Finite Group Databasebeta

← #72308147

Subgroups — #72308147

Order 1024.

11145 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×#106C211yesmaximal, characteristic
2C2×#32931C211yesmaximal, characteristic
3C2×#23028C213yesmaximal, complemented
4#520272C212yesmaximal
5#392621C212yesmaximal
6#374418C213yesmaximal, complemented
7#373850C213yesmaximal, complemented
8#368433C216yesmaximal, complemented
9#368010C216yesmaximal, complemented
10#356221C213yesmaximal, complemented
11#354781C211yesmaximal, characteristic
12C25×C8C2211yescharacteristic
13C24×C4⋊C4C2212yes—
14C22×#467C2213yes—
15C22×#441C2211yescharacteristic, complemented
16C23×#106—224no—
17C23×#106C2216yes—
18C23×#106C2213yes—
19C23×#106C2213yes—
20C2×#2848C2212yes—
21C2×#2360C2216yes—
22#32931C4116yescomplemented
23#23028—224no—
24C2×#1971C2213yescomplemented
25C2×#1943C2213yescomplemented
26C2×#1927C2216yescomplemented
27C2×#1892C2216yes—
28C2×#1872C2216yes—
29C2×#1872C2216yes—
30C2×#1865C2216yescomplemented
31C2×#1808C2213yes—
32C2×#1779C2211yescharacteristic
33C2×#1776C2213yescomplemented
34C2×#1752C2213yes—
35C2×#1752C2213yes—
36#14693—22no—
37#14139C2216yes—
38#14130C2216yes—
39#14043—26no—
40#14039—26no—
41C22⋊C4×C4⋊C4C2216yes—
42C4×#379—22no—
43C4×#379C2211yescharacteristic
44#11327C2216yescomplemented
45#11257C2216yescomplemented
46#11122C22112yescomplemented
47#10908C22112yescomplemented
48#10863C2216yescomplemented
49#10730C2216yes—
50#10730C2216yes—
51#10714C2213yes—
52#10714C2213yes—
53C25×C4—22no—
54C25×C4C2311yescharacteristic
55C23×Q16C2311yescharacteristic
56C24×C8—212no—
57C24×C8—212no—
58C24×C8C2×C418yes—
59C24×C8C2313yes—
60C24×C8C2313yes—
61C22×C22⋊Q8C2316yes—
62C22×C4×Q8C2312yes—
63C23×C4⋊C4—224no—
64C23×C4⋊C4—26no—
65C23×C4⋊C4—412no—
66C23×C4⋊C4C2316yes—
67C23×C4⋊C4C2316yes—
68C2×#467—212no—
69C2×#467—212no—
70C2×#467C2×C4124yescomplemented
71C2×#441—212no—
72C22×#107C2313yes—
73C22×#106—2128no—
74C22×#106—248no—
75C22×#106—224no—
76C22×#106—224no—
77C22×#106—212no—
78C22×#106—224no—
79C22×#106D4124yes—
80C22×#106C2316yes—
81C22×#106C2316yes—
82C22×#106C2313yes—
83C22×#106C2311yescharacteristic
84C22×#103C2312yes—
85C22×#103C2316yes—
86C22×#103C2316yes—
87C22×C22⋊C8C2313yes—
88C22×C22⋊C8C2313yes—
89C22×#97C2313yes—
90C22×#97C2311yescharacteristic
91#2848—28no—
92#2848C2×C4116yes—
93#2360—224no—
94#2360—412no—
95C2×#391C2316yes—
96C2×#389—26no—
