Finite Group Databasebeta

← #72258795

Subgroups — #72258795

Order 1536.

922 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#71295C211yesmaximal, characteristic
2#10813660C211yesmaximal, characteristic, complemented
3#10713698C211yesmaximal, characteristic, complemented
4C2×#67443C211yesmaximal, characteristic, complemented
5#10682358C211yesmaximal, characteristic, complemented
6#10682135C211yesmaximal, characteristic, complemented
7C3×#9837C211yesmaximal, characteristic, complemented
8#136868—31nomaximal, Sylow
9C2×C3×#493C2211yescharacteristic
10C2×#5147—21no—
11#79116—21no—
12#79106—21no—
13C2×#4700C2211yescharacteristic
14C3×#1957—22no—
15C3×#1844C2211yescharacteristic
16C2×C3×#322C2211yescharacteristic
17C2×#4563—21no—
18C2×#4563—21no—
19#72033—21no—
20#71295—22no—
21#71295C414yescomplemented
22#70592—21no—
23#70572—21no—
24#69779C2211yescharacteristic
25C2×#4463C2211yescharacteristic
26C2×#4463C2211yescharacteristic
27#69592—22no—
28#67443—22no—
29C2×#2213—31no—
30#22689—31no—
31#13261—31no—
32C2×#1336—31no—
33#9837D311yescharacteristic, complemented
34#7268—31no—
35#7182—31no—
36C22×C3×D8—21no—
37C22×C3×D8—21no—
38C2×C3×#127—21no—
39C2×#904D411yescharacteristic
40C2×#901—21no—
41#5504—22no—
42C3×#493—21no—
43C3×#493—21no—
44C3×#493—21no—
45C3×#493C2×C412yescomplemented
46C2×C3×#106—21no—
47C2×C3×#106—21no—
48C2×C3×#102—44no—
49C2×C3×C4×C8—21no—
50C2×C3×C4×C8C2311yescharacteristic
51C3×#384D412yes—
52C2×#827—41no—
53#5147—24no—
54#5147—42no—
55C2×#826—41no—
56C2×#819D411yescharacteristic
57C2×#819D411yescharacteristic
58#5106D411yescharacteristic
59#5101—21no—
60C2×#812—21no—
61C2×#812—21no—
62#4966—22no—
63#4964—22no—
64C2×#805—21no—
65#4869—41no—
66#4861—42no—
67#4860—41no—
68D3×#106—41no—
69#4853—41no—
70#4852—42no—
71#4700—21no—
72#4700—21no—
73#4700C2×C412yes—
74#4700D412yes—
75C3×#322—21no—
76C3×#322—21no—
77#4563—24no—
78#4563—42no—
79#4562—42no—
80#4530—22no—
81#4523—21no—
82#4463—21no—
83#4463—21no—
84#4463—21no—
85#4463C2×C412yescomplemented
86C2×#509—61no—
87C2×#493C2×D311yescharacteristic
88C2×#443—31no—
89#2952—61no—
90#2912—61no—
91#2213—34no—
92#2213—62no—
93C2×#364—31no—
94C2×#364—31no—
95C2×#361—61no—
96C2×#361—61no—
97#2140—61no—
98#1957—22no—
99#1951—61no—
100#1859—62no—
101#1844C2×D311yescharacteristic
102#1815—61no—
103#1772—31no—
104C2×#322C2×D311yescharacteristic
105#1336—62no—
106C22×C3×D4—42no—
107C22×C3×D4—41no—
108C2×#235—21no—
109C2×C3×D8—22no—
110C2×C3×D8—22no—
111C2×C3×D8—41no—
112C2×C3×D8—41no—
113C2×C3×D8—41no—
114C2×C3×D8—41no—
115C2×C3×D8—41no—
116C2×C3×D8—41no—
117C2×C3×D8—41no—
118C2×C3×D8—41no—
119C2×C3×D8—41no—
120C2×C3×D8—41no—
121C22×C3×C8—22no—
122C22×C3×C8C2×D411yescharacteristic
123C22×C3×C8C2×D411yescharacteristic
124C3×#127—22no—
125C3×#127—42no—
126C3×#127—41no—
127C2×C3×C4⋊C4—22no—
128C2×C3×C4⋊C4—42no—
129C2×C3×C42C2×D411yescharacteristic
130C4×#67—82no—
131#926—21no—
132C2×#208—21no—
133C2×#208—21no—
134C2×Dic12D811yescharacteristic
135#904—21no—
136#904—21no—
137#904C22⋊C412yes—
138#904SD1612yes—
139C2×D24D811yescharacteristic
140C2×#204—21no—
141#901—22no—
142#901—22no—
143#901—41no—
144#901—41no—
145C2×ΓL(1,𝔽25)—41no—
146C2×D3×C8—41no—
147#893—42no—
148#892—42no—
