Finite Group Databasebeta

← #25102515

Subgroups — #25102515

Order 11008.

1486 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583067C211yesmaximal, characteristic
2C4×#13063350C212yesmaximal, complemented
3#19137317C211yesmaximal, characteristic, complemented
4#19134993C211yesmaximal, characteristic, complemented
5#19134971C212yesmaximal, complemented
6#19134828C212yesmaximal, complemented
7#19134805C211yesmaximal, characteristic, complemented
8#19134549C211yesmaximal, characteristic, complemented
9#19134405C212yesmaximal, complemented
10C4×#13063242C211yesmaximal, characteristic, complemented
11C4×H(5,𝔽2)×C43C211yesmaximal, characteristic, complemented
12C2×#13063350C2212yes—
13C2×#13063321C2211yescharacteristic
14C2×C4×D172C2211yescharacteristic, complemented
15#14583067C418yescomplemented
16#14583009—22no—
17#14582997—22no—
18C4×#10621884—22no—
19C4×#10621882—22no—
20#14582942C2212yescomplemented
21C2×#13063252C2212yes—
22C2×#13063252C2212yes—
23#14582903C2212yescomplemented
24#14582900C2212yes—
25C2×#13063242C2211yescharacteristic
26C2×#13063242C2211yescharacteristic
27C4×#10621877C2212yescomplemented
28#14582802C2211yescharacteristic
29#14582800C2211yescharacteristic, complemented
30#14582778—22no—
31#14582775—22no—
32#14582764C2211yescharacteristic, complemented
33#14582745C2211yescharacteristic, complemented
34#14582668C2212yescomplemented
35#14582658C2212yescomplemented
36#14582658C2212yescomplemented
37C2×H(5,𝔽2)×C43C2211yescharacteristic
38C43×#520—22no—
39C43×#520C2212yes—
40C43×#518—22no—
41C43×#518—22no—
42C43×#518C2211yescharacteristic
43C4×#41×C43—22no—
44C4×#41×C43C2212yes—
45C2×C4×D4×C43—22no—
46C2×C4×D4×C43—22no—
47C2×C4×D4×C43C2211yescharacteristic
48C22×D172C2311yescharacteristic
49C22×C22⋊D43—22no—
50C22×C4×D43—21no—
51#13063350—24no—
52#13063350C2×C418yescomplemented
53C2×#10621884—22no—
54C2×#10621882—22no—
55C2×#10621877C2312yes—
56C2×#10621877C2312yes—
57#13063321—22no—
58#13063321C2×C414yescomplemented
59C4×D172—22no—
60C4×D172—22no—
61C2×C4⋊Dic43C2311yescharacteristic
62C2×C22⋊Dic43—22no—
63C2×(C2×D43)⋊C4—21no—
64C2×(C2×D43)⋊C4—22no—
65C2×C4×Dic43C2311yescharacteristic
66C2×C4×Dic43C2311yescharacteristic
67#13063258—22no—
68#13063256—22no—
69#13063252—24no—
70#13063252C2×C418yescomplemented
71#13063249—22no—
72#13063246—22no—
73#13063245—22no—
74#13063245—22no—
75#13063242—22no—
76#13063242C2×C414yescomplemented
77#13063236—22no—
78C4×C43⋊C8—22no—
79C43⋊#95—21no—
80C43⋊#95—21no—
81C43⋊#95—22no—
82H(5,𝔽2)×C43—24no—
83H(5,𝔽2)×C43—22no—
84H(5,𝔽2)×C43C2×C414yescomplemented
85C2×#41×C43—22no—
86C2×#41×C43C2312yes—
87C22×D4×C43—22no—
88C22×D4×C43—22no—
89C22×D4×C43C2311yescharacteristic
90C23×C4×C43—22no—
91C23×C4×C43D412yes—
92C4×Q8×C43—22no—
93C4×Q8×C43D412yescomplemented
