Finite Group Databasebeta

← #10731955

Subgroups — #10731955

Order 768.

1710 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×#4987C211yesmaximal, characteristic
2C2×#4987C211yesmaximal, characteristic
3C2×#4975C211yesmaximal, characteristic, complemented
4C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
5C2×#4675C214yesmaximal
6C3×C4⋊#95C211yesmaximal, characteristic, complemented
7#71563C211yesmaximal, characteristic, complemented
8#71535C211yesmaximal, characteristic, complemented
9#69421C214yesmaximal, complemented
10#15150—31nomaximal, Sylow
11C23×C4×D3C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2214yes—
21C22×C4×Dic3C2211yescharacteristic
22C2×C3×#95C2214yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2214yes—
28#4987—24no—
29#4987—24no—
30#4975—24no—
31C4×(C2×D3)⋊C4—24no—
32#4675—216no—
33C2×#397—34no—
34C2×#384—31no—
35C2×#380—31no—
36C2×#380—31no—
37C2×C4×C22⋊C4—31no—
38#1445—34no—
39C4⋊#95—31no—
40C4⋊#95—31no—
41C4⋊#95D311yescharacteristic, complemented
42C24×D3C2311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2311yescharacteristic
50C23×Dic3C2312yes—
51C22×C4×D3—22no—
52C22×C4×D3—22no—
53C22×C4×D3—22no—
54C22×C4×D3—22no—
55C22×C4×D3—22no—
56C22×C4×D3—22no—
57C22×C4×D3D414yes—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C42—22no—
65C2×C3×C42—22no—
66C2×C3×C42—22no—
67C2×(C2×D3)⋊C4—24no—
68C2×(C2×D3)⋊C4—24no—
69C2×(C2×D3)⋊C4—28no—
70C2×(C2×D3)⋊C4—28no—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—24no—
73C2×(C2×D3)⋊C4D418yes—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×Dic3⋊C4—28no—
81C2×Dic3⋊C4—28no—
82C2×Dic3⋊C4—28no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3D414yes—
86C3×#95—28no—
87C3⋊#95—28no—
88C3⋊#95—28no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C24×C4—31no—
93C22×C4⋊C4—34no—
94C22×C4⋊C4—31no—
95C22×C4⋊C4—31no—
96C22×C4⋊C4C2×D311yescharacteristic
97C22×C4⋊C4C2×D311yescharacteristic
98C22×C22⋊C4—32no—
99C22×C22⋊C4—32no—
100C22×C22⋊C4—32no—
101C22×C42—31no—
102C22×C42C2×D311yescharacteristic
103#397—616no—
104#384—64no—
105#380—64no—
106#380—64no—
107C4×C22⋊C4—64no—
108C2×#95—34no—
109C2×#95—34no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95C2×D314yes—
114C24×C3C2411yescharacteristic
115C23×D3—21no—
116C23×D3—22no—
117C23×D3—21no—
118C23×D3—21no—
119C23×D3—21no—
120C23×D3—21no—
121C23×D3—21no—
122C23×D3—22no—
123C23×D3—22no—
124C23×D3C2×D414yes—
125C23×D3C2×D412yes—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4—22no—
137C22×C3×C4—24no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4C2×D414yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4#4112yes—
146C22×C3×C4#4112yes—
147C22×C3×C4#4112yes—
148C22×C3×C4#4112yes—
149C22×C3×C4#4112yes—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—24no—
161C22×Dic3—24no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—24no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—24no—
170C22×Dic3C2×D412yes—
171C22×Dic3C2×D414yes—
172C22×Dic3C2×D414yes—
173C22×Dic3#4114yes—
174C2×C4×D3—28no—
175C2×C4×D3—28no—
176C2×C4×D3—28no—
177C2×C4×D3—44no—
178C2×C4×D3—44no—
179C2×C4×D3—44no—
180C2×C4×D3—44no—
181C3×C4⋊C4—44no—
182C3×C4⋊C4—44no—
183C3×C42—44no—
184(C2×D3)⋊C4—216no—
185(C2×D3)⋊C4—48no—
186(C2×D3)⋊C4—48no—
187C4⋊Dic3—44no—
188C4⋊Dic3—44no—
189Dic3⋊C4—416no—
190C4×Dic3—28no—
191C25—31no—
192C23×C4—32no—
193C23×C4—31no—
194C23×C4—34no—
195C23×C4—32no—
196C23×C4—32no—
197C23×C4—62no—
