Finite Group Databasebeta

← #6497534

Subgroups — #6497534

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C211yesmaximal, characteristic
2C2×#3233C211yesmaximal, characteristic
3#35281C211yesmaximal, characteristic, complemented
4#34084C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#33201C211yesmaximal, characteristic, complemented
7#33196C211yesmaximal, characteristic, complemented
8#33195C211yesmaximal, characteristic, complemented
9#29867C211yesmaximal, characteristic, complemented
10#29866C211yesmaximal, characteristic, complemented
11#29865C211yesmaximal, characteristic, complemented
12#29864C211yesmaximal, characteristic, complemented
13#28730C214yesmaximal, complemented
14#28698C214yesmaximal, complemented
15#27656C214yesmaximal, complemented
16#27654C214yesmaximal
17#21839C211yesmaximal, characteristic, complemented
18#21833C211yesmaximal, characteristic, complemented
19#20815C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#529C2211yescharacteristic
23#3239—28no—
24#3233—28no—
25C2×#501C2212yes—
26C2×#499C2212yes—
27C2×#489C2212yes—
28C2×#488C2212yes—
29C2×#484C2212yes—
30C2×#481C2212yes—
31C2×#485C2212yes—
32C2×#482C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#476C2211yescharacteristic
38C2×#474C2211yescharacteristic
39C2×#474C2211yescharacteristic
40C2×#474C2211yescharacteristic
41C2×#432C2211yescharacteristic
42C2×#431C2211yescharacteristic
43#2868C2211yescharacteristic
44#2868C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2660C2214yes—
48#2567—22no—
49#2546—21no—
50#2546—21no—
51#2502—22no—
52#2434—21no—
53#2434—21no—
54#2335—22no—
55#2334—22no—
56#2330C2212yes—
57#2326C2212yes—
58#2324C2212yes—
59#2323C2212yes—
60#2317C2214yes—
61#2298—21no—
62#2296—21no—
63#2295C2211yescharacteristic
64#2293—21no—
65#2293C2211yescharacteristic
66#2290—21no—
67#2285C2211yescharacteristic
68C42×Q8C2211yescharacteristic
69C42×D4C2211yescharacteristic
70C4×#117C2211yescharacteristic
71#2085C2214yes—
72#2085C2214yes—
73#2084C2214yes—
74#2083C2214yes—
75#2074C2214yes—
76#2073C2214yes—
77#2072C2214yes—
78#2071C2214yes—
79#1992C2214yes—
80#1990C2214yes—
81#1988C2211yescharacteristic
82#1988C2211yescharacteristic
83#1988C2211yescharacteristic
84#1988C2211yescharacteristic
85#1988C2211yescharacteristic
86#1988C2211yescharacteristic
87#1987C2211yescharacteristic
88#1987C2211yescharacteristic
89#1949C2214yes—
90#1949C2214yes—
91#1948C2214yes—
92#1947C2214yes—
93#1842C2211yescharacteristic
94#1842C2211yescharacteristic
95#1831C2211yescharacteristic
96#1831C2211yescharacteristic
97#1825C2214yes—
98#1823C2214yes—
99#1805C2214yes—
100#1804C2214yes—
101#1803C2214yes—
102#1802C2214yes—
103#1798C2214yes—
104#1768C2214yes—
105C23×Q8—21no—
106#533—24no—
107#533—24no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8—24no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121#529—24no—
122#529D418yes—
123C2×#126—21no—
124C2×#125C2312yes—
125C2×#125C2311yescharacteristic
126C2×#124C2311yescharacteristic
127C2×#123—21no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8C2311yescharacteristic
131C2×#121C2311yescharacteristic
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4C2312yes—
138C2×C4×D4C2311yescharacteristic
139C2×#117—21no—
140C2×#117—21no—
141C2×#117—21no—
142C2×#117C2312yes—
143C2×#117C2312yes—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2312yes—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#501—28no—
149#499—28no—
150#489—28no—
151#488—28no—
152#485—28no—
153#484—28no—
154#482—28no—
155#481—28no—
156#477—24no—
157#476—24no—
158#475—24no—
159#475—24no—
160#475—24no—
161#474—24no—
162#474—24no—
163#474—24no—
164#432—24no—
165#431—24no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2314yes—
179C2×C4⋊C8C2314yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×#97C2312yes—
191C2×C4×C8C2312yes—
192#398—24no—
193#398—24no—
194#395—24no—
195#395—24no—
196#391—22no—
197#391—22no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391C2312yes—
203#390—22no—
204#390C2312yes—
205#389—24no—
206#389—24no—
207#389—22no—
208#389—22no—
209#389—22no—
210#389C2312yes—
211#389C2312yes—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#388—22no—
217#388—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#386—22no—
223#385—22no—
224#385—24no—
225#385—24no—
226#385—22no—
227#385—22no—
228#384—22no—
229#384—22no—
230#384C2314yes—
231#384C2314yes—
232#384C2312yes—
233#382—21no—
234#382C2314yes—
235#382C2314yes—
236#380—21no—
237#380C2312yes—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4D412yes—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4—22no—
250C4×C22⋊C4—21no—
251C4×C22⋊C4—21no—
252C4×C22⋊C4D412yes—
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255#376—21no—
256#376C2312yes—
257C43—21no—
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2314yes—
269#336C2314yes—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8—44no—
