Finite Group Databasebeta

← #10832455

Subgroups — #10832455

Order 768.

1476 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#949C211yesmaximal, characteristic, complemented
2C3×#3006C211yesmaximal, characteristic, complemented
3#84075C211yesmaximal, characteristic, complemented
4C2×#5199C212yesmaximal, complemented
5C2×#5036C211yesmaximal, characteristic, complemented
6#72095C212yesmaximal, complemented
7#71683C211yesmaximal, characteristic, complemented
8#70307C212yesmaximal, complemented
9#70303C212yesmaximal, complemented
10#69988C211yesmaximal, characteristic, complemented
11#69984C211yesmaximal, characteristic, complemented
12#24398—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)C2211yescharacteristic, complemented
16C2×C3×AGL(1,ℤ/8)C2211yescharacteristic, complemented
17C2×C3×#121C2211yescharacteristic
18C2×#949—21no—
19C2×#949—21no—
20C2×#949—28no—
21C2×#949C2212yes—
22C2×#949C2212yes—
23C2×#948C2212yescomplemented
24C22×#215—21no—
25C22×#213—21no—
26C22×#213C2212yescomplemented
27C22×#208C2211yescharacteristic
28C3×#449—21no—
29C3×#449—21no—
30C3×#448—21no—
31C3×#447—21no—
32C3×#417C2211yescharacteristic, complemented
33C3×#417C2211yescharacteristic, complemented
34C3×#407C2211yescharacteristic
35#5234—22no—
36#5234C2212yescomplemented
37C2×#833C2212yescomplemented
38#5207—21no—
39#5207C2211yescharacteristic
40C2×#830—21no—
41C2×#830C2211yescharacteristic
42#5199—28no—
43C2×#823C2212yescomplemented
44C2×#823C2212yescomplemented
45C2×#822C2212yescomplemented
46#5036—21no—
47#5036—21no—
48#5036—22no—
49C2×#809C2212yescomplemented
50#4558C2212yescomplemented
51#4530C2212yescomplemented
52C22×AGL(1,ℤ/8)—31no—
53C2×#472—32no—
54#3006D311yescharacteristic, complemented
55C2×#447—31no—
56#2880—31no—
57#2051—32no—
58#2018—32no—
59#1907—32no—
60#1900—31no—
61#1855—31no—
62#1783—31no—
63C24×D3—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—21no—
74C22×C3×D4—21no—
75C22×C3×D4D411yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×D12—21no—
80C22×D12—21no—
81C22×D12—24no—
82C22×D12—21no—
83C22×D12—21no—
84C22×D12—21no—
85C22×D12—42no—
86C22×D12C2312yes—
87C3×AGL(1,ℤ/8)—24no—
88C3×AGL(1,ℤ/8)—42no—
89C2×C3×SD16—21no—
90C2×C3×SD16—21no—
91C2×C3×D8—21no—
92C2×C3×D8—21no—
93C2×C3×M16C2311yescharacteristic
94C2×C3×M16C2311yescharacteristic
95C3×#121—21no—
96C3×#121—21no—
97C3×#121—21no—
98C3×#121—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101#949—28no—
102#949—28no—
103#949—24no—
104#949—24no—
105#949—24no—
106#949D418yescomplemented
107#948—24no—
108#948—42no—
109C2×#215—28no—
110C2×#215—22no—
111C2×#215—22no—
112C2×#213—28no—
113C2×#213—22no—
114C2×#213—22no—
115C2×#213—24no—
116C2×#213—22no—
117C2×#213—22no—
118C2×#213—42no—
119C2×#208—21no—
120C2×#208—24no—
121C2×#208—22no—
122C2×#208C2312yes—
123C2×#208C2312yes—
124C22×C3⋊C8—21no—
