Finite Group Databasebeta

← #10700607

Subgroups — #10700607

Order 768.

1866 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic
2C22×C3×#99C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic
4#71544C212yesmaximal, complemented
5C2×#4527C212yesmaximal, complemented
6#10761—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C24×Dic3C411yescharacteristic, complemented
9C24×Dic3C2211yescharacteristic
10C22×C3×C22⋊C4C2212yes—
11C22×#217C411yescharacteristic, complemented
12C22×#217C2212yes—
13C22×#217C2211yescharacteristic
14C2×D4×Dic3—22no—
15C2×D4×Dic3—22no—
16C2×D4×Dic3—24no—
17C2×D4×Dic3—24no—
18C22×C4⋊Dic3C411yescharacteristic, complemented
19C22×C4×Dic3C411yescharacteristic, complemented
20C2×C3×#99—22no—
21C2×C3×#99—22no—
22C2×C3×#99C414yescomplemented
23C2×#834—22no—
24C2×#834—22no—
25C2×#834C414yescomplemented
26C2×C3⋊#95—24no—
27#4527—24no—
28C22×C4×D4—31no—
29C22×#99—31no—
30C22×#99D311yescharacteristic
31C2×#342—32no—
32C22⋊#95—32no—
33C25×C3C2×C411yescharacteristic
34C25×C3C2311yescharacteristic
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4—22no—
38C22×C3×D4—22no—
39C22×C3×D4C2×C412yes—
40C22×C3×D4C2×C412yes—
41C22×C3×D4D412yes—
42C22×C3×D4D412yes—
43C22×C3×D4D412yes—
44C22×C3×D4Q812yes—
45C23×C3×C4—22no—
46C23×C3×C4C2×C411yescharacteristic
47C23×Dic3—22no—
48C23×Dic3—21no—
49C23×Dic3—22no—
50C23×Dic3—21no—
51C23×Dic3—22no—
52C23×Dic3—22no—
53C23×Dic3—22no—
54C23×Dic3—44no—
55C23×Dic3—44no—
56C23×Dic3C2×C411yescharacteristic
57C23×Dic3C2×C411yescharacteristic
58C23×Dic3C2×C411yescharacteristic
59C23×Dic3C2×C411yescharacteristic
60C23×Dic3D414yes—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4C2×C414yescomplemented
67C2×#217—22no—
68C2×#217—21no—
69C2×#217—22no—
70C2×#217—21no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217—22no—
78C2×#217C2×C414yescomplemented
79C2×#217D414yes—
80D4×Dic3—216no—
81D4×Dic3—216no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C3×#99—28no—
89#834—28no—
90C3⋊#95—48no—
91C23×D4C2×D311yescharacteristic
92C24×C4—31no—
93C24×C4—31no—
94C2×C4×D4—64no—
95C2×C4×D4—64no—
96C2×C4×D4—62no—
97C2×C4×D4—62no—
98C22×C4⋊C4—31no—
99C22×C22⋊C4—31no—
100C22×C22⋊C4—32no—
101C22×C22⋊C4—31no—
102C22×C22⋊C4C2×D312yes—
103C22×C42—31no—
104C2×#99—22no—
105C2×#99—22no—
106C2×#99—34no—
107C2×#99—62no—
108C2×#99—62no—
109C2×#99Dic314yescomplemented
110C2×#95—64no—
111#342—64no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3—42no—
119C24×C3—42no—
120C24×C3C4212yes—
121C24×C3C22⋊C412yes—
122C24×C3C4⋊C412yes—
123C24×C3C22×C411yescharacteristic
124C24×C3C22×C411yescharacteristic
125C24×C3C22×C411yescharacteristic
126C24×C3C2×D411yescharacteristic
127C24×C3C2×D412yes—
128C24×C3C2×D412yes—
129C24×C3C2×D411yescharacteristic
130C24×C3C2×D411yescharacteristic
131C24×C3C2×Q811yescharacteristic
132C24×C3#4112yes—
133C24×C3#4112yes—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4C22⋊C414yes—
147C2×C3×D4C22⋊C414yes—
148C2×C3×D4C4⋊C414yes—
149C2×C3×D4C4⋊C414yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—24no—
155C22×C3×C4—44no—
156C22×C3×C4—44no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4C4212yes—
160C22×C3×C4C22⋊C412yes—
161C22×C3×C4C4⋊C412yes—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—24no—
165C22×Dic3—28no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—24no—
173C22×Dic3—24no—
174C22×Dic3—28no—
175C22×Dic3—24no—
176C22×Dic3—42no—
177C22×Dic3—44no—
178C22×Dic3—44no—
179C22×Dic3—48no—
180C22×Dic3—44no—
181C22×Dic3—48no—
182C22×Dic3—48no—
183C22×Dic3—44no—
184C22×Dic3—44no—
185C22×Dic3—42no—
186C22×Dic3—42no—
187C22×Dic3—44no—
188C22×Dic3—44no—
189C22×Dic3—42no—
190C22×Dic3—42no—
191C22×Dic3C22⋊C412yes—
192C22×Dic3C22⋊C412yes—
193C22×Dic3C22⋊C412yes—
194C22×Dic3C22⋊C412yes—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—42no—
