Finite Group Databasebeta

← #10725792

Subgroups — #10725792

Order 768.

1550 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4991C211yesmaximal, characteristic
2C3×#1445C211yesmaximal, characteristic, complemented
3#71564C211yesmaximal, characteristic, complemented
4#69429C212yesmaximal, complemented
5#69426C212yesmaximal
6#69408C212yesmaximal, complemented
7#69408C212yesmaximal, complemented
8#69408C212yesmaximal, complemented
9#69402C212yesmaximal, complemented
10#14617—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×(C2×D3)⋊C4C2212yes—
14C22×(C2×D3)⋊C4C2212yes—
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4×Dic3C2212yes—
20C22×C4×Dic3C2211yescharacteristic
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32#4991C418yescomplemented
33C2×#394—31no—
34#1445D311yescharacteristic, complemented
35#1440—32no—
36#1438—32no—
37#1437—32no—
38#1437—32no—
39#1437—32no—
40#1436—31no—
41#1434—32no—
42C24×D3C2312yes—
43C23×C3×C4C2312yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C22×D12—42no—
52C22×D12—42no—
53C22×D12—41no—
54C22×D12—41no—
55C22×D12C2×C414yes—
56C2×C3×C4⋊C4—41no—
57C2×C3×C4⋊C4—41no—
58C2×C3×C4⋊C4C2×C414yes—
59C2×(C2×D3)⋊C4—24no—
60C2×(C2×D3)⋊C4—24no—
61C2×(C2×D3)⋊C4—24no—
62C2×(C2×D3)⋊C4—24no—
63C2×(C2×D3)⋊C4—24no—
64C2×(C2×D3)⋊C4—24no—
65C2×(C2×D3)⋊C4—42no—
66C2×(C2×D3)⋊C4—42no—
67C2×(C2×D3)⋊C4—42no—
68C2×(C2×D3)⋊C4—42no—
69C2×(C2×D3)⋊C4—42no—
70C2×(C2×D3)⋊C4—42no—
71C2×(C2×D3)⋊C4—42no—
72C2×(C2×D3)⋊C4—42no—
73C2×(C2×D3)⋊C4—42no—
74C2×(C2×D3)⋊C4—42no—
75C2×(C2×D3)⋊C4C2×C418yes—
76C2×(C2×D3)⋊C4C2×C418yes—
77C2×C4×Dic3—24no—
78C2×C4×Dic3—42no—
79C2×C4×Dic3—42no—
80C2×C4×Dic3—42no—
81C2×C4×Dic3—41no—
82C3×#95—42no—
83C3×#95—41no—
84C3×#95—42no—
85C3×#95—41no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—42no—
89C3⋊#95—42no—
90C3⋊#95—42no—
91C3⋊#95—41no—
92C3⋊#95C2×C414yes—
93C23×D4—31no—
94C22×C4⋊C4C2×D311yescharacteristic
95C22×C22⋊C4—32no—
96C22×C22⋊C4—32no—
97C22×C22⋊C4—32no—
98C22×C22⋊C4—32no—
99C22×C22⋊C4—32no—
100C22×C22⋊C4—32no—
101C22×C42—32no—
102C22×C42—31no—
103#394—38no—
104C2×#95—31no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—31no—
111C2×#95C2×D312yes—
112C2×#95C2×D312yes—
113C2×#95C2×D311yescharacteristic
114C2×#95C2×D311yescharacteristic
115C24×C3C2411yescharacteristic
116C23×D3—22no—
117C23×D3—22no—
118C23×D3—22no—
119C23×D3—22no—
120C23×D3—22no—
121C23×D3—22no—
122C23×D3—22no—
123C23×D3—22no—
124C23×D3—22no—
125C23×D3—22no—
126C23×D3—22no—
127C23×D3—22no—
128C23×D3—22no—
129C23×D3—22no—
130C23×D3C22×C414yes—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C22×C412yes—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C22×C412yes—
159C22×C3×C4#4114yes—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3C22×C414yes—
179C22×Dic3#4114yes—
180C22×Dic3#4114yes—
181C2×D12—44no—
182C2×D12—48no—
