Finite Group Databasebeta

← #10827345

Subgroups — #10827345

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×#5040C211yesmaximal, characteristic, complemented
3#82349C211yesmaximal, characteristic, complemented
4C22×#810C211yesmaximal, characteristic, complemented
5#71794C212yesmaximal, complemented
6#71685C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#71612C212yesmaximal, complemented
9#71610C211yesmaximal, characteristic, complemented
10#71559C212yesmaximal, complemented
11#71553C211yesmaximal, characteristic, complemented
12#23961—31nomaximal, Sylow
13C22×C3×#41—21no—
14C23×Dic6C2211yescharacteristic
15C3×#547—24no—
16C3×#547—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×#121C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C22×#214C2212yes—
28C2×#924C2212yes—
29C23×C3⋊C8C2211yescharacteristic
30#5220—21no—
31C2×#832C2212yes—
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2212yes—
35#5040—24no—
36#4979—21no—
37C2×#810—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41#4947—21no—
42#4946—21no—
43#4929C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#547D311yescharacteristic
49C2×#448—31no—
50C22×#103—31no—
51#2409—31no—
52#2024—32no—
53#2000—32no—
54#1893—31no—
55#1877—32no—
56#1872—32no—
57#1869—31no—
58#1823—31no—
59C2×C3×#41—28no—
60C2×C3×#41—44no—
61C2×C3×#41—44no—
62C22×C3×Q8—21no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4D411yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×Dic3—21no—
73C22×Dic6—22no—
74C22×Dic6—24no—
75C22×Dic6—22no—
76C22×Dic6—21no—
77C22×Dic6—42no—
78C22×Dic6C2312yes—
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8—22no—
81C3×#123—28no—
82C3×#123—28no—
83C3×C22⋊Q8—28no—
84C3×#121—22no—
85C3×#121—22no—
86C3×#121—22no—
87C3×#121D414yes—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4D414yes—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×#217—21no—
110C2×#217—22no—
111C2×#214—24no—
112C2×#214—24no—
113C2×#214—42no—
114C2×C4⋊Dic3C2312yes—
115#924—24no—
116#924—42no—
117C2×Dic3⋊C4—22no—
118C22×C3⋊C8—22no—
119C22×C3⋊C8—42no—
120C22×C3⋊C8C2312yes—
121C22×C3⋊C8C2312yes—
122#832—42no—
123C3⋊#95—22no—
124C3⋊#95—22no—
125C3⋊#95—21no—
126#830—22no—
127#830—42no—
128#810—24no—
129#810—24no—
130#810—24no—
131C22×#41—21no—
132C23×Q8—31no—
133C22×SD16—32no—
134C23×C8—31no—
135#547—24no—
136#547—24no—
137C2×C4⋊Q8C2×D311yescharacteristic
138C2×#123—21no—
139C2×#123—21no—
140C2×C22⋊Q8—21no—
141C2×C22⋊Q8—32no—
142C2×#121C2×D311yescharacteristic
143C2×#121C2×D311yescharacteristic
144C2×C4×D4C2×D311yescharacteristic
145C2×C4×D4C2×D311yescharacteristic
146C22×C4⋊C4C2×D311yescharacteristic
147C22×C4⋊C4C2×D311yescharacteristic
148#448—64no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—32no—
152C2×#103—64no—
153C2×#102—32no—
154C2×C22⋊C8—31no—
155C2×C22⋊C8—32no—
156#395—61no—
157#393—61no—
158#391—32no—
159#386—61no—
160#385—61no—
161#380—31no—
162#340—32no—
163#340—32no—
164#340—32no—
165#336—32no—
166C24×C3—21no—
167C24×C3C2×D411yescharacteristic
168C24×C3C2×D411yescharacteristic
169C3×#41—432no—
170C2×C3×Q8—24no—
171C2×C3×Q8—42no—
172C2×C3×Q8—42no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4SD1614yescomplemented
185C2×C3×D4C2×D412yes—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4—42no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—44no—
214C22×Dic3—42no—
215C2×Dic6—28no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—48no—
222C2×Dic6—44no—
223C2×Dic6—44no—
224C2×Dic6—42no—
225C2×Dic6#4114yescomplemented
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4C2×D412yes—
234C3×C4⋊C4C2×D412yes—
235C3×C4⋊C4C2×D412yes—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—22no—
241C3×C42—21no—
242C3×C42—21no—
243#217—44no—
244#217—42no—
245#214—48no—
246C4⋊Dic3—22no—
247C4⋊Dic3—22no—
248Dic3⋊C4—44no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—44no—
256C2×#41—28no—
257C2×#41—44no—
258C2×#41—44no—
259C22×Q8—21no—
260C22×Q8—32no—
261C22×Q8—64no—
262C22×Q8—61no—
263C22×Q8—62no—
264C22×Q8—62no—
265C22×Q8—122no—
266C22×D4—21no—
267C22×D4—21no—
268C22×D4#6711yescharacteristic
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—61no—
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D311yescharacteristic
276C2×SD16—64no—
277C2×SD16—64no—
278C2×SD16—122no—
279C22×C8—32no—
280C22×C8—32no—
281C22×C8—62no—
282C22×C8—122no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285#123—28no—
286#123—28no—
287C22⋊Q8—28no—
288C22⋊Q8—64no—
289C22⋊Q8—122no—
290#121—22no—
291#121—22no—
292#121—22no—
293#121#6714yes—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4#6711yescharacteristic
307C2×C4⋊C4#6714yes—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—62no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4#6711yescharacteristic
