Finite Group Databasebeta

← #10832465

Subgroups — #10832465

Order 768.

1450 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic, complemented
3#84075C211yesmaximal, characteristic, complemented
4C2×#5218C211yesmaximal, characteristic, complemented
5C2×#5200C212yesmaximal, complemented
6#72129C211yesmaximal, characteristic, complemented
7#72097C212yesmaximal, complemented
8#72093C212yesmaximal, complemented
9#72091C212yesmaximal, complemented
10#72003C211yesmaximal, characteristic, complemented
11#72001C211yesmaximal, characteristic, complemented
12#24399—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)—21no—
16C2×C3×AGL(1,ℤ/8)—21no—
17C2×C3×AGL(1,ℤ/8)—28no—
18C2×C3×AGL(1,ℤ/8)C2212yes—
19C2×C3×AGL(1,ℤ/8)C2212yes—
20C22×C3×SD16—21no—
21C22×C3×D8—21no—
22C22×C3×M16C2211yescharacteristic
23C2×#951C2211yescharacteristic, complemented
24C2×#951C2211yescharacteristic, complemented
25C2×#948C2212yescomplemented
26C22×#214C2212yescomplemented
27C2×#924C2211yescharacteristic
28#5239—21no—
29#5239—21no—
30#5234—22no—
31#5234C2212yescomplemented
32#5232C2211yescharacteristic, complemented
33#5232C2211yescharacteristic, complemented
34#5228—21no—
35#5218—21no—
36#5218—21no—
37#5218—21no—
38#5218—22no—
39C2×#833C2212yescomplemented
40C2×#833C2212yescomplemented
41C2×#832C2212yescomplemented
42#5207—21no—
43#5207C2211yescharacteristic
44#5207C2211yescharacteristic
45C2×#830—21no—
46C2×#830C2211yescharacteristic
47#5200—28no—
48C2×#824C2212yescomplemented
49C2×#822C2212yescomplemented
50#4558C2212yescomplemented
51#4554C2212yescomplemented
52C22×AGL(1,ℤ/8)D311yescharacteristic
53C2×#471—32no—
54#3007—31no—
55C2×#450—31no—
56#2880—31no—
57#2051—32no—
58#2019—32no—
59#1907—32no—
60#1901—31no—
61#1855—31no—
62#1783—31no—
63C25×C3—21no—
64C2×C3×#41—21no—
65C2×C3×#41—21no—
66C2×C3×#41—24no—
67C2×C3×#41—42no—
68C2×C3×#41D411yescharacteristic
69C2×C3×#41D411yescharacteristic
70C2×C3×#41C2311yescharacteristic
71C2×C3×#41C2311yescharacteristic
72C22×C3×Q8D411yescharacteristic
73C22×C3×D4—24no—
74C22×C3×D4—21no—
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4—21no—
80C22×C3×D4—42no—
81C22×C3×D4D411yescharacteristic
82C22×C3×D4C2312yes—
83C23×C3×C4—21no—
84C23×C3×C4C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C3×AGL(1,ℤ/8)—28no—
87C3×AGL(1,ℤ/8)—28no—
88C3×AGL(1,ℤ/8)—24no—
89C3×AGL(1,ℤ/8)—24no—
90C3×AGL(1,ℤ/8)—24no—
91C3×AGL(1,ℤ/8)D418yes—
92C2×C3×SD16—28no—
93C2×C3×SD16—22no—
94C2×C3×SD16—22no—
95C2×C3×D8—28no—
96C2×C3×D8—22no—
97C2×C3×D8—22no—
98C2×C3×M16—21no—
99C2×C3×M16—24no—
100C2×C3×M16C2312yes—
101C22×C3×C8—21no—
102C2×#217—21no—
103#951—24no—
104#951—42no—
105#948—24no—
106#948—42no—
107C2×#216—21no—
108C2×#216—21no—
109C2×#214—24no—
110C2×#214—21no—
111C2×#214—21no—
