Finite Group Databasebeta

← #10827335

Subgroups — #10827335

Order 768.

1690 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5611C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic, complemented
3#84053C211yesmaximal, characteristic, complemented
4C22×#832C211yesmaximal, characteristic, complemented
5#72043C212yesmaximal, complemented
6#72037C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#70200C212yesmaximal, complemented
9#70163C212yesmaximal, complemented
10#69952C211yesmaximal, characteristic, complemented
11#69924C211yesmaximal, characteristic, complemented
12#23960—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#237—21no—
15C22×C3×D8C2212yes—
16C2×C3×#121C2211yescharacteristic
17C2×#953—21no—
18C2×#938C2212yes—
19C2×#938C2211yescharacteristic, complemented
20C2×#935—21no—
21C2×D4×Dic3C2211yescharacteristic, complemented
22C2×C4×#67C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25#5611—24no—
26#5611—24no—
27C2×D3×C4⋊C4C2211yescharacteristic, complemented
28C2×#888C2211yescharacteristic, complemented
29C2×#883—21no—
30C2×#878—21no—
31C3×#447—24no—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2211yescharacteristic
34#5247—21no—
35#5223—21no—
36#5222—21no—
37#5221C2212yes—
38C2×#832—24no—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×#832C2212yescomplemented
42#5212C2211yescharacteristic
43C2×(C2×D3)⋊C8C2212yescomplemented
44#4551C2212yescomplemented
45#4551C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C2×#547—31no—
49C2×#447D311yescharacteristic, complemented
50C22×#102—31no—
51#2408—31no—
52#2023—32no—
53#1999—32no—
54#1893—31no—
55#1879—32no—
56#1874—32no—
57#1869—31no—
58#1823—31no—
59C25×C3—21no—
60C22×C3×D4—24no—
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—42no—
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67—22no—
71C22×#67D411yescharacteristic
72C23×Dic3—21no—
73C23×Dic3—21no—
74C2×#237—28no—
75C2×#237—44no—
76C2×#237—44no—
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×Dic6—21no—
80C2×C3×D8—24no—
81C2×C3×D8—24no—
82C2×C3×D8—42no—
83C22×C3×C8C2312yes—
84C3×#121—22no—
85C3×#121D414yescomplemented
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88#953—44no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—22no—
93C2×#217—22no—
94C2×#217—22no—
95C2×#217D411yescharacteristic
96#938—22no—
97#938—24no—
98#938—22no—
99#938—42no—
100#935—28no—
101D4×Dic3—22no—
102D4×Dic3—22no—
103C4×#67—22no—
104C4×#67—22no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—22no—
108C2×C4⋊Dic3D414yescomplemented
109C2×C4⋊Dic3C2312yes—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8—22no—
117C22×C3⋊C8—42no—
118C22×C3⋊C8C2312yes—
119D3×C4⋊C4—22no—
120D3×C4⋊C4—22no—
121#888—22no—
122#888—22no—
123#883—28no—
124#878—28no—
125C3×#102—24no—
126C3×C22⋊C8—22no—
127#832—24no—
128#832—24no—
129#832—42no—
130C3⋊#95—21no—
131(C2×D3)⋊C8—42no—
132C22×#41—61no—
133C23×D4C2×D311yescharacteristic
134C22×D8C2×D312yes—
135C23×C8—31no—
136#547—64no—
137#547—64no—
138C2×C4⋊Q8—31no—
139C2×#123—61no—
140C2×#123—61no—
141C2×C22⋊Q8—61no—
142C2×#121—31no—
143C2×#121—32no—
144C2×#121C2×D311yescharacteristic
145C2×#120—61no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C22×C4⋊C4—31no—
149C22×C4⋊C4—31no—
150#447—24no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×#102—64no—
155C2×#102C2×D312yes—
156C2×C22⋊C8—32no—
157C2×C22⋊C8C2×D311yescharacteristic
158#394—61no—
159#392—61no—
160#390—32no—
161#380—31no—
162#379—61no—
163#340—32no—
164#340—32no—
165#340—32no—
166#336—32no—
167C24×C3—22no—
168C24×C3—21no—
169C24×C3—44no—
170C24×C3—42no—
171C24×C3—42no—
172C24×C3—42no—
173C24×C3—82no—
174C24×C3C2×D411yescharacteristic
175C23×D3C2×D411yescharacteristic
176C2×C3×D4—28no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—48no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C2×C3×D4C2×D412yes—
193C2×C3×D4#4114yescomplemented
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2411yescharacteristic
201C2×#67—22no—