317C2×C22⋊C4#6711yescharacteristic
318C2×C42C22×D311yescharacteristic
319#103—64no—
320#103—64no—
321#103—64no—
322#102—122no—
323C22⋊C8—62no—
324C22⋊C8—122no—
325#95—61no—
326#95—62no—
327#95—62no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3—42no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—42no—
339C23×C3#12012yes—
340C23×C3C2×SD1612yes—
341C23×C3C2×SD1612yes—
342C23×C3AGL(1,ℤ/8)12yes—
343C23×C3C22×D411yescharacteristic
344C3×Q8—48no—
345C3×D4—28no—
346C3×D4—24no—
347C3×D4—48no—
348C3×D4—48no—
349C3×D4—41no—
350C3×D4—41no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—48no—
365C2×C3×C4—48no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—48no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—42no—
376C2×C3×C4#12012yes—
377C2×C3×C4C2×SD1612yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4112yes—
381C2×C3×C4H(5,𝔽2)12yes—
382C2×Dic3—24no—
383C2×Dic3—48no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389Dic6—24no—
390Dic6—24no—
391Dic6—48no—
392Dic6—42no—
393Dic6—42no—
394C3⋊C8—42no—
395C3⋊C8—42no—
396C24—21no—
397C24C2×#6711yescharacteristic
398C24C2×#6711yescharacteristic
399#41—432no—
400C2×Q8—24no—
401C2×Q8—34no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—68no—
405C2×Q8—62no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—62no—
409C2×Q8—62no—
410C2×Q8—124no—
411C2×Q8—128no—
412C2×Q8—124no—
413C2×Q8—122no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4#21414yescomplemented
426C2×D4C2×#6712yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—124no—
448C22×C4—122no—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6712yes—
454C22×C4C2×#6712yes—
455C22×C4C23×D311yescharacteristic
456SD16—128no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—124no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4—124no—
474C4⋊C4C2×#6712yes—
475C4⋊C4C2×#6712yes—
476C4⋊C4C2×#6712yes—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—22no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—124no—
483C22⋊C4—122no—
484C42—21no—
485C42—21no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—22no—
491C22×C3—24no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3#45014yes—
501C22×C3C2×#12011yescharacteristic
502C22×C3#54612yes—
503C22×C3C22×SD1611yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3#57812yes—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—41no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—41no—
513C3×C4#52511yescharacteristic
514C3×C4#53511yescharacteristic
515C3×C4#54612yes—
516Dic3—44no—
517Dic3—82no—
518C23—22no—
519C23—22no—
520C23—21no—
521C23—22no—
522C23—22no—
523C23—21no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23#93212yes—
530C23C2×#21412yes—
531C23C2×#21412yes—
532C23#95312yes—
533C23C22×#6711yescharacteristic
534Q8—48no—
535Q8—64no—
536Q8—64no—
537Q8—128no—
538Q8—122no—
539Q8—122no—
540D4—28no—
541D4—24no—
542D4—48no—
543D4—48no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—64no—
572C2×C4—128no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4C2×#21412yes—
579C2×C4#95312yes—
580C2×C4D3×#4112yes—
581C2×C4#100912yes—
582C2×C4C22×#6711yescharacteristic
583C2×C4C22×#6711yescharacteristic
584C8—122no—
585C8—122no—
586C2×C3—24no—
587C2×C3—42no—
588C2×C3—82no—
589C2×C3#240411yescharacteristic
590C2×C3C2×#45011yescharacteristic
591C2×C3#301111yescharacteristic
592C2×C3#329812yes—
593C2×C3#330812yes—
594C22—24no—
595C22—24no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—24no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—82no—
607C22—82no—
608C22#521814yes—
609C22C2×#93211yescharacteristic
610C22C22×#21411yescharacteristic
611C22#582912yes—
612C22#583812yes—
613C22C2×#95311yescharacteristic
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—124no—
622C4—242no—
623C4#582811yescharacteristic
624C4#582912yes—
625C4#583011yescharacteristic
626C3#2396111yescharacteristic, complemented, Sylow
627C2—24no—
628C2—42no—
629C2—82no—
630C2#8353112yes—
631C2#8353212yes—
632C2C2×#521811yescharacteristic
633C2#8409611yescharacteristic
634C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#547C2
socleC23×C3C22×D4
hypercenterC2×#547D3
hypocenterC3#23961
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C2×#5040C2semidirect
3. #82349C2semidirect
4. C22×#810C2semidirect
5. #71794C2semidirect
6. #71685C2semidirect
7. #71681C2semidirect
8. #71612C2semidirect
9. #71610C2semidirect
10. #71559C2semidirect
11. #71553C2semidirect
184. C2×C3×D4SD16semidirect
225. C2×Dic6#41semidirect
425. C2×D4#214semidirect
626. C3#23961semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×#50402
#823492
C22×#8102
#717942
#716852
#716812
#716122
#716102
#715592
#715532
#239613