Finite Group Databasebeta

← #10944094

Subgroups — #10944094

Order 768.

1546 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5545C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C2×#5169C212yesmaximal
4#83261C211yesmaximal, characteristic, complemented
5C2×#5083C211yesmaximal, characteristic, complemented
6#72047C212yesmaximal, complemented
7#71605C211yesmaximal, characteristic, complemented
8#70480C211yesmaximal, characteristic, complemented
9#70476C211yesmaximal, characteristic, complemented
10#70276C212yesmaximal, complemented
11#70269C212yesmaximal, complemented
12#32914—31nomaximal, Sylow
13C22×D3×D4C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic, complemented
16C2×C3×C4×D4C2211yescharacteristic
17C2×#946C2212yes—
18C22×#215C2212yes—
19C2×D4×Dic3C2211yescharacteristic
20C2×C4×#67—21no—
21C22×(C2×D3)⋊C4—21no—
22#5545—24no—
23#5545—24no—
24C2×#893C2211yescharacteristic
25C2×#892C2211yescharacteristic
26C2×#67⋊C4—21no—
27C2×D3×C22⋊C4—21no—
28C2×C4×D12C2211yescharacteristic
29C2×#872C2211yescharacteristic
30C3×#461—24no—
31C2×C3×C4⋊C8C2211yescharacteristic, complemented
32C2×C3×#103C2212yescomplemented
33#5229C2212yescomplemented
34C2×#835C2212yescomplemented
35C2×#832C2212yes—
36#5169—28no—
37C2×#821C2212yes—
38C2×(C2×D3)⋊C8C2212yes—
39#5083—24no—
40#4992—21no—
41#4992—21no—
42#4983—21no—
43#4983—21no—
44#4936C2211yescharacteristic, complemented
45C2×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
46#4551C2212yescomplemented
47#4550C2212yescomplemented
48#4529C2212yescomplemented
49C2×#518—31no—
50C2×#461—31no—
51C2×#461D311yescharacteristic, complemented
52C2×#450—32no—
53#2621—31no—
54#1877—32no—
55#1877—32no—
56#1873—32no—
57#1804—31no—
58#1804—31no—
59#1798—31no—
60C24×D3—21no—
61C22×C3×Q8C2312yes—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C22×#67—21no—
65C23×Dic3—21no—
66C2×D3×D4—22no—
67C2×D3×D4—22no—
68C2×D3×D4—22no—
69C2×D3×D4—22no—
70C2×D3×D4—22no—
71C2×D3×D4D414yes—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—21no—
75C22×C4×D3C2311yescharacteristic
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—22no—
82C3×C4×D4—22no—
83C3×C4×D4D414yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90#946—24no—
91#946—24no—
92C2×#215—24no—
93C2×#215—24no—
94C2×#215—24no—
95D4×Dic3—22no—
96D4×Dic3—22no—
97C4×#67—28no—
98C2×(C2×D3)⋊C4—28no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—22no—
103C2×(C2×D3)⋊C4D411yescharacteristic
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—22no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113#893—22no—
114#893—22no—
115#892—22no—
116#892—22no—
117#67⋊C4—28no—
118D3×C22⋊C4—28no—
119C4×D12—22no—
120C4×D12D414yes—
121#872—22no—
122#872—22no—
123C3×C4⋊C8—22no—
124C3×#103—24no—
125#835—24no—
126#832—24no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—22no—
130C3⋊#95—22no—
131#821—24no—
132(C2×D3)⋊C8—24no—
133C4⋊(C3⋊C8)—22no—
134C23×D4—31no—
135C22×SD16—32no—
136C22×SD16C2×D312yes—
137C2×C4⋊Q8C2×D311yescharacteristic, complemented
138C2×C22⋊Q8—32no—
139#518—64no—
140#518—64no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×C4×D4—61no—
145C2×C4×D4—61no—
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—31no—
149C22×C22⋊C4—61no—
150C22×C22⋊C4—61no—
151#461—24no—
152#461—64no—
153#450—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic, complemented
156C2×#103—32no—
157C2×#103C2×D312yescomplemented
158C2×#102—32no—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#398—61no—
162#398—61no—
163#397—61no—
164#397—61no—
165#391—32no—
166#382—31no—
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3—21no—
171C23×D3—21no—
172C23×D3—24no—
173C23×D3—24no—
174C23×D3—24no—
175C23×D3—82no—
176C23×D3—82no—
177C23×D3C2×D411yescharacteristic
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4C2×D412yes—
188C2×C3×D4C2×D412yes—
189C2×C3×D4#4112yes—