125C22×C3⋊C8—22no—
126C22×C3⋊C8C2312yes—
127C3×#103—21no—
128C3×#103—21no—
129C3×#102—21no—
130C3×#102—21no—
131C3×C22⋊C8—21no—
132C3×C22⋊C8—21no—
133#833—24no—
134#830—21no—
135#830—21no—
136#830—22no—
137#830—22no—
138#830D411yescharacteristic
139#830D411yescharacteristic
140#830D412yes—
141#823—24no—
142#823—42no—
143#822—24no—
144#809—22no—
145#809—22no—
146C22×#41C2×D311yescharacteristic
147C23×D4—31no—
148C2×AGL(1,ℤ/8)—32no—
149C2×AGL(1,ℤ/8)—32no—
150C2×AGL(1,ℤ/8)—68no—
151C2×AGL(1,ℤ/8)—61no—
152C2×AGL(1,ℤ/8)—61no—
153C2×AGL(1,ℤ/8)C2×D311yescharacteristic, complemented
154C2×AGL(1,ℤ/8)C2×D311yescharacteristic, complemented
155C22×SD16—61no—
156C22×D8—32no—
157C22×D8—61no—
158C2×#131—32no—
159C22×M16—31no—
160C2×#121C2×D311yescharacteristic
161#472—68no—
162#449—21no—
163#449—21no—
164#448—21no—
165#447—21no—
166#447—61no—
167#447—61no—
168#447—62no—
169C2×#108—32no—
170#417C2×D311yescharacteristic, complemented
171#417C2×D311yescharacteristic, complemented
172C2×#102—32no—
173C2×#100—32no—
174C2×#100—32no—
175C2×#100—32no—
176#408—32no—
177#408—62no—
178#407—31no—
179#407—61no—
180#407C2×D311yescharacteristic
181C2×C22⋊C8—31no—
182C2×C22⋊C8—61no—
183#343—32no—
184#341—32no—
185C24×C3—21no—
186C24×C3—21no—
187C24×C3#4111yescharacteristic
188C23×D3—22no—
189C23×D3—44no—
190C23×D3—41no—
191C23×D3—41no—
192C23×D3—82no—
193C23×D3—82no—
194C3×#41—22no—
195C3×#41—22no—
196C3×#41—22no—
197C3×#41—22no—
198C3×#41—22no—
199C3×#41—44no—
200C3×#41—41no—
201C3×#41—41no—
202C3×#41—44no—
203C3×#41C2×D414yes—
204C2×C3×Q8—21no—
205C2×C3×Q8—22no—
206C2×C3×Q8—41no—
207C2×C3×Q8C2×D411yescharacteristic
208C2×C3×Q8C2×D411yescharacteristic
209C2×C3×D4—21no—
210C2×C3×D4—22no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—22no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—22no—
219C2×C3×D4—41no—
220C2×C3×D4—44no—
221C2×C3×D4—41no—
222C2×C3×D4—41no—
223C2×C3×D4—82no—
224C2×C3×D4C2×D411yescharacteristic
225C2×C3×D4C2×D411yescharacteristic
226C22×C3×C4—21no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—44no—
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4#4111yescharacteristic
240C22×C3×C4C2411yescharacteristic
241C2×D12—24no—
242C2×D12—24no—
243C2×D12—22no—
244C2×D12—22no—
245C2×D12—22no—
246C2×D12—22no—
247C2×D12—22no—
248C2×D12—44no—
249C2×D12—44no—
250C2×D12—44no—
251C2×D12—42no—
252C2×D12—42no—
253C2×D12—41no—
254C2×D12—41no—
255C2×D12C2×D414yes—
256C3×SD16—42no—
257C3×SD16—42no—
258C3×D8—42no—
259C3×D8—42no—
260C3×M16—22no—
261C3×M16—22no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C22⋊C4—41no—
267C3×C22⋊C4—41no—
268#215—28no—
269#215—28no—
270#213—28no—
271#213—28no—
272#213—44no—
273#213—44no—
274#208—24no—
275#208—22no—
276#208—22no—
277#208—22no—
278#208—44no—
279#208C2×D414yes—
280C2×C3⋊C8—24no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8—22no—