97C2×#389C2316yes—
98C2×#387—26no—
99C2×#385—26no—
100C2×#385—26no—
101C2×#385—26no—
102C2×#380C2316yes—
103C2×#379—22no—
104C2×#379D412yes—
105C2×C4×C4⋊C4C2316yes—
106C2×C4×C22⋊C4—26no—
107C2×C4×C22⋊C4—26no—
108C2×C4×C22⋊C4C2313yes—
109C2×C4×C22⋊C4C2313yes—
110#1971—212no—
111#1943—212no—
112#1927—224no—
113#1892C2×C4148yescomplemented
114#1872—224no—
115#1872—412no—
116#1865—224no—
117#1808C2×C4124yescomplemented
118#1779—24no—
119#1776—212no—
120#1752—212no—
121#1752—212no—
122C2×#340C2316yes—
123C2×#340C2316yes—
124C2×#340C23112yes—
125C2×#340C2316yes—
126C2×#336C2313yes—
127C2×#336C2313yes—
128C2×#336C2316yes—
129#1444—26no—
130#1442—26no—
131#1441—212no—
132#1436—212no—
133C4⋊#95C23112yes—
134C22⋊#95—26no—
135C22⋊#95—26no—
136C22⋊#95—26no—
137C22⋊#95—26no—
138C22⋊#95D416yes—
139C26C2×D411yescharacteristic
140C23×Q8C2412yes—
141C24×C4—26no—
142C24×C4—26no—
143C24×C4—26no—
144C24×C4—26no—
145C24×C4—26no—
146C24×C4—26no—
147C24×C4—424no—
148C24×C4—48no—
149C24×C4—812no—
150C24×C4C22×C414yes—
151C24×C4C2×D413yes—
152C24×C4#4114yes—
153C24×C4C2413yes—
154C22×Q16—26no—
155C22×Q16—26no—
156C22×Q16C22×C414yescomplemented
157C23×C8—232no—
158C23×C8—296no—
159C23×C8—26no—
160C23×C8—26no—
161C23×C8—296no—
162C23×C8—212no—
163C23×C8—212no—
164C23×C8—212no—
165C23×C8—212no—
166C23×C8C22×C4112yes—
167C23×C8C22×C4112yes—
168C23×C8C2×D4112yes—
169C23×C8#41112yes—
170C23×C8C2411yescharacteristic
171C23×C8C2413yes—
172C23×C8C2413yes—
173C23×C8C2411yescharacteristic
174C2×C22⋊Q8—212no—
175C2×C22⋊Q8—212no—
176C2×C22⋊Q8—212no—
177C2×C22⋊Q8—212no—
178C2×C22⋊Q8—212no—
179C2×C22⋊Q8C22×C4124yes—
180C2×C4×Q8—212no—
181C2×C4×Q8—46no—
182C22×C4⋊C4—264no—
183C22×C4⋊C4—212no—
184C22×C4⋊C4—224no—
185C22×C4⋊C4—224no—
186C22×C4⋊C4—26no—
187C22×C4⋊C4—26no—
188C22×C4⋊C4—212no—
189C22×C4⋊C4—212no—
190C22×C4⋊C4—26no—
191C22×C4⋊C4—22no—
192C22×C4⋊C4—26no—
193C22×C4⋊C4—412no—
194C22×C4⋊C4—412no—
195C22×C4⋊C4—448no—
196C22×C4⋊C4—424no—
197C22×C4⋊C4—496no—
198C22×C4⋊C4C2×D4124yes—
199C22×C4⋊C4C2416yes—
200C22×C4⋊C4C2416yes—
201C22×C4⋊C4C2412yes—
202C22×C22⋊C4—26no—
203C22×C22⋊C4—26no—
204C22×C22⋊C4C2×D416yes—
205C22×C22⋊C4C2×D416yes—
206C22×C42—22no—
207C22×C42—26no—
208C22×C42C2411yescharacteristic
209C22×C42C2413yes—
210#467—296no—
211#441—416no—
212C2×#107—26no—
213C2×#107—26no—
214C2×#107C22×C4112yescomplemented
215C2×#106—296no—
216C2×#106—2192no—
217C2×#106—212no—
