Finite Group Databasebeta

← #25130916

Subgroups — #25130916

Order 11008.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583456C211yesmaximal, characteristic, complemented
2C2×#14583190C211yesmaximal, characteristic, complemented
3C2×#14583105C211yesmaximal, characteristic, complemented
4C4×#13063299C211yesmaximal, characteristic
5#19135792C212yesmaximal, complemented
6#19135600C212yesmaximal
7#19135542C211yesmaximal, characteristic, complemented
8#19135517C211yesmaximal, characteristic, complemented
9#19134949C212yesmaximal, complemented
10#19134866C212yesmaximal, complemented
11C43×#1797C211yesmaximal, characteristic, complemented
12C22×#10621873C2212yes—
13#14583456—24no—
14#14583456—24no—
15C2×#13063325C2211yescharacteristic
16C2×#13063325C2211yescharacteristic
17C2×#13063317—21no—
18C2×#13063304—21no—
19C2×#13063304C2212yes—
20C2×#13063299C2211yescharacteristic
21C2×C4×Dic86—21no—
22C2×C4×Dic86C2212yes—
23C2×C4×D172C2211yescharacteristic
24C2×C4×C22⋊D43—21no—
25C2×#13063284—21no—
26#14583190—24no—
27#14583105—24no—
28C2×#13063257C2212yes—
29C2×#13063250C2212yes—
30C2×#13063250C2212yes—
31C2×#13063250C2212yes—
32C2×#13063247C2212yes—
33#14582801C2211yescharacteristic
34#14582797C2211yescharacteristic
35#14582790—21no—
36#14582778—21no—
37#14582759C2211yescharacteristic
38C4×C4⋊Dic43C2211yescharacteristic
39C4×(C2×D43)⋊C4—21no—
40#14582662C2212yes—
41#14582662C2212yes—
42C22×C42×C43C2211yescharacteristic
43C2×C4⋊C8×C43C2211yescharacteristic
44C2×C4⋊C8×C43C2211yescharacteristic
45C2×C22⋊C8×C43C2212yes—
46C43×#336C2212yes—
47C22×Dic86—21no—
48C22×Dic86C2312yes—
49C22×D172C2311yescharacteristic
50C22×C22⋊D43—21no—
51C22×C4×D43—21no—
52C2×#10621873—24no—
53C2×#10621873—24no—
54C2×#10621873—42no—
55#13063325—22no—
56#13063325—22no—
57#13063325D414yescomplemented
58#13063325D414yescomplemented
59#13063317—28no—
60#13063304—28no—
61#13063304—24no—
62#13063304—24no—
63#13063299—22no—
64#13063299—22no—
65C4×Dic86—28no—
66C4×Dic86—24no—
67C4×Dic86—24no—
68C4×D172—22no—
69C4×D172—22no—
70C4×C22⋊D43—44no—
71#13063284—28no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2312yes—
78C2×C4⋊Dic43C2311yescharacteristic
79C2×Dic43⋊C4—22no—
80C2×Dic43⋊C4—22no—
81C2×Dic43⋊C4—21no—
82C2×C22⋊Dic43—22no—
83C2×C22⋊Dic43—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×C4×Dic43—22no—
89C2×C4×Dic43—21no—
90#13063257—24no—
91#13063250—24no—
92#13063250—24no—
93#13063250—42no—
94#13063247—24no—
95C43⋊#95—21no—
96C43⋊#95—21no—
97C43⋊#95—21no—
98C23×C4×C43D411yescharacteristic
99C23×C4×C43D411yescharacteristic
100C23×C4×C43C2311yescharacteristic
101C22×C8×C43C2312yes—
102C22×C8×C43C2312yes—
103C2×C42×C43—22no—
104C2×C42×C43—22no—
105C2×C42×C43D411yescharacteristic
106C2×C42×C43D411yescharacteristic
107C2×C42×C43C2311yescharacteristic
108C2×C42×C43C2311yescharacteristic
109C4⋊C8×C43—22no—
110C4⋊C8×C43—22no—
111C22⋊C8×C43—42no—
112C24×C43C2×D411yescharacteristic
113C23×D43—21no—
114C22×C4×C43—22no—
115C22×C4×C43—22no—
116C22×C4×C43—22no—
117C22×C4×C43—21no—
118C22×C4×C43—21no—
119C22×C4×C43—21no—
120C22×C4×C43—21no—
121C22×C4×C43—21no—
122C22×C4×C43—21no—
123C22×C4×C43—41no—
124C22×C4×C43—41no—
125C22×C4×C43SD1614yescomplemented
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43C2×D411yescharacteristic