283C2×C3⋊C8—22no—
284C2×C3⋊C8—22no—
285C2×C3⋊C8—44no—
286C2×C3⋊C8C2×D412yes—
287C2×C3⋊C8C2×D412yes—
288C25—61no—
289C2×#41—21no—
290C2×#41—21no—
291C2×#41—24no—
292C2×#41—42no—
293C2×#41#6711yescharacteristic
294C2×#41#6711yescharacteristic
295C2×#41C22×D311yescharacteristic
296C2×#41C22×D311yescharacteristic
297C22×Q8#6711yescharacteristic
298C22×D4—21no—
299C22×D4—21no—
300C22×D4—32no—
301C22×D4—64no—
302C22×D4—61no—
303C22×D4—61no—
304C22×D4—61no—
305C22×D4—61no—
306C22×D4—61no—
307C22×D4—122no—
308C22×D4#6711yescharacteristic
309C22×D4C22×D311yescharacteristic
310C23×C4—21no—
311C23×C4C22×D311yescharacteristic
312AGL(1,ℤ/8)—24no—
313AGL(1,ℤ/8)—38no—
314AGL(1,ℤ/8)—42no—
315AGL(1,ℤ/8)—64no—
316AGL(1,ℤ/8)—64no—
317AGL(1,ℤ/8)—68no—
318AGL(1,ℤ/8)—68no—
319AGL(1,ℤ/8)—64no—
320C2×SD16—21no—
321C2×SD16—21no—
322C2×SD16—68no—
323C2×SD16—62no—
324C2×SD16—62no—
325C2×D8—21no—
326C2×D8—21no—
327C2×D8—62no—
328C2×D8—62no—
329C2×D8—68no—
330C2×D8—62no—
331C2×D8—62no—
332C2×D8—64no—
333C2×D8—122no—
334#131—64no—
335#131—122no—
336C2×M16—32no—
337C2×M16—32no—
338C2×M16—62no—
339C2×M16—64no—
340C2×M16—61no—
341C2×M16C22×D311yescharacteristic
342C2×M16C22×D311yescharacteristic
343C22×C8—32no—
344C22×C8—62no—
345C22×C8—61no—
346#121—21no—
347#121—21no—
348#121—21no—
349#121—21no—
350C2×C4⋊C4C22×D311yescharacteristic
351C2×C22⋊C4—21no—
352#108—64no—
353#103—21no—
354#103—21no—
355#102—21no—
356#102—21no—
357#102—62no—
358#102—62no—
359#100—64no—
360#100—64no—
361#100—122no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364C22⋊C8—31no—
365C22⋊C8—31no—
366C22⋊C8—32no—
367C22⋊C8—62no—
368C22⋊C8—62no—
369C22⋊C8—61no—
370C22⋊C8—61no—
371C23×C3—21no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—21no—
375C23×C3—41no—
376C23×C3—42no—
377C23×C3—42no—
378C23×C3—41no—
379C23×C3—42no—
380C23×C3—81no—
381C23×C3—81no—
382C23×C3—81no—
383C23×C3#12111yescharacteristic
384C23×C3#12111yescharacteristic
385C23×C3C2×#4111yescharacteristic
386C22×D3—24no—
387C22×D3—44no—
388C22×D3—44no—
389C22×D3—42no—
390C22×D3—42no—
391C22×D3—82no—
392C22×D3—82no—
393C22×D3—82no—
394C22×D3—816no—
395C22×D3—82no—
396C22×D3—82no—
397C22×D3—82no—
398C3×Q8—21no—
399C3×Q8—21no—
400C3×Q8—21no—
401C3×Q8—42no—
402C3×Q8#12012yes—
403C3×D4—24no—
404C3×D4—21no—
405C3×D4—21no—
406C3×D4—21no—
407C3×D4—42no—
408C3×D4—42no—
409C3×D4—42no—
410C3×D4—42no—
411C3×D4—41no—
412C3×D4—41no—
413C3×D4—42no—
414C3×D4—82no—
415C3×D4—82no—
416C3×D4#12012yes—
417C2×C3×C4—21no—
418C2×C3×C4—22no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—24no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—21no—
427C2×C3×C4—42no—
428C2×C3×C4—42no—
429C2×C3×C4—42no—
430C2×C3×C4—42no—
431C2×C3×C4—41no—
432C2×C3×C4—81no—
433C2×C3×C4#12111yescharacteristic
434C2×C3×C4#12111yescharacteristic
435C2×C3×C4C22×D412yes—