202C2×#67—22no—
203C2×#67—24no—
204C2×#67—42no—
205C2×#67—42no—
206C2×#67—44no—
207C2×#67—44no—
208C2×#67—82no—
209C2×#67D814yescomplemented
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—22no—
215C22×Dic3—42no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222#237—432no—
223C2×C4×D3—24no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×Dic6—24no—
231C2×Dic6—42no—
232C2×Dic6—42no—
233C3×D8—48no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4C2×D412yes—
239C3×C22⋊C4—24no—
240#217—24no—
241#217—24no—
242#217—22no—
243#217—44no—
244#217—42no—
245#217—44no—
246#217—82no—
247(C2×D3)⋊C4—22no—
248C4⋊Dic3—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—22no—
252C4⋊Dic3—22no—
253C4⋊Dic3C2×D412yes—
254C4⋊Dic3C2×D412yes—
255Dic3⋊C4—24no—
256Dic3⋊C4—24no—
257Dic3⋊C4—22no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—44no—
264C25—21no—
265C2×#41—68no—
266C2×#41—124no—
267C2×#41—124no—
268C22×Q8—61no—
269C22×D4—24no—
270C22×D4—21no—
271C22×D4—22no—
272C22×D4—21no—
273C22×D4—22no—
274C22×D4—31no—
275C22×D4—42no—
276C22×D4—61no—
277C22×D4—62no—
278C22×D4C22×D312yes—
279C22×D4C22×D311yescharacteristic
280C23×C4—31no—
281C23×C4—61no—
282C23×C4—61no—
283C23×C4—61no—
284C23×C4C22×D311yescharacteristic
285C2×D8—24no—
286C2×D8—24no—
287C2×D8—42no—
288C22×C8—32no—
289C22×C8—62no—
290C22×C8—122no—
291C22×C8C22×D312yes—
292C4⋊Q8—62no—
293C4⋊Q8—62no—
294#123—68no—
295#123—68no—
296C22⋊Q8—68no—
297#121—22no—
298#121—62no—
299#121—62no—
300#121—64no—
301#121—122no—
302#121#6714yescomplemented
303#120—124no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4—62no—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—62no—
326C2×C22⋊C4—62no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—62no—
329C2×C42—31no—
330#102—24no—
331#102—64no—
332#102—64no—
333#102—122no—
334C22⋊C8—22no—
335C22⋊C8—122no—
336#95—61no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—22no—
340C23×C3—24no—
341C23×C3—48no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3—42no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—88no—
353C23×C3—88no—
354C23×C3—84no—
355C23×C3—84no—
356C23×C3#12012yes—
357C23×C3C2×D812yes—
358C23×C3AGL(1,ℤ/8)12yes—
359C23×C3C22×D411yescharacteristic
360C22×D3—21no—
361C22×D3—22no—
362C22×D3—21no—
363C22×D3—22no—
364C22×D3C2×D812yes—
365C3×D4—24no—
366C3×D4—24no—
367C3×D4—24no—
368C3×D4—48no—
369C3×D4—48no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—42no—
373C3×D4—42no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4#12012yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4112yes—
385C2×C3×C4H(5,𝔽2)12yes—
386#67—28no—
387#67—48no—
388#67—88no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—48no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—48no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3C2×D812yes—
407C4×D3—48no—
408C4×D3—41no—
409C4×D3—41no—
410C4×D3—42no—
411Dic6—48no—
412C3×C8—42no—
413C3⋊C8—42no—
414C24—21no—
415C24—22no—
416C24—31no—
417C24—44no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—82no—
422C24D3×D411yescharacteristic
423#41—1232no—
424C2×Q8—64no—
425C2×Q8—122no—
426C2×Q8—122no—
427C2×D4—28no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—34no—
438C2×D4—48no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—62no—
445C2×D4—62no—
446C2×D4—64no—
447C2×D4—122no—
448C2×D4—122no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—242no—
452C2×D4#23714yescomplemented
453C2×D4C2×#6712yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—61no—
469C22×C4—62no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4—122no—
476C22×C4D3×D411yescharacteristic
477C22×C4D3×D412yes—
478C22×C4C2×#6712yes—
479C22×C4C23×D311yescharacteristic
480D8—48no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—62no—
485C2×C8—62no—
486C2×C8—62no—
487C2×C8—124no—
488C4⋊C4—21no—
489C4⋊C4—32no—
490C4⋊C4—32no—
491C4⋊C4—64no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C4⋊C4—62no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—62no—
