D4311yescharacteristic
579C4—21no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—432no—
584C4—861no—
585C4—861no—
586C4—1721no—
587C4—1721no—
588C4—1721no—
589C4C2×#1306333111yescharacteristic
590C4#1458366411yescharacteristic
591C4C2×SD16×D4311yescharacteristic
592C4#135×D4311yescharacteristic
593C2—81no—
594C2—862no—
595C2—3441no—
596C2#1914331511yescharacteristic
597C2C4×SD16×D4311yescharacteristic
598C2#1914350011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC42C23×D43
FittingC8×SD16×C43C2
socleC22×C43C2×C4×D4
prime radical#14583446C22
cosocleC42×C43C24
hypercenterC8×SD16D43
hypocenterC43#17997
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. Q8×#10621872C2semidirect
4. #19133488C2semidirect
5. #19133487C2semidirect
6. #19133439C2semidirect
7. #19133438C2semidirect
8. #19133431C2semidirect
9. #19133430C2semidirect
10. #19133392C2semidirect
11. #19133364C2semidirect
13. Q8⋊C8×D43C2semidirect
14. C8×#10621872C2semidirect
15. C8×SD16×C43C2semidirect
211. #10621872SD16semidirect
219. Q8×C43#131semidirect
304. C8×SD16D43semidirect
366. C43#17997semidirect
523. Q8#13063331semidirect

Maximal subgroups

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

subgroupindex
#191446242
Q8×#106218722
#191433342
#191334882
#191334872
#191334392
#191334382
#191334312
#191334302
#191333922
#191333642
#191330442
Q8⋊C8×D432
C8×#106218722
C8×SD16×C432
#1799743