190C22×C3×C4—21no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—82no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×#67—24no—
198C2×#67—24no—
199C2×#67—24no—
200C22×Dic3—21no—
201C22×Dic3—24no—
202C22×Dic3—22no—
203C22×Dic3—82no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207D3×D4—28no—
208D3×D4—28no—
209D3×D4—28no—
210D3×D4—28no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12C2×D412yes—
215C2×D12C2×D412yes—
216C2×D12#4112yes—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—24no—
222C2×C4×D3—24no—
223C2×C4×D3—82no—
224C2×C4×D3—82no—
225C2×C4×D3C2×D412yes—
226C2×C4×D3#4112yes—
227C3×SD16—48no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4C2×D412yes—
235C3×C22⋊C4—24no—
236C3×C42C2×D412yes—
237C3×C42C2×D412yes—
238#217—24no—
239#215—48no—
240(C2×D3)⋊C4—24no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—44no—
246C4⋊Dic3—21no—
247C4⋊Dic3—22no—
248C4⋊Dic3—22no—
249C4⋊Dic3C2×D412yes—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—44no—
253C4×Dic3—24no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C25—61no—
260C22×Q8C22×D312yes—
261C22×D4—34no—
262C22×D4—31no—
263C22×D4—62no—
264C22×D4—61no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—62no—
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C23×C4—31no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—64no—
279C2×SD16—64no—
280C2×SD16—64no—
281C22×C8—32no—
282C22×C8C22×D312yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C22⋊Q8—64no—
286C22⋊Q8—64no—
287C4×D4—22no—
288C4×D4—34no—
289C4×D4—68no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—68no—
293C4×D4—62no—
294C4×D4—62no—
295C4×D4—62no—
296C4×D4#6714yes—
297#117—62no—
298#117—62no—
299#117—62no—
300#117—62no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4—68no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—68no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—62no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42C22×D311yescharacteristic
322C4⋊C8—22no—
323C4⋊C8—62no—
324#103—24no—
325#103—64no—
326#102—64no—
327#102—64no—
328C22⋊C8—64no—
329#95—62no—
330#95—62no—
331#95—61no—
332#95—61no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3C22×D411yescharacteristic
339C22×D3—21no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—21no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—216no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—88no—
350C22×D3—88no—
351C22×D3—84no—
352C22×D3—84no—
353C22×D3#12112yes—
354C3×Q8—42no—
355C3×Q8—42no—
356C3×Q8—42no—
357C3×Q8—42no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4#12114yes—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—84no—
370C2×C3×C4#12012yes—
371C2×C3×C4#12012yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378#67—216no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—84no—
391C2×Dic3#12112yes—
392D12—22no—
393D12—22no—
394D12—22no—
395D12#12114yes—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—84no—
401C4×D3—84no—
402C3×C8—42no—
403C3⋊C8—42no—
404C24—21no—
405C24—31no—
406C24—64no—
407C24—64no—
408C24—64no—
409C24—61no—
410C24—242no—
411C24—242no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—32no—
422C2×D4—32no—
423C2×D4—32no—
424C2×D4—64no—
425C2×D4—68no—
426C2×D4—68no—
427C2×D4—68no—
428C2×D4—64no—
429C2×D4—64no—
430C2×D4—68no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4D3×D412yes—
435C2×D4#23712yes—
436C2×D4C2×#6712yes—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—31no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—64no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—62no—
454C22×C4—82no—
455C22×C4—242no—
456C22×C4—242no—
457C22×C4—242no—
458C22×C4D3×D411yescharacteristic
459C22×C4D3×D411yescharacteristic
460C22×C4C23×D311yescharacteristic
461SD16—48no—
462SD16—128no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—62no—
467C2×C8—62no—
