Finite Group Databasebeta

← #26222196

Subgroups — #26222196

Order 12032.

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×#15689291C211yesmaximal, characteristic
2C4×#13094049C212yesmaximal, complemented
3#20246110C211yesmaximal, characteristic, complemented
4#20243786C211yesmaximal, characteristic, complemented
5#20243764C212yesmaximal, complemented
6#20243621C212yesmaximal, complemented
7#20243598C211yesmaximal, characteristic, complemented
8#20243342C211yesmaximal, characteristic, complemented
9#20243198C212yesmaximal, complemented
10C4×#13093941C211yesmaximal, characteristic, complemented
11C4×H(5,𝔽2)×C47C211yesmaximal, characteristic, complemented
12C2×#13094049C2212yes—
13C2×#13094020C2211yescharacteristic
14C2×C4×D188C2211yescharacteristic, complemented
15#15689291C418yescomplemented
16#15689233—22no—
17#15689221—22no—
18C4×#10625336—22no—
19C4×#10625334—22no—
20#15689166C2212yescomplemented
21C2×#13093951C2212yes—
22C2×#13093951C2212yes—
23#15689127C2212yescomplemented
24#15689124C2212yes—
25C2×#13093941C2211yescharacteristic
26C2×#13093941C2211yescharacteristic
27C4×#10625329C2212yescomplemented
28#15689026C2211yescharacteristic
29#15689024C2211yescharacteristic, complemented
30#15689002—22no—
31#15688999—22no—
32#15688988C2211yescharacteristic, complemented
33#15688969C2211yescharacteristic, complemented
34#15688892C2212yescomplemented
35#15688882C2212yescomplemented
36#15688882C2212yescomplemented
37C2×H(5,𝔽2)×C47C2211yescharacteristic
38C47×#520—22no—
39C47×#520C2212yes—
40C47×#518—22no—
41C47×#518—22no—
42C47×#518C2211yescharacteristic
43C4×#41×C47—22no—
44C4×#41×C47C2212yes—
45C2×C4×D4×C47—22no—
46C2×C4×D4×C47—22no—
47C2×C4×D4×C47C2211yescharacteristic
48C22×D188C2311yescharacteristic
49C22×C22⋊D47—22no—
50C22×C4×D47—21no—
51#13094049—24no—
52#13094049C2×C418yescomplemented
53C2×#10625336—22no—
54C2×#10625334—22no—
55C2×#10625329C2312yes—
56C2×#10625329C2312yes—
57#13094020—22no—
58#13094020C2×C414yescomplemented
59C4×D188—22no—
60C4×D188—22no—
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C22⋊Dic47—22no—
63C2×(C2×D47)⋊C4—21no—
64C2×(C2×D47)⋊C4—22no—
65C2×C4×Dic47C2311yescharacteristic
66C2×C4×Dic47C2311yescharacteristic
67#13093957—22no—
68#13093955—22no—
69#13093951—24no—
70#13093951C2×C418yescomplemented
71#13093948—22no—
72#13093945—22no—
73#13093944—22no—
74#13093944—22no—
75#13093941—22no—
76#13093941C2×C414yescomplemented
77#13093935—22no—
78C4×C47⋊C8—22no—
79C47⋊#95—21no—
80C47⋊#95—21no—
81C47⋊#95—22no—
82H(5,𝔽2)×C47—24no—
83H(5,𝔽2)×C47—22no—
84H(5,𝔽2)×C47C2×C414yescomplemented
85C2×#41×C47—22no—
86C2×#41×C47C2312yes—
87C22×D4×C47—22no—
88C22×D4×C47—22no—
89C22×D4×C47C2311yescharacteristic
90C23×C4×C47—22no—
91C23×C4×C47D412yes—
92C4×Q8×C47—22no—
93C4×Q8×C47D412yescomplemented
94C4×Q8×C47D412yescomplemented
95C4×D4×C47—24no—
96C4×D4×C47—24no—
97C4×D4×C47—24no—
98C4×D4×C47—24no—
99C4×D4×C47—22no—
100C4×D4×C47—22no—
101C4×D4×C47—24no—
102C4×D4×C47—22no—
103C4×D4×C47—22no—
104C4×D4×C47—22no—
105C4×D4×C47—22no—
106C4×D4×C47—22no—
107C4×D4×C47D412yescomplemented
108C4×D4×C47D412yescomplemented
109#117×C47—24no—
110#117×C47—22no—
111#117×C47—22no—
112#117×C47C2312yes—
113C2×C4⋊C4×C47—22no—
114C2×C4⋊C4×C47—22no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C22⋊C4×C47—22no—
117C2×C22⋊C4×C47—24no—
118C2×C22⋊C4×C47—22no—
