Finite Group Databasebeta

← #10868451

Subgroups — #10868451

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82466C211yesmaximal, characteristic
2#74173C211yesmaximal, characteristic, complemented
3C3×#1805C211yesmaximal, characteristic, complemented
4#71632C211yesmaximal, characteristic, complemented
5#70482C211yesmaximal, characteristic
6#70447C211yesmaximal, characteristic, complemented
7#70278C211yesmaximal, characteristic
8#70258C211yesmaximal, characteristic, complemented
9#70245C211yesmaximal, characteristic
10#70241C211yesmaximal, characteristic, complemented
11#69558C211yesmaximal, characteristic
12#68896C211yesmaximal, characteristic
13#68821C211yesmaximal, characteristic, complemented
14#68793C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic
16#27070—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#382C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#4990—21no—
32#4989—21no—
33#4984C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4940—21no—
44#4939C2211yescharacteristic
45C4×(C2×D3)⋊C4—21no—
46C4×Dic3⋊C4C2211yescharacteristic
47#4929C2211yescharacteristic
48C3⋊#401—21no—
49#4660—21no—
50#4654—21no—
51#4651—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2514—31no—
61#2367—31no—
62#2092—31no—
63#2055—31no—
64#2042—31no—
65#2015—31no—
66#1969—31no—
67#1927—31no—
68#1927—31no—
69#1921—31no—
70#1916—31no—
71#1878—31no—
72#1877—31no—
73#1805D311yescharacteristic, complemented
74C4×#102—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×Q8—22no—
82C3×C4×Q8—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89#947—22no—
90#947—22no—
91C2×#216—22no—
92C2×#216—22no—
93C2×#216—41no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4D411yescharacteristic
103C2×Dic3⋊C4D411yescharacteristic
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×#204—22no—
111C2×#204—41no—
112C2×#204—41no—
113#893—22no—
114#893—41no—
115C3×C4⋊C8—41no—
116C3×#103—41no—
117C3×#103—41no—
118C3×#95—21no—
119C3×#95—21no—
120#835—22no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#821—22no—
135#816—41no—
136C8×Dic3—41no—
137#810—22no—
138#809—22no—
139#809—41no—
140#808—41no—
141#807—22no—
142C22×Q16—31no—
143C22×SD16—31no—
144C2×#124—31no—
145C2×C4×Q8C2×D311yescharacteristic
146C2×C4×D4—31no—
147C2×#107—31no—
148C2×#106—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#103C2×D311yescharacteristic
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×C4×C8—31no—
159#401—61no—
160#400—61no—
161#391—31no—
162#391—31no—
163#391—31no—
164#391—31no—
165#390—31no—
166#389—31no—
167#388—61no—
168#388—61no—
169#387—61no—
170#385—61no—
171#383—61no—
172#382C2×D311yescharacteristic
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C22⋊C4—61no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C23×D3—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8#4112yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12#4112yes—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—81no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4#4112yes—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—42no—
237#216—44no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—81no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248Dic3⋊C4—22no—
249Dic3⋊C4—41no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—42no—
255C4×Dic3—81no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8#4112yes—
262#204—44no—
263C22×Q8—31no—
264C22×Q8C22×D311yescharacteristic
265C22×D4—31no—
266C23×C4—61no—
267C2×Q16—62no—
268C2×Q16—62no—
269C2×Q16—121no—
270C2×SD16—62no—
271C2×SD16—121no—
272C2×SD16—121no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277#124—62no—
278#124—62no—
279C4×Q8—22no—
280C4×Q8—41no—
281C4×D4—62no—
282C4×D4—121no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—21no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—62no—
304#106—121no—
305C4⋊C8—41no—
306#103—41no—
