Finite Group Databasebeta

← #10731954

Subgroups — #10731954

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×C3×#380C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic
3C2×C4×#217C211yesmaximal, characteristic
4C2×#4975C211yesmaximal, characteristic, complemented
5C2×#4952C214yesmaximal
6#71563C211yesmaximal, characteristic, complemented
7#71563C211yesmaximal, characteristic, complemented
8#71563C211yesmaximal, characteristic, complemented
9#69421C214yesmaximal, complemented
10#15150—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×#217C2212yes—
16C22×#217C2212yes—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×Dic3⋊C4C2214yes—
20C22×C4×Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C3×#380—24no—
23C2×C3×#95C2212yes—
24#5212—24no—
25C4×#217—24no—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31#4975—24no—
32#4952—216no—
33C2×#397—34no—
34C2×#384—31no—
35C2×#380—31no—
36C2×#380D311yescharacteristic
37C2×C4×C22⋊C4—31no—
38#1445—34no—
39C4⋊#95—31no—
40C4⋊#95—31no—
41C4⋊#95—31no—
42C25×C3C2311yescharacteristic
43C23×C3×C4—22no—
44C23×C3×C4—22no—
45C23×C3×C4—22no—
46C23×C3×C4—22no—
47C23×C3×C4—22no—
48C23×C3×C4—22no—
49C23×C3×C4D414yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
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×C22⋊C4—28no—
65C2×C3×C22⋊C4—24no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—28no—
69C2×#217—24no—
70C2×#217D418yes—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×Dic3⋊C4—28no—
78C2×Dic3⋊C4—28no—
79C2×Dic3⋊C4—28no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3—22no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3D414yes—
86C3×#95—24no—
87C3⋊#95—24no—
88C3⋊#95—28no—
89C3⋊#95—28no—
90C3⋊#95—28no—
91C3⋊#95—24no—
92C24×C4C2×D311yescharacteristic
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⋊C4C2×D312yes—
101C22×C42—31no—
102C22×C42—31no—
103#397—616no—
104#384—64no—
105#380—24no—
106#380—64no—
107C4×C22⋊C4—64no—
108C2×#95—34no—
109C2×#95—34no—
110C2×#95—34no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95C2×D312yes—
114C24×C3—21no—
115C24×C3—22no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3C2×D414yes—
124C24×C3C2×D412yes—
125C24×C3C2411yescharacteristic
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—28no—
133C22×C3×C4—28no—
134C22×C3×C4—28no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—44no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4—44no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4#4112yes—
153C22×C3×C4#4112yes—
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4112yes—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—24no—
168C22×Dic3—24no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—24no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—24no—
177C22×Dic3C2×D414yes—
178C22×Dic3C2×D414yes—
179C22×Dic3C2×D414yes—
180C22×Dic3#4114yes—
181C3×C4⋊C4—44no—
182C3×C4⋊C4—44no—
183C3×C22⋊C4—48no—
184#217—216no—
185#217—48no—
186C4⋊Dic3—44no—
187C4⋊Dic3—44no—
188Dic3⋊C4—416no—
189C4×Dic3—28no—
190C4×Dic3—44no—
191C25C22×D311yescharacteristic
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—32no—
199C23×C4—32no—
200C23×C4—32no—
201C23×C4—32no—
202C23×C4#6714yes—
203C23×C4C22×D312yes—
204C23×C4C22×D312yes—
205C23×C4C22×D311yescharacteristic
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—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—38no—
225C2×C22⋊C4—64no—
226C2×C22⋊C4—64no—
227C2×C22⋊C4—68no—
228C2×C22⋊C4—64no—
229C2×C42—34no—
230C2×C42—62no—
231C2×C42—62no—
232C2×C42—62no—
233C2×C42—62no—
234C2×C42—62no—
235#95—24no—
236#95—64no—
237#95—68no—
238#95—68no—
239#95—68no—
240#95—64no—
241C23×C3—24no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—24no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3#12714yes—
264C23×C3C22×D411yescharacteristic
265C23×C3C22×D412yes—
266C23×C3C22×D411yescharacteristic
267C23×C3C2×#4111yescharacteristic
268C23×C3C2×#4111yescharacteristic
269C23×C3C2×#4111yescharacteristic
270C23×C3C2×#4111yescharacteristic
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4112yes—
273C23×C32−1+412yes—
274C23×C32−1+412yes—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—416no—
283C2×C3×C4—44no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—44no—
290C2×C3×C4—42no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4#12114yes—
294C2×C3×C4#12314yes—
295C2×C3×C4#12314yes—
296C2×C3×C4#12314yes—
297C2×C3×C4#12414yes—
298C2×C3×C4#12614yes—
