Finite Group Databasebeta

← #6477445

Subgroups — #6477445

Order 512.

3390 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2C2×#3187C211yesmaximal, characteristic, complemented
3C22×#484C211yesmaximal, characteristic, complemented
4#35010C211yesmaximal, characteristic, complemented
5#33009C212yesmaximal, complemented
6#32984C212yesmaximal, complemented
7#29643C212yesmaximal, complemented
8#29598C212yesmaximal, complemented
9#25535C218yesmaximal, complemented
10#24092C214yesmaximal, complemented
11#23964C214yesmaximal, complemented
12#21758C211yesmaximal, characteristic, complemented
13#21738C211yesmaximal, characteristic, complemented
14C2×#1824C211yesmaximal, characteristic, complemented
15C2×#553—28no—
16C2×#553—22no—
17C2×#553—22no—
18C2×#553—28no—
19C2×#553C2214yes—
20C2×#553C2214yes—
21C22×#126—21no—
22C22×#125C2211yescharacteristic
23C22×#123—21no—
24C22×C22⋊Q8C2211yescharacteristic
25C22×#121—21no—
26C22×#121C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×D4—21no—
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3187—28no—
34C2×#484—28no—
35C2×#484C2212yes—
36C2×#484C2212yes—
37C2×#484C2214yescomplemented
38C2×#481C2212yescomplemented
39C2×#483C2212yes—
40C2×#465C2214yescomplemented
41C2×#448C2212yescomplemented
42C2×#450C2212yes—
43C22×#107C2212yescomplemented
44C22×#103C2212yescomplemented
45C22×C22⋊C8C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2627—21no—
50#2626—21no—
51#2579—22no—
52#2564—22no—
53#2487C2212yescomplemented
54#2417C2214yescomplemented
55#2409C2212yescomplemented
56#2315—21no—
57#2308—21no—
58C2×#380C2211yescharacteristic
59#2059C2218yescomplemented
60#2033C2218yescomplemented
61#2026C2218yescomplemented
62#2000C2218yescomplemented
63#1893C2214yescomplemented
64#1877C2218yescomplemented
65#1874C2214yescomplemented
66#1873C2218yescomplemented
67#1869C2214yescomplemented
68#1869C2214yescomplemented
69#1862C2218yescomplemented
70#1854C2218yescomplemented
71#1824C2212yes—
72#1824C2212yes—
73#1824C2214yescomplemented
74#1823C2212yes—
75#1817C2212yes—
76#1786C2212yes—
77#1779C2212yes—
78C2×#340C2214yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4—21no—
82C23×D4D411yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4—21no—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C22×SD16C2314yes—
89C23×C8C2312yes—
90C23×C8C2312yes—
91#553—232no—
92#553—232no—
93#553—216no—
94#553—216no—
95#553—216no—
96#553—216no—
97C2×#126—216no—
98C2×#126—24no—
99C2×#126—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125—22no—
103C2×#125C2314yes—
104C2×#123—216no—
105C2×#123—24no—
106C2×#123—24no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2314yes—
114C2×#121—216no—
115C2×#121—24no—
116C2×#121—24no—
117C2×#121—24no—
118C2×#121—24no—
119C2×#121—22no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121—24no—
123C2×#121D418yes—
124C2×#121C2314yes—
125C2×#121C2312yes—
126C2×#121C2312yes—
127C2×C4×D4—216no—
128C2×C4×D4—24no—
129C2×C4×D4—24no—
130C2×C4×D4—24no—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—22no—
135C2×C4×D4D412yes—
136C2×C4×D4D412yes—
137C2×C4×D4C2314yes—
138C2×C4×D4C2312yes—
139C2×C4×D4C2312yes—
140C2×#117—22no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117C2314yes—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—24no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4D411yescharacteristic
152C22×C4⋊C4D418yes—
153C22×C4⋊C4C2311yescharacteristic
154C22×C4⋊C4C2311yescharacteristic
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2312yes—
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2312yes—
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4D411yescharacteristic
167C22×C22⋊C4D411yescharacteristic
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170#484—28no—
171#484—28no—
172#484—216no—
173#483—28no—
174#481—28no—
