Finite Group Databasebeta

← #10720194

Subgroups — #10720194

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#896C211yesmaximal, characteristic
2C22×#872C211yesmaximal, characteristic
3C2×#4646C212yesmaximal, complemented
4C2×#4666C211yesmaximal, characteristic
5C3×#1436C211yesmaximal, characteristic, complemented
6#71564C211yesmaximal, characteristic, complemented
7#71536C211yesmaximal, characteristic, complemented
8#69429C212yesmaximal
9#69429C212yesmaximal
10#69405C211yesmaximal, characteristic, complemented
11Dic3⋊#95C212yesmaximal, complemented
12#14138—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C2×#896—24no—
27C2×#896—24no—
28C2×#896C418yescomplemented
29C2×#872—24no—
30C2×#872—28no—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41#4666—24no—
42#4646—28no—
43C22×C22⋊Q8—31no—
44C22×#117—31no—
45C2×#391—32no—
46C2×C4×C22⋊C4—31no—
47#1440—32no—
48#1440—32no—
49#1436—31no—
50#1436—31no—
51#1436—31no—
52#1436D311yescharacteristic, complemented
53C4⋊#95—32no—
54C24×D3C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C22×C4×D3—24no—
66C22×C4×D3—24no—
67C22×C4×D3C2×C414yes—
68C22×C4×D3D414yes—
69C22×C4×D3D414yes—
70C22×Dic6—24no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—24no—
74C22×Dic6C2×C414yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4C2×C414yes—
78C2×C3×C42—24no—
79C2×C3×C42—22no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4C2×C414yes—
91C2×(C2×D3)⋊C4C2×C414yes—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4C2×C414yes—
99C2×Dic3⋊C4C2×C414yes—
100C2×Dic3⋊C4D418yes—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—24no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3—24no—
105C2×C4×Dic3—24no—
106#896—232no—
107#872—232no—
108C3×#95—24no—
109C3×#95—22no—
110C3×#95—24no—
111C3⋊#95—24no—
112C3⋊#95—22no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116C3⋊#95—24no—
117C3⋊#95—24no—
118C23×Q8—31no—
119C24×C4—31no—
120C2×C22⋊Q8—38no—
121C2×C22⋊Q8—64no—
122C2×C22⋊Q8—64no—
123C2×#117—68no—
124C2×#117—64no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4C2×D311yescharacteristic
129C22×C22⋊C4—31no—
130C22×C22⋊C4—31no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4—32no—
133C22×C42—31no—
134C22×C42—32no—
135C22×C42C2×D311yescharacteristic
136#391—68no—
137C4×C22⋊C4—64no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95—31no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95C2×D312yes—
146C2×#95C2×D311yescharacteristic
147C2×#95C2×D312yes—
148C24×C3C2411yescharacteristic
149C23×D3—22no—
150C23×D3—22no—
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—22no—
154C23×D3—22no—
155C23×D3C22×C412yes—
156C23×D3C2×D412yes—
157C23×D3C2×D412yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4C22×C412yes—
181C22×C3×C4C22×C412yes—
182C22×C3×C4C22×C412yes—
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4#4114yes—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3C22×C412yes—
208C22×Dic3C22×C412yes—
209C22×Dic3C22×C412yes—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D412yes—
212C22×Dic3C2×D412yes—
213C22×Dic3C2×D414yes—
214C22×Dic3#4114yes—
215C22×Dic3#4114yes—
216C2×C4×D3—216no—
217C2×C4×D3—28no—
218C2×C4×D3—216no—
219C2×C4×D3—28no—
220C2×C4×D3C22⋊C4116yes—
221C2×Dic6—28no—
222C2×Dic6—216no—
223C2×Dic6—44no—
224C2×Dic6—48no—
225C2×Dic6—44no—
226C3×C4⋊C4—28no—
227C3×C42—28no—
228(C2×D3)⋊C4—216no—
229(C2×D3)⋊C4—216no—
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—28no—
232Dic3⋊C4—28no—
233Dic3⋊C4—28no—
234Dic3⋊C4—216no—
235C4×Dic3—28no—
236C4×Dic3—48no—
237C25—31no—
238C22×Q8—34no—
239C22×Q8—62no—
240C22×Q8—62no—
241C22×Q8—64no—
242C22×Q8—64no—
243C23×C4—31no—
244C23×C4—34no—
245C23×C4—34no—
246C23×C4—34no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—32no—
250C23×C4—32no—
251C23×C4—64no—
252C23×C4—64no—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D312yes—
256C23×C4C22×D312yes—
257C23×C4C22×D311yescharacteristic
258C22⋊Q8—632no—
259#117—632no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—38no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4C4×D314yes—
272C2×C22⋊C4—34no—
273C2×C22⋊C4—34no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—64no—
287C2×C42—62no—
288C2×C42—64no—
289C2×C42—64no—
290C2×C42—64no—
291#95—24no—
292#95—22no—
293#95—24no—
294#95—64no—
295#95—64no—
296#95—64no—
297#95—64no—
298#95—64no—
299#95—62no—
300#95—64no—
301C23×C3C23×C411yescharacteristic
302C23×C3C22×D411yescharacteristic
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D412yes—
305C23×C3C2×#4112yes—
306C23×C3C2×#4112yes—
307C23×C3C2×#4112yes—
308C23×C32−1+412yes—
309C23×C32−1+412yes—
310C22×D3—24no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—24no—
314C22×D3—22no—
315C22×D3—24no—
