Finite Group Databasebeta

← #10823948

Subgroups — #10823948

Order 768.

1226 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4770C212yesmaximal
2D3×#390C211yesmaximal, characteristic
3#74318C211yesmaximal, characteristic, complemented
4C3×#2071C211yesmaximal, characteristic, complemented
5#70166C212yesmaximal, complemented
6#69640C211yesmaximal, characteristic, complemented
7#69585C211yesmaximal, characteristic, complemented
8#68903C212yesmaximal, complemented
9#68867C212yesmaximal, complemented
10#68660C211yesmaximal, characteristic, complemented
11#68635C211yesmaximal, characteristic, complemented
12#23628—31nomaximal, Sylow
13C22×D3×D4C2211yescharacteristic
14C22×ΓL(1,𝔽25)C2212yes—
15C2×#892C2212yes—
16C2×D3×C4⋊C4C2211yescharacteristic
17C2×D3×C22⋊C4—21no—
18C2×D3×C22⋊C4—21no—
19C2×C42×D3C2211yescharacteristic
20C2×C3×#102C2212yes—
21C3×#390C2211yescharacteristic
22C3×#384C2211yescharacteristic, complemented
23#5221C2211yescharacteristic
24C2×#832C2212yes—
25C2×#821C2212yes—
26#4988C2211yescharacteristic
27#4987—21no—
28#4987—21no—
29#4977C2212yescomplemented
30Dic3×C4⋊C4—21no—
31Dic3×C4⋊C4—21no—
32C2×#809C2212yes—
33#4957—21no—
34#4957—21no—
35#4939C2211yescharacteristic
36#4935C2211yescharacteristic, complemented
37#4770—28no—
38C3×#340C2212yescomplemented
39C3×#336C2211yescharacteristic, complemented
40#4551C2212yescomplemented
41#4550C2212yescomplemented
42#4529C2212yescomplemented
43#4529C2212yescomplemented
44#4524C2211yescharacteristic, complemented
45C2×#418—32no—
46C2×#390—31no—
47#2285—31no—
48#2073—32no—
49#2073—32no—
50#2071—31no—
51#2071—31no—
52#2071—31no—
53#2071D311yescharacteristic, complemented
54#1736—32no—
55#1728—31no—
56C24×D3—21no—
57C22×C3×D4C2311yescharacteristic
58C22×#67—21no—
59C2×D3×D4—24no—
60C2×D3×D4—24no—
61C2×D3×D4—22no—
62C2×D3×D4D414yes—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×C4×D3D411yescharacteristic
66C22×C4×D3D411yescharacteristic
67C22×C4×D3C2311yescharacteristic
68C22×C3×C8C2312yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—21no—
77C2×#217—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—21no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93C2×ΓL(1,𝔽25)—24no—
94C2×ΓL(1,𝔽25)—24no—
95C2×ΓL(1,𝔽25)—24no—
96#892—24no—
97#892—24no—
98D3×C4⋊C4—24no—
99D3×C22⋊C4—44no—
100D3×C22⋊C4—44no—
101C42×D3—24no—
102C3×#102—24no—
103#832—24no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106#821—24no—
107#809—24no—
108C23×D4—31no—
109C22×M16—32no—
110C2×#117—32no—
111C22×C4⋊C4—31no—
112C22×C22⋊C4—61no—
113C22×C22⋊C4—61no—
114C22×C42—31no—
115#418—68no—
116C2×#102—32no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102C2×D312yes—
120#390—31no—
121#390—31no—
122#390—31no—
123#390—61no—
124#390—61no—
125#390C2×D311yescharacteristic
126#384—31no—
127#384C2×D311yescharacteristic, complemented
128#382—32no—
129#380—61no—
130#380—61no—
131C4×C4⋊C4—61no—
132C4×C4⋊C4—61no—
133#340—32no—
134#340—32no—
135#340—32no—
136#340C2×D312yescomplemented
137#336—32no—
138#336—31no—
139#336C2×D311yescharacteristic, complemented
140C24×C3—21no—
141C23×D3—21no—
142C23×D3—24no—
143C23×D3—42no—
144C23×D3—42no—
145C23×D3—42no—
146C23×D3—42no—
147C23×D3—42no—
148C23×D3—42no—
149C23×D3C2×D411yescharacteristic
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—21no—
153C2×C3×D4C2×D412yes—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C2×#67—24no—
160C2×#67—42no—
161C2×#67—42no—
162C22×Dic3—22no—
163C22×Dic3—21no—
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167D3×D4—216no—