183C2×D12—44no—
184C2×D12—48no—
185C2×D12—44no—
186C3×C4⋊C4—44no—
187(C2×D3)⋊C4—48no—
188(C2×D3)⋊C4—48no—
189(C2×D3)⋊C4—48no—
190(C2×D3)⋊C4—48no—
191(C2×D3)⋊C4—48no—
192(C2×D3)⋊C4—48no—
193C4×Dic3—48no—
194C4×Dic3—44no—
195C25—32no—
196C22×D4—34no—
197C22×D4—122no—
198C22×D4—121no—
199C22×D4—121no—
200C22×D4—122no—
201C23×C4—32no—
202C23×C4—32no—
203C23×C4—32no—
204C23×C4C22×D312yes—
205C23×C4C22×D311yescharacteristic
206C23×C4C22×D311yescharacteristic
207C23×C4C22×D311yescharacteristic
208C23×C4C22×D312yes—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C4×D314yes—
212C2×C22⋊C4—38no—
213C2×C22⋊C4—38no—
214C2×C22⋊C4—64no—
215C2×C22⋊C4—64no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—64no—
218C2×C22⋊C4—64no—
219C2×C22⋊C4—64no—
220C2×C22⋊C4—122no—
221C2×C22⋊C4—122no—
222C2×C22⋊C4—122no—
223C2×C22⋊C4—122no—
224C2×C22⋊C4—122no—
225C2×C22⋊C4—122no—
226C2×C22⋊C4—122no—
227C2×C22⋊C4—122no—
228C2×C22⋊C4—122no—
229C2×C22⋊C4—122no—
230C2×C42—64no—
231C2×C42—122no—
232C2×C42—122no—
233C2×C42—121no—
234C2×C42—122no—
235#95—34no—
236#95—42no—
237#95—41no—
238#95—42no—
239#95—41no—
240#95—64no—
241#95—64no—
242#95—121no—
243#95—122no—
244#95—122no—
245#95—122no—
246C23×C3C23×C411yescharacteristic
247C23×C3C2×#4112yes—
248C23×C3C2×#4112yes—
249C23×C3C2×#4112yes—
250C23×C3H(5,𝔽2)11yescharacteristic
251C23×C3H(5,𝔽2)12yes—
252C23×C3H(5,𝔽2)11yescharacteristic
253C23×C3H(5,𝔽2)12yes—
254C23×C3H(5,𝔽2)12yes—
255C22×D3—24no—
256C22×D3—24no—
257C22×D3—24no—
258C22×D3—24no—
259C22×D3—24no—
260C22×D3—24no—
261C22×D3—24no—
262C22×D3—42no—
263C22×D3—42no—
264C22×D3—42no—
265C22×D3—42no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340D12—416no—
341C24—34no—
342C24—62no—
343C24—62no—
344C24—62no—
345C24—62no—
346C24—62no—
347C24—62no—
348C24—62no—
349C24—62no—
350C24—62no—
351C24—62no—
352C24—62no—
353C24—62no—
354C24—62no—
355C24—62no—
356C24C23×D311yescharacteristic
357C2×D4—124no—
358C2×D4—128no—
359C2×D4—124no—
360C2×D4—124no—
361C2×D4—128no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—34no—
388C22×C4—34no—
389C22×C4—34no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4C2×C4×D312yes—
409C22×C4C2×C4×D312yes—
410C22×C4C2×C4×D312yes—
411C22×C4#23914yes—
412C4⋊C4—44no—
413C22⋊C4—128no—
414C22⋊C4—128no—
415C22⋊C4—128no—
416C22⋊C4—128no—
417C22⋊C4—128no—
418C22⋊C4—128no—
419C42—128no—
420C42—124no—
421C22×C3C4×#4111yescharacteristic
422C22×C3C4×#4112yes—
423C22×C3#51811yescharacteristic
424C22×C3#51811yescharacteristic
425C22×C3#51812yes—
426C22×C3#53812yes—
427C22×C3#53812yes—
428C22×C3#53812yes—
429C22×C3#54012yes—
430C22×C3#54012yes—
431C22×C3#54012yes—
432C22×C3#55112yes—
433C22×C3#55112yes—
434C22×C3#55312yes—
435C22×C3#55312yes—
436C22×C3#55511yescharacteristic
437C22×C3#55511yescharacteristic
438C22×C3#55912yes—
439C22×C3#56011yescharacteristic
