Finite Group Databasebeta

← #6436708

Subgroups — #6436708

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3116C211yesmaximal, characteristic
2C2×#3107C211yesmaximal, characteristic
3#35322C211yesmaximal, characteristic, complemented
4#34060C211yesmaximal, characteristic
5#29867C211yesmaximal, characteristic, complemented
6#29865C211yesmaximal, characteristic, complemented
7#29831C211yesmaximal, characteristic, complemented
8#29830C211yesmaximal, characteristic, complemented
9#29829C211yesmaximal, characteristic, complemented
10#29828C211yesmaximal, characteristic, complemented
11#29799C211yesmaximal, characteristic, complemented
12#29798C211yesmaximal, characteristic, complemented
13#29766C214yesmaximal, complemented
14#29726C214yesmaximal
15#23691C214yesmaximal, complemented
16#23689C214yesmaximal, complemented
17#21394C211yesmaximal, characteristic, complemented
18#21385C211yesmaximal, characteristic, complemented
19#20815C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23#3116—28no—
24#3107—28no—
25C2×#485C2212yes—
26C2×#485C2212yes—
27C2×#477C2211yescharacteristic
28C2×#475C2211yescharacteristic
29C2×#476C2211yescharacteristic
30C2×#474C2211yescharacteristic
31C2×#462C2211yescharacteristic
32C2×#462C2211yescharacteristic
33C2×#461C2211yescharacteristic
34C2×#461C2211yescharacteristic
35C2×#452C2212yes—
36C2×#452C2212yes—
37C2×#441C2212yes—
38C2×#440C2212yes—
39C2×C4×Q16C2212yes—
40C2×C4×SD16C2212yes—
41C2×#432C2211yescharacteristic
42C2×#431C2211yescharacteristic
43#2868C2211yescharacteristic
44#2862C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2621C2214yes—
48#2565—22no—
49#2565—21no—
50#2565—21no—
51#2499—22no—
52#2499—21no—
53#2499—21no—
54#2327C2214yes—
55#2320C2212yes—
56#2319C2212yes—
57#2318—22no—
58#2317C2212yes—
59#2314—22no—
60#2311C2212yes—
61#2295C2211yescharacteristic
62#2295C2211yescharacteristic
63#2293C2211yescharacteristic
64#2293C2211yescharacteristic
65#2290—21no—
66#2290—21no—
67#2285—21no—
68#2283—21no—
69C4×#117C2211yescharacteristic
70C2×C4×C4⋊C4C2211yescharacteristic
71#2085C2214yes—
72#2085C2214yes—
73#2074C2214yes—
74#2073C2214yes—
75#2068C2214yes—
76#2068C2214yes—
77#2066C2214yes—
78#2066C2214yes—
79#2056C2214yes—
80#2056C2214yes—
81#2055C2214yes—
82#2055C2214yes—
83#2043C2214yes—
84#2043C2214yes—
85#2016C2214yes—
86#2013C2214yes—
87#1988C2211yescharacteristic
88#1988C2211yescharacteristic
89#1987C2211yescharacteristic
90#1987C2211yescharacteristic
91#1981C2211yescharacteristic
92#1981C2211yescharacteristic
93#1981C2211yescharacteristic
94#1981C2211yescharacteristic
95#1872C2214yes—
96#1872C2214yes—
97#1842C2211yescharacteristic
98#1842C2211yescharacteristic
99#1831C2211yescharacteristic
100#1831C2211yescharacteristic
101#1798C2214yes—
102#1768C2214yes—
103#1763C2214yes—
104C4×#103C2214yes—
105C22×#41C2311yescharacteristic
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110#533—24no—
111#533—24no—
112#533—24no—
113#533—24no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2314yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×#125—21no—
120C2×#125—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125C4×#41—24no—
126C4×#41D418yes—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2311yescharacteristic
132C2×C4×Q8C2311yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—21no—
140C2×#117—21no—
141C2×#117—21no—
142C2×#117C2311yescharacteristic
143C2×#117C2312yes—
144C22×C4⋊C4C2312yes—
145C22×C42—21no—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#485—28no—
149#485—28no—
150#477—24no—
151#476—24no—
152#475—24no—
153#474—24no—
154#462—24no—
155#462—24no—
156#461—24no—
157#461—24no—
158#452—28no—
159#452—28no—
160#441—28no—
161#440—28no—
162C4×Q16—28no—
163C4×SD16—28no—
164#432—24no—
165#431—24no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2314yes—
175C2×C4⋊C8C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×#97C2312yes—
191C2×C4×C8C2312yes—
192#398—24no—
193#398—24no—
194#397—24no—
195#397—24no—
196#391—21no—
197#391—21no—
198#391—22no—
199#391—21no—
200#391—21no—
201#391—22no—
202#391C2314yes—
203#391C2314yes—
204#391C2312yes—
205#391C2312yes—
206#390C2312yes—
207#390C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#387—22no—
217#387—22no—
218#387—24no—
219#387—24no—
220#387—24no—
221#387—24no—
222#387—22no—
223#387—22no—
224#386—22no—
225#386—22no—
226#385—22no—
227#385—22no—
228#384—22no—
229#384—21no—
230#384—21no—
231#384—22no—
232#383—22no—
233#382—22no—
234#382—21no—
235#382—21no—
236#382—22no—
237#382—22no—
238#382C2314yes—
239#381—22no—
240#380—22no—
241#380—22no—
242#380—22no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4D412yes—
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C22⋊C4—21no—
254C4×C22⋊C4—21no—
255C4×C22⋊C4D412yes—
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258#376—21no—
259#376—21no—
260#376C2312yes—
261C2×#95—21no—
262C43C2311yescharacteristic
263C43C2311yescharacteristic
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2314yes—
269#336C2314yes—
270C2×#41—22no—
271C2×#41—42no—
272C2×#41—42no—
273C2×#41C2×D414yes—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8C2412yes—