175#465—216no—
176#450—28no—
177#448—28no—
178C2×#107—28no—
179C2×#107C2314yes—
180C2×#107C2314yes—
181C2×#106C2314yes—
182C2×#103—28no—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#102C2314yes—
187C2×#102C2318yes—
188C2×C22⋊C8—24no—
189C2×C22⋊C8—24no—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2318yes—
195#400—22no—
196#399—22no—
197#398—24no—
198#396—24no—
199#395—22no—
200#393—22no—
201#391C2314yes—
202#389C2314yes—
203#387—24no—
204#386—24no—
205#386—22no—
206#386—22no—
207#385—24no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#381—22no—
213#380—21no—
214#380—21no—
215#380—24no—
216#380C2312yes—
217#380C2312yes—
218#380C2314yes—
219#379—22no—
220C2×#95—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C25C2×D411yescharacteristic
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—44no—
232C22×Q8C2412yes—
233C22×D4—24no—
234C22×D4—28no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—28no—
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—84no—
247C22×D4C2×D412yes—
248C22×D4C2×D412yes—
249C22×D4C2×D414yes—
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—28no—
255C23×C4—28no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—24no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4—84no—
267C23×C4—84no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D412yes—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D414yes—
276C23×C4C2×D414yes—
277C23×C4C2412yes—
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×SD16—28no—
282C2×SD16—48no—
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—44no—
286C22×C8—44no—
287C22×C8C2414yes—
288C22×C8C2414yes—
289#126—232no—
290#126—232no—
291#125—216no—
292#125—28no—
293#125—28no—
294#123—232no—
295#123—232no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—48no—
303#121—232no—
304#121—232no—
305#121—216no—
306#121—28no—
307#121—28no—
308#121—28no—
309#121—24no—
310#121—24no—
311#121—28no—
312#121C2×D418yes—
313#121C2×D418yes—
314C4×D4—232no—
315C4×D4—232no—
316C4×D4—216no—
317C4×D4—28no—
318C4×D4—28no—
319C4×D4—28no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—28no—
325#117—216no—
326#117—28no—
327#117—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2×D418yes—
361C2×C4⋊C4C2×D418yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—28no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—44no—
386C2×C22⋊C4—44no—
387C2×C22⋊C4C2×D412yes—
388C2×C22⋊C4C2×D412yes—
389C2×C22⋊C4C2×D412yes—
390C2×C22⋊C4C2×D412yes—
391C2×C42—28no—
392C2×C42—24no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42C2×D412yes—
396C2×C42C2412yes—
397#107—28no—
398#107—28no—
399#106—28no—
400#103—28no—
401#103—28no—
402#103—28no—
403#102—28no—
404#102—48no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409C22⋊C8—48no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416C24—22no—
417C24—24no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—44no—
426C24—44no—
427C24—42no—
428C24—42no—
429C24#12014yes—
430C24#13514yes—
431C24#13714yes—
432C24C22×D412yes—
433C24C22×D411yescharacteristic
434C24C22×D411yescharacteristic
435C24C22×D411yescharacteristic
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—48no—
440C2×Q8—42no—
441C2×Q8—82no—
442C2×Q8—84no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—28no—
446C2×D4—28no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—216no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—28no—
455C2×D4—48no—
456C2×D4—44no—
457C2×D4—48no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—416no—
463C2×D4—432no—
464C2×D4—88no—
465C2×D4—816no—
466C2×D4C22×D414yes—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—216no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—28no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—44no—
508C22×C4—88no—