112C2×#214—22no—
113C2×#214—22no—
114C2×#214—42no—
115C2×C4⋊Dic3C2311yescharacteristic
116#924—21no—
117#924—21no—
118#924—21no—
119#924—21no—
120C2×Dic3⋊C4—21no—
121C2×#208—22no—
122C2×#208C2311yescharacteristic
123C2×#208C2312yes—
124C2×#208C2311yescharacteristic
125C22×C3⋊C8—22no—
126C22×C3⋊C8C2312yes—
127#835—21no—
128#835—21no—
129#833—24no—
130#833—42no—
131#832—21no—
132#832—21no—
133#832—22no—
134#832—22no—
135#830—21no—
136#830—21no—
137#830—21no—
138#830—21no—
139#830—22no—
140#830—22no—
141#830D411yescharacteristic
142#830D411yescharacteristic
143#830D412yes—
144#824—24no—
145#822—24no—
146C22×#41C2×D311yescharacteristic
147C23×D4C2×D311yescharacteristic
148C2×#137—31no—
149C2×#137—31no—
150C2×AGL(1,ℤ/8)—21no—
151C2×AGL(1,ℤ/8)—21no—
152C2×AGL(1,ℤ/8)—28no—
153C2×AGL(1,ℤ/8)C2×D312yes—
154C2×AGL(1,ℤ/8)C2×D312yes—
155C22×SD16—21no—
156C22×SD16—32no—
157C22×D8—21no—
158C2×#131—32no—
159C22×M16C2×D311yescharacteristic
160C2×C22⋊Q8—31no—
161#471—68no—
162#452—61no—
163#452—61no—
164#451—61no—
165#450—61no—
166#450—61no—
167#450—62no—
168#450—61no—
169C2×#108—32no—
170#417—31no—
171#417—31no—
172C2×#102—32no—
173C2×#101—32no—
174C2×#100—32no—
175C2×#100—32no—
176#408—32no—
177#408—62no—
178#407—31no—
179#407—31no—
180#407—61no—
181C2×C22⋊C8—31no—
182C2×C22⋊C8—61no—
183#343—32no—
184#341—32no—
185C24×C3—21no—
186C24×C3—22no—
187C24×C3—44no—
188C24×C3—41no—
189C24×C3—41no—
190C24×C3—82no—
191C24×C3—82no—
192C24×C3#4111yescharacteristic
193C3×#41—22no—
194C3×#41—22no—
195C3×#41—22no—
196C3×#41—22no—
197C3×#41—22no—
198C3×#41—44no—
199C3×#41—41no—
200C3×#41—41no—
201C3×#41—44no—
202C3×#41C2×D414yes—
203C2×C3×Q8—21no—
204C2×C3×Q8—22no—
205C2×C3×Q8—41no—
206C2×C3×Q8C2×D411yescharacteristic
207C2×C3×Q8C2×D411yescharacteristic
208C2×C3×D4—21no—
209C2×C3×D4—24no—
210C2×C3×D4—24no—
211C2×C3×D4—22no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—22no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—41no—
222C2×C3×D4—44no—
223C2×C3×D4—44no—
224C2×C3×D4—44no—
225C2×C3×D4—42no—
226C2×C3×D4—42no—
227C2×C3×D4—44no—
228C2×C3×D4—41no—
229C2×C3×D4—41no—
230C2×C3×D4C2×D411yescharacteristic
231C2×C3×D4C2×D414yes—
232C2×C3×D4C2×D411yescharacteristic
233C22×C3×C4—21no—
234C22×C3×C4—22no—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—44no—
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D411yescharacteristic
243C22×C3×C4C2×D411yescharacteristic
244C22×C3×C4C2×D411yescharacteristic
245C22×C3×C4#4111yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C22×Dic3—21no—
248C22×Dic3—21no—
249C2×Dic6—22no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—22no—
253C3×SD16—28no—
254C3×SD16—28no—
255C3×D8—28no—
256C3×D8—28no—
257C3×M16—24no—
258C3×M16—22no—
259C3×M16C2×D414yes—