198C23×C4—62no—
199C23×C4—62no—
200C23×C4—62no—
201C23×C4—62no—
202C23×C4—62no—
203C23×C4C22×D312yes—
204C23×C4C22×D312yes—
205C23×C4C22×D312yes—
206C23×C4C22×D311yescharacteristic
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—68no—
214C2×C4⋊C4—68no—
215C2×C4⋊C4—68no—
216C2×C4⋊C4—62no—
217C2×C4⋊C4—62no—
218C2×C4⋊C4—62no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—62no—
222C2×C22⋊C4—38no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—64no—
225C2×C22⋊C4—64no—
226C2×C22⋊C4—64no—
227C2×C22⋊C4—68no—
228C2×C22⋊C4—68no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—34no—
233C2×C42—62no—
234C2×C42—62no—
235#95—28no—
236#95—64no—
237#95—64no—
238#95—64no—
239#95—68no—
240#95—68no—
241C23×C3C22×D411yescharacteristic
242C23×C3C22×D411yescharacteristic
243C23×C3C22×D412yes—
244C23×C3C2×#4111yescharacteristic
245C23×C3C2×#4111yescharacteristic
246C23×C3C2×#4111yescharacteristic
247C23×C3C2×#4111yescharacteristic
248C23×C3C2×#4111yescharacteristic
249C23×C3C2×#4112yes—
250C23×C32−1+412yes—
251C23×C32−1+412yes—
252C22×D3—24no—
253C22×D3—22no—
254C22×D3—22no—
255C22×D3—22no—
256C22×D3—22no—
257C22×D3—24no—
258C22×D3—22no—
259C22×D3—22no—
260C22×D3—24no—
261C22×D3—24no—
262C22×D3—24no—
263C22×D3—24no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3#12714yes—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—44no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—44no—
289C2×C3×C4—42no—
290C2×C3×C4—44no—
291C2×C3×C4—44no—
292C2×C3×C4#12114yes—
293C2×C3×C4#12314yes—
294C2×C3×C4#12314yes—
295C2×C3×C4#12314yes—
296C2×C3×C4#12414yes—
297C2×C3×C4#12614yes—
298C2×C3×C4#12614yes—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—44no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—44no—
319C2×Dic3—42no—
320C2×Dic3—44no—
321C2×Dic3—44no—
322C2×Dic3#12118yes—
323C4×D3—416no—
324C24—34no—
325C24—32no—
326C24—61no—
327C24—61no—
328C24—61no—
329C24—61no—
330C24—61no—
331C24—61no—
332C24—62no—
333C24—62no—
334C24—62no—
335C24C23×D311yescharacteristic
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—34no—
353C22×C4—34no—
354C22×C4—34no—
355C22×C4—32no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—68no—
367C22×C4—68no—
368C22×C4—68no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—64no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—64no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—64no—
378C22×C4—64no—
379C22×C4—124no—
380C22×C4—124no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4#23512yes—
384C22×C4#23512yes—
385C22×C4#23512yes—
386C22×C4D3×D414yes—
387C22×C4D3×D412yes—
388C22×C4D3×D412yes—
389C22×C4#23712yes—
390C22×C4#23912yes—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—1216no—
394C4⋊C4—124no—
395C4⋊C4—124no—
396C22⋊C4—616no—
397C22⋊C4—128no—
398C22⋊C4—128no—
399C42—44no—
400C42—68no—
401C22×C3C2×#12111yescharacteristic
402C22×C3C2×#12112yes—
403C22×C3C2×#12311yescharacteristic
404C22×C3C2×#12311yescharacteristic
405C22×C3C2×#12311yescharacteristic
406C22×C3C2×#12411yescharacteristic
407C22×C3C2×#12611yescharacteristic
408C22×C3C2×#12611yescharacteristic
409C22×C3C2×#12711yescharacteristic
410C22×C3#53211yescharacteristic
411C22×C3#53212yes—
412C22×C3#53611yescharacteristic
413C22×C3#53611yescharacteristic
414C22×C3#53612yes—
415C22×C3#54112yes—
416C22×C3#54712yes—
417C22×C3#54712yes—
418C22×C3#54712yes—
419C22×C3#54712yes—
420C22×C3#55212yes—
421C22×C3#55212yes—