199#217—28no—
200#217—24no—
201#217—24no—
202#217—24no—
203#217—24no—
204#217—42no—
205#217—42no—
206C4⋊Dic3—28no—
207C4×Dic3—28no—
208C25C4×D311yescharacteristic
209C25C22×D311yescharacteristic
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4Dic612yes—
215C22×D4C4×D312yes—
216C22×D4D1212yes—
217C22×D4C2×Dic312yes—
218C22×D4#6712yes—
219C22×D4#6712yes—
220C23×C4—22no—
221C23×C4—31no—
222C23×C4—34no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—62no—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4—61no—
230C23×C4—62no—
231C23×C4—61no—
232C23×C4—62no—
233C23×C4—124no—
234C23×C4—124no—
235C23×C4C4×D311yescharacteristic
236C4×D4—616no—
237C4×D4—616no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—34no—
247C2×C22⋊C4—34no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—62no—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4C2×Dic314yescomplemented
260C2×C42—62no—
261C2×C42—62no—
262C2×C42—62no—
263#99—28no—
264#99—68no—
265#95—128no—
266C23×C3—24no—
267C23×C3—22no—
268C23×C3—24no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—41no—
277C23×C3—42no—
278C23×C3—41no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—44no—
287C23×C3—44no—
288C23×C3—42no—
289C23×C3—44no—
290C23×C3—42no—
291C23×C3—44no—
292C23×C3—42no—
293C23×C3—44no—
294C23×C3—44no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3#9514yescomplemented
299C23×C3C2×C4211yescharacteristic
300C23×C3C2×C22⋊C411yescharacteristic
301C23×C3C2×C22⋊C411yescharacteristic
302C23×C3C2×C22⋊C411yescharacteristic
303C23×C3C2×C22⋊C411yescharacteristic
304C23×C3C2×C22⋊C411yescharacteristic
305C23×C3C2×C4⋊C411yescharacteristic
306C23×C3C2×C4⋊C411yescharacteristic
307C23×C3C2×C4⋊C411yescharacteristic
308C23×C3#11711yescharacteristic
309C23×C3#11711yescharacteristic
310C23×C3C4×D412yes—
311C23×C3C4×D412yes—
312C23×C3C4×D412yes—
313C23×C3C4×D412yes—
314C23×C3#12012yes—
315C23×C3#12012yes—
316C23×C3#12112yes—
317C23×C3#12112yes—
318C23×C3C22⋊Q812yes—
319C23×C3C22⋊Q812yes—
320C23×C3#12312yes—
321C23×C3#12312yes—
322C3×D4—28no—
323C3×D4—28no—
324C3×D4—28no—
325C3×D4—28no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—48no—
333C2×C3×C4—44no—
334C2×C3×C4—84no—
335C2×C3×C4—82no—
336C2×C3×C4—82no—
337C2×C3×C4#9514yescomplemented
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—48no—
347C2×Dic3—48no—
348C2×Dic3—416no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—48no—
358C2×Dic3—44no—
359C2×Dic3—84no—
360C2×Dic3—82no—
361C2×Dic3—82no—
362C2×Dic3#9914yescomplemented
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24C4×Dic312yes—
372C24Dic3⋊C412yes—
373C24(C2×D3)⋊C412yes—
374C24C2×Dic611yescharacteristic
375C24C2×C4×D311yescharacteristic
376C24C2×C4×D311yescharacteristic
377C24C2×D1211yescharacteristic
378C24D3×D412yes—
379C24D3×D412yes—
380C24#23712yes—
381C24#23712yes—
382C24C22×Dic311yescharacteristic
383C24C2×#6711yescharacteristic
384C24C2×#6711yescharacteristic
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4Dic3⋊C414yes—
398C2×D4C4⋊Dic314yes—
399C2×D4(C2×D3)⋊C414yes—
400C2×D4#21714yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—64no—
415C22×C4—68no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—68no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—64no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—124no—
429C22×C4—124no—
430C22×C4—122no—
431C22×C4—124no—
432C22×C4—128no—
433C22×C4—124no—
434C22×C4—128no—
435C22×C4—128no—
436C22×C4—124no—
437C22×C4—122no—
438C22×C4—122no—
439C22×C4—124no—
440C22×C4—124no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4C4×Dic312yes—
444C22×C4Dic3⋊C412yes—
445C22×C4(C2×D3)⋊C412yes—
446C4⋊C4—68no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—68no—
452C22⋊C4—64no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—64no—