260C2×C3×C8—24no—
261C2×C3×C8—22no—
262#217—41no—
263#217—41no—
264#216—42no—
265#216—42no—
266#214—44no—
267#214—44no—
268#214—42no—
269#214—42no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272Dic3⋊C4—41no—
273Dic3⋊C4—41no—
274#208—22no—
275#208—22no—
276#208—22no—
277#208—22no—
278#208—44no—
279C2×C3⋊C8—22no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—44no—
285C2×C3⋊C8C2×D412yes—
286C2×C3⋊C8C2×D412yes—
287C25—21no—
288C2×#41—21no—
289C2×#41—21no—
290C2×#41—24no—
291C2×#41—42no—
292C2×#41#6711yescharacteristic
293C2×#41#6711yescharacteristic
294C2×#41C22×D311yescharacteristic
295C2×#41C22×D311yescharacteristic
296C22×Q8—31no—
297C22×Q8#6711yescharacteristic
298C22×D4—24no—
299C22×D4—21no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—42no—
306C22×D4#6711yescharacteristic
307C22×D4C22×D312yes—
308C23×C4—21no—
309C23×C4C22×D311yescharacteristic
310#137—64no—
311#137—122no—
312AGL(1,ℤ/8)—28no—
313AGL(1,ℤ/8)—28no—
314AGL(1,ℤ/8)—24no—
315AGL(1,ℤ/8)—24no—
316AGL(1,ℤ/8)—24no—
317AGL(1,ℤ/8)#6718yes—
318C2×Q16—61no—
319C2×Q16—61no—
320C2×SD16—28no—
321C2×SD16—22no—
322C2×SD16—22no—
323C2×SD16—62no—
324C2×SD16—62no—
325C2×SD16—64no—
326C2×SD16—61no—
327C2×SD16—61no—
328C2×SD16—122no—
329C2×D8—28no—
330C2×D8—22no—
331C2×D8—22no—
332#131—64no—
333#131—122no—
334C2×M16—21no—
335C2×M16—24no—
336C2×M16—32no—
337C2×M16—31no—
338C2×M16—31no—
339C2×M16—62no—
340C2×M16C22×D312yes—
341C22×C8—21no—
342C22×C8—32no—
343C22×C8—62no—
344C22⋊Q8—61no—
345C22⋊Q8—61no—
346C22⋊Q8—61no—
347C22⋊Q8—61no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—61no—
350C2×C22⋊C4—61no—
351#108—64no—
352#103—61no—
353#103—61no—
354#102—62no—
355#102—62no—
356#102—61no—
357#102—61no—
358#101—64no—
359#100—64no—
360#100—122no—
361C22⋊C8—31no—
362C22⋊C8—31no—
363C22⋊C8—32no—
364C22⋊C8—62no—
365C22⋊C8—62no—
366C22⋊C8—61no—
367C22⋊C8—61no—
368C22⋊C8—61no—
369C22⋊C8—61no—
370C23×C3—21no—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—21no—
374C23×C3—24no—
375C23×C3—41no—
376C23×C3—42no—
377C23×C3—44no—
378C23×C3—44no—
379C23×C3—42no—
380C23×C3—42no—
381C23×C3—42no—
382C23×C3—81no—
383C23×C3—82no—
384C23×C3—82no—
385C23×C3—82no—
386C23×C3—82no—
387C23×C3—82no—
388C23×C3—816no—
389C23×C3—82no—
390C23×C3#12111yescharacteristic
391C23×C3#12111yescharacteristic
392C23×C3C2×#4111yescharacteristic
393C3×Q8—21no—
394C3×Q8—21no—
395C3×Q8—21no—
396C3×Q8—42no—
397C3×Q8#12012yes—
398C3×D4—24no—
399C3×D4—21no—
400C3×D4—24no—
401C3×D4—21no—
402C3×D4—21no—
403C3×D4—24no—
404C3×D4—42no—
405C3×D4—42no—
406C3×D4—44no—
407C3×D4—44no—
408C3×D4—42no—
409C3×D4—42no—
410C3×D4—42no—
411C3×D4—42no—
412C3×D4#12012yes—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—24no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—21no—