119C2×C22⋊C4×C47D412yes—
120C2×C42×C47—22no—
121C2×C42×C47—22no—
122C2×C42×C47C2311yescharacteristic
123C24×C47—22no—
124C24×C47C2×D412yes—
125C23×D47—21no—
126#41×C47—28no—
127#41×C47—28no—
128#41×C47—22no—
129#41×C47—22no—
130#41×C47C22×C414yes—
131#41×C47#4114yes—
132C2×Q8×C47C2×D412yes—
133C2×D4×C47—24no—
134C2×D4×C47—28no—
135C2×D4×C47—28no—
136C2×D4×C47—28no—
137C2×D4×C47—22no—
138C2×D4×C47—28no—
139C2×D4×C47—22no—
140C2×D4×C47—22no—
141C2×D4×C47C22×C412yes—
142C2×D4×C47C2×D412yes—
143C2×D4×C47#4112yes—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—24no—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—22no—
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—82no—
154C22×C4×C47—84no—
155C22×C4×C47C2×D412yes—
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2411yescharacteristic
159C2×C22⋊D47—28no—
160C2×C22⋊D47—82no—
161C2×C22⋊D47—82no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47—21no—
165C2×D188—22no—
166C2×D188—22no—
167C2×D188—21no—
168C2×D188C22×C412yescomplemented
169C2×C4×D47—42no—
170C2×C4×D47—82no—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47C2×D412yes—
176C4⋊C4×C47C2×D412yes—
177C22⋊C4×C47—24no—
178C22⋊C4×C47—24no—
179C22⋊C4×C47—22no—
180C22⋊C4×C47—22no—
181C22⋊C4×C47—22no—
182C22⋊C4×C47—24no—
183C42×C47—24no—
184C42×C47—22no—
185C42×C47—22no—
186C42×C47—22no—
187C42×C47C2×D412yes—
188C42×C47C2×D412yes—
189C22⋊Dic47—82no—
190#10625336—28no—
191#10625334—28no—
192(C2×D47)⋊C4—41no—
193(C2×D47)⋊C4—41no—
194(C2×D47)⋊C4—82no—
195C4⋊Dic47—22no—
196C4×Dic47—21no—
197C4×Dic47—22no—
198C4×Dic47C22×C412yescomplemented
199#10625329—22no—
200#10625329—22no—
201#10625329—22no—
202#10625329C22×C414yescomplemented
203C2×C47⋊C8—22no—
204C2×C47⋊C8—22no—
205C23×C47—28no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47—22no—
209C23×C47—42no—
210C23×C47—42no—
211C23×C47—82no—
212C23×C47—82no—
213C23×C47C4×D414yes—
214C23×C47C22×D411yescharacteristic
215C22×D47—22no—
216C22×D47—42no—
217C22×D47—42no—
218C22×D47—41no—
219C22×D47—41no—
220Q8×C47—22no—
221Q8×C47C4×D414yes—
222D4×C47—216no—
223D4×C47—24no—
224D4×C47—24no—
225D4×C47—24no—
226D4×C47—22no—
227D4×C47—24no—
228D4×C47C4×D414yes—
229C2×C4×C47—28no—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—44no—
239C2×C4×C47—44no—
240C2×C4×C47—42no—
241C2×C4×C47—82no—
242C2×C4×C47—82no—
243C2×C4×C47—82no—
244C2×C4×C47—84no—
245C2×C4×C47—82no—
246C2×C4×C47#12011yescharacteristic
247C2×C4×C47#12011yescharacteristic
248C2×C4×C47#12012yes—
249C2×C4×C47C23×C411yescharacteristic
250C2×C4×C47C22×D412yes—
251C2×C4×C47C2×#4112yes—
252C2×C4×C47C2×#4111yescharacteristic
253C22⋊D47—88no—
254C2×Dic47—22no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—42no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262D188—24no—
263D188—42no—
264C4×D47—84no—
265C47⋊C8—24no—
266C47⋊C8—42no—
267#21473—471nomaximal, Sylow
268C22×C47—21no—
269C22×C47—21no—
270C22×C47—24no—
271C22×C47—24no—
272C22×C47—44no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47—42no—
276C22×C47—42no—
277C22×C47—82no—
278C22×C47—88no—
279C22×C47C2×C4×D411yescharacteristic
280C22×C47C2×#12011yescharacteristic
281C22×C47#52511yescharacteristic
282C2×D47—44no—
283C2×D47—42no—
284C2×D47—82no—
285C2×D47—81no—
286C2×D47—81no—
287C4×C47—21no—
288C4×C47—21no—
289C4×C47—22no—
290C4×C47—24no—