218C2×#106—212no—
219C2×#106—224no—
220C2×#106—224no—
221C2×#106—296no—
222C2×#106C2×D4124yes—
223C2×#106C2×D4112yes—
224C2×#106C2×D4112yes—
225C2×#103—212no—
226C2×#103—212no—
227C2×#103—212no—
228C2×#103—212no—
229C2×#103—412no—
230C2×#103C22×C4124yescomplemented
231C2×C22⋊C8—26no—
232C2×C22⋊C8—26no—
233C2×C22⋊C8—26no—
234C2×C22⋊C8—26no—
235C2×C22⋊C8—26no—
236C2×C22⋊C8—26no—
237C2×#97—26no—
238C2×#97—26no—
239C2×#97—212no—
240#391—212no—
241#391—46no—
242#389—212no—
243#389—424no—
244#389—46no—
245#387—424no—
246#385—248no—
247#385—424no—
248#385—812no—
249#380—212no—
250#380C22×C4124yes—
251#379—216no—
252#379—28no—
253C4×C4⋊C4—212no—
254C4×C4⋊C4—46no—
255C4×C22⋊C4—26no—
256C4×C22⋊C4—26no—
257C4×C22⋊C4—26no—
258C4×C22⋊C4—26no—
259C4×C22⋊C4—424no—
260C4×C22⋊C4—812no—
261C2×#95—26no—
262C2×#95—26no—
263C2×#95—26no—
264C2×#95—26no—
265C2×#95—26no—
266C2×#95—212no—
267C2×#95—212no—
268C2×#95C2×D416yes—
269C2×#95C2×D416yes—
270#340—212no—
271#340—212no—
272#340—412no—
273#340C22×C4124yescomplemented
274#336—26no—
275#336—26no—
276#336—212no—
277C25—26no—
278C25—26no—
279C25—212no—
280C25C4×D414yes—
281C25#13718yes—
282C25C22×D413yes—
283C22×Q8—22no—
284C22×Q8—26no—
285C22×Q8—26no—
286C22×Q8—26no—
287C22×Q8—26no—
288C22×Q8C23×C414yes—
289C23×C4—28no—
290C23×C4—224no—
291C23×C4—224no—
292C23×C4—224no—
293C23×C4—248no—
294C23×C4—26no—
295C23×C4—26no—
296C23×C4—224no—
297C23×C4—224no—
298C23×C4—212no—
299C23×C4—212no—
300C23×C4—224no—
301C23×C4—224no—
302C23×C4—26no—
303C23×C4—26no—
304C23×C4—26no—
305C23×C4—26no—
306C23×C4—22no—
307C23×C4—22no—
308C23×C4—26no—
309C23×C4—26no—
310C23×C4—224no—
311C23×C4—424no—
312C23×C4—424no—
313C23×C4—464no—
314C23×C4—412no—
315C23×C4—412no—
316C23×C4—412no—
317C23×C4—412no—
318C23×C4—412no—
319C23×C4—812no—
320C23×C4—812no—
321C23×C4—896no—
322C23×C4—896no—
323C23×C4C4×D4112yes—
324C23×C4C23×C416yes—
325C23×C4C22×D412yes—
326C23×C4C22×D416yes—
327C23×C4C22×D413yes—
328C23×C4C22×D416yes—
329C23×C4C2×#4116yes—
330C23×C4C2×#4116yes—
331C23×C4H(5,𝔽2)16yes—
332C23×C42−1+416yes—
333C23×C4C2511yescharacteristic
334C2×Q16—224no—
335C2×Q16—44no—
336C2×Q16—412no—
337C22×C8—248no—
338C22×C8—224no—
339C22×C8—2256no—
340C22×C8—224no—
341C22×C8—224no—
342C22×C8—26no—
343C22×C8—26no—
344C22×C8—248no—
345C22×C8—248no—
346C22×C8—224no—
347C22×C8—224no—
348C22×C8—248no—
349C22×C8—23no—
350C22×C8—23no—
351C22×C8—26no—
352C22×C8—21no—