423C2×C3×C4—42no—
424C2×C3×C4—42no—
425C2×C3×C4—42no—
426C2×C3×C4—42no—
427C2×C3×C4#12111yescharacteristic
428C2×C3×C4#12111yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C22×D411yescharacteristic
431C2×C3×C4C22×D412yes—
432C2×C3×C4H(5,𝔽2)12yes—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—42no—
436C2×Dic3—81no—
437Dic6—41no—
438Dic6—41no—
439Dic6—42no—
440C3×C8—24no—
441C3⋊C8—22no—
442C3⋊C8—22no—
443C3⋊C8—41no—
444C3⋊C8—41no—
445C3⋊C8—42no—
446C24—21no—
447C24—22no—
448C24—44no—
449C24—41no—
450C24—41no—
451C24—82no—
452C24—82no—
453C24#23711yescharacteristic
454#41—22no—
455#41—22no—
456#41—22no—
457#41—22no—
458#41—22no—
459#41—44no—
460#41—41no—
461#41—41no—
462#41—44no—
463#41C2×#6714yes—
464C2×Q8—21no—
465C2×Q8—22no—
466C2×Q8—41no—
467C2×Q8—61no—
468C2×Q8—62no—
469C2×Q8—62no—
470C2×Q8—61no—
471C2×Q8C2×#6711yescharacteristic
472C2×Q8C2×#6711yescharacteristic
473C2×D4—21no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—41no—
487C2×D4—44no—
488C2×D4—44no—
489C2×D4—44no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—44no—
493C2×D4—41no—
494C2×D4—41no—
495C2×D4C2×#6714yes—
496C2×D4C2×#6711yescharacteristic
497C2×D4C2×#6711yescharacteristic
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—44no—
506C22×C4—61no—
507C22×C4—61no—
508C22×C4D3×D411yescharacteristic
509C22×C4D3×D411yescharacteristic
510C22×C4#23711yescharacteristic
511C22×C4C2×#6711yescharacteristic
512C22×C4C2×#6711yescharacteristic
513C22×C4C23×D311yescharacteristic
514Q16—122no—
515Q16—122no—
516SD16—28no—
517SD16—28no—
518SD16—124no—
519SD16—124no—
520SD16—122no—
521SD16—122no—
522D8—28no—
523D8—28no—
524M16—24no—
525M16—22no—
526M16—62no—
527M16—62no—
528M16—62no—
529M16—62no—
530M16—124no—
531M16C2×#6714yes—
532C2×C8—24no—
533C2×C8—22no—
534C2×C8—32no—
535C2×C8—32no—
536C2×C8—62no—
537C2×C8—62no—
538C2×C8—62no—
539C2×C8—61no—
540C2×C8—61no—
541C2×C8—124no—
542C4⋊C4—61no—
543C4⋊C4—61no—
544C4⋊C4—121no—
545C4⋊C4—121no—
546C22⋊C4—121no—
547C22⋊C4—121no—
548C22×C3—21no—
549C22×C3—21no—
550C22×C3—21no—
551C22×C3—21no—
552C22×C3—21no—
553C22×C3—22no—
554C22×C3—21no—
555C22×C3—21no—
556C22×C3—44no—
557C22×C3—41no—
558C22×C3—41no—
559C22×C3—41no—
560C22×C3—44no—
561C22×C3—41no—
562C22×C3—41no—
563C22×C3—84no—
564C22×C3—84no—
565C22×C3—88no—
566C22×C3—84no—
567C22×C3—88no—
568C22×C3—82no—
569C22×C3—82no—
570C22×C3#47112yescomplemented
571C22×C3C2×#12111yescharacteristic
572C22×C3#54612yes—
573C3×C4—21no—
574C3×C4—21no—
575C3×C4—22no—
576C3×C4—41no—
577C3×C4C2×#12011yescharacteristic
578C3×C4#53411yescharacteristic
579C3×C4D4212yes—
580Dic3—81no—
581Dic3—81no—
582C23—22no—
583C23—21no—
584C23—22no—
585C23—21no—
586C23—24no—
587C23—41no—
588C23—42no—
589C23—44no—