291C4×C47—42no—
292C4×C47—42no—
293C4×C47—42no—
294C4×C47C2×C4×D412yes—
295C4×C47#52512yes—
296Dic47—42no—
297Dic47—81no—
298Dic47—81no—
299C4×H(5,𝔽2)D4711yescharacteristic, complemented
300C2×#453—471no—
301C4×AGL(1,ℤ/8)—472no—
302#2328—471no—
303#1887—471no—
304#1883—472no—
305#1860—472no—
306#1856—471no—
307#1792—471no—
308#1759—472no—
309C4×#100—471no—
310C2×C47—21no—
311C2×C47—21no—
312C2×C47—44no—
313C2×C47—44no—
314C2×C47C4×#12011yescharacteristic
315C2×C47#301911yescharacteristic
316C2×C47#302111yescharacteristic
317D47—82no—
318C2×H(5,𝔽2)C2×D4711yescharacteristic
319C2×AGL(1,ℤ/8)—472no—
320C2×#127—471no—
321#520—22no—
322#520C2×D4712yes—
323#518—22no—
324#518—22no—
325#518C2×D4711yescharacteristic
326C4×#41—22no—
327C4×#41C2×D4712yes—
328C2×C4×D4—22no—
329C2×C4×D4—22no—
330C2×C4×D4—471no—
331C2×C4×D4C2×D4711yescharacteristic
332#453—478no—
333#442—942no—
334#440—942no—
335C4×SD16—942no—
336C4×D8—942no—
337#428—472no—
338C2×#104—472no—
339C2×#104—472no—
340#418—472no—
341#417—472no—
342C2×#100—471no—
343C2×#100—471no—
344C4×M16—472no—
345#390—471no—
346#390—471no—
347#386—942no—
348#386—942no—
349#382—471no—
350#376—471no—
351#343—472no—
352#339—472no—
353#339—472no—
354C47#2147311yescharacteristic, complemented, Sylow
355H(5,𝔽2)—24no—
356H(5,𝔽2)—22no—
357H(5,𝔽2)C4×D4714yescomplemented
358C2×#41—22no—
359C2×#41C22×D4712yes—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—471no—
363C22×D4—942no—
364C22×D4C22×D4711yescharacteristic
365C23×C4—22no—
366C23×C4—941no—
367C23×C4C22⋊D4712yes—
368AGL(1,ℤ/8)—478no—
369AGL(1,ℤ/8)—944no—
370C2×SD16—942no—
371C2×D8—942no—
372C2×M16—472no—
373C2×M16—472no—
374#127—474no—
375#127—942no—
376C4×Q8—22no—
377C4×Q8C22⋊D4712yescomplemented
378C4×Q8C22⋊D4712yescomplemented
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—942no—
392C4×D4—942no—
393C4×D4C22⋊D4712yescomplemented
394C4×D4C22⋊D4712yescomplemented
395#117—24no—
396#117—22no—
397#117—22no—
398#117C22×D4712yes—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—942no—
407C2×C22⋊C4—942no—
408C2×C22⋊C4—941no—
409C2×C22⋊C4C22⋊D4712yes—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42—471no—
413C2×C42—471no—
414C2×C42C22×D4711yescharacteristic
415#107—942no—
416#106—942no—
417#104—478no—
418#104—944no—
419#103—942no—
420#102—942no—
421#102—942no—
422#102—942no—
423#100—474no—
424#100—942no—
425#97—942no—
426C4×C8—942no—
427#95—941no—
428#95—941no—
429#95—942no—
430C24—22no—
431C24—941no—
432C24C2×C22⋊D4712yes—
433#41—28no—
434#41—28no—
435#41—22no—
436#41—22no—
437#41C2×C4×D4714yes—
438#41#1062535014yes—
439C2×Q8C2×C22⋊D4712yes—
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—472no—
449C2×D4—948no—
450C2×D4—942no—
451C2×D4—942no—
452C2×D4—941no—
453C2×D4—3762no—
454C2×D4—3762no—
455C2×D4C2×C4×D4712yes—
456C2×D4#1062535012yes—
457C2×D4C2×C22⋊D4712yes—
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—941no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—3762no—
473C22×C4—1882no—
474C22×C4C2×C22⋊D4712yes—
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C2×C22⋊D4711yescharacteristic
477C22×C4C23×D4711yescharacteristic
478SD16—948no—
479D8—948no—
480M16—474no—
481M16—942no—
482M16—942no—
483M16—942no—
484C2×C8—942no—
485C2×C8—942no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—942no—
491C4⋊C4C2×C22⋊D4712yes—