276C22×Q8—82no—
277C22×Q8—82no—
278C22×Q8C2411yescharacteristic
279C22×D4—21no—
280C22×D4C2411yescharacteristic
281C23×C4—21no—
282C23×C4—22no—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—24no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—24no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303C4⋊Q8—82no—
304C4⋊Q8—82no—
305C4⋊Q8—82no—
306C4⋊Q8—82no—
307#126—28no—
308#125—24no—
309#125—22no—
310#125—24no—
311#125C2×D414yes—
312#124—22no—
313#124C2×D414yes—
314#123—28no—
315C22⋊Q8—22no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C22⋊Q8C2×D414yes—
319#121—22no—
320#121C2×D414yes—
321C4×Q8—22no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8C2×D414yes—
327C4×D4—28no—
328C4×D4—22no—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4C2×D414yes—
332#117—28no—
333#117—24no—
334#117—24no—
335#117—24no—
336#117—24no—
337#117—82no—
338#117—82no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2×D412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2411yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D412yes—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
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×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2412yes—
407#107—24no—
408#107—24no—
409#107—24no—
410#107—24no—
411#106—24no—
412#106—24no—
413#106—24no—
414#106—24no—
415C4⋊C8—22no—
416C4⋊C8—22no—
417C4⋊C8—22no—
418C4⋊C8—22no—
419C4⋊C8—28no—
420C4⋊C8—28no—
421#103—24no—
422#103—24no—
423#103—24no—
424#103—24no—
425#102—24no—
426#102—24no—
427#102—24no—
428#102—24no—
429C22⋊C8—24no—
430C22⋊C8—24no—
431#97—24no—
432C4×C8—24no—
433#95—22no—
434#95—22no—
435#95—21no—
436#95—22no—
437#95—22no—
438#95—21no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—24no—
443#95—24no—
444#95—24no—
445#95—24no—
446#95C2×D412yes—
447#95C2×D412yes—
448#95C2×D412yes—
449#95C2×D412yes—
450C24—21no—
451C24C22×D411yescharacteristic
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—44no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—82no—
468C2×Q8—82no—
469C2×Q8—816no—
470C2×Q8C22×D412yes—
471C2×D4—24no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—82no—
476C2×D4C22×D412yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4#12114yes—
501C22×C4#12114yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C2×#4112yes—
513C22×C4C2511yescharacteristic
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—84no—
556C4⋊C4—84no—
557C4⋊C4—84no—
558C4⋊C4—84no—
559C4⋊C4C22×D412yes—
560C4⋊C4C22×D412yes—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C42—24no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—42no—
581C42—42no—
582C42—44no—
583C42—44no—
584C42—42no—
585C42—42no—
586C42—82no—
587C42—82no—
588C42#12114yes—
589C42#12114yes—
590C42C22×D412yes—
591C42C22×D412yes—
592C42C2×#4112yes—
593C42C2×#4112yes—
594C422−1+414yes—
595C422−1+414yes—
596C23—22no—
597C23—22no—
598C23—21no—
599C23—21no—
600C23—22no—
601C23—42no—
602C23—42no—
603C23—41no—
604C23—41no—
605C23C2×#12112yes—
606C23C23×D411yescharacteristic
607Q8—22no—
608Q8—22no—
609Q8—42no—
610Q8—84no—
611Q8—88no—
612Q8—84no—
613Q8—84no—
614Q8—84no—
615D4—22no—
616D4—22no—
617D4—42no—
618D4—84no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—48no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—88no—
665C2×C4—88no—
666C2×C4C2×#12112yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#12112yes—
669C2×C4C2×#12112yes—
670C2×C4C2×#12112yes—
671C2×C4#53614yes—
672C2×C4#53614yes—
673C2×C4#57612yes—
674C2×C4#57712yes—
675C2×C4C23×D411yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C2×C4C2×2−1+412yes—
678C2×C4C2×2−1+412yes—
679C8—42no—
680C8—42no—
681C8—42no—
682C8—42no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—82no—
692C22—82no—
693C22C22×#12111yescharacteristic
694C22C2×#53612yes—
695C22C2×#57611yescharacteristic
696C22C2×#57711yescharacteristic
697C22#358612yes—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—24no—
702C4—24no—
703C4—22no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4#304514yes—
716C4#349911yescharacteristic
717C4#349911yescharacteristic
718C4#350912yes—
719C2—42no—
720C2—82no—
721C2#3531611yescharacteristic
722C2C2×#304511yescharacteristic
723C2#6042711yescharacteristic
724C2#6088812yes—
725C2#6088912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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. #35281C2semidirect
5. #33202C2semidirect
6. #33201C2semidirect
7. #33196C2semidirect
8. #33195C2semidirect
9. #29867C2semidirect
10. #29866C2semidirect
11. #29865C2semidirect
12. #29864C2semidirect
13. #28730C2semidirect
14. #28698C2semidirect
15. #27656C2semidirect
17. #21839C2semidirect
18. #21833C2semidirect
19. #20815C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
C2×#32332
#352812
#340842
#332022
#332012
#331962
#331952
#298672
#298662
#298652
#298642
#287302
#286982
#276562
#276542
#218392
#218332
#208152