Finite Group Databasebeta

← #10720191

Subgroups — #10720191

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#878C211yesmaximal, characteristic
2C22×#877C211yesmaximal, characteristic
3C2×Dic3×C22⋊C4C211yesmaximal, characteristic
4C2×#4646C212yesmaximal, complemented
5C3×#1440C211yesmaximal, characteristic
6#71549C211yesmaximal, characteristic
7#71549C211yesmaximal, characteristic
8#71548C211yesmaximal, characteristic
9#69407C212yesmaximal, complemented
10#69406C212yesmaximal, complemented
11Dic3⋊#95C212yesmaximal, complemented
12#14138—31nomaximal, Sylow
13C24×Dic3C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×#217C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2212yes—
26C22×C4×Dic3C2212yes—
27C2×#878—24no—
28C2×#878—24no—
29C2×#878C418yescomplemented
30C2×#877—24no—
31C2×#877—28no—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41Dic3×C22⋊C4—24no—
42#4646—28no—
43C22×C22⋊Q8—31no—
44C22×#117—31no—
45C2×#391—32no—
46C2×C4×C22⋊C4—31no—
47#1440—31no—
48#1440—31no—
49#1440—31no—
50#1440D311yescharacteristic
51#1436—32no—
52#1436—32no—
53C4⋊#95—32no—
54C25×C3C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×Dic3—24no—
60C23×Dic3—24no—
61C23×Dic3C2×C414yes—
62C23×Dic3D414yes—
63C23×Dic3D414yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2312yes—
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C22×Dic6—24no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—24no—
74C22×Dic6C2×C414yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4C2×C414yes—
82C2×#217—22no—
83C2×#217—22no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217—24no—
87C2×#217—24no—
88C2×#217C2×C414yes—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3C2×C414yes—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4C2×C414yes—
99C2×Dic3⋊C4C2×C414yes—
100C2×Dic3⋊C4D418yes—
101C2×C4×Dic3—24no—
102C2×C4×Dic3—24no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3—24no—
105C2×C4×Dic3—24no—
106C2×C4×Dic3—24no—
107#878—232no—
108#877—232no—
109C3×#95—24no—
110C3×#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
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⋊C4—31no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—31no—
131C22×C22⋊C4—31no—
132C22×C22⋊C4C2×D311yescharacteristic
133C22×C22⋊C4C2×D311yescharacteristic
134C22×C22⋊C4C2×D311yescharacteristic
135C22×C42—32no—
136C22×C42—32no—
137#391—68no—
138C4×C22⋊C4—64no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95C2×D312yes—
147C2×#95C2×D312yes—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3C22×C412yes—
155C24×C3C2×D412yes—
156C24×C3C2×D412yes—
157C24×C3C2411yescharacteristic
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
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×C4C22×C412yes—
179C22×C3×C4C22×C412yes—
180C22×C3×C4C2×D414yes—
181C22×C3×C4#4114yes—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—216no—
185C22×Dic3—28no—
186C22×Dic3—216no—
187C22×Dic3—28no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
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×Dic3—22no—
208C22×Dic3C22⋊C4116yes—
209C22×Dic3C22×C412yes—
210C22×Dic3C22×C412yes—
211C22×Dic3C22×C412yes—
212C22×Dic3C22×C412yes—
213C22×Dic3C2×D414yes—
214C22×Dic3C2×D412yes—
215C22×Dic3C2×D412yes—
216C22×Dic3C2×D412yes—
217C22×Dic3C2×D412yes—
218C22×Dic3C2×D414yes—
219C22×Dic3#4114yes—
220C22×Dic3#4114yes—
221C2×Dic6—28no—
222C2×Dic6—216no—
223C2×Dic6—44no—
224C2×Dic6—48no—
225C2×Dic6—44no—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—28no—
228C3×C22⋊C4—28no—
229#217—28no—
230#217—28no—
231#217—28no—
232C4⋊Dic3—28no—
233Dic3⋊C4—28no—
234Dic3⋊C4—28no—
235Dic3⋊C4—216no—
236C4×Dic3—216no—
237C4×Dic3—48no—
238C25C22×D311yescharacteristic
239C22×Q8—34no—
240C22×Q8—62no—
241C22×Q8—62no—
242C22×Q8—64no—
243C22×Q8—64no—
244C23×C4—31no—
245C23×C4—34no—
246C23×C4—34no—
247C23×C4—34no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—31no—
251C23×C4—32no—
252C23×C4—32no—
253C23×C4—64no—
254C23×C4—64no—
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D312yes—
258C23×C4C22×D312yes—
259C22⋊Q8—632no—
260#117—632no—
261C2×C4⋊C4—34no—
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⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—34no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4C4×D314yes—
287C2×C42—64no—
288C2×C42—64no—
289C2×C42—64no—
290C2×C42—64no—
291C2×C42—64no—
292C2×C42—64no—
293#95—24no—
294#95—24no—
295#95—64no—
296#95—64no—
297#95—64no—
298#95—64no—
299#95—64no—
300#95—64no—
301#95—64no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—24no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3C2×C22⋊C414yes—
323C23×C3C23×C411yescharacteristic
324C23×C3C22×D411yescharacteristic
