Finite Group Databasebeta

← #10728987

Subgroups — #10728987

Order 768.

1886 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#117C211yesmaximal, characteristic
2C2×#5210C212yesmaximal
3C2×#4981C211yesmaximal, characteristic, complemented
4C2×#4953C211yesmaximal, characteristic
5C2×#4928C211yesmaximal, characteristic, complemented
6#71566C212yesmaximal, complemented
7#71565C211yesmaximal, characteristic, complemented
8#71549C212yesmaximal
9#71546C211yesmaximal, characteristic
10#71536C212yesmaximal, complemented
11#71534C211yesmaximal, characteristic, complemented
12#14907—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C2×C3×#117—24no—
15C2×C3×#117—24no—
16C2×C3×#117—24no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2212yes—
29#5210—28no—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4981—24no—
41#4953—24no—
42#4928—24no—
43C22×#117D311yescharacteristic
44C2×#398—31no—
45C2×#397—31no—
46C2×#389—31no—
47C2×#385—32no—
48#1441—31no—
49#1441—32no—
50#1440—32no—
51#1439—31no—
52#1436—32no—
53#1434—31no—
54C25×C3C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4—22no—
58C23×C3×C4—22no—
59C23×C3×C4—22no—
60C23×C3×C4—22no—
61C23×C3×C4D414yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2312yes—
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C23×Dic3C2312yes—
72C3×#117—232no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4D414yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4D414yes—
85C2×C3×C42—22no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42D414yes—
91C2×#217—24no—
92C2×#217—24no—
93C2×#217—24no—
94C2×#217—24no—
95C2×#217—24no—
96C2×#217—24no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—24no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—22no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—24no—
105C2×Dic3⋊C4—24no—
106C2×Dic3⋊C4—24no—
107C2×Dic3⋊C4—24no—
108C3⋊#95—24no—
109C3⋊#95—22no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—22no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116C3⋊#95—24no—
117C3⋊#95—24no—
118C24×C4C2×D311yescharacteristic
119C2×#117—24no—
120C2×#117—24no—
121C2×#117—24no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C4⋊C4C2×D311yescharacteristic
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4C2×D311yescharacteristic
131C22×C22⋊C4C2×D311yescharacteristic
132C22×C42C2×D311yescharacteristic
133C22×C42C2×D311yescharacteristic
134#398—64no—
135#397—64no—
136#389—64no—
137#385—68no—
138C2×#95—31no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—32no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—21no—
153C24×C3—21no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3C2×D412yes—
157C24×C3C2×D412yes—
158C24×C3#4112yes—
159C24×C3C2411yescharacteristic
160C22×C3×C4—28no—
161C22×C3×C4—28no—
162C22×C3×C4—28no—
163C22×C3×C4—28no—
164C22×C3×C4—28no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—28no—
176C22×C3×C4—28no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4#4112yes—
200C22×C3×C4#4112yes—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—22no—
221C22×Dic3—22no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—22no—
225C22×Dic3—22no—
226C22×Dic3—22no—
227C22×Dic3#4114yes—
228C22×Dic3#4114yes—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—28no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—28no—
233C3×C42—28no—
234C3×C42—28no—
235#217—216no—
236#217—48no—
237C4⋊Dic3—28no—
238C4⋊Dic3—44no—
239Dic3⋊C4—48no—
240Dic3⋊C4—48no—
241C25C22×D311yescharacteristic
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—32no—
249C23×C4—32no—
250C23×C4—32no—
251C23×C4—32no—
252C23×C4D1214yes—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259#117—232no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—64no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4#6714yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4#6714yes—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42#6714yes—
295#95—64no—
296#95—64no—
297#95—64no—
298#95—64no—
299#95—64no—
300#95—64no—
301#95—62no—
302#95—64no—
303#95—62no—
304#95—64no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—24no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—41no—
318C23×C3—42no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3#12114yes—
328C23×C3C22×D411yescharacteristic
329C23×C3C22×D411yescharacteristic
330C23×C3C22×D411yescharacteristic
331C23×C3C22×D411yescharacteristic
332C23×C3C2×#4111yescharacteristic
333C23×C3C2×#4111yescharacteristic