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4H(5,𝔽2)12yes—
439D12—24no—
440D12—24no—
441D12—44no—
442D12—44no—
443D12—42no—
444D12—42no—
445C3×C8—41no—
446C3×C8—41no—
447C3⋊C8—24no—
448C3⋊C8—22no—
449C3⋊C8—22no—
450C3⋊C8—42no—
451C24—21no—
452C24—21no—
453C24—62no—
454C24—124no—
455C24—121no—
456C24—121no—
457C24—242no—
458C24—242no—
459C24#23711yescharacteristic
460#41—22no—
461#41—22no—
462#41—22no—
463#41—22no—
464#41—22no—
465#41—44no—
466#41—41no—
467#41—41no—
468#41—44no—
469#41D3×D414yes—
470C2×Q8—21no—
471C2×Q8—22no—
472C2×Q8—41no—
473C2×Q8C2×#6711yescharacteristic
474C2×Q8C2×#6711yescharacteristic
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—34no—
486C2×D4—41no—
487C2×D4—44no—
488C2×D4—41no—
489C2×D4—41no—
490C2×D4—64no—
491C2×D4—64no—
492C2×D4—62no—
493C2×D4—62no—
494C2×D4—62no—
495C2×D4—62no—
496C2×D4—62no—
497C2×D4—82no—
498C2×D4—124no—
499C2×D4—124no—
500C2×D4—124no—
501C2×D4—122no—
502C2×D4—122no—
503C2×D4—121no—
504C2×D4—121no—
505C2×D4C2×#6711yescharacteristic
506C2×D4C2×#6711yescharacteristic
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—44no—
516C22×C4D3×D411yescharacteristic
517C22×C4D3×D411yescharacteristic
518C22×C4#23711yescharacteristic
519C22×C4C2×#6711yescharacteristic
520C22×C4C2×#6711yescharacteristic
521C22×C4C23×D311yescharacteristic
522SD16—42no—
523SD16—42no—
524SD16—68no—
525SD16—68no—
526D8—42no—
527D8—42no—
528D8—68no—
529D8—68no—
530D8—124no—
531D8—124no—
532M16—22no—
533M16—22no—
534M16—34no—
535M16—64no—
536M16—62no—
537M16—62no—
538M16—62no—
539M16—124no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—32no—
543C2×C8—32no—
544C2×C8—64no—
545C2×C8—62no—
546C2×C8—62no—
547C2×C8—62no—
548C2×C8—62no—
549C2×C8—124no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22×C3—21no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—21no—
559C22×C3—22no—
560C22×C3—21no—
561C22×C3—21no—
562C22×C3—44no—
563C22×C3—41no—
564C22×C3—41no—
565C22×C3—41no—
566C22×C3—41no—
567C22×C3—41no—
568C22×C3—84no—
569C22×C3—81no—
570C22×C3—81no—
571C22×C3—81no—
572C22×C3—162no—
573C22×C3#47212yescomplemented
574C22×C3C2×#12111yescharacteristic
575C22×C3#54612yes—
576C2×D3—44no—
577C2×D3—88no—
578C2×D3—84no—
579C2×D3—88no—
580C2×D3—82no—
581C2×D3—82no—
582C2×D3—84no—
583C3×C4—21no—
584C3×C4—21no—
585C3×C4—22no—
586C3×C4—41no—
587C3×C4—81no—
588C3×C4C2×#12011yescharacteristic
589C3×C4#53411yescharacteristic
590C3×C4D4212yes—
591C23—22no—
592C23—21no—
593C23—22no—
594C23—21no—
595C23—41no—
596C23—42no—
597C23—42no—
598C23—41no—
599C23—42no—
600C23—64no—
601C23—81no—
602C23—81no—
603C23—81no—
604C23—124no—
605C23—124no—
606C23—122no—
607C23—122no—
608C23—242no—
609C23—242no—
610C23—242no—
611C23—242no—
612C23—242no—
613C23—2416no—
614C23—242no—
615C23#93911yescharacteristic
616C23#93911yescharacteristic
617C23C2×#23711yescharacteristic
618Q8—21no—