325C23×C3C22×D411yescharacteristic
326C23×C3C22×D412yes—
327C23×C3C2×#4112yes—
328C23×C3C2×#4112yes—
329C23×C3C2×#4112yes—
330C23×C32−1+412yes—
331C23×C32−1+412yes—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
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×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—24no—
359C2×Dic3—22no—
360C2×Dic3—232no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—24no—
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—41no—
376C2×Dic3—41no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3C2×C22⋊C414yes—
386C2×Dic3C2×C22⋊C414yes—
387C2×Dic3C4×D418yes—
388C2×Dic3#12014yes—
389C2×Dic3#12118yes—
390C2×Dic3#12314yes—
391Dic6—416no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24C2×C4×D312yes—
399C24D3×D412yes—
400C24D3×D412yes—
401C24C23×D311yescharacteristic
402C2×Q8—68no—
403C2×Q8—616no—
404C2×Q8—124no—
405C2×Q8—124no—
406C2×Q8—128no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
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—32no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—32no—
431C22×C4—34no—
432C22×C4—316no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—34no—
439C22×C4—34no—
440C22×C4—68no—
441C22×C4—68no—
442C22×C4—616no—
443C22×C4—616no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4C2×C4×D312yes—
467C22×C4C2×C4×D312yes—
468C22×C4D3×D414yes—
469C22×C4#23714yes—
470C4⋊C4—616no—
471C4⋊C4—68no—
472C4⋊C4—68no—
473C4⋊C4—68no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—68no—
478C22⋊C4—68no—
479C22⋊C4—68no—
480C42—616no—
481C42—128no—
482C22×C3—24no—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3C22×C22⋊C411yescharacteristic
492C22×C3C2×C4×D412yes—
493C22×C3C4×#4112yes—
494C22×C3#51911yescharacteristic
495C22×C3#51911yescharacteristic
496C22×C3C2×#12011yescharacteristic
497C22×C3C2×#12112yes—
498C22×C3C2×#12311yescharacteristic
499C22×C3#52511yescharacteristic
500C22×C3#52511yescharacteristic
501C22×C3#53212yes—
502C22×C3#53612yes—
503C22×C3#53612yes—
504C22×C3#54712yes—
505C22×C3#54712yes—
506C22×C3#54812yes—
507C22×C3#54812yes—
508C22×C3#55212yes—
509C22×C3#55212yes—
510C22×C3#55612yes—
511C22×C3#55612yes—
512C3×C4—44no—
513C3×C4—44no—
514C3×C4—42no—
515C3×C4—42no—
516Dic3—24no—
517Dic3—44no—
518Dic3—44no—
519Dic3—42no—
520Dic3—42no—
521Dic3#38618yes—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23D3×C22⋊C414yes—
543C23C22×C4×D311yescharacteristic
544C23C2×D3×D411yescharacteristic
545C23C2×D3×D411yescharacteristic
546C23C2×D3×D412yes—
547C23C2×#23712yes—
548C23D3×#4112yes—
549C23D3×#4112yes—
550C23#101012yes—
551C23#101012yes—
552Q8—1216no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—34no—
564C2×C4—34no—
565C2×C4—38no—
566C2×C4—34no—
567C2×C4—38no—
568C2×C4—34no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—632no—
584C2×C4—62no—
585C2×C4—64no—
586C2×C4—62no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—121no—
601C2×C4—122no—
602C2×C4—121no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C3—44no—
613C2×C3C2×#38611yescharacteristic
614C2×C3#229711yescharacteristic
615C2×C3#231011yescharacteristic
616C2×C3#236212yes—
617C2×C3#236212yes—
618C2×C3#241512yes—
619C2×C3#241512yes—
620C2×C3#243212yes—
621C2×C3#249912yes—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22C2×D3×C22⋊C411yescharacteristic
632C22#551611yescharacteristic
633C22#551611yescharacteristic
634C22C4×#23712yes—
635C22C4×D3×D412yes—
636C22#558611yescharacteristic
637C22D3×#12011yescharacteristic
638C22#559612yes—
639C22#559812yes—
640C22D3×#12112yes—
641C22#561112yes—
642C22#564812yes—
643C22#564812yes—
644C22#565012yes—
645C22D3×#12311yescharacteristic
646C22#565512yes—
647C22#565611yescharacteristic
648C22#567112yes—
649C22#567712yes—
650C22#570212yes—
651C22#570812yes—
652C4—38no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—64no—
658C4—124no—
659C4—122no—
660C4—124no—
661C4—122no—
662C3#1413811yescharacteristic, complemented, Sylow
663C2—44no—
664C2#7405311yescharacteristic
665C2#7465811yescharacteristic
666C2D3×#38611yescharacteristic
667C2#7547812yes—
668C2#7547912yes—
669C2#7689712yes—
670C2#7690412yes—
671C2#7746212yes—
672C2#7749112yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1440C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1440D3
hypocenterC3#14138
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
4. C2×#4646C2semidirect
9. #69407C2semidirect
10. #69406C2semidirect
11. Dic3⋊#95C2semidirect
29. C2×#878C4semidirect
662. C3#14138semidirect

Maximal subgroups

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

subgroupindex
C22×#8782
C22×#8772
C2×Dic3×C22⋊C42
C2×#46462
C3×#14402
#715492
#715482
#694072
#694062
Dic3⋊#952
#141383