149D3×C4⋊C4—42no—
150#887—42no—
151C3×#106—41no—
152C3×#106—41no—
153C3×#106—41no—
154C3×#106—41no—
155C3×#102—88no—
156C3×C4×C8—41no—
157C3×C4×C8—41no—
158C3×C4×C8C22×C411yescharacteristic
159C3×C4×C8C2×D411yescharacteristic
160C3×C4×C8#4111yescharacteristic
161C3×C4×C8#4111yescharacteristic
162#827—44no—
163#826—44no—
164#819—21no—
165#819—21no—
166#819—21no—
167#819C22⋊C412yes—
168#818D812yes—
169#817—21no—
170#815—41no—
171C8×Dic3—41no—
172#812—22no—
173#812—41no—
174#812—41no—
175#812—41no—
176C2×C3⋊C16—41no—
177C2×C3⋊C16—41no—
178#810—42no—
179#809—42no—
180#808—42no—
181#807—42no—
182#805—22no—
183#805—22no—
184C2×#135—31no—
185C22×D8—21no—
186C22×D8—21no—
187C2×#131—61no—
188C2×#127—21no—
189#520—62no—
190#509—64no—
191#509—122no—
192C2×SD32—121no—
193C2×D16—121no—
194C2×M32—61no—
195C2×M32—61no—
196#493—21no—
197#493—21no—
198#493—21no—
199#493C4×D312yescomplemented
200#443—32no—
201#443—32no—
202#443—61no—
203#443—61no—
204#442—121no—
205#441—121no—
206C4×SD16—122no—
207C2×#108—31no—
208C2×#108—31no—
209#428—62no—
210#428—122no—
211#427—31no—
212#427—121no—
213C2×#106—21no—
214C2×#106—21no—
215C2×#106—121no—
216C2×#104—61no—
217#417—62no—
218C2×#102—44no—
219#405—61no—
220C2×C4×C8—21no—
221C2×C4×C8C22×D311yescharacteristic
222#384#6712yes—
223#364—32no—
224#364—61no—
225#364—61no—
226#364—61no—
227#361—64no—
228#361—122no—
229#357—122no—
230#343—62no—
231#339—61no—
232#322—21no—
233#322—21no—
234C24×C3—81no—
235C2×C3×D4—44no—
236C2×C3×D4—42no—
237C2×C3×D4—84no—
238C2×C3×D4—84no—
239C2×C3×D4—82no—
240C2×C3×D4—81no—
241C2×C3×D4—81no—
242C2×C3×D4—81no—
243C2×C3×D4—81no—
244C2×C3×D4—81no—
245C22×C3×C4—21no—
246C22×C3×C4—42no—
247C22×C3×C4#12011yescharacteristic
248C2×#67—41no—
249#235—22no—
250#235—22no—
251#235—41no—
252#235—41no—
253C2×D12—21no—
254C2×C4×D3—41no—
255C2×C4×D3—82no—
256C2×Dic6—21no—
257C3×D8—42no—
258C3×D8—42no—
259C3×D8—42no—
260C3×D8—42no—
261C3×D8—81no—
262C3×D8—81no—
263C3×D8—81no—
264C3×D8—81no—
265C2×C3×C8—42no—
266C2×C3×C8—44no—
267C2×C3×C8C2×C22⋊C411yescharacteristic
268C2×C3×C8C4×D411yescharacteristic
269C2×C3×C8#12111yescharacteristic
270C2×C3×C8#12311yescharacteristic
271C2×C3×C8C2×D811yescharacteristic
272C2×C3×C8C2×D811yescharacteristic
273C2×C3×C8C2×SD1611yescharacteristic
274C2×C3×C8#13511yescharacteristic
275C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
276C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
277C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
278C2×C3×C8#13711yescharacteristic
279C3×C4⋊C4—42no—
280C3×C4⋊C4—42no—
281C3×C4⋊C4—84no—
282C3×C42—21no—
283C3×C42C4×D411yescharacteristic
284C3×C42#12111yescharacteristic
285#217—41no—
286(C2×D3)⋊C4—82no—
287C4⋊Dic3—22no—
288Dic3⋊C4—82no—
289C4×Dic3—41no—
290C4×Dic3—82no—
291#208—22no—
292#208—41no—
293#208—41no—
294#208—41no—
295C2×C3⋊C8—41no—
296C2×C3⋊C8—41no—
297Dic12—21no—
298Dic12#10212yescomplemented
299D24—21no—
300D24#10212yescomplemented
301#204—22no—
302#204—22no—
303ΓL(1,𝔽25)—42no—
304ΓL(1,𝔽25)—42no—
305D3×C8—42no—
306D3×C8—42no—
307C3⋊C16—42no—
308C3⋊C16—81no—
309C2×#41—61no—
310C22×D4—42no—
311C22×D4—41no—
312#135—32no—
313#135—32no—
314#135—61no—
315#135—61no—
316C2×Q16—31no—
317C2×SD16—61no—
318C2×D8—22no—
319C2×D8—22no—