168D3×D4—48no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—21no—
172C2×D12C2×D412yes—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—41no—
178C2×C4×D3—41no—
179C2×C4×D3—42no—
180C2×C4×D3—42no—
181C2×C4×D3—42no—
182C2×C4×D3C2×D412yes—
183C2×C4×D3#4114yes—
184C2×C4×D3#4114yes—
185C2×C4×D3#4112yes—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—42no—
196C3×C42—22no—
197#217—42no—
198#217—42no—
199(C2×D3)⋊C4—42no—
200(C2×D3)⋊C4—42no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—42no—
203(C2×D3)⋊C4—42no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—42no—
208C4⋊Dic3—42no—
209Dic3⋊C4—42no—
210C4×Dic3—22no—
211C4×Dic3—22no—
212C4×Dic3—22no—
213C4×Dic3—44no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217ΓL(1,𝔽25)—48no—
218C25—61no—
219C22×D4—34no—
220C22×D4—31no—
221C22×D4—64no—
222C22×D4—64no—
223C22×D4—62no—
224C22×D4—61no—
225C22×D4C22×D311yescharacteristic
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—61no—
230C2×M16—64no—
231C2×M16—64no—
232C2×M16—64no—
233C22×C8—32no—
234C22×C8C22×D312yes—
235#117—64no—
236#117—64no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—61no—
245C2×C4⋊C4C22×D311yescharacteristic
246C2×C4⋊C4C22×D312yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—61no—
250C2×C22⋊C4—61no—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—61no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—124no—
257C2×C22⋊C4—124no—
258C2×C42—31no—
259C2×C42—31no—
260C2×C42—31no—
261C2×C42—64no—
262C2×C42—62no—
263C2×C42C22×D311yescharacteristic
264#102—24no—
265#102—64no—
266#102—64no—
267#102—64no—
268#95—62no—
269#95—62no—
270C23×C3—22no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3C22×D411yescharacteristic
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—21no—
282C22×D3—22no—
283C22×D3—48no—
284C22×D3—48no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—48no—
292C22×D3—84no—
293C22×D3—84no—
294C22×D3—84no—
295C22×D3—84no—
296C22×D3#12112yes—
297C22×D3#12312yes—
298C22×D3#12412yes—
299C3×D4—24no—
300C3×D4—42no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4AGL(1,ℤ/8)12yes—
312C2×C3×C4AGL(1,ℤ/8)12yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4112yes—
316C2×C3×C4C2×#4111yescharacteristic
317#67—48no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3#12112yes—
331C2×Dic3#12312yes—
332C2×Dic3#12412yes—
333D12—24no—
334D12—42no—
335C4×D3—44no—
336C4×D3—44no—
337C4×D3—84no—
338C4×D3#12318yes—
339C4×D3#12618yes—
340C3×C8—42no—
341C3⋊C8—42no—
342C24—21no—
343C24—31no—
344C24—64no—
345C24—61no—
346C24—122no—
347C24—122no—
348C24—122no—
349C24—122no—
350C24—122no—
351C24—122no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—32no—
356C2×D4—616no—
357C2×D4—64no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—61no—
361C2×D4—128no—
362C2×D4—122no—
363C2×D4—122no—
364C2×D4D3×D412yes—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—31no—
368C22×C4—32no—
369C22×C4—34no—
370C22×C4—34no—
371C22×C4—32no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4—121no—
381C22×C4—121no—
382C22×C4—122no—
383C22×C4—122no—
384C22×C4—122no—
385C22×C4D3×D411yescharacteristic
386C22×C4D3×D411yescharacteristic
387C22×C4C23×D311yescharacteristic
388M16—128no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—122no—
404C4⋊C4—122no—
405C4⋊C4—122no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—122no—
409C22⋊C4—122no—
410C22⋊C4—124no—
411C22⋊C4—122no—
412C22⋊C4—122no—
413C22⋊C4—122no—
414C22⋊C4—122no—
415C42—22no—
416C42—62no—
417C42—62no—
418C42—62no—
419C42—124no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3C2×#12111yescharacteristic