94C4×Q8×C43D412yescomplemented
95C4×D4×C43—24no—
96C4×D4×C43—24no—
97C4×D4×C43—24no—
98C4×D4×C43—24no—
99C4×D4×C43—22no—
100C4×D4×C43—22no—
101C4×D4×C43—24no—
102C4×D4×C43—22no—
103C4×D4×C43—22no—
104C4×D4×C43—22no—
105C4×D4×C43—22no—
106C4×D4×C43—22no—
107C4×D4×C43D412yescomplemented
108C4×D4×C43D412yescomplemented
109#117×C43—24no—
110#117×C43—22no—
111#117×C43—22no—
112#117×C43C2312yes—
113C2×C4⋊C4×C43—22no—
114C2×C4⋊C4×C43—22no—
115C2×C4⋊C4×C43C2311yescharacteristic
116C2×C22⋊C4×C43—22no—
117C2×C22⋊C4×C43—24no—
118C2×C22⋊C4×C43—22no—
119C2×C22⋊C4×C43D412yes—
120C2×C42×C43—22no—
121C2×C42×C43—22no—
122C2×C42×C43C2311yescharacteristic
123C24×C43—22no—
124C24×C43C2×D412yes—
125C23×D43—21no—
126#41×C43—28no—
127#41×C43—28no—
128#41×C43—22no—
129#41×C43—22no—
130#41×C43C22×C414yes—
131#41×C43#4114yes—
132C2×Q8×C43C2×D412yes—
133C2×D4×C43—24no—
134C2×D4×C43—28no—
135C2×D4×C43—28no—
136C2×D4×C43—28no—
137C2×D4×C43—22no—
138C2×D4×C43—22no—
139C2×D4×C43—28no—
140C2×D4×C43—22no—
141C2×D4×C43C22×C412yes—
142C2×D4×C43C2×D412yes—
143C2×D4×C43#4112yes—
144C22×C4×C43—22no—
145C22×C4×C43—22no—
146C22×C4×C43—22no—
147C22×C4×C43—22no—
148C22×C4×C43—24no—
149C22×C4×C43—22no—
150C22×C4×C43—22no—
151C22×C4×C43—22no—
152C22×C4×C43—22no—
153C22×C4×C43—82no—
154C22×C4×C43—84no—
155C22×C4×C43C2×D412yes—
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2411yescharacteristic
159C2×C22⋊D43—28no—
160C2×C22⋊D43—82no—
161C2×C22⋊D43—82no—
162C22×Dic43—21no—
163C22×Dic43—21no—
164C22×Dic43—21no—
165C2×D172—21no—
166C2×D172—22no—
167C2×D172—22no—
168C2×D172C22×C412yescomplemented
169C2×C4×D43—42no—
170C2×C4×D43—82no—
171C4⋊C4×C43—24no—
172C4⋊C4×C43—22no—
173C4⋊C4×C43—22no—
174C4⋊C4×C43—22no—
175C4⋊C4×C43C2×D412yes—
176C4⋊C4×C43C2×D412yes—
177C22⋊C4×C43—24no—
178C22⋊C4×C43—24no—
179C22⋊C4×C43—22no—
180C22⋊C4×C43—22no—
181C22⋊C4×C43—24no—
182C22⋊C4×C43—22no—
183C42×C43—24no—
184C42×C43—22no—
185C42×C43—22no—
186C42×C43—22no—
187C42×C43C2×D412yes—
188C42×C43C2×D412yes—
189C22⋊Dic43—82no—
190#10621884—28no—
191#10621882—28no—
192(C2×D43)⋊C4—41no—
193(C2×D43)⋊C4—41no—
194(C2×D43)⋊C4—82no—
195C4⋊Dic43—22no—
196C4×Dic43—21no—
197C4×Dic43—22no—
198C4×Dic43C22×C412yescomplemented
199#10621877—22no—
200#10621877—22no—
201#10621877—22no—
202#10621877C22×C414yescomplemented
203C2×C43⋊C8—22no—
204C2×C43⋊C8—22no—
205C23×C43—28no—
206C23×C43—22no—
207C23×C43—22no—
208C23×C43—22no—
209C23×C43—42no—
210C23×C43—42no—
211C23×C43—82no—
212C23×C43—82no—
213C23×C43C4×D414yes—
214C23×C43C22×D411yescharacteristic
215C22×D43—22no—
216C22×D43—42no—
217C22×D43—42no—
218C22×D43—41no—
219C22×D43—41no—
220Q8×C43—22no—
221Q8×C43C4×D414yes—
222D4×C43—216no—
223D4×C43—24no—
224D4×C43—24no—
225D4×C43—24no—
226D4×C43—22no—
227D4×C43—24no—
228D4×C43C4×D414yes—