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2411yescharacteristic
132C2×C22⋊D43—24no—
133C2×C22⋊D43—42no—
134C2×C22⋊D43—42no—
135C22×Dic43—21no—
136C22×Dic43—22no—
137C22×Dic43—22no—
138C22×Dic43—21no—
139C22×Dic43—21no—
140C2×D172—21no—
141C2×D172—21no—
142C2×D172—21no—
143C2×D172—21no—
144C2×D172—21no—
145C2×D172#4112yes—
146C2×C4×D43—42no—
147C2×C4×D43—42no—
148C2×C4×D43—42no—
149C2×Dic86—22no—
150C2×Dic86—22no—
151C2×Dic86—22no—
152C2×Dic86—22no—
153C2×Dic86—24no—
154C2×Dic86—22no—
155C2×Dic86—42no—
156C2×Dic86—42no—
157C2×Dic86C2×D414yes—
158C2×C8×C43—22no—
159C2×C8×C43—22no—
160C2×C8×C43—22no—
161C2×C8×C43—22no—
162C2×C8×C43—22no—
163C2×C8×C43—22no—
164C42×C43—22no—
165C42×C43—22no—
166C42×C43—21no—
167C42×C43—21no—
168C42×C43C2×D412yes—
169C42×C43C2×D412yes—
170C22⋊Dic43—28no—
171C22⋊Dic43—42no—
172(C2×D43)⋊C4—24no—
173(C2×D43)⋊C4—42no—
174(C2×D43)⋊C4—42no—
175(C2×D43)⋊C4—42no—
176C4⋊Dic43—24no—
177C4⋊Dic43—24no—
178C4⋊Dic43—24no—
179C4⋊Dic43—22no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43—21no—
184C4⋊Dic43—21no—
185C4⋊Dic43C2×D414yes—
186C4⋊Dic43#4112yes—
187Dic43⋊C4—28no—
188Dic43⋊C4—28no—
189Dic43⋊C4—42no—
190C4×Dic43—28no—
191C4×Dic43—42no—
192#10621873—48no—
193C23×C43—21no—
194C23×C43—21no—
195C23×C43—21no—
196C23×C43—21no—
197C23×C43—21no—
198C23×C43—21no—
199C23×C43C2×SD1612yes—
200C23×C43C22×D411yescharacteristic
201C22×D43—22no—
202C22×D43—41no—
203C22×D43—41no—
204C22×D43—41no—
205C22×D43—41no—
206C22×D43—41no—
207C22×D43—41no—
208C2×C4×C43—21no—
209C2×C4×C43—21no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—28no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—44no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43#12012yes—
225C2×C4×C43#12012yes—
226C2×C4×C43C2×SD1612yes—
227C2×C4×C43C2×SD1612yes—
228C2×C4×C43#13712yes—
229C2×C4×C43C22×D411yescharacteristic
230C2×C4×C43C22×D411yescharacteristic
231C2×C4×C43C2×#4111yescharacteristic
232C2×C4×C43C2×#4111yescharacteristic
233C2×C4×C43H(5,𝔽2)11yescharacteristic
234C2×C4×C432−1+411yescharacteristic
235C22⋊D43—48no—
236C2×Dic43—24no—
237C2×Dic43—24no—
238C2×Dic43—22no—
239C2×Dic43—42no—
240C2×Dic43—42no—
241C2×Dic43—41no—
242C2×Dic43—42no—
243C2×Dic43—42no—
244C2×Dic43—41no—
245C2×Dic43—41no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251D172—22no—
252D172—22no—
253D172—41no—
254D172—41no—
255C4×D43—84no—
256Dic86—24no—
257Dic86—24no—
258Dic86—48no—
259Dic86—42no—
260Dic86—42no—
261C8×C43—42no—
262C8×C43—42no—
263#24151—431nomaximal, Sylow
264C22×C43—22no—
265C22×C43—22no—
266C22×C43—22no—
267C22×C43—41no—
268C22×C43—41no—
269C22×C43—41no—
270C22×C43—41no—
271C22×C43C2×#12011yescharacteristic
272C22×C43#52511yescharacteristic
273C22×C43#53711yescharacteristic
274C22×C43C22×SD1611yescharacteristic
275C22×C43C2×#13711yescharacteristic
276C22×C43#57511yescharacteristic
277C22×C43#57511yescharacteristic
278C2×D43—42no—
279C2×D43—42no—
280C2×D43—42no—
281C2×D43—81no—
282C2×D43—81no—
283C2×D43—81no—
284C2×D43—81no—
285C4×C43—22no—
286C4×C43—22no—
287C4×C43—41no—
288C4×C43—41no—
289C4×C43#44814yes—
290C4×C43#54611yescharacteristic
291C4×C43#54711yescharacteristic
292C4×C43#54811yescharacteristic
293C4×C43#55011yescharacteristic
294Dic43—42no—
295Dic43—44no—
296Dic43—44no—
297Dic43—81no—
298Dic43—81no—
299C2×#533—431no—
300C2×#476—431no—