353C22×C8—23no—
354C22×C8—26no—
355C22×C8—23no—
356C22×C8—23no—
357C22×C8C4×D4148yes—
358C22×C8C23×C416yes—
359C22×C8C23×C412yes—
360C22×C8C22×D416yes—
361C22×C8C22×D416yes—
362C22×C8C2×#4116yes—
363C22×C8C2×#4116yes—
364C22⋊Q8—248no—
365C22⋊Q8—248no—
366C22⋊Q8—424no—
367C22⋊Q8—424no—
368C4×Q8—48no—
369C4×Q8—812no—
370C2×C4⋊C4—26no—
371C2×C4⋊C4—248no—
372C2×C4⋊C4—212no—
373C2×C4⋊C4—26no—
374C2×C4⋊C4—26no—
375C2×C4⋊C4—212no—
376C2×C4⋊C4—26no—
377C2×C4⋊C4—26no—
378C2×C4⋊C4—248no—
379C2×C4⋊C4—224no—
380C2×C4⋊C4—224no—
381C2×C4⋊C4—26no—
382C2×C4⋊C4—26no—
383C2×C4⋊C4—424no—
384C2×C4⋊C4—448no—
385C2×C4⋊C4—448no—
386C2×C4⋊C4—424no—
387C2×C4⋊C4—424no—
388C2×C4⋊C4—4192no—
389C2×C4⋊C4—424no—
390C2×C4⋊C4—412no—
391C2×C4⋊C4—412no—
392C2×C4⋊C4—412no—
393C2×C4⋊C4—412no—
394C2×C4⋊C4—412no—
395C2×C4⋊C4—812no—
396C2×C4⋊C4C23×C4112yes—
397C2×C4⋊C4C22×D4112yes—
398C2×C4⋊C4C22×D4112yes—
399C2×C22⋊C4—224no—
400C2×C22⋊C4—224no—
401C2×C22⋊C4—212no—
402C2×C22⋊C4—212no—
403C2×C22⋊C4—212no—
404C2×C22⋊C4—212no—
405C2×C22⋊C4—212no—
406C2×C22⋊C4—212no—
407C2×C22⋊C4—412no—
408C2×C22⋊C4—412no—
409C2×C22⋊C4—412no—
410C2×C22⋊C4—412no—
411C2×C42—23no—
412C2×C42—23no—
413C2×C42—26no—
414C2×C42—23no—
415C2×C42—23no—
416C2×C42—26no—
417C2×C42—412no—
418C2×C42—412no—
419C2×C42—812no—
420#107—224no—
421#106—248no—
422#106—224no—
423#106—248no—
424#106#120116yes—
425#103—248no—
426#103—48no—
427#103—812no—
428C22⋊C8—412no—
429C22⋊C8—412no—
430#97—44no—
431#97—412no—
432#95—224no—
433#95—212no—
434#95—212no—
435#95—412no—
436#95—412no—
437#95—412no—
438#95—412no—
439#95—812no—
440#95—812no—
441C24—28no—
442C24—224no—
443C24—248no—
444C24—248no—
445C24—248no—
446C24—224no—
447C24—26no—
448C24—224no—
449C24—26no—
450C24—248no—
451C24—212no—
452C24#441116yescomplemented
453C24C2×C4×D416yes—
454C24D4216yes—
455C24#54716yes—
456C24C2×#137112yes—
457C24#577112yes—
458C24C23×D411yescharacteristic
459C2×Q8—212no—
460C2×Q8—212no—
461C2×Q8—46no—
462C2×Q8—46no—
463C2×Q8—42no—
464C2×Q8—46no—
465C2×Q8—46no—
466C2×Q8—46no—
467C2×Q8—412no—
468C22×C4—224no—
469C22×C4—232no—
470C22×C4—248no—
471C22×C4—224no—
472C22×C4—296no—
473C22×C4—232no—
474C22×C4—296no—
475C22×C4—224no—
476C22×C4—26no—
477C22×C4—26no—
478C22×C4—248no—
479C22×C4—212no—
480C22×C4—212no—
481C22×C4—26no—
482C22×C4—26no—
483C22×C4—212no—
484C22×C4—212no—
485C22×C4—212no—
486C22×C4—26no—
487C22×C4—212no—
488C22×C4—26no—