299C2×C3×C4#12614yes—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—44no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—44no—
320C2×Dic3—42no—
321C2×Dic3—44no—
322C2×Dic3—44no—
323C2×Dic3#12118yes—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24D3×D414yes—
334C24C2×#6712yes—
335C24C23×D311yescharacteristic
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—34no—
356C22×C4—34no—
357C22×C4—34no—
358C22×C4—34no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—64no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—64no—
380C22×C4—64no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4#23512yes—
384C22×C4#23512yes—
385C22×C4D3×D412yes—
386C22×C4#23712yes—
387C22×C4#23712yes—
388C22×C4#23712yes—
389C22×C4C2×#6712yes—
390C22×C4C2×#6712yes—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—1216no—
394C4⋊C4—124no—
395C4⋊C4—124no—
396C22⋊C4—48no—
397C22⋊C4—616no—
398C22⋊C4—128no—
399C42—68no—
400C42—124no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—44no—
412C22×C3C2×#12111yescharacteristic
413C22×C3C2×#12112yes—
414C22×C3C2×#12311yescharacteristic
415C22×C3C2×#12311yescharacteristic
416C22×C3C2×#12311yescharacteristic
417C22×C3C2×#12411yescharacteristic
418C22×C3C2×#12611yescharacteristic
419C22×C3C2×#12611yescharacteristic
420C22×C3C2×#12711yescharacteristic
421C22×C3#53211yescharacteristic
422C22×C3#53212yes—
423C22×C3#53611yescharacteristic
424C22×C3#53611yescharacteristic
425C22×C3#53612yes—
426C22×C3#54112yes—
427C22×C3#54712yes—
428C22×C3#54712yes—
429C22×C3#54712yes—
430C22×C3#54712yes—
431C22×C3#55212yes—
432C22×C3#55212yes—
433C22×C3#55612yes—
434C22×C3#55612yes—
435C3×C4—42no—
436C3×C4—82no—
437C3×C4—82no—
438C3×C4#39718yescomplemented
439Dic3—28no—
440Dic3—44no—
441Dic3—82no—
442Dic3—82no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23#94014yes—
466C23C2×#23511yescharacteristic
467C23C2×#23511yescharacteristic
468C23C2×D3×D411yescharacteristic
469C23C2×D3×D412yes—
470C23C2×#23711yescharacteristic
471C23C2×#23711yescharacteristic
472C23C2×#23711yescharacteristic
473C23#100712yes—
474C23D3×#4112yes—
475C23#101012yes—
476C23C22×#6711yescharacteristic
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—38no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—416no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—124no—
510C2×C4—124no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—124no—
518C2×C4—124no—
519C2×C4#87914yes—
520C2×C4#87914yes—
521C2×C4#88314yes—
522C2×C4#88514yes—
523C2×C4#92914yes—
524C2×C4#93514yes—
525C2×C4#93914yes—
526C2×C3—44no—
527C2×C3C2×#39711yescharacteristic
528C2×C3#248612yes—
529C2×C3#249612yes—
530C2×C3#249612yes—
531C2×C3#249911yescharacteristic
532C2×C3#249911yescharacteristic
533C2×C3#250111yescharacteristic
534C2×C3#250111yescharacteristic
535C2×C3#254712yes—
536C2×C3#262111yescharacteristic
537C2×C3#285611yescharacteristic
538C22—24no—
539C22—24no—
540C22—24no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—44no—
545C22—42no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22C2×#87911yescharacteristic
550C22C2×#87911yescharacteristic
551C22C2×#88311yescharacteristic
552C22C2×#88511yescharacteristic
553C22#550712yes—
554C22#550712yes—
555C22#552912yes—
556C22#553312yes—
557C22#559612yes—
558C22D3×#12112yes—
559C22#561112yes—
560C22#561112yes—
561C22#562512yes—
562C22#562812yes—
563C22#563912yes—
564C22#564112yes—
565C22C2×#92911yescharacteristic
566C22C2×#93511yescharacteristic
567C22C2×#93911yescharacteristic
568C22C2×#94011yescharacteristic
569C22#581611yescharacteristic
570C22#582511yescharacteristic
571C22#583211yescharacteristic
572C4—42no—
573C4—68no—
574C4—82no—
575C4—82no—
576C4—124no—
577C4—242no—
578C4—242no—
579C4#495218yescomplemented
580C3#1515011yescharacteristic, complemented, Sylow
581C2—44no—
582C2#7391112yes—
583C2#7392212yes—
584C2#7392312yes—
585C2#7393712yes—
586C2C2×#495211yescharacteristic
587C2#8217711yescharacteristic
588C2#8225611yescharacteristic
589C2#8266411yescharacteristic
590C2#8270011yescharacteristic
591C2#8277411yescharacteristic
592C2#8278611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#380C2
prime radicalC25×C3C23
hypercenterC2×#380D3
hypocenterC3#15150
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. C2×#4975C2semidirect
6. #71563C2semidirect
7. #71563C2semidirect
8. #71563C2semidirect
9. #69421C2semidirect
438. C3×C4#397semidirect
579. C4#4952semidirect
580. C3#15150semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3802
C2×#52122
C2×C4×#2172
C2×#49752
C2×#49522
#715632
#694212
#151503