320C2×D8—31no—
321C2×D8—41no—
322C2×D8—41no—
323C2×D8—41no—
324C2×D8—41no—
325C2×D8—41no—
326C2×D8—41no—
327C2×D8—41no—
328C2×D8—41no—
329C2×D8—41no—
330C2×D8—41no—
331#131—62no—
332#131—62no—
333#131—121no—
334#131—121no—
335C2×M16—61no—
336C2×M16—61no—
337C2×M16—121no—
338C22×C8—22no—
339C22×C8—121no—
340C22×C8D3×D411yescharacteristic
341C22×C8D3×D411yescharacteristic
342#127—22no—
343#127—42no—
344#127—41no—
345C4×Q8—122no—
346C4×D4—122no—
347C4×D4—242no—
348#117—61no—
349#117—122no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—122no—
353C2×C42C2×#6711yescharacteristic
354SD32—124no—
355D16—124no—
356M32—62no—
357M32—121no—
358M32—121no—
359M32—121no—
360C2×C16—121no—
361C2×C16—121no—
362#108—32no—
363#108—61no—
364#108—61no—
365#108—61no—
366#107—32no—
367#107—122no—
368#106—41no—
369#106—41no—
370#106—41no—
371#106—41no—
372#106—61no—
373#106—122no—
374#104—62no—
375#104—62no—
376#103—122no—
377#102—88no—
378#102—122no—
379#97—121no—
380C4×C8—41no—
381C4×C8—41no—
382C4×C8—121no—
383C4×C8C2×C4×D311yescharacteristic
384C4×C8#23511yescharacteristic
385C4×C8D3×D411yescharacteristic
386C4×C8#23711yescharacteristic
387C23×C3—82no—
388C23×C3—162no—
389C23×C3—162no—
390C23×C3—161no—
391C23×C3—161no—
392C23×C3#45311yescharacteristic
393C22×D3—41no—
394C3×D4—88no—
395C3×D4—82no—
396C3×D4—82no—
397C3×D4—164no—
398C3×D4—161no—
399C3×D4—161no—
400C2×C3×C4—24no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—82no—
404C2×C3×C4—162no—
405C2×C3×C4#38611yescharacteristic
406C2×C3×C4#44711yescharacteristic
407C2×C3×C4#44711yescharacteristic
408C2×C3×C4#44811yescharacteristic
409C2×C3×C4#44911yescharacteristic
410C2×C3×C4#45411yescharacteristic
411#67—42no—
412#67—42no—
413C2×Dic3—41no—
414C2×Dic3—42no—
415D12—22no—
416D12—41no—
417C4×D3—42no—
418C4×D3—42no—
419C4×D3—322no—
420Dic6—22no—
421Dic6—41no—
422C3×C8—82no—
423C3×C8C2×#10211yescharacteristic
424C3×C8#41711yescharacteristic
425C3×C8C4×D811yescharacteristic
426C3×C8#44211yescharacteristic
427C3×C8#45911yescharacteristic
428C3×C8#46311yescharacteristic
429C3×C8#48011yescharacteristic
430C3×C8#48111yescharacteristic
431C3⋊C8—42no—
432C3⋊C8—81no—
433C24—81no—
434#41—62no—
435#41—62no—
436#41—121no—
437#41—121no—
438C2×Q8—61no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—61no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—82no—
445C2×D4—81no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4—121no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—121no—
454C22×C4—242no—
455C22×C4#93711yescharacteristic
456Q16—32no—
457Q16—61no—
458SD16—62no—
459SD16—62no—
460D8—32no—
461D8—42no—
462D8—42no—
463D8—42no—
464D8—42no—
465D8—61no—
466D8—81no—
467D8—81no—
468D8—81no—
469D8—81no—
470M16—62no—
471M16—121no—
472M16—122no—
473M16—122no—
474M16—121no—
475M16—121no—
476C2×C8—42no—
477C2×C8—44no—
478C2×C8—121no—
479C2×C8—122no—
480C2×C8—122no—
481C2×C8—121no—
482C2×C8D3×C22⋊C411yescharacteristic
483C2×C8#67⋊C411yescharacteristic
484C2×C8#88311yescharacteristic
485C2×C8#88411yescharacteristic
486C2×C8D3×D811yescharacteristic
487C2×C8D3×D811yescharacteristic
488C2×C8#91111yescharacteristic
489C2×C8#91111yescharacteristic
490C2×C8D3×SD1611yescharacteristic
491C2×C8#91411yescharacteristic