492C4⋊C4C2×C22⋊D4712yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—24no—
499C22⋊C4—3762no—
500C22⋊C4—3762no—
501C22⋊C4—1881no—
502C22⋊C4—1881no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—472no—
508C42—941no—
509C42—942no—
510C42C2×C22⋊D4712yes—
511C42C2×C22⋊D4712yes—
512C23—28no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—42no—
517C23—42no—
518C23—82no—
519C23—82no—
520C23—942no—
521C23—1882no—
522C23—1882no—
523C23—1881no—
524C23—1881no—
525C23C4×C22⋊D4714yes—
526C23C22×C22⋊D4711yescharacteristic
527Q8—22no—
528Q8C4×C22⋊D4714yes—
529D4—216no—
530D4—24no—
531D4—24no—
532D4—24no—
533D4—24no—
534D4—22no—
535D4—944no—
536D4—3768no—
537D4—1882no—
538D4C4×C22⋊D4714yes—
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—82no—
554C2×C4—84no—
555C2×C4—82no—
556C2×C4—942no—
557C2×C4—3764no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1882no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4#1309399311yescharacteristic
566C2×C4#1309399312yes—
567C2×C4#1309399311yescharacteristic
568C2×C4C22×C4×D4711yescharacteristic
569C2×C4C22×C22⋊D4712yes—
570C2×C4C2×#1062535012yes—
571C2×C4C2×#1062535011yescharacteristic
572C8—944no—
573C8—1882no—
574C22—21no—
575C22—21no—
576C22—24no—
577C22—24no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—82no—
584C22—88no—
585C22—3762no—
586C22—3761no—
587C22—3761no—
588C22—1884no—
589C22—1882no—
590C22C2×C4×C22⋊D4711yescharacteristic
591C22C2×#1309399311yescharacteristic
592C22#1568963411yescharacteristic
593C4—21no—
594C4—21no—
595C4—22no—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—3761no—
601C4—3761no—
602C4—1882no—
603C4C2×C4×C22⋊D4712yes—
604C4#1568963412yes—
605C2—21no—
606C2—21no—
607C2—44no—
608C2—44no—
609C2—3762no—
610C2C4×#1309399311yescharacteristic
611C2#2025274511yescharacteristic
612C2#2025276011yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×H(5,𝔽2)×C47C2
socleC22×C47C2×#120
prime radicalC2×#15689291C2
cosocleC22×C4×C47C24
hypercenterC4×H(5,𝔽2)D47
hypocenterC47#21473
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C4×#13094049C2semidirect
3. #20246110C2semidirect
4. #20243786C2semidirect
5. #20243764C2semidirect
6. #20243621C2semidirect
7. #20243598C2semidirect
8. #20243342C2semidirect
9. #20243198C2semidirect
10. C4×#13093941C2semidirect
11. C4×H(5,𝔽2)×C47C2semidirect
14. C2×C4×D188C22semidirect
15. #15689291C4semidirect
20. #15689166C22semidirect
23. #15689127C22semidirect
27. C4×#10625329C22semidirect
29. #15689024C22semidirect
32. #15688988C22semidirect
33. #15688969C22semidirect
34. #15688892C22semidirect
35. #15688882C22semidirect
36. #15688882C22semidirect
52. #13094049C2×C4semidirect
58. #13094020C2×C4semidirect
70. #13093951C2×C4semidirect
76. #13093941C2×C4semidirect
84. H(5,𝔽2)×C47C2×C4semidirect
93. C4×Q8×C47D4semidirect
94. C4×Q8×C47D4semidirect
107. C4×D4×C47D4semidirect
108. C4×D4×C47D4semidirect
168. C2×D188C22×C4semidirect
198. C4×Dic47C22×C4semidirect
202. #10625329C22×C4semidirect
299. C4×H(5,𝔽2)D47semidirect
354. C47#21473semidirect
357. H(5,𝔽2)C4×D47semidirect
377. C4×Q8C22⋊D47semidirect
378. C4×Q8C22⋊D47semidirect
393. C4×D4C22⋊D47semidirect
394. C4×D4C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892912
C4×#130940492
#202461102
#202437862
#202437642
#202436212
#202435982
#202433422
#202431982
C4×#130939412
C4×H(5,𝔽2)×C472
#2147347