301C2×#461—431no—
302C4×#121—431no—
303#2035—432no—
304#2000—432no—
305#1987—431no—
306#1981—431no—
307#1880—432no—
308#1865—432no—
309#1797D4311yescharacteristic, complemented
310C2×C43—42no—
311C2×C43#240711yescharacteristic
312C2×C43C2×#44811yescharacteristic
313C2×C43#300511yescharacteristic
314C2×C43#310111yescharacteristic
315C2×C43#311611yescharacteristic
316C2×C43#324611yescharacteristic
317C2×C43#325211yescharacteristic
318D43—82no—
319C22×SD16—432no—
320#533—864no—
321#533—864no—
322C2×C4⋊Q8—431no—
323C2×C4⋊Q8—431no—
324C2×#125—861no—
325C2×C22⋊Q8—432no—
326C2×C22⋊Q8—861no—
327C2×#121—431no—
328C2×C4×Q8—432no—
329C2×C4×Q8—861no—
330C2×C4×D4—431no—
331C2×C4×D4—861no—
332C2×#117—861no—
333C22×C42C2×D4311yescharacteristic
334#476—864no—
335#461—864no—
336C2×#106—432no—
337C2×C4⋊C8C2×D4311yescharacteristic
338C2×C4⋊C8C2×D4311yescharacteristic
339C2×#103—432no—
340C2×#103—432no—
341C2×#103—432no—
342C2×#102—432no—
343C2×C22⋊C8C2×D4312yes—
344#390—431no—
345#389—431no—
346#388—861no—
347#386—861no—
348#380—431no—
349C4×C4⋊C4—431no—
350C4×C22⋊C4—861no—
351#340—432no—
352#340—432no—
353#336C2×D4312yes—
354C43#2415111yescharacteristic, complemented, Sylow
355C22×Q8—432no—
356C22×Q8—861no—
357C22×D4—431no—
358C22×D4—861no—
359C23×C4—861no—
360C23×C4D17211yescharacteristic
361C23×C4D17211yescharacteristic
362C23×C4C22×D4311yescharacteristic
363C2×SD16—864no—
364C2×SD16—864no—
365C2×SD16—1722no—
366C22×C8C22×D4312yes—
367C22×C8C22×D4312yes—
368C4⋊Q8—434no—
369C4⋊Q8—434no—
370C4⋊Q8—862no—
371C4⋊Q8—862no—
372#125—868no—
373C22⋊Q8—864no—
374C22⋊Q8—864no—
375C22⋊Q8—868no—
376#121—862no—
377#121—862no—
378C4×Q8—864no—
379C4×Q8—864no—
380C4×Q8—868no—
381C4×D4—862no—
382C4×D4—862no—
383C4×D4—1724no—
384#117—868no—
385C2×C4⋊C4—432no—
386C2×C4⋊C4—431no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—431no—
389C2×C4⋊C4—861no—
390C2×C4⋊C4—861no—
391C2×C4⋊C4—862no—
392C2×C4⋊C4—862no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4—861no—
395C2×C22⋊C4—862no—
396C2×C22⋊C4—861no—
397C2×C22⋊C4—861no—
398C2×C22⋊C4—861no—
399C2×C22⋊C4—861no—
400C2×C22⋊C4—861no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—862no—
404C2×C42—861no—
405C2×C42D17211yescharacteristic
406C2×C42D17211yescharacteristic
407C2×C42C22×D4311yescharacteristic
408C2×C42C22×D4311yescharacteristic
409#106—864no—
410C4⋊C8—22no—
411C4⋊C8—22no—
412#103—864no—
413#103—864no—
414#103—1722no—
415#102—864no—
416C22⋊C8—42no—
417#95—861no—
418#95—861no—
419#95—861no—
420C24—861no—
421C24C2×D17211yescharacteristic
422C2×Q8—434no—
423C2×Q8—864no—
424C2×Q8—862no—
425C2×Q8—862no—
426C2×Q8—862no—
427C2×Q8—862no—
428C2×Q8—862no—
429C2×Q8—1722no—
430C2×Q8—1722no—
431C2×D4—432no—
432C2×D4—864no—
433C2×D4—861no—
434C2×D4—861no—
435C2×D4—861no—
436C2×D4—861no—
437C2×D4—861no—
438C2×D4—1722no—
439C2×D4—1722no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—861no—
452C22×C4—862no—
453C22×C4—862no—
454C22×C4—861no—
455C22×C4—861no—
456C22×C4—1722no—
457C22×C4—1722no—
458C22×C4—1722no—
459C22×C4#1062187314yescomplemented
460C22×C4C2×D17211yescharacteristic
461C22×C4C2×D17211yescharacteristic
462C22×C4C2×D17211yescharacteristic
463C22×C4C2×D17211yescharacteristic
464C22×C4C2×D17211yescharacteristic
465C22×C4C23×D4311yescharacteristic