422C22×C3#55612yes—
423C22×C3#55612yes—
424C2×D3—24no—
425C2×D3—24no—
426C2×D3—24no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—44no—
430C2×D3—44no—
431C2×D3—44no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—44no—
435C3×C4—44no—
436C3×C4—82no—
437C3×C4—82no—
438C3×C4#39718yescomplemented
439Dic3—28no—
440Dic3—42no—
441Dic3—82no—
442Dic3—82no—
443C23—34no—
444C23—62no—
445C23—62no—
446C23—62no—
447C23—64no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—64no—
452C23—64no—
453C23—64no—
454C23—64no—
455C23—64no—
456C23—122no—
457C23—122no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23—122no—
466C23C2×#23511yescharacteristic
467C23C2×#23511yescharacteristic
468C23C2×#23511yescharacteristic
469C23C2×D3×D411yescharacteristic
470C23C2×D3×D412yes—
471C23C2×D3×D411yescharacteristic
472C23C2×#23711yescharacteristic
473C23C2×#23911yescharacteristic
474C23#100712yes—
475C23D3×#4112yes—
476C23#101012yes—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—38no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—1216no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—124no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—124no—
517C2×C4—124no—
518C2×C4—124no—
519C2×C4#87614yes—
520C2×C4#88314yes—
521C2×C4#88414yes—
522C2×C4#88514yes—
523C2×C4#89414yes—
524C2×C4#89414yes—
525C2×C4#89814yes—
526C2×C3C2×#39711yescharacteristic
527C2×C3#248612yes—
528C2×C3#249612yes—
529C2×C3#249612yes—
530C2×C3#249911yescharacteristic
531C2×C3#249911yescharacteristic
532C2×C3#250111yescharacteristic
533C2×C3#250111yescharacteristic
534C2×C3#254712yes—
535C2×C3#262111yescharacteristic
536C2×C3#285611yescharacteristic
537D3—44no—
538C22—64no—
539C22—64no—
540C22—64no—
541C22—124no—
542C22—122no—
543C22—122no—
544C22—122no—
545C22—124no—
546C22—124no—
547C22—124no—
548C22—122no—
549C22C2×#87611yescharacteristic
550C22C2×#88311yescharacteristic
551C22C2×#88411yescharacteristic
552C22C2×#88511yescharacteristic
553C22C2×#89411yescharacteristic
554C22C2×#89411yescharacteristic
555C22#550712yes—
556C22C2×#89811yescharacteristic
557C22#553312yes—
558C22#554812yes—
559C22#555112yes—
560C22#558112yes—
561C22#558312yes—
562C22D3×#12112yes—
563C22#561112yes—
564C22#562812yes—
565C22#564112yes—
566C22#566811yescharacteristic
567C22#567511yescharacteristic
568C22#569412yes—
569C22D3×#12711yescharacteristic
570C22#572411yescharacteristic
571C22#572612yes—
572C4—44no—
573C4—68no—
574C4—82no—
575C4—82no—
576C4—122no—
577C4—242no—
578C4—242no—
579C4#467518yescomplemented
580C3#1515011yescharacteristic, complemented, Sylow
581C2—124no—
582C2C2×#467511yescharacteristic
583C2#7379212yes—
584C2#7393712yes—
585C2#7416511yescharacteristic
586C2#7432811yescharacteristic
587C2#7623112yes—
588C2#7623712yes—
589C2#7633511yescharacteristic
590C2#7647611yescharacteristic
591C2#7651811yescharacteristic
592C2#7664111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C4⋊#95C2
prime radicalC24×D3C23
hypercenterC4⋊#95D3
hypocenterC3#15150
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#4975C2semidirect
6. C3×C4⋊#95C2semidirect
7. #71563C2semidirect
8. #71535C2semidirect
9. #69421C2semidirect
41. C4⋊#95D3semidirect
438. C3×C4#397semidirect
579. C4#4675semidirect
580. C3#15150semidirect

Maximal subgroups

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

subgroupindex
C2×#49872
C2×#49752
C2×C4×(C2×D3)⋊C42
C2×#46752
C3×C4⋊#952
#715632
#715352
#694212
#151503