229C2×C4×C43—28no—
230C2×C4×C43—22no—
231C2×C4×C43—24no—
232C2×C4×C43—24no—
233C2×C4×C43—22no—
234C2×C4×C43—22no—
235C2×C4×C43—22no—
236C2×C4×C43—41no—
237C2×C4×C43—41no—
238C2×C4×C43—44no—
239C2×C4×C43—44no—
240C2×C4×C43—42no—
241C2×C4×C43—82no—
242C2×C4×C43—82no—
243C2×C4×C43—82no—
244C2×C4×C43—84no—
245C2×C4×C43—82no—
246C2×C4×C43#12011yescharacteristic
247C2×C4×C43#12011yescharacteristic
248C2×C4×C43#12012yes—
249C2×C4×C43C23×C411yescharacteristic
250C2×C4×C43C22×D412yes—
251C2×C4×C43C2×#4112yes—
252C2×C4×C43C2×#4111yescharacteristic
253C22⋊D43—88no—
254C2×Dic43—22no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—42no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262D172—24no—
263D172—42no—
264C4×D43—84no—
265C43⋊C8—24no—
266C43⋊C8—42no—
267#21473—431nomaximal, Sylow
268C22×C43—21no—
269C22×C43—21no—
270C22×C43—24no—
271C22×C43—24no—
272C22×C43—44no—
273C22×C43—42no—
274C22×C43—42no—
275C22×C43—42no—
276C22×C43—42no—
277C22×C43—82no—
278C22×C43—88no—
279C22×C43C2×C4×D411yescharacteristic
280C22×C43C2×#12011yescharacteristic
281C22×C43#52511yescharacteristic
282C2×D43—44no—
283C2×D43—42no—
284C2×D43—82no—
285C2×D43—81no—
286C2×D43—81no—
287C4×C43—21no—
288C4×C43—21no—
289C4×C43—22no—
290C4×C43—24no—
291C4×C43—42no—
292C4×C43—42no—
293C4×C43—42no—
294C4×C43C2×C4×D412yes—
295C4×C43#52512yes—
296Dic43—42no—
297Dic43—81no—
298Dic43—81no—
299C4×H(5,𝔽2)D4311yescharacteristic, complemented
300C2×#453—431no—
301C4×AGL(1,ℤ/8)—432no—
302#2328—431no—
303#1887—431no—
304#1883—432no—
305#1860—432no—
306#1856—431no—
307#1792—431no—
308#1759—432no—
309C4×#100—431no—
310C2×C43—21no—
311C2×C43—21no—
312C2×C43—44no—
313C2×C43—44no—
314C2×C43C4×#12011yescharacteristic
315C2×C43#301911yescharacteristic
316C2×C43#302111yescharacteristic
317D43—82no—
318C2×H(5,𝔽2)C2×D4311yescharacteristic
319C2×AGL(1,ℤ/8)—432no—
320C2×#127—431no—
321#520—22no—
322#520C2×D4312yes—
323#518—22no—
324#518—22no—
325#518C2×D4311yescharacteristic
326C4×#41—22no—
327C4×#41C2×D4312yes—
328C2×C4×D4—22no—
329C2×C4×D4—22no—
330C2×C4×D4—431no—
331C2×C4×D4C2×D4311yescharacteristic
332#453—438no—
333#442—862no—
334#440—862no—
335C4×SD16—862no—
336C4×D8—862no—
337#428—432no—
338C2×#104—432no—
339C2×#104—432no—
340#418—432no—
341#417—432no—
342C2×#100—431no—
343C2×#100—431no—
344C4×M16—432no—
345#390—431no—
346#390—431no—
347#386—862no—
348#386—862no—
349#382—431no—
350#376—431no—
351#343—432no—
352#339—432no—
353#339—432no—
354C43#2147311yescharacteristic, complemented, Sylow
355H(5,𝔽2)—24no—
356H(5,𝔽2)—22no—
357H(5,𝔽2)C4×D4314yescomplemented
358C2×#41—22no—
359C2×#41C22×D4312yes—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—431no—
363C22×D4—862no—
364C22×D4C22×D4311yescharacteristic
365C23×C4—22no—
366C23×C4—861no—
367C23×C4C22⋊D4312yes—