466SD16—1728no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C4⋊C4—434no—
474C4⋊C4—432no—
475C4⋊C4—868no—
476C4⋊C4—868no—
477C4⋊C4—864no—
478C4⋊C4—864no—
479C4⋊C4—864no—
480C4⋊C4—862no—
481C4⋊C4—861no—
482C4⋊C4—861no—
483C4⋊C4—861no—
484C4⋊C4—861no—
485C4⋊C4—861no—
486C4⋊C4—1722no—
487C22⋊C4—864no—
488C22⋊C4—868no—
489C22⋊C4—1722no—
490C22⋊C4—1722no—
491C22⋊C4—1722no—
492C22⋊C4—1722no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—868no—
498C42—1722no—
499C42D4×D4312yes—
500C42D4×D4312yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—862no—
508C23—1721no—
509C23—1721no—
510C23—1721no—
511C23—1721no—
512C23—1721no—
513C23—1721no—
514C23C2×#1062187312yes—
515C23C22×D17211yescharacteristic
516Q8—864no—
517Q8—864no—
518Q8—1728no—
519Q8—1722no—
520Q8—1722no—
521D4—862no—
522D4—862no—
523D4—1728no—
524D4—1721no—
525D4—1721no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—864no—
543C2×C4—864no—
544C2×C4—862no—
545C2×C4—3444no—
546C2×C4—1721no—
547C2×C4—1721no—
548C2×C4—1721no—
549C2×C4—1721no—
550C2×C4—1721no—
551C2×C4—1721no—
552C2×C4—1722no—
553C2×C4—1721no—
554C2×C4—1721no—
555C2×C4—1722no—
556C2×C4—1722no—
557C2×C4—1722no—
558C2×C4#1306329512yes—
559C2×C4#1306329512yes—
560C2×C4C2×#1062187312yes—
561C2×C4C2×#1062187312yes—
562C2×C4#1306335812yes—
563C2×C4C2×D4×D4311yescharacteristic
564C2×C4C2×D4×D4311yescharacteristic
565C2×C4#41×D4311yescharacteristic
566C2×C4#41×D4311yescharacteristic
567C2×C4#1306337011yescharacteristic
568C2×C4#1306337311yescharacteristic
569C8—42no—
570C8—42no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—3441no—
579C22—3441no—
580C22—3441no—
581C22—3441no—
582C22—1722no—
583C22—1722no—
584C22—1722no—
585C22C2×#1306329511yescharacteristic
586C22#1458341111yescharacteristic
587C22#1458348011yescharacteristic
588C22C22×#1062187311yescharacteristic
589C22C2×#1306335811yescharacteristic
590C22#1458369911yescharacteristic
591C22#1458369911yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—3441no—
597C4—3441no—
598C4—1724no—
599C4—1722no—
600C4—1724no—
601C4#1458304014yes—
602C4#1458354611yescharacteristic
603C4#1458355111yescharacteristic
604C4#1458355911yescharacteristic
605C4#1458356711yescharacteristic
606C2—42no—
607C2—3442no—
608C2#1913794111yescharacteristic
609C2C2×#1458304011yescharacteristic
610C2#1914384011yescharacteristic
611C2#1914460411yescharacteristic
612C2#1914475811yescharacteristic
613C2#1914591211yescharacteristic
614C2#1914596911yescharacteristic
subgroupG/H
centerC23C22×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1797C2
socleC23×C43C22×D4
prime radicalC2×#13063299C22
hypercenter#1797D43
hypocenterC43#24151
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C2×#14583456C2semidirect
2. C2×#14583190C2semidirect
3. C2×#14583105C2semidirect
5. #19135792C2semidirect
7. #19135542C2semidirect
8. #19135517C2semidirect
9. #19134949C2semidirect
10. #19134866C2semidirect
11. C43×#1797C2semidirect
57. #13063325D4semidirect
58. #13063325D4semidirect
125. C22×C4×C43SD16semidirect
309. #1797D43semidirect
354. C43#24151semidirect
459. C22×C4#10621873semidirect

Maximal subgroups

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

subgroupindex
C2×#145834562
C2×#145831902
C2×#145831052
C4×#130632992
#191357922
#191356002
#191355422
#191355172
#191349492
#191348662
C43×#17972
#2415143