498C4⋊C4—62no—
499C4⋊C4C2×#6712yes—
500C22⋊C4—24no—
501C22⋊C4—64no—
502C22⋊C4—64no—
503C22⋊C4—62no—
504C22⋊C4—62no—
505C22⋊C4—124no—
506C22⋊C4—124no—
507C22⋊C4—122no—
508C22⋊C4—242no—
509C42—61no—
510C42—61no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—24no—
514C22×C3—24no—
515C22×C3—44no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—44no—
520C22×C3—44no—
521C22×C3—816no—
522C22×C3—82no—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3—82no—
526C22×C3—82no—
527C22×C3—82no—
528C22×C3#44714yes—
529C22×C3C2×#12011yescharacteristic
530C22×C3#54612yes—
531C22×C3C22×D811yescharacteristic
532C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
533C22×C3#57712yes—
534C2×D3—22no—
535C2×D3—24no—
536C2×D3—41no—
537C2×D3—41no—
538C2×D3—42no—
539C3×C4—22no—
540C3×C4—42no—
541C3×C4#52511yescharacteristic
542C3×C4#53511yescharacteristic
543C3×C4#54612yes—
544Dic3—22no—
545Dic3—41no—
546Dic3—42no—
547Dic3—42no—
548Dic3—41no—
549Dic3—82no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—32no—
555C23—48no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—61no—
567C23—62no—
568C23—61no—
569C23—62no—
570C23—88no—
571C23—88no—
572C23—84no—
573C23—84no—
574C23D3×D812yes—
575C23#91112yes—
576C23#93712yes—
577C23C2×D3×D411yescharacteristic
578Q8—128no—
579D4—24no—
580D4—24no—
581D4—24no—
582D4—48no—
583D4—48no—
584D4—42no—
585D4—41no—
586D4—42no—
587D4—41no—
588D4—68no—
589D4—128no—
590D4—248no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—32no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—62no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—128no—
608C2×C4—128no—
609C2×C4—128no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—122no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4#93712yes—
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×#23712yes—
623C2×C4#100412yes—
624C2×C4C22×#6711yescharacteristic
625C8—42no—
626C8—122no—
627C2×C3—24no—
628C2×C3—82no—
629C2×C3—84no—
630C2×C3#240411yescharacteristic
631C2×C3C2×#44711yescharacteristic
632C2×C3#301011yescharacteristic
633C2×C3#329412yes—
634C2×C3#330612yes—
635D3—42no—
636C22—24no—
637C22—24no—
638C22—22no—
639C22—22no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—42no—
646C22—62no—
647C22—64no—
648C22—816no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—122no—
656C22—121no—
657C22—121no—
658C22#515614yes—
659C22#559112yes—
660C22C2×D3×D811yescharacteristic
661C22C2×#91111yescharacteristic
662C22#577512yes—
663C22C2×#93711yescharacteristic
664C4—22no—
665C4—42no—
666C4—62no—
667C4—122no—
668C4—122no—
669C4—121no—
670C4—121no—
671C4—242no—
672C4#558611yescharacteristic
673C4#559211yescharacteristic
674C4#580612yes—
675C3#2396011yescharacteristic, complemented, Sylow
676C2—24no—
677C2—84no—
678C2—82no—
679C2—122no—
680C2#8001412yes—
681C2#8001612yes—
682C2#8258111yescharacteristic
683C2C2×#515611yescharacteristic
684C2#8394811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#447C2
socleC23×C3C22×D4
hypercenterC2×#447D3
hypocenterC3#23960
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C2×#5611C2semidirect
2. C2×C3×#447C2semidirect
3. #84053C2semidirect
4. C22×#832C2semidirect
5. #72043C2semidirect
6. #72037C2semidirect
7. #71681C2semidirect
8. #70200C2semidirect
9. #70163C2semidirect
10. #69952C2semidirect
11. #69924C2semidirect
19. C2×#938C22semidirect
21. C2×D4×Dic3C22semidirect
27. C2×D3×C4⋊C4C22semidirect
28. C2×#888C22semidirect
41. C2×#832C22semidirect
43. C2×(C2×D3)⋊C8C22semidirect
44. #4551C22semidirect
45. #4551C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
49. C2×#447D3semidirect
85. C3×#121D4semidirect
108. C2×C4⋊Dic3D4semidirect
193. C2×C3×D4#41semidirect
209. C2×#67D8semidirect
302. #121#67semidirect
452. C2×D4#237semidirect
675. C3#23960semidirect

Maximal subgroups

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

subgroupindex
C2×#56112
C2×C3×#4472
#840532
C22×#8322
#720432
#720372
#716812
#702002
#701632
#699522
#699242
#239603