368AGL(1,ℤ/8)—438no—
369AGL(1,ℤ/8)—864no—
370C2×SD16—862no—
371C2×D8—862no—
372C2×M16—432no—
373C2×M16—432no—
374#127—434no—
375#127—862no—
376C4×Q8—22no—
377C4×Q8C22⋊D4312yescomplemented
378C4×Q8C22⋊D4312yescomplemented
379C4×D4—24no—
380C4×D4—24no—
381C4×D4—24no—
382C4×D4—24no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—24no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—862no—
392C4×D4—862no—
393C4×D4C22⋊D4312yescomplemented
394C4×D4C22⋊D4312yescomplemented
395#117—24no—
396#117—22no—
397#117—22no—
398#117C22×D4312yes—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—431no—
402C2×C4⋊C4C22×D4311yescharacteristic
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—862no—
407C2×C22⋊C4—862no—
408C2×C22⋊C4—861no—
409C2×C22⋊C4C22⋊D4312yes—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42—431no—
413C2×C42—431no—
414C2×C42C22×D4311yescharacteristic
415#107—862no—
416#106—862no—
417#104—438no—
418#104—864no—
419#103—862no—
420#102—862no—
421#102—862no—
422#102—862no—
423#100—434no—
424#100—862no—
425#97—862no—
426C4×C8—862no—
427#95—861no—
428#95—861no—
429#95—862no—
430C24—22no—
431C24—861no—
432C24C2×C22⋊D4312yes—
433#41—28no—
434#41—28no—
435#41—22no—
436#41—22no—
437#41C2×C4×D4314yes—
438#41#1062189814yes—
439C2×Q8C2×C22⋊D4312yes—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—28no—
444C2×D4—22no—
445C2×D4—28no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—432no—
449C2×D4—868no—
450C2×D4—862no—
451C2×D4—862no—
452C2×D4—861no—
453C2×D4—3442no—
454C2×D4—3442no—
455C2×D4C2×C4×D4312yes—
456C2×D4#1062189812yes—
457C2×D4C2×C22⋊D4312yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—82no—
468C22×C4—84no—
469C22×C4—861no—
470C22×C4—861no—
471C22×C4—861no—
472C22×C4—3442no—
473C22×C4—1722no—
474C22×C4C2×C22⋊D4312yes—
475C22×C4C2×C22⋊D4311yescharacteristic
476C22×C4C2×C22⋊D4311yescharacteristic
477C22×C4C23×D4311yescharacteristic
478SD16—868no—
479D8—868no—
480M16—434no—
481M16—862no—
482M16—862no—
483M16—862no—
484C2×C8—862no—
485C2×C8—862no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—862no—
491C4⋊C4C2×C22⋊D4312yes—
492C4⋊C4C2×C22⋊D4312yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—24no—
499C22⋊C4—3442no—
500C22⋊C4—3442no—
501C22⋊C4—1721no—
502C22⋊C4—1721no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—432no—
508C42—861no—
509C42—862no—
510C42C2×C22⋊D4312yes—
511C42C2×C22⋊D4312yes—
512C23—28no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—42no—
517C23—42no—
518C23—82no—
519C23—82no—
520C23—862no—
521C23—1722no—
522C23—1722no—
523C23—1721no—
524C23—1721no—
525C23C4×C22⋊D4314yes—
526C23C22×C22⋊D4311yescharacteristic
527Q8—22no—
528Q8C4×C22⋊D4314yes—
529D4—216no—
530D4—24no—
531D4—24no—
532D4—24no—
533D4—24no—
534D4—22no—
535D4—864no—