277C22×Q8C2412yes—
278C22×Q8C2411yescharacteristic
279C22×D4—21no—
280C22×D4C2411yescharacteristic
281C23×C4—21no—
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—22no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288C2×Q16—24no—
289C2×Q16—24no—
290C2×Q16—44no—
291C2×Q16—44no—
292C2×SD16—24no—
293C2×SD16—24no—
294C2×SD16—44no—
295C2×SD16—44no—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—44no—
309C4⋊Q8—44no—
310#125—44no—
311#125—44no—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8C2×D414yes—
327C4×Q8C2×D414yes—
328C4×D4—28no—
329C4×D4—28no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4C2×D414yes—
333C4×D4C2×D414yes—
334#117—22no—
335#117—28no—
336#117—42no—
337#117—42no—
338#117—44no—
339#117—44no—
340#117—82no—
341#117—82no—
342#117C2×D414yes—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4C2×D412yes—
365C2×C4⋊C4C2×D412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2411yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—28no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—22no—
392C2×C42—82no—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D412yes—
398C2×C42C2×D412yes—
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D414yes—
402C2×C42#4114yes—
403C2×C42C2412yes—
404C2×C42C2412yes—
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#107—24no—
408#107—24no—
409#106—24no—
410#106—24no—
411C4⋊C8—22no—
412C4⋊C8—22no—
413C4⋊C8—22no—
414C4⋊C8—22no—
415C4⋊C8—44no—
416C4⋊C8—44no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—24no—
421#103—24no—
422#103—24no—
423#103—44no—
424#103—44no—
425#103—44no—
426#103—44no—
427#102—24no—
428#102—24no—
429C22⋊C8—24no—
430C22⋊C8—24no—
431#97—24no—
432C4×C8—24no—
433#95—22no—
434#95—21no—
435#95—21no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—24no—
445#95—24no—
446#95—24no—
447#95—24no—
448#95—24no—
449#95—24no—
450#95C2×D412yes—
451#95C2×D412yes—
452#95C2×D412yes—
453#95C2×D412yes—
454C24—21no—
455C24C22×D411yescharacteristic
456#41—44no—
457#41—44no—
458#41—48no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—24no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×Q8—24no—
470C2×Q8—44no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—42no—
475C2×Q8C22×D412yes—
476C2×D4—24no—
477C2×D4—21no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—44no—
481C2×D4C22×D412yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—44no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4#12114yes—
507C22×C4#12114yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C2×#4112yes—
519C22×C4C2511yescharacteristic
520Q16—48no—
521Q16—48no—
522SD16—48no—
523SD16—48no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4C22×D412yes—
564C4⋊C4C22×D412yes—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C42—24no—
577C42—24no—
578C42—24no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—42no—
588C42—42no—
589C42—44no—
590C42—44no—
591C42—42no—
592C42—42no—
593C42—82no—
594C42#12114yes—
595C42#12114yes—
596C42C22×D412yes—
597C42C22×D412yes—
598C42C2×#4112yes—
599C42C2×#4112yes—
600C23—22no—
601C23—22no—
602C23—21no—
603C23—21no—
604C23—22no—
605C23—42no—
606C23—42no—
607C23—41no—
608C23—41no—
609C23C2×#12112yes—
610C23C23×D411yescharacteristic
611Q8—22no—
612Q8—22no—
613Q8—42no—
614Q8—42no—
615Q8—42no—
616Q8—42no—
617Q8—42no—
618Q8—44no—
619Q8—44no—
620Q8—84no—
621Q8—84no—
622D4—22no—
623D4—22no—
624D4—48no—
625D4—42no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—28no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—44no—
651C2×C4—48no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—48no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—84no—
673C2×C4—88no—
674C2×C4C2×#12112yes—
675C2×C4C2×#12112yes—
676C2×C4C2×#12112yes—
677C2×C4C2×#12112yes—
678C2×C4C2×#12112yes—
679C2×C4#57712yes—
680C2×C4#57812yes—
681C2×C4C23×D411yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4#58512yes—
684C2×C4#58512yes—
685C8—42no—
686C8—42no—
687C8—42no—
688C8—42no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—82no—
698C22—82no—
699C22C22×#12111yescharacteristic
700C22#344912yes—
701C22C2×#57711yescharacteristic
702C22C2×#57811yescharacteristic
703C22#358612yes—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4#304614yes—
722C4#344911yescharacteristic
723C4#344911yescharacteristic
724C4#345512yes—
725C2—42no—
726C2—82no—
727C2#3463311yescharacteristic
728C2C2×#304611yescharacteristic
729C2#6042711yescharacteristic
730C2#6099112yes—
731C2#6099512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #35322C2semidirect
5. #29867C2semidirect
6. #29865C2semidirect
7. #29831C2semidirect
8. #29830C2semidirect
9. #29829C2semidirect
10. #29828C2semidirect
11. #29799C2semidirect
12. #29798C2semidirect
13. #29766C2semidirect
15. #23691C2semidirect
16. #23689C2semidirect
17. #21394C2semidirect
18. #21385C2semidirect
19. #20815C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31162
C2×#31072
#353222
#340602
#298672
#298652
#298312
#298302
#298292
#298282
#297992
#297982
#297662
#297262
#236912
#236892
#213942
#213852
#208152