468C2×C8—62no—
469C4⋊C4—21no—
470C4⋊C4—32no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—62no—
474C4⋊C4—62no—
475C4⋊C4—61no—
476C4⋊C4—62no—
477C4⋊C4—62no—
478C4⋊C4—124no—
479C4⋊C4C2×#6712yes—
480C22⋊C4—24no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—64no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—64no—
487C22⋊C4—124no—
488C42—64no—
489C42—61no—
490C42—61no—
491C42D3×D412yes—
492C42C2×#6712yes—
493C22×C3—24no—
494C22×C3—44no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—44no—
498C22×C3C2×#12011yescharacteristic
499C22×C3C2×#12111yescharacteristic
500C22×C3#52511yescharacteristic
501C22×C3#57712yes—
502C22×C3#57712yes—
503C2×D3—22no—
504C2×D3—24no—
505C2×D3—24no—
506C2×D3—42no—
507C2×D3—44no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—44no—
513C2×D3—816no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—82no—
518C3×C4C2×#12111yescharacteristic
519C3×C4C2×#12111yescharacteristic
520C3×C4#52511yescharacteristic
521C3×C4#52511yescharacteristic
522Dic3—22no—
523Dic3—41no—
524Dic3—42no—
525Dic3—41no—
526Dic3—82no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—32no—
531C23—42no—
532C23—42no—
533C23—616no—
534C23—62no—
535C23—62no—
536C23—61no—
537C23—62no—
538C23—62no—
539C23—61no—
540C23—62no—
541C23—122no—
542C23—122no—
543C23—248no—
544C23—244no—
545C23—248no—
546C23—244no—
547C23C2×D3×D411yescharacteristic
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4—34no—
556D4—616no—
557D4—62no—
558D4—62no—
559D4—62no—
560D4#93814yes—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—32no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—84no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—244no—
586C2×C4—244no—
587C2×C4—244no—
588C2×C4#93712yes—
589C2×C4#93712yes—
590C2×C4C2×D3×D411yescharacteristic
591C2×C4C2×D3×D411yescharacteristic
592C2×C4C2×#23711yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4C22×#6711yescharacteristic
596C8—42no—
597C8—122no—
598C2×C3—44no—
599C2×C3#284711yescharacteristic
600C2×C3#300911yescharacteristic
601C2×C3#300911yescharacteristic
602C2×C3#304212yes—
603C2×C3#312312yes—
604D3—42no—
605D3—44no—
606C22—24no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—62no—
612C22—64no—
613C22—64no—
614C22—122no—
615C22—121no—
616C22—122no—
617C22—124no—
618C22—124no—
619C22—122no—
620C22—121no—
621C22—2416no—
622C22D3×#12111yescharacteristic
623C22#560611yescharacteristic
624C22#576212yes—
625C22#577812yes—
626C22C2×#93711yescharacteristic
627C4—22no—
628C4—22no—
629C4—42no—
630C4—62no—
631C4—82no—
632C4—121no—
633C4—122no—
634C4—121no—
635C4—242no—
636C4D3×#12111yescharacteristic
637C4#560611yescharacteristic
638C4C2×#93811yescharacteristic
639C4#582811yescharacteristic
640C3#3291411yescharacteristic, complemented, Sylow
641C2—44no—
642C2—122no—
643C2—124no—
644C2#8041312yes—
645C2#8043012yes—
646C2#8272311yescharacteristic
647C2#8385911yescharacteristic
648C2#8398211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenterC2×#461D3
hypocenterC3#32914
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. C2×C3×#461C2semidirect
4. #83261C2semidirect
5. C2×#5083C2semidirect
6. #72047C2semidirect
7. #71605C2semidirect
8. #70480C2semidirect
9. #70476C2semidirect
10. #70276C2semidirect
11. #70269C2semidirect
15. C2×C3×C4⋊Q8C22semidirect
31. C2×C3×C4⋊C8C22semidirect
32. C2×C3×#103C22semidirect
33. #5229C22semidirect
34. C2×#835C22semidirect
44. #4936C22semidirect
45. C2×C4⋊(C3⋊C8)C22semidirect
46. #4551C22semidirect
47. #4550C22semidirect
48. #4529C22semidirect
51. C2×#461D3semidirect
137. C2×C4⋊Q8C2×D3semidirect
155. C2×C4⋊C8C2×D3semidirect
157. C2×#103C2×D3semidirect
640. C3#32914semidirect

Maximal subgroups

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

subgroupindex
C2×#55452
C2×C3×#4612
C2×#51692
#832612
C2×#50832
#720472
#716052
#704802
#704762
#702762
#702692
#329143