Finite Group Databasebeta

← #10729118

Subgroups — #10729118

Order 768.

1846 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5210C211yesmaximal, characteristic
2C22×C3⋊#95C211yesmaximal, characteristic
3C2×#4950C211yesmaximal, characteristic
4C2×#4942C211yesmaximal, characteristic
5C3×C22⋊#95C211yesmaximal, characteristic
6#71544C211yesmaximal, characteristic
7#71544C211yesmaximal, characteristic
8#69421C212yesmaximal, complemented
9#69412C212yesmaximal, complemented
10#69411C212yesmaximal, complemented
11#69409C212yesmaximal, complemented
12#14920—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26#5210—24no—
27C2×C3⋊#95—24no—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4950—24no—
39#4942—24no—
40C2×#385—31no—
41C2×#385—31no—
42C2×#380—31no—
43C22×#95—31no—
44#1445—32no—
45#1442—32no—
46#1441—32no—
47#1441—32no—
48C22⋊#95—31no—
49C22⋊#95—31no—
50C22⋊#95D311yescharacteristic
51C25×C3C2311yescharacteristic
52C23×C3×C4—22no—
53C23×C3×C4—24no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—42no—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×Dic3—22no—
62C23×Dic3—22no—
63C23×Dic3—22no—
64C23×Dic3—22no—
65C23×Dic3—24no—
66C23×Dic3—42no—
67C23×Dic3—42no—
68C23×Dic3—42no—
69C23×Dic3D414yes—
70C23×Dic3Q814yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2312yes—
75C23×Dic3C2312yes—
76C23×Dic3C2311yescharacteristic
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4Q814yes—
83C2×#217—22no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217D414yes—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—42no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—42no—
98C2×Dic3⋊C4—42no—
99C3×#95—24no—
100C3×#95—42no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C3⋊#95—42no—
110C3⋊#95—42no—
111C24×C4—31no—
112C24×C4—31no—
113C24×C4C2×D311yescharacteristic
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4—31no—
118C22×C22⋊C4—31no—
119C22×C22⋊C4—31no—
120C22×C22⋊C4C2×D311yescharacteristic
121C22×C22⋊C4C2×D311yescharacteristic
122#385—64no—
123#385—64no—
124#380—64no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—64no—
136C2×#95C2×D312yes—
137C2×#95C2×D312yes—
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3C2×D412yes—
143C24×C3C2×D412yes—
144C24×C3C2×Q812yes—
145C24×C3C2×Q812yes—
146C24×C3#4112yes—
147C24×C3#4112yes—
148C24×C3#4112yes—
149C24×C3C2411yescharacteristic
150C22×C3×C4—28no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
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—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—48no—
173C22×C3×C4—48no—
174C22×C3×C4—44no—
175C22×C3×C4—44no—
176C22×C3×C4C2×Q812yes—
177C22×C3×C4C2×Q812yes—
178C22×C3×C4#4114yes—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—28no—
183C22×Dic3—28no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—28no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—44no—
212C22×Dic3—48no—
213C22×Dic3—48no—
214C22×Dic3—48no—
215C22×Dic3—44no—
216C22×Dic3—44no—
217C22×Dic3—48no—
218C22×Dic3C2×D412yes—
219C22×Dic3C2×D414yes—
220C22×Dic3C2×D412yes—
221C22×Dic3C2×Q814yes—
222C22×Dic3#4114yes—
223C3×C22⋊C4—28no—
224C3×C22⋊C4—44no—
225#217—28no—
226#217—44no—
227C4⋊Dic3—44no—
228C4⋊Dic3—44no—
229Dic3⋊C4—48no—
230Dic3⋊C4—48no—
231C25C22×D311yescharacteristic
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4—32no—
240C23×C4—34no—
241C23×C4—34no—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—42no—
245C23×C4—64no—
246C23×C4—62no—
247C23×C4—62no—
248C23×C4—62no—
249C23×C4—62no—
250C23×C4—122no—
251C23×C4—122no—
252C23×C4—122no—
253C23×C4C22×D312yes—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D312yes—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—122no—
265C2×C4⋊C4—122no—
266C2×C4⋊C4—122no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—34no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4Dic614yes—
279#95—24no—
280#95—42no—
281#95—64no—
282#95—64no—
283#95—64no—
284#95—62no—
285#95—62no—
286#95—64no—
287#95—122no—
288#95—122no—
289#95—122no—
290#95—122no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3#12114yes—
308C23×C3C22⋊Q814yes—
309C23×C3C22⋊Q814yes—
310C23×C3#12314yes—
311C23×C3#12514yes—
312C23×C3#12514yes—
313C23×C3C4⋊Q814yes—
314C23×C3C22×D411yescharacteristic
315C23×C3C22×D411yescharacteristic
316C23×C3C22×Q811yescharacteristic
317C23×C3C22×Q811yescharacteristic
318C23×C3C2×#4111yescharacteristic
319C23×C3C2×#4111yescharacteristic
320C23×C3C2×#4111yescharacteristic
321C23×C3C2×#4112yes—
322C23×C3C2×#4112yes—
323C23×C3H(5,𝔽2)12yes—
324C23×C3H(5,𝔽2)12yes—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—416no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—416no—
364C2×Dic3—416no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3#12314yes—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24C2×Dic612yes—
391C24#23512yes—