307#103—41no—
308#103—62no—
309#103—62no—
310#103—121no—
311#102—62no—
312#102—62no—
313#102—121no—
314C4×C8—121no—
315#95—21no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×C3×C4H(5,𝔽2)11yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3#13512yes—
387D12—22no—
388D12—22no—
389D12—41no—
390D12—41no—
391C4×D3—84no—
392Dic6—41no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396C3×C8—41no—
397C3×C8—41no—
398C3⋊C8—22no—
399C3⋊C8—41no—
400C24—61no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8#23912yes—
413C2×D4—32no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—241no—
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4C23×D311yescharacteristic
436Q16—124no—
437SD16—124no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—32no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—62no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—121no—
462C4⋊C4—122no—
463C4⋊C4#23512yes—
464C22⋊C4—64no—
465C22⋊C4—64no—
466C22⋊C4—122no—
467C22⋊C4—122no—
468C22⋊C4—241no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42—122no—
477C42—241no—
478C22×C3#52511yescharacteristic
479C22×C3#54611yescharacteristic
480C22×C3#55011yescharacteristic
481C22×C3C2×#13511yescharacteristic
482C22×C3#57511yescharacteristic
483C22×C3#57711yescharacteristic
484C22×C3#57811yescharacteristic
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#52911yescharacteristic
499C3×C4#53811yescharacteristic
500C3×C4#55111yescharacteristic
501C3×C4#55311yescharacteristic
502Dic3—22no—
503Dic3—41no—
504Dic3—41no—
505Dic3—41no—
506Dic3—81no—
507Dic3—81no—
508Dic3—81no—
509C23—62no—
510C23—121no—
511C23—121no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23C2×D3×D411yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521Q8—121no—
522Q8—121no—
523Q8—121no—
524Q8—121no—
525D4—62no—
526D4—62no—
527D4—121no—
528D4—121no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—32no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—244no—
569C2×C4C2×#23511yescharacteristic
570C2×C4#100411yescharacteristic
571C2×C4#100411yescharacteristic
572C2×C4C2×#23911yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C8—41no—
576C8—41no—
577C8—62no—
578C8—121no—
579C2×C3#246711yescharacteristic
580C2×C3#310211yescharacteristic
581C2×C3#312011yescharacteristic
582C2×C3#329611yescharacteristic
583C2×C3#331311yescharacteristic
584C2×C3#339311yescharacteristic
585C2×C3#340111yescharacteristic
586D3—82no—
587C22—122no—
588C22—122no—
589C22—122no—
590C22—241no—
591C22—241no—
592C22—241no—
593C22—241no—
594C22#549011yescharacteristic
595C22#559911yescharacteristic
596C22#562711yescharacteristic
597C22D3×#13511yescharacteristic
598C22#577011yescharacteristic
599C22#577711yescharacteristic
600C22#577911yescharacteristic
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—62no—
606C4—81no—
607C4—81no—
608C4—121no—
609C4—121no—
610C4—121no—
611C4—241no—
612C4—241no—
613C4—241no—
614C4#551411yescharacteristic
615C4#552811yescharacteristic
616C4#561311yescharacteristic
617C4#563711yescharacteristic
618C3#2707011yescharacteristic, complemented, Sylow
619C2—242no—
620C2#7390611yescharacteristic
621C2#7884511yescharacteristic
622C2#7896711yescharacteristic
623C2#8058811yescharacteristic
624C2#8059311yescharacteristic
625C2#8120411yescharacteristic
626C2#8120911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1805D3
hypocenterC3#27070
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. #74173C2semidirect
3. C3×#1805C2semidirect
4. #71632C2semidirect
6. #70447C2semidirect
8. #70258C2semidirect
10. #70241C2semidirect
13. #68821C2semidirect
14. #68793C2semidirect
73. #1805D3semidirect
618. C3#27070semidirect

Maximal subgroups

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

subgroupindex
#824662
#741732
C3×#18052
#716322
#704822
#704472
#702782
#702582
#702452
#702412
#695582
#688962
#688212
#687932
#687802
#270703