316C22×D3—22no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—24no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C22×D3C2×C22⋊C414yes—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4C2×C22⋊C414yes—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—24no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—41no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3C2×C22⋊C414yes—
385C2×Dic3C4×D418yes—
386C2×Dic3#12014yes—
387C2×Dic3#12118yes—
388C2×Dic3#12314yes—
389C4×D3—232no—
390Dic6—416no—
391C24—32no—
392C24—32no—
393C24—32no—
394C24—62no—
395C24—62no—
396C24—62no—
397C24—62no—
398C24—62no—
399C24—62no—
400C24C23×D311yescharacteristic
401C2×Q8—68no—
402C2×Q8—616no—
403C2×Q8—124no—
404C2×Q8—124no—
405C2×Q8—128no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—34no—
431C22×C4—32no—
432C22×C4—34no—
433C22×C4—316no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—68no—
439C22×C4—68no—
440C22×C4—616no—
441C22×C4—616no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4C2×C4×D312yes—
463C22×C4C2×C4×D312yes—
464C22×C4C2×C4×D312yes—
465C22×C4#23514yes—
466C22×C4D3×D414yes—
467C22×C4D3×D412yes—
468C22×C4D3×D412yes—
469C4⋊C4—28no—
470C4⋊C4—616no—
471C4⋊C4—68no—
472C4⋊C4—68no—
473C22⋊C4—68no—
474C22⋊C4—68no—
475C22⋊C4—616no—
476C22⋊C4—616no—
477C42—28no—
478C42—68no—
479C42—128no—
480C22×C3C22×C22⋊C411yescharacteristic
481C22×C3C2×C4×D412yes—
482C22×C3C4×#4111yescharacteristic
483C22×C3C4×#4111yescharacteristic
484C22×C3#51912yes—
485C22×C3C2×#12011yescharacteristic
486C22×C3C2×#12112yes—
487C22×C3C2×#12311yescharacteristic
488C22×C3#52512yes—
489C22×C3#53212yes—
490C22×C3#53611yescharacteristic
491C22×C3#53611yescharacteristic
492C22×C3#53612yes—
493C22×C3#54712yes—
494C22×C3#54712yes—
495C22×C3#54812yes—
496C22×C3#54812yes—
497C22×C3#55212yes—
498C22×C3#55212yes—
499C22×C3#55612yes—
500C22×C3#55612yes—
501C2×D3—24no—
502C2×D3—24no—
503C2×D3—24no—
504C2×D3—44no—
505C2×D3—44no—
506C2×D3—44no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—44no—
510C3×C4—24no—
511C3×C4—44no—
512C3×C4—44no—
513C3×C4—42no—
514C3×C4—42no—
515Dic3—44no—
516Dic3—44no—
517Dic3—42no—
518Dic3—42no—
519Dic3#38618yes—
520C23—34no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—62no—
526C23—64no—
527C23—64no—
528C23—64no—
529C23—64no—
530C23—62no—
531C23—64no—
532C23—64no—
533C23—122no—
534C23—122no—
535C23—122no—
536C23—122no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—122no—
541C23C22×C4×D311yescharacteristic
542C23C2×#23512yes—
543C23C2×D3×D412yes—
544C23C2×D3×D411yescharacteristic
545C23C2×D3×D411yescharacteristic
546C23#100712yes—
547C23D3×#4112yes—
548C23D3×#4112yes—
549C23#101012yes—
550Q8—1216no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—34no—
567C2×C4—34no—
568C2×C4—38no—
569C2×C4—34no—
570C2×C4—38no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—632no—
585C2×C4—64no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—121no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4D3×C22⋊C414yes—
610C2×C3C2×#38611yescharacteristic
611C2×C3#229711yescharacteristic
612C2×C3#231011yescharacteristic
613C2×C3#236212yes—
614C2×C3#236212yes—
615C2×C3#241512yes—
616C2×C3#241512yes—
617C2×C3#243212yes—
618C2×C3#249912yes—
619D3—44no—
620C22—64no—
621C22—64no—
622C22—64no—
623C22—124no—
624C22—124no—
625C22—124no—
626C22—124no—
627C22—124no—
628C22—124no—
629C22C4×#23511yescharacteristic
630C22C2×D3×C22⋊C411yescharacteristic
631C22#550712yes—
632C22#552211yescharacteristic
633C22#553112yes—
634C22C4×D3×D412yes—
635C22#554712yes—
636C22#554812yes—
637C22#555212yes—
638C22#556812yes—
639C22#557212yes—
640C22D3×#12011yescharacteristic
641C22D3×#12112yes—
642C22#562812yes—
643C22#563312yes—
644C22D3×#12311yescharacteristic
645C22#565512yes—
646C22#567511yescharacteristic
647C22#569312yes—
648C22#570212yes—
649C22#572411yescharacteristic
650C4—24no—
651C4—38no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—122no—
657C4—122no—
658C4—124no—
659C4—124no—
660C3#1413811yescharacteristic, complemented, Sylow
661C2—124no—
662C2#7356311yescharacteristic
663C2#7364812yes—
664C2#7441112yes—
665C2D3×#38611yescharacteristic
666C2#7496612yes—
667C2#7502512yes—
668C2#7549612yes—
669C2#7599512yes—
670C2#7661011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1436C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenter#1436D3
hypocenterC3#14138
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
3. C2×#4646C2semidirect
5. C3×#1436C2semidirect
6. #71564C2semidirect
7. #71536C2semidirect
10. #69405C2semidirect
11. Dic3⋊#95C2semidirect
28. C2×#896C4semidirect
52. #1436D3semidirect
660. C3#14138semidirect

Maximal subgroups

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

subgroupindex
C22×#8962
C22×#8722
C2×#46462
C2×#46662
C3×#14362
#715642
#715362
#694292
#694052
Dic3⋊#952
#141383