492C2×C8#91511yescharacteristic
493C2×C8#91611yescharacteristic
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—62no—
497C4⋊C4—84no—
498C4⋊C4—242no—
499C22⋊C4—121no—
500C22⋊C4—242no—
501C42—21no—
502C42—121no—
503C42—242no—
504C42C4×#6711yescharacteristic
505C42#93811yescharacteristic
506C16—122no—
507C16—241no—
508C22×C3—22no—
509C22×C3—164no—
510C22×C3—162no—
511C22×C3—321no—
512C22×C3—321no—
513C22×C3—322no—
514C22×C3#161911yescharacteristic
515C22×C3#162211yescharacteristic
516C22×C3#188311yescharacteristic
517C2×D3—42no—
518C2×D3—82no—
519C3×C4—22no—
520C3×C4—22no—
521C3×C4—162no—
522C3×C4#162511yescharacteristic
523C3×C4#162711yescharacteristic
524C3×C4#187411yescharacteristic
525C3×C4#187911yescharacteristic
526Dic3—42no—
527Dic3—82no—
528C23—82no—
529C23—121no—
530C23—162no—
531C23—162no—
532C23—161no—
533C23—161no—
534C23#507711yescharacteristic
535Q8—62no—
536Q8—121no—
537D4—62no—
538D4—88no—
539D4—82no—
540D4—82no—
541D4—122no—
542D4—122no—
543D4—121no—
544D4—164no—
545D4—161no—
546D4—161no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—82no—
551C2×C4—121no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—162no—
556C2×C4—962no—
557C2×C4#495611yescharacteristic
558C2×C4#506211yescharacteristic
559C2×C4#515611yescharacteristic
560C2×C4#515611yescharacteristic
561C2×C4#517011yescharacteristic
562C2×C4#517211yescharacteristic
563C8—82no—
564C8—122no—
565C8—241no—
566C8#475611yescharacteristic
567C8D3×#10211yescharacteristic
568C8#477111yescharacteristic
569C8#477411yescharacteristic
570C8#477711yescharacteristic
571C8#478511yescharacteristic
572C8#478611yescharacteristic
573C8#478711yescharacteristic
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—322no—
577C2×C3#897711yescharacteristic
578C2×C3#1231412yes—
579D3—82no—
580C22—22no—
581C22—122no—
582C22—164no—
583C22—162no—
584C22—242no—
585C22—321no—
586C22—321no—
587C22—322no—
588C22#6966311yescharacteristic
589C22#7126411yescharacteristic
590C22#7128111yescharacteristic
591C4—22no—
592C4—22no—
593C4—122no—
594C4—162no—
595C4—242no—
596C4#7020011yescharacteristic
597C4#7020811yescharacteristic
598C4#7119611yescharacteristic
599C4#7120511yescharacteristic
600C3#13686811yescharacteristic, complemented, Sylow
601C2—21no—
602C2—21no—
603C2—242no—
604C2—322no—
605C2#1068994511yescharacteristic
606C2#1070859812yes—
subgroupG/H
centerC22#71281
commutatorC3×C4×C8C22×C4
FrattiniC2×C4×C8C22×D3
FittingC3×#9837C2
socleC22×C3#1619
prime radicalC2×#71295C2
cosocleC2×C3×C4×C8C23
hypercenter#9837D3
hypocenterC3#136868
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #9837

Complements

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

NH ≅ G/Ntype
2. #10813660C2semidirect
3. #10713698C2semidirect
4. C2×#67443C2semidirect
5. #10682358C2semidirect
6. #10682135C2semidirect
7. C3×#9837C2semidirect
21. #71295C4semidirect
33. #9837D3semidirect
45. C3×#493C2×C4semidirect
85. #4463C2×C4semidirect
199. #493C4×D3semidirect
298. Dic12#102semidirect
300. D24#102semidirect
600. C3#136868semidirect

Maximal subgroups

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

subgroupindex
C2×#712952
#108136602
#107136982
C2×#674432
#106823582
#106821352
C3×#98372
#1368683