440C22×C3#56012yes—
441C22×C3H(3,𝔽4)11yescharacteristic
442C2×D3—44no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—44no—
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—82no—
450C2×D3—82no—
451C2×D3—82no—
452C2×D3—82no—
453C2×D3—82no—
454C2×D3—82no—
455C2×D3—82no—
456C2×D3—82no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—44no—
461C3×C4—82no—
462Dic3—44no—
463Dic3—44no—
464Dic3—44no—
465C23—64no—
466C23—64no—
467C23—64no—
468C23—64no—
469C23—64no—
470C23—64no—
471C23—64no—
472C23—122no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23—122no—
483C23—122no—
484C23—122no—
485C23—122no—
486C23—122no—
487C23—122no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23C22×C4×D311yescharacteristic
501C23#100411yescharacteristic
502C23#100412yes—
503C23#100411yescharacteristic
504C23C2×#23912yes—
505C23D3×#4112yes—
506C23D3×#4112yes—
507C23#100912yes—
508C23#100912yes—
509D4—1216no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C3#233311yescharacteristic
561C2×C3#233312yes—
562C2×C3#237211yescharacteristic
563C2×C3#237412yes—
564C2×C3#237511yescharacteristic
565C2×C3#275212yes—
566C2×C3#275312yes—
567C2×C3#275612yes—
568C2×C3#280312yes—
569D3—84no—
570C22—124no—
571C22—124no—
572C22—124no—
573C22—124no—
574C22—124no—
575C22—124no—
576C22—124no—
577C22—242no—
578C22—242no—
579C22—242no—
580C22—242no—
581C22—242no—
582C22—242no—
583C22—242no—
584C22—242no—
585C22#548611yescharacteristic
586C22#552111yescharacteristic
587C22#552211yescharacteristic
588C22#554512yes—
589C22C4×#23912yes—
590C22#562012yes—
591C22#563712yes—
592C22#563712yes—
593C22#564412yes—
594C22#565712yes—
595C22#566312yes—
596C22#566312yes—
597C22#566412yes—
598C22#567011yescharacteristic
599C22#568311yescharacteristic
600C22#569512yes—
601C22#569712yes—
602C22#570412yes—
603C22#570912yes—
604C22#572011yescharacteristic
605C22#573411yescharacteristic
606C4—42no—
607C4—42no—
608C4—42no—
609C4—44no—
610C4—82no—
611C4—124no—
612C4—124no—
613C4—124no—
614C3#1461711yescharacteristic, complemented, Sylow
615C2—244no—
616C2#7405811yescharacteristic
617C2#7495212yes—
618C2#7497512yes—
619C2#7643411yescharacteristic
620C2#7649511yescharacteristic
621C2#7764412yes—
622C2#7787812yes—
623C2#7807812yes—
624C2#7825212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1445C2
socle, cosocleC24×C3C24
prime radicalC23×D12C22
hypercenter#1445D3
hypocenterC3#14617
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C3×#1445C2semidirect
3. #71564C2semidirect
4. #69429C2semidirect
6. #69408C2semidirect
7. #69408C2semidirect
8. #69408C2semidirect
9. #69402C2semidirect
32. #4991C4semidirect
34. #1445D3semidirect
614. C3#14617semidirect

Maximal subgroups

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

subgroupindex
C2×#49912
C3×#14452
#715642
#694292
#694262
#694082
#694022
#146173