456C22⋊C4—122no—
457C22⋊C4—122no—
458C42—68no—
459C22×C3—24no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—24no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3C2×#9511yescharacteristic
495C22×C3C4×C22⋊C411yescharacteristic
496C22×C3C4×C22⋊C411yescharacteristic
497C22×C3#37911yescharacteristic
498C22×C3#38011yescharacteristic
499C22×C3#38011yescharacteristic
500C22×C3#38111yescharacteristic
501C22×C3#38512yes—
502C22×C3#38512yes—
503C22×C3#38612yes—
504C22×C3#38612yes—
505C22×C3C2×#9912yes—
506C22×C3#41012yes—
507C3×C4—24no—
508C3×C4—84no—
509Dic3—24no—
510Dic3—42no—
511Dic3—42no—
512Dic3—42no—
513Dic3—84no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23C3⋊#9514yescomplemented
547C23#87711yescharacteristic
548C23#87711yescharacteristic
549C23#87812yes—
550C23#87812yes—
551C23D3×C22⋊C411yescharacteristic
552C23D3×C22⋊C411yescharacteristic
553C23#67⋊C412yes—
554C23#67⋊C412yes—
555C23#88212yes—
556C23#88612yes—
557C23C2×C4×Dic311yescharacteristic
558C23C2×Dic3⋊C411yescharacteristic
559C23C2×Dic3⋊C411yescharacteristic
560C23C2×C4⋊Dic311yescharacteristic
561C23C2×(C2×D3)⋊C411yescharacteristic
562C23C2×(C2×D3)⋊C411yescharacteristic
563C23D4×Dic312yes—
564C23D4×Dic312yes—
565C23#93512yes—
566C23#93712yes—
567C23#93912yes—
568C23#93912yes—
569C23C2×#21711yescharacteristic
570D4—28no—
571D4—28no—
572D4—28no—
573D4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—34no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—64no—
591C2×C4—82no—
592C2×C4—84no—
593C2×C4—82no—
594C2×C4—128no—
595C2×C4—128no—
596C2×C4—1216no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—128no—
606C2×C4—124no—
607C2×C4—242no—
608C2×C4—244no—
609C2×C4—242no—
610C2×C4C3⋊#9514yescomplemented
611C2×C3—22no—
612C2×C3—22no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—22no—
618C2×C3—22no—
619C2×C3—44no—
620C2×C3C22⋊#9511yescharacteristic
621C2×C3C4×#9912yes—
622C2×C3#178111yescharacteristic
623C2×C3#178211yescharacteristic
624C2×C3#179912yes—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—21no—
631C22—21no—
632C22—22no—
633C22—24no—
634C22—22no—
635C22—22no—
636C22—24no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—41no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—44no—
659C22—42no—
660C22#474012yes—
661C22D3×#9912yes—
662C22Dic3×C22⋊C411yescharacteristic
663C22Dic3×C22⋊C411yescharacteristic
664C22#494211yescharacteristic
665C22#494211yescharacteristic
666C22#494311yescharacteristic
667C22#494612yes—
668C22#495012yes—
669C22#495511yescharacteristic
670C22#495812yes—
671C22C2×C3⋊#9511yescharacteristic
672C22#522012yes—
673C4—24no—
674C4—64no—
675C4—84no—
676C4—122no—
677C4—122no—
678C4—122no—
679C4—244no—
680C3#1076111yescharacteristic, complemented, Sylow
681C2—22no—
682C2—22no—
683C2—21no—
684C2—21no—
685C2—21no—
686C2—21no—
687C2—22no—
688C2—22no—
689C2—44no—
690C2Dic3×#9912yes—
691C2#7002711yescharacteristic
692C2#7002812yes—
693C2#7003511yescharacteristic
694C2#7154411yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC23×C3C2×C42
FrattiniC25C22×D3
FittingC22×C3×#99C2
socleC23×C3C2×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenterC22×#99D3
hypocenterC3#10761
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
4. #71544C2semidirect
5. C2×#4527C2semidirect
8. C24×Dic3C4semidirect
11. C22×#217C4semidirect
18. C22×C4⋊Dic3C4semidirect
19. C22×C4×Dic3C4semidirect
22. C2×C3×#99C4semidirect
25. C2×#834C4semidirect
66. C2×C3×C22⋊C4C2×C4semidirect
78. C2×#217C2×C4semidirect
109. C2×#99Dic3semidirect
259. C2×C22⋊C4C2×Dic3semidirect
298. C23×C3#95semidirect
337. C2×C3×C4#95semidirect
362. C2×Dic3#99semidirect
546. C23C3⋊#95semidirect
610. C2×C4C3⋊#95semidirect
680. C3#10761semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C22×C3×#992
C22×#8342
#715442
C2×#45272
#107613