536D4—3448no—
537D4—1722no—
538D4C4×C22⋊D4314yes—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—84no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—862no—
557C2×C4—3444no—
558C2×C4—1721no—
559C2×C4—1721no—
560C2×C4—1722no—
561C2×C4—1721no—
562C2×C4—1721no—
563C2×C4—1721no—
564C2×C4—1721no—
565C2×C4#1306329411yescharacteristic
566C2×C4#1306329412yes—
567C2×C4#1306329411yescharacteristic
568C2×C4C22×C4×D4311yescharacteristic
569C2×C4C22×C22⋊D4312yes—
570C2×C4C2×#1062189812yes—
571C2×C4C2×#1062189811yescharacteristic
572C8—864no—
573C8—1722no—
574C22—21no—
575C22—21no—
576C22—24no—
577C22—24no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—82no—
584C22—88no—
585C22—3442no—
586C22—3441no—
587C22—3441no—
588C22—1724no—
589C22—1722no—
590C22C2×C4×C22⋊D4311yescharacteristic
591C22C2×#1306329411yescharacteristic
592C22#1458341011yescharacteristic
593C4—21no—
594C4—21no—
595C4—22no—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—3441no—
601C4—3441no—
602C4—1722no—
603C4C2×C4×C22⋊D4312yes—
604C4#1458341012yes—
605C2—21no—
606C2—21no—
607C2—44no—
608C2—44no—
609C2—3442no—
610C2C4×#1306329411yescharacteristic
611C2#1914395211yescharacteristic
612C2#1914396711yescharacteristic
subgroupG/H
centerC22C2×#13063294
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC4×H(5,𝔽2)×C43C2
socleC22×C43C2×#120
prime radicalC2×#14583067C2
cosocleC22×C4×C43C24
hypercenterC4×H(5,𝔽2)D43
hypocenterC43#21473
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C4×#13063350C2semidirect
3. #19137317C2semidirect
4. #19134993C2semidirect
5. #19134971C2semidirect
6. #19134828C2semidirect
7. #19134805C2semidirect
8. #19134549C2semidirect
9. #19134405C2semidirect
10. C4×#13063242C2semidirect
11. C4×H(5,𝔽2)×C43C2semidirect
14. C2×C4×D172C22semidirect
15. #14583067C4semidirect
20. #14582942C22semidirect
23. #14582903C22semidirect
27. C4×#10621877C22semidirect
29. #14582800C22semidirect
32. #14582764C22semidirect
33. #14582745C22semidirect
34. #14582668C22semidirect
35. #14582658C22semidirect
36. #14582658C22semidirect
52. #13063350C2×C4semidirect
58. #13063321C2×C4semidirect
70. #13063252C2×C4semidirect
76. #13063242C2×C4semidirect
84. H(5,𝔽2)×C43C2×C4semidirect
93. C4×Q8×C43D4semidirect
94. C4×Q8×C43D4semidirect
107. C4×D4×C43D4semidirect
108. C4×D4×C43D4semidirect
168. C2×D172C22×C4semidirect
198. C4×Dic43C22×C4semidirect
202. #10621877C22×C4semidirect
299. C4×H(5,𝔽2)D43semidirect
354. C43#21473semidirect
357. H(5,𝔽2)C4×D43semidirect
377. C4×Q8C22⋊D43semidirect
378. C4×Q8C22⋊D43semidirect
393. C4×D4C22⋊D43semidirect
394. C4×D4C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145830672
C4×#130633502
#191373172
#191349932
#191349712
#191348282
#191348052
#191345492
#191344052
C4×#130632422
C4×H(5,𝔽2)×C432
#2147343