334C23×C3C2×#4111yescharacteristic
335C23×C3C2×#4112yes—
336C23×C3C2×#4112yes—
337C23×C32−1+412yes—
338C23×C32−1+412yes—
339C2×C3×C4—232no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4#12014yes—
372C2×C3×C4#12114yes—
373C2×C3×C4#12114yes—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C24—21no—
403C24—21no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24—21no—
410C24C2×D1212yes—
411C24#23512yes—
412C24C2×#6712yes—
413C24C23×D311yescharacteristic
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—34no—
444C22×C4—34no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
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×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4C2×D1212yes—
472C22×C4C2×D1212yes—
473C22×C4#23512yes—
474C22×C4#23512yes—
475C22×C4C2×#6712yes—
476C22×C4C2×#6712yes—
477C22×C4C2×#6712yes—
478C22×C4C2×#6712yes—
479C22×C4C2×#6712yes—
480C22×C4C2×#6712yes—
481C22×C4C2×#6712yes—
482C22×C4C2×#6712yes—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—68no—
486C4⋊C4—128no—
487C4⋊C4—128no—
488C4⋊C4—124no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—616no—
492C22⋊C4—128no—
493C42—28no—
494C42—28no—
495C22×C3—24no—
496C22×C3—28no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—44no—
503C22×C3—42no—
504C22×C3—44no—
505C22×C3—44no—
506C22×C3C2×#12011yescharacteristic
507C22×C3C2×#12111yescharacteristic
508C22×C3C2×#12111yescharacteristic
509C22×C3C2×#12111yescharacteristic
510C22×C3#52512yes—
511C22×C3#52511yescharacteristic
512C22×C3#52511yescharacteristic
513C22×C3#52511yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3#52912yes—
516C22×C3#53211yescharacteristic
517C22×C3#53612yes—
518C22×C3#53611yescharacteristic
519C22×C3#53611yescharacteristic
520C22×C3#54112yes—
521C22×C3#54712yes—
522C22×C3#54712yes—
523C22×C3#54712yes—
524C22×C3#54812yes—
525C22×C3#55212yes—
526C22×C3#55212yes—
527C22×C3#55212yes—
528C22×C3#55612yes—
529C3×C4—24no—
530C3×C4—24no—
531C3×C4—24no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—42no—
535Dic3—44no—
536Dic3—44no—
537Dic3—82no—
538Dic3—82no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23#93014yes—
562C23C22×D1211yescharacteristic
563C23C2×#23511yescharacteristic
564C23C2×#23511yescharacteristic
565C23C2×#23511yescharacteristic
566C23D3×#4112yes—
567C23D3×#4112yes—
568C23#101012yes—
569C23#101012yes—
570C23C22×#6711yescharacteristic
571C23C22×#6711yescharacteristic
572C23C22×#6711yescharacteristic
573C2×C4—232no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4#93014yes—
634C2×C4#93014yes—
635C2×C4#95314yes—
636C2×C3—44no—
637C2×C3#239211yescharacteristic
638C2×C3#240611yescharacteristic
639C2×C3#241312yes—
640C2×C3#241512yes—
641C2×C3#243211yescharacteristic
642C2×C3#247512yes—
643C2×C3#247812yes—
644C2×C3#259911yescharacteristic
645C2×C3#261111yescharacteristic
646C2×C3#262111yescharacteristic
647C2×C3#284711yescharacteristic
648C22—24no—
649C22—28no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22#551812yes—
660C22#552412yes—
661C22#552612yes—
662C22#552812yes—
663C22#552912yes—
664C22#553112yes—
665C22#553112yes—
666C22#553312yes—
667C22C2×#93011yescharacteristic
668C22C2×#93011yescharacteristic
669C22C2×#93011yescharacteristic
670C22#579111yescharacteristic
671C22#579111yescharacteristic
672C22#579111yescharacteristic
673C22#582412yes—
674C22#582412yes—
675C22#582412yes—
676C22#582511yescharacteristic
677C22#582511yescharacteristic
678C22#582811yescharacteristic
679C22#583112yes—
680C22#583211yescharacteristic
681C22C2×#95311yescharacteristic
682C4—24no—
683C4—24no—
684C4—24no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—124no—
689C4—124no—
690C4—242no—
691C4—242no—
692C3#1490711yescharacteristic, complemented, Sylow
693C2—44no—
694C2#8224612yes—
695C2#8225112yes—
696C2#8225212yes—
697C2#8225811yescharacteristic
698C2#8226111yescharacteristic
699C2#8226211yescharacteristic
700C2#8227311yescharacteristic
701C2#8227712yes—
702C2#8403211yescharacteristic
703C2#8409011yescharacteristic
704C2#8409311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#117C2
prime radicalC25×C3C23
hypercenterC22×#117D3
hypocenterC3#14907
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#4981C2semidirect
5. C2×#4928C2semidirect
6. #71566C2semidirect
7. #71565C2semidirect
10. #71536C2semidirect
11. #71534C2semidirect
692. C3#14907semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C2×#52102
C2×#49812
C2×#49532
C2×#49282
#715662
#715652
#715492
#715462
#715362
#715342
#149073