489C22×C4—212no—
490C22×C4—212no—
491C22×C4—212no—
492C22×C4—212no—
493C22×C4—212no—
494C22×C4—212no—
495C22×C4—24no—
496C22×C4—26no—
497C22×C4—26no—
498C22×C4—212no—
499C22×C4—212no—
500C22×C4—26no—
501C22×C4—22no—
502C22×C4—26no—
503C22×C4—212no—
504C22×C4—26no—
505C22×C4—448no—
506C22×C4—42no—
507C22×C4—46no—
508C22×C4—448no—
509C22×C4—46no—
510C22×C4—46no—
511C22×C4—412no—
512C22×C4—46no—
513C22×C4—46no—
514C22×C4—448no—
515C22×C4—46no—
516C22×C4—412no—
517C22×C4—46no—
518C22×C4—412no—
519C22×C4—412no—
520C22×C4—824no—
521C22×C4—848no—
522C22×C4—848no—
523C22×C4—848no—
524C22×C4—848no—
525C22×C4—8256no—
526C22×C4—824no—
527C22×C4—824no—
528C22×C4C2×C4×D416yes—
529C22×C4C2×C4×D4112yes—
530C22×C4#520112yes—
531C22×C4D4216yes—
532C22×C4#54716yes—
533C22×C4#548112yes—
534C22×C4#553112yes—
535C22×C4C24×C411yescharacteristic
536C22×C4C23×D413yes—
537C22×C4C22×#4113yes—
538C22×C4C22×#4111yescharacteristic
539C22×C4C2×H(5,𝔽2)13yes—
540C22×C4C2×2−1+413yes—
541Q16—416no—
542C2×C8—296no—
543C2×C8—212no—
544C2×C8—26no—
545C2×C8—26no—
546C2×C8—212no—
547C2×C8—296no—
548C2×C8—41no—
549C2×C8—43no—
550C2×C8—43no—
551C2×C8—43no—
552C2×C8—46no—
553C2×C8—43no—
554C2×C8—43no—
555C2×C8—46no—
556C2×C8C2×C4×D4112yes—
557C2×C8C2×C4×D4112yes—
558C2×C8C2×#12014yes—
559C2×C8#525112yes—
560C4⋊C4—212no—
561C4⋊C4—212no—
562C4⋊C4—212no—
563C4⋊C4—212no—
564C4⋊C4—212no—
565C4⋊C4—448no—
566C4⋊C4—448no—
567C4⋊C4—448no—
568C4⋊C4—46no—
569C4⋊C4—46no—
570C4⋊C4—412no—
571C4⋊C4—88no—
572C4⋊C4—1612no—
573C4⋊C4C2×#12018yes—
574C22⋊C4—424no—
575C22⋊C4—448no—
576C22⋊C4—448no—
577C22⋊C4—424no—
578C42—41no—
579C42—43no—
580C42—43no—
581C42—43no—
582C42—46no—
583C42—88no—
584C42—1612no—
585C23—232no—
586C23—264no—
587C23—2192no—
588C23—296no—
589C23—212no—
590C23—26no—
591C23—212no—
592C23—224no—
593C23—212no—
594C23—224no—
595C23—212no—
596C23—224no—
597C23—212no—
598C23—224no—
599C23—296no—
600C23—26no—
601C23#2361112yes—
602C23C2×#441112yes—
603C23#2952112yes—
604C23#3266124yes—
605C23#3344124yes—
606C23C22×C4×D411yescharacteristic
607C23C2×D4213yes—
608C23C2×#54713yes—
609C23C22×#13712yes—
610C23C2×#57716yes—
611Q8—412no—
612Q8—44no—
613Q8—82no—
614Q8—86no—
615C2×C4—296no—
616C2×C4—26no—
617C2×C4—212no—
618C2×C4—212no—
619C2×C4—26no—
620C2×C4—212no—
621C2×C4—224no—
622C2×C4—212no—
623C2×C4—212no—
624C2×C4—224no—
625C2×C4—212no—
626C2×C4—28no—
627C2×C4—412no—
628C2×C4—412no—
629C2×C4—424no—