590C23—44no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—81no—
595C23—82no—
596C23—82no—
597C23—82no—
598C23—82no—
599C23—82no—
600C23—816no—
601C23—82no—
602C23#93911yescharacteristic
603C23#93911yescharacteristic
604C23C2×#23711yescharacteristic
605Q8—21no—
606Q8—21no—
607Q8—21no—
608Q8—42no—
609Q8—122no—
610Q8—121no—
611Q8—121no—
612Q8#95312yes—
613D4—24no—
614D4—24no—
615D4—21no—
616D4—21no—
617D4—21no—
618D4—24no—
619D4—42no—
620D4—42no—
621D4—44no—
622D4—44no—
623D4—42no—
624D4—42no—
625D4—42no—
626D4—42no—
627D4#95312yes—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—122no—
645C2×C4—241no—
646C2×C4#93911yescharacteristic
647C2×C4#93911yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4#100412yes—
650C2×C4C22×#6712yes—
651C2×C4C22×#6711yescharacteristic
652C8—24no—
653C8—62no—
654C8—62no—
655C8—122no—
656C8—121no—
657C8—121no—
658C2×C3—22no—
659C2×C3—22no—
660C2×C3—41no—
661C2×C3—42no—
662C2×C3—81no—
663C2×C3—84no—
664C2×C3#241011yescharacteristic
665C2×C3C2×#47111yescharacteristic
666C2×C3#306811yescharacteristic
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—22no—
673C22—21no—
674C22—21no—
675C22—44no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—44no—
680C22—41no—
681C22—41no—
682C22—84no—
683C22—84no—
684C22—88no—
685C22—84no—
686C22—88no—
687C22—82no—
688C22—82no—
689C22#520012yescomplemented
690C22C2×#93911yescharacteristic
691C22#580612yes—
692C4—21no—
693C4—21no—
694C4—22no—
695C4—41no—
696C4—241no—
697C4—241no—
698C4D4×#6712yes—
699C4#580411yescharacteristic
700C4C2×#95311yescharacteristic
701C3#2439911yescharacteristic, complemented, Sylow
702C2—22no—
703C2—22no—
704C2—41no—
705C2—42no—
706C2—81no—
707C2—84no—
708C2C2×#520011yescharacteristic
709C2#8394311yescharacteristic
710C2#8404811yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×AGL(1,ℤ/8)C2
socleC22×C3C2×#121
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#24399
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #84084C2semidirect
3. #84075C2semidirect
4. C2×#5218C2semidirect
5. C2×#5200C2semidirect
6. #72129C2semidirect
7. #72097C2semidirect
8. #72093C2semidirect
9. #72091C2semidirect
10. #72003C2semidirect
11. #72001C2semidirect
23. C2×#951C22semidirect
24. C2×#951C22semidirect
25. C2×#948C22semidirect
26. C22×#214C22semidirect
31. #5234C22semidirect
32. #5232C22semidirect
33. #5232C22semidirect
39. C2×#833C22semidirect
40. C2×#833C22semidirect
41. C2×#832C22semidirect
48. C2×#824C22semidirect
49. C2×#822C22semidirect
50. #4558C22semidirect
51. #4554C22semidirect
570. C22×C3#471semidirect
689. C22#5200semidirect
701. C3#24399semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
#840842
#840752
C2×#52182
C2×#52002
#721292
#720972
#720932
#720912
#720032
#720012
#243993