392C24#23512yes—
393C24D3×D412yes—
394C24D3×D412yes—
395C24#23712yes—
396C24D3×Q812yes—
397C24C23×D311yescharacteristic
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
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—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—32no—
421C22×C4—34no—
422C22×C4—32no—
423C22×C4—34no—
424C22×C4—34no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—68no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—62no—
433C22×C4—68no—
434C22×C4—68no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
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—62no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—124no—
462C22×C4—128no—
463C22×C4—128no—
464C22×C4—128no—
465C22×C4—124no—
466C22×C4—128no—
467C22×C4—124no—
468C22×C4C2×Dic612yes—
469C22×C4C2×Dic612yes—
470C22×C4#23714yes—
471C4⋊C4—128no—
472C4⋊C4—124no—
473C4⋊C4—128no—
474C4⋊C4—124no—
475C22⋊C4—28no—
476C22⋊C4—44no—
477C22⋊C4—68no—
478C22⋊C4—124no—
479C22×C3—28no—
480C22×C3—24no—
481C22×C3—24no—
482C22×C3—24no—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—28no—
486C22×C3—24no—
487C22×C3—28no—
488C22×C3—28no—
489C22×C3#39818yes—
490C22×C3C2×#12111yescharacteristic
491C22×C3C2×C22⋊Q811yescharacteristic
492C22×C3C2×C22⋊Q811yescharacteristic
493C22×C3C2×#12311yescharacteristic
494C22×C3C2×#12311yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3C2×#12511yescharacteristic
497C22×C3C2×#12511yescharacteristic
498C22×C3#52912yes—
499C22×C3C2×C4⋊Q811yescharacteristic
500C22×C3#53312yes—
501C22×C3#53411yescharacteristic
502C22×C3#53511yescharacteristic
503C22×C3#53812yes—
504C22×C3#54312yes—
505C22×C3#54612yes—
506C22×C3#54612yes—
507C22×C3#55112yes—
508C22×C3#55112yes—
509C22×C3#55312yes—
510C22×C3#55312yes—
511C22×C3#55412yes—
512C22×C3#55412yes—
513C3×C4—42no—
514C3×C4—44no—
515C3×C4—42no—
516C3×C4—82no—
517Dic3—24no—
518Dic3—44no—
519Dic3—42no—
520Dic3—44no—
521Dic3—81no—
522Dic3—81no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23#87014yes—
540C23#87814yes—
541C23#88314yes—
542C23#88414yes—
543C23#88814yes—
544C23#88914yes—
545C23#89614yes—
546C23C22×Dic611yescharacteristic
547C23C2×#23511yescharacteristic
548C23C2×#23511yescharacteristic
549C23C2×D3×D411yescharacteristic
550C23C2×D3×D411yescharacteristic
551C23C2×#23711yescharacteristic
552C23C2×#23712yes—
553C23#100412yes—
554C23C2×D3×Q811yescharacteristic
555C23D3×#4112yes—
556C23#100912yes—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—34no—
565C2×C4—416no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—62no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—62no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—1216no—
597C2×C4—1216no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—122no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—121no—
616C2×C4—122no—
617C2×C4—121no—
618C2×C3—28no—
619C2×C3C2×#39811yescharacteristic
620C2×C3#241111yescharacteristic
621C2×C3#241211yescharacteristic
622C2×C3#244412yes—
623C2×C3#248711yescharacteristic
624C2×C3#250512yes—
625C2×C3#254912yes—
626C2×C3#256912yes—
627C2×C3#257911yescharacteristic
628C2×C3#262611yescharacteristic
629C2×C3#285511yescharacteristic
630C22—24no—
631C22—28no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—28no—
637C22—28no—
638C22—28no—
639C22—24no—
640C22#465718yes—
641C22C2×#87011yescharacteristic
642C22C2×#87811yescharacteristic
643C22C2×#88311yescharacteristic
644C22C2×#88411yescharacteristic
645C22#549512yes—
646C22C2×#88811yescharacteristic
647C22C2×#88911yescharacteristic
648C22C2×#89611yescharacteristic
649C22#552712yes—
650C22#553812yes—
651C22#553912yes—
652C22#554312yes—
653C22#556012yes—
654C22#558611yescharacteristic
655C22#559112yes—
656C22#559211yescharacteristic
657C22#560112yes—
658C22#561812yes—
659C22#562212yes—
660C22#563412yes—
661C22D3×#12311yescharacteristic
662C22#565311yescharacteristic
663C22#565412yes—
664C4—42no—
665C4—44no—
666C4—42no—
667C4—64no—
668C4—82no—
669C4—124no—
670C4—122no—
671C4—124no—
672C4—241no—
673C4—241no—
674C3#1492011yescharacteristic, complemented, Sylow
675C2—28no—
676C2C2×#465711yescharacteristic
677C2#7352012yes—
678C2#7382511yescharacteristic
679C2#7394412yes—
680C2#7412611yescharacteristic
681C2#7447212yes—
682C2#7458911yescharacteristic
683C2#7462111yescharacteristic
684C2#7470512yes—
685C2#7473811yescharacteristic
686C2#7476911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C22⋊#95C2
prime radicalC25×C3C23
hypercenterC22⋊#95D3
hypocenterC3#14920
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
8. #69421C2semidirect
9. #69412C2semidirect
10. #69411C2semidirect
11. #69409C2semidirect
674. C3#14920semidirect

Maximal subgroups

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

subgroupindex
C2×#52102
C22×C3⋊#952
C2×#49502
C2×#49422
C3×C22⋊#952
#715442
#694212
#694122
#694112
#694092
#149203