630C2×C4—46no—
631C2×C4—412no—
632C2×C4—412no—
633C2×C4—412no—
634C2×C4—412no—
635C2×C4—46no—
636C2×C4—46no—
637C2×C4—412no—
638C2×C4—46no—
639C2×C4—46no—
640C2×C4—412no—
641C2×C4—42no—
642C2×C4—412no—
643C2×C4—896no—
644C2×C4—896no—
645C2×C4—86no—
646C2×C4—812no—
647C2×C4—812no—
648C2×C4—86no—
649C2×C4—86no—
650C2×C4—812no—
651C2×C4—82no—
652C2×C4#2361112yes—
653C2×C4C22×C4×D413yes—
654C2×C4C2×#52013yes—
655C2×C4C22×#12011yescharacteristic
656C2×C4C2×#52513yes—
657C2×C4C2×#54813yes—
658C2×C4C2×#55313yes—
659C2×C4#348413yes—
660C2×C4#348613yes—
661C2×C4#349216yes—
662C8—212no—
663C8—81no—
664C8—83no—
665C8C4×#12018yes—
666C22—2128no—
667C22—212no—
668C22—224no—
669C22—248no—
670C22—224no—
671C22—212no—
672C22—224no—
673C22—224no—
674C22#23028124yescomplemented
675C22C2×#236113yes—
676C22D4×#12011yescharacteristic
677C22#3610313yes—
678C22C22×#44111yescharacteristic
679C22C2×#295213yes—
680C22C2×#326616yes—
681C22#6105016yes—
682C22C2×#334416yes—
683C22#6125916yes—
684C4—212no—
685C4—48no—
686C4—84no—
687C4—86no—
688C4—812no—
689C4—82no—
690C4—812no—
691C4C2×C4×#12011yescharacteristic
692C4#3420013yes—
693C4#3657511yescharacteristic
694C4#3658113yes—
695C2—224no—
696C2#52027011yescharacteristic
697C2C2×#2302813yes—
698C2#638194513yes—
699C2#657348212yes—
700C2#657906216yes—
subgroupG/H
center, socleC24C23×D4
commutatorC22×C4C24×C4
Frattini, cosocleC23×C4C25
prime radicalC26C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#379 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#23028C2semidirect
6. #374418C2semidirect
7. #373850C2semidirect
8. #368433C2semidirect
9. #368010C2semidirect
10. #356221C2semidirect
15. C22×#441C22semidirect
22. #32931C4semidirect
24. C2×#1971C22semidirect
25. C2×#1943C22semidirect
26. C2×#1927C22semidirect
30. C2×#1865C22semidirect
33. C2×#1776C22semidirect
44. #11327C22semidirect
45. #11257C22semidirect
46. #11122C22semidirect
47. #10908C22semidirect
48. #10863C22semidirect
70. C2×#467C2×C4semidirect
113. #1892C2×C4semidirect
117. #1808C2×C4semidirect
156. C22×Q16C22×C4semidirect
214. C2×#107C22×C4semidirect
230. C2×#103C22×C4semidirect
273. #340C22×C4semidirect
452. C24#441semidirect
674. C22#23028semidirect

Maximal subgroups

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

subgroupindex
C24×#1062
C2×#329312
C2×#230282
#5202722
#3926212
#3744182
#3738502
#3684332
#3680102
#3562212
#3547812