428C22×C3C2×#12311yescharacteristic
429C22×C3C2×#12411yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
432C22×C3#57511yescharacteristic
433C22×C3#57511yescharacteristic
434C2×D3—22no—
435C2×D3—22no—
436C2×D3—22no—
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—816no—
444C2×D3—81no—
445C2×D3—81no—
446C2×D3—81no—
447C2×D3—81no—
448C2×D3#39714yes—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—81no—
452C3×C4—82no—
453C3×C4C2×#12312yes—
454C3×C4C2×#12612yes—
455Dic3—41no—
456Dic3—41no—
457Dic3—81no—
458Dic3—82no—
459Dic3#39714yes—
460C23—22no—
461C23—32no—
462C23—32no—
463C23—32no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—61no—
471C23—61no—
472C23—61no—
473C23—61no—
474C23—62no—
475C23—128no—
476C23—128no—
477C23—128no—
478C23—121no—
479C23—121no—
480C23—121no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—244no—
485C23—244no—
486C23—244no—
487C23—244no—
488C23C2×D3×D411yescharacteristic
489D4—24no—
490D4—42no—
491D4—64no—
492D4—128no—
493D4—122no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—32no—
501C2×C4—38no—
502C2×C4—38no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—124no—
516C2×C4—124no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—244no—
524C2×C4#91112yes—
525C2×C4#91412yes—
526C2×C4C2×D3×D411yescharacteristic
527C2×C4D3×#4112yes—
528C2×C4D3×#4112yes—
529C2×C4D3×#4111yescharacteristic
530C8—42no—
531C8—122no—
532C2×C3—82no—
533C2×C3C2×#39711yescharacteristic
534C2×C3#303611yescharacteristic
535C2×C3#303611yescharacteristic
536C2×C3#309411yescharacteristic
537C2×C3#309411yescharacteristic
538C2×C3#313811yescharacteristic
539C2×C3#314011yescharacteristic
540D3—24no—
541D3—82no—
542C22—34no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—62no—
547C22—62no—
548C22—62no—
549C22—62no—
550C22—62no—
551C22—62no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—122no—
557C22—122no—
558C22—122no—
559C22—2416no—
560C22—241no—
561C22—241no—
562C22—241no—
563C22—241no—
564C22D3×#12111yescharacteristic
565C22D3×#12311yescharacteristic
566C22D3×#12411yescharacteristic
567C22C2×#91111yescharacteristic
568C22C2×#91411yescharacteristic
569C22#577311yescharacteristic
570C22#577711yescharacteristic
571C4—22no—
572C4—22no—
573C4—34no—
574C4—81no—
575C4—82no—
576C4—121no—
577C4—121no—
578C4—242no—
579C4—241no—
580C4D3×#12312yes—
581C4D3×#12612yes—
582C3#2362811yescharacteristic, complemented, Sylow
583C2—64no—
584C2—82no—
585C2—242no—
586C2D3×#39711yescharacteristic
587C2#8033211yescharacteristic
588C2#8042811yescharacteristic
589C2#8128311yescharacteristic
590C2#8132611yescharacteristic
591C2#8156411yescharacteristic
592C2#8168211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenter#2071D3
hypocenterC3#23628
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
3. #74318C2semidirect
4. C3×#2071C2semidirect
5. #70166C2semidirect
6. #69640C2semidirect
7. #69585C2semidirect
8. #68903C2semidirect
9. #68867C2semidirect
10. #68660C2semidirect
11. #68635C2semidirect
22. C3×#384C22semidirect
29. #4977C22semidirect
36. #4935C22semidirect
38. C3×#340C22semidirect
39. C3×#336C22semidirect
40. #4551C22semidirect
41. #4550C22semidirect
42. #4529C22semidirect
43. #4529C22semidirect
44. #4524C22semidirect
53. #2071D3semidirect
127. #384C2×D3semidirect
136. #340C2×D3semidirect
139. #336C2×D3semidirect
582. C3#23628semidirect

Maximal subgroups

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

subgroupindex
C2×#47702
D3×#3902
#743182
C3×#20712
#701662
#696402
#695852
#689032
#688672
#686602
#686352
#236283