619Q8—21no—
620Q8—21no—
621Q8—42no—
622Q8#93712yes—
623D4—24no—
624D4—21no—
625D4—21no—
626D4—21no—
627D4—42no—
628D4—42no—
629D4—42no—
630D4—42no—
631D4—41no—
632D4—41no—
633D4—42no—
634D4—64no—
635D4—64no—
636D4—82no—
637D4—82no—
638D4—124no—
639D4—124no—
640D4—122no—
641D4—122no—
642D4#93712yes—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—21no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—81no—
659C2×C4#93911yescharacteristic
660C2×C4#93911yescharacteristic
661C2×C4C2×D3×D412yes—
662C2×C4C2×D3×D411yescharacteristic
663C2×C4#100412yes—
664C2×C4C22×#6711yescharacteristic
665C8—41no—
666C8—41no—
667C8—64no—
668C8—62no—
669C8—62no—
670C8—122no—
671C2×C3—22no—
672C2×C3—22no—
673C2×C3—41no—
674C2×C3—42no—
675C2×C3—81no—
676C2×C3—161no—
677C2×C3#241011yescharacteristic
678C2×C3C2×#47211yescharacteristic
679C2×C3#306811yescharacteristic
680D3—84no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—22no—
688C22—21no—
689C22—44no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—84no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—124no—
700C22—162no—
701C22—244no—
702C22—248no—
703C22—244no—
704C22—248no—
705C22—242no—
706C22—242no—
707C22#519912yescomplemented
708C22#559112yes—
709C22C2×#93911yescharacteristic
710C4—21no—
711C4—21no—
712C4—22no—
713C4—41no—
714C4—81no—
715C4#559012yes—
716C4C2×#93711yescharacteristic
717C4#580411yescharacteristic
718C3#2439811yescharacteristic, complemented, Sylow
719C2—22no—
720C2—22no—
721C2—41no—
722C2—42no—
723C2—81no—
724C2—161no—
725C2—244no—
726C2#8260611yescharacteristic
727C2C2×#519911yescharacteristic
728C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3006C2
socleC22×C3C2×#121
hypercenter#3006D3
hypocenterC3#24398
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
1. C22×#949C2semidirect
2. C3×#3006C2semidirect
3. #84075C2semidirect
4. C2×#5199C2semidirect
5. C2×#5036C2semidirect
6. #72095C2semidirect
7. #71683C2semidirect
8. #70307C2semidirect
9. #70303C2semidirect
10. #69988C2semidirect
11. #69984C2semidirect
15. C2×C3×AGL(1,ℤ/8)C22semidirect
16. C2×C3×AGL(1,ℤ/8)C22semidirect
23. C2×#948C22semidirect
26. C22×#213C22semidirect
32. C3×#417C22semidirect
33. C3×#417C22semidirect
36. #5234C22semidirect
37. C2×#833C22semidirect
43. C2×#823C22semidirect
44. C2×#823C22semidirect
45. C2×#822C22semidirect
49. C2×#809C22semidirect
50. #4558C22semidirect
51. #4530C22semidirect
54. #3006D3semidirect
106. #949D4semidirect
153. C2×AGL(1,ℤ/8)C2×D3semidirect
154. C2×AGL(1,ℤ/8)C2×D3semidirect
170. #417C2×D3semidirect
171. #417C2×D3semidirect
573. C22×C3#472semidirect
707. C22#5199semidirect
718. C3#24398semidirect

Maximal subgroups

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

subgroupindex
C22×#9492
C3×#30062
#840752
C2×#51992
C2×#50362
#720952
#716832
#703072
#703032
#699882
#699842
#243983