509C22×C4—82no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—82no—
513C22×C4—816no—
514C22×C4—816no—
515C22×C4—82no—
516C22×C4—82no—
517C22×C4#12014yes—
518C22×C4#12014yes—
519C22×C4#12014yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C2511yescharacteristic
533SD16—88no—
534C2×C8—24no—
535C2×C8—24no—
536C2×C8—28no—
537C2×C8—28no—
538C2×C8—48no—
539C2×C8—84no—
540C4⋊C4—216no—
541C4⋊C4—216no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—28no—
545C4⋊C4—28no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—28no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—84no—
559C4⋊C4—82no—
560C4⋊C4—84no—
561C4⋊C4C22×D414yes—
562C4⋊C4C22×D414yes—
563C4⋊C4C22×D414yes—
564C22⋊C4—216no—
565C22⋊C4—216no—
566C22⋊C4—216no—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—28no—
570C22⋊C4—28no—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C42—216no—
580C42—24no—
581C42—22no—
582C42—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—24no—
590C23—22no—
591C23—24no—
592C23—24no—
593C23—22no—
594C23—22no—
595C23—44no—
596C23—44no—
597C23—48no—
598C23—48no—
599C23—44no—
600C23—48no—
601C23—48no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—48no—
607C23—42no—
608C23—42no—
609C23—84no—
610C23—84no—
611C23—88no—
612C23—88no—
613C23#452116yes—
614C23C2×#12012yes—
615C23C2×#12012yes—
616C23C2×#12012yes—
617C23C2×#13512yes—
618C23C2×#13512yes—
619C23C2×#13512yes—
620C23C2×#13712yes—
621C23C2×#13712yes—
622C23C2×#13712yes—
623C23C23×D411yescharacteristic
624Q8—44no—
625Q8—44no—
626Q8—84no—
627D4—28no—
628D4—48no—
629D4—48no—
630D4—48no—
631D4—416no—
632D4—416no—
633D4—44no—
634D4—816no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—48no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—88no—
678C2×C4—88no—
679C2×C4—816no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—164no—
683C2×C4C2×#12012yes—
684C2×C4C2×#12012yes—
685C2×C4C2×#12012yes—
686C2×C4C23×D411yescharacteristic
687C2×C4C23×D411yescharacteristic
688C2×C4#58514yes—
689C8—44no—
690C8—44no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—416no—
704C22—44no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—42no—
712C22—88no—
713C22—816no—
714C22—84no—
715C22—84no—
716C22C2×#45214yes—
717C22C2×#45214yes—
718C22C2×#45214yes—
719C22C22×#12011yescharacteristic
720C22#348512yes—
721C22C22×#13511yescharacteristic
722C22C22×#13711yescharacteristic
723C22#358512yes—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—42no—
731C4—44no—
732C4—44no—
733C4—84no—
734C4—82no—
735C4—82no—
736C4#344912yes—
737C4#348512yes—
738C2—24no—
739C2—24no—
740C2—24no—
741C2—82no—
742C2—84no—
743C2#3508811yescharacteristic
744C2C22×#45211yescharacteristic
745C2#6036711yescharacteristic
746C2#6169912yes—
747C2#6170812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#553C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3187C2semidirect
3. C22×#484C2semidirect
4. #35010C2semidirect
5. #33009C2semidirect
6. #32984C2semidirect
7. #29643C2semidirect
8. #29598C2semidirect
9. #25535C2semidirect
10. #24092C2semidirect
11. #23964C2semidirect
12. #21758C2semidirect
13. #21738C2semidirect
14. C2×#1824C2semidirect
37. C2×#484C22semidirect
38. C2×#481C22semidirect
40. C2×#465C22semidirect
41. C2×#448C22semidirect
43. C22×#107C22semidirect
44. C22×#103C22semidirect
53. #2487C22semidirect
54. #2417C22semidirect
55. #2409C22semidirect
59. #2059C22semidirect
60. #2033C22semidirect
61. #2026C22semidirect
62. #2000C22semidirect
63. #1893C22semidirect
64. #1877C22semidirect
65. #1874C22semidirect
66. #1873C22semidirect
67. #1869C22semidirect
68. #1869C22semidirect
69. #1862C22semidirect
70. #1854C22semidirect
73. #1824C22semidirect
78. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
C2×#31872
C22×#4842
#350102
#330092
#329842
#296432
#295982
#255352
#240922
#239642
#217582
#217382
C2×#18242