Finite Group Databasebeta

← #6526702

Subgroups — #6526702

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3238C211yesmaximal, characteristic, complemented
2C2×#3114C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic, complemented
4#34224C211yesmaximal, characteristic, complemented
5#28120C212yesmaximal, complemented
6#28114C212yesmaximal
7#27647C212yesmaximal, complemented
8#27644C212yesmaximal, complemented
9#25465C214yesmaximal, complemented
10#25334C214yesmaximal, complemented
11#24829C212yesmaximal, complemented
12#24827C212yesmaximal, complemented
13#24694C212yesmaximal, complemented
14#24692C212yesmaximal, complemented
15#21955C211yesmaximal, characteristic, complemented
16#21840C211yesmaximal, characteristic, complemented
17#20739C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C22⋊Q8—22no—
21C22×C4×D4C2211yescharacteristic
22#3238—28no—
23C2×#501C2212yescomplemented
24C2×#489C2212yescomplemented
25#3114—28no—
26C2×#483C2211yescharacteristic
27C2×#483C2212yes—
28C2×#480C2211yescharacteristic
29C2×#480C2212yes—
30C2×#477C2212yescomplemented
31C2×#478C2212yes—
32C2×#476C2212yescomplemented
33C2×#463C2212yes—
34C2×#450C2211yescharacteristic
35C2×#447C2211yescharacteristic
36C2×#442C2212yes—
37C2×#440C2212yes—
38C2×#432C2211yescharacteristic
39C2×#432C2211yescharacteristic
40#2542—21no—
41#2542—21no—
42#2542—22no—
43#2542C2212yes—
44#2542C2212yes—
45#2542C2214yes—
46C2×#399—21no—
47C2×#399—21no—
48#2373—22no—
49#2370C2212yescomplemented
50#2367C2212yescomplemented
51#2357C2211yescharacteristic, complemented
52#2356C2211yescharacteristic, complemented
53#2351C2211yescharacteristic, complemented
54#2349C2211yescharacteristic, complemented
55#2347—21no—
56#2343—21no—
57#2322—22no—
58#2318—21no—
59C4×C22⋊Q8C2212yes—
60C4×#121C2212yes—
61C2×#390C2212yes—
62C2×#384C2212yes—
63#2283C2211yescharacteristic
64#2275—21no—
65C2×C4×C4⋊C4C2211yescharacteristic
66#2084C2214yescomplemented
67#2083C2214yescomplemented
68#2072C2214yescomplemented
69#2071C2214yescomplemented
70#2067C2214yes—
71#2065C2214yes—
72#2055C2214yes—
73#2054C2214yes—
74#1993C2212yescomplemented
75#1993C2212yescomplemented
76#1987C2214yescomplemented
77#1948C2214yescomplemented
78#1947C2214yescomplemented
79#1940C2212yescomplemented
80#1940C2212yescomplemented
81#1940C2214yescomplemented
82#1926C2214yes—
83#1925C2214yes—
84#1880C2214yes—
85#1879C2214yes—
86#1823C2212yes—
87#1823C2212yes—
88#1823C2212yes—
89#1804C2212yes—
90#1804C2212yes—
91#1803C2214yescomplemented
92#1802C2214yescomplemented
93#1797C2212yes—
94#1797C2212yes—
95#1786C2212yes—
96#1762C2214yes—
97C23×Q8—21no—
98C23×D4C2311yescharacteristic
99C24×C4—21no—
100C22×SD16C2312yes—
101C22×D8C2312yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108#529—24no—
109#529D418yes—
110C2×#126—21no—
111C2×#125C2311yescharacteristic
112C2×#124C2311yescharacteristic
113C2×#123—21no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—416no—
119C2×C22⋊Q8C2311yescharacteristic
120C2×#121C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—24no—
128C2×C4×D4—24no—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4D412yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C22⋊C4—22no—
143C22×C42C2312yes—
144C22×C42C2311yescharacteristic
145#501—28no—
146#489—28no—
147#483—24no—
148#483—28no—
149#480—24no—
150#480—28no—
151#478—28no—
152#477—28no—
153#476—28no—
154#463—28no—
155#450—24no—
156#447—24no—
157#442—28no—
158#440—28no—
159#432—24no—
160#432—24no—
161C2×#107C2312yes—
162C2×#107C2314yes—
163C2×#106C2312yes—
164C2×#106C2314yes—
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2314yes—
168C2×C4⋊C8C2314yes—
169C2×#103C2314yes—
170C2×#103C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2314yes—
174C2×#102C2314yes—
175C2×#102C2312yes—
176C2×#102C2314yes—
177C2×C22⋊C8C2312yes—
178C2×C22⋊C8C2311yescharacteristic
179C2×C22⋊C8C2311yescharacteristic
180C2×#97C2312yes—
181C2×#97C2312yes—
182#399—22no—
183#399—22no—
184#399—24no—
185#399—22no—
186#399—22no—
187#399—24no—
188#391C2312yes—
189#390—22no—
190#390—22no—
191#390C2312yes—
192#390C2314yes—
193#390C2314yes—
194#389—22no—
195#389—22no—
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#386—22no—
207#385—22no—
208#385—22no—
209#385—22no—
210#384—22no—
211#384—22no—
212#384C2314yes—
213#384C2314yes—
214#383—21no—
215#383—21no—
216#382—21no—
217#382C2311yescharacteristic
218#382C2311yescharacteristic
219#381—21no—
220#381—21no—
221#380—21no—
222#380C2312yes—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D412yes—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C4⋊C4C2312yes—
235C4×C22⋊C4—21no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—21no—
239C4×C22⋊C4—22no—
240#376—21no—
241#376—21no—
242C2×#95—21no—
243C2×#95D411yescharacteristic
244C2×#95D411yescharacteristic
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2314yes—
252#336C2314yes—
253C25—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—48no—
257C22×Q8—44no—
258C22×Q8C2411yescharacteristic
259C22×D4—21no—
260C22×D4—24no—
261C22×D4—21no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—42no—
265C22×D4C2411yescharacteristic
266C22×D4C2412yes—
267C23×C4—21no—
268C23×C4—22no—
269C23×C4—21no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—48no—
274C23×C4—44no—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D412yes—
279C23×C4C2411yescharacteristic
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—42no—
283C2×D8—24no—
284C2×D8—24no—
285C2×D8—42no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2414yes—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295#126—28no—
296#125—22no—
297#125C2×D414yes—
298#124—22no—
299#124C2×D414yes—
300#123—28no—
301C22⋊Q8—22no—
302C22⋊Q8—416no—
303C22⋊Q8—416no—
304C22⋊Q8—416no—
305C22⋊Q8—416no—
306C22⋊Q8C2×D414yes—
307#121—22no—
308#121C2×D414yes—
309C4×Q8—22no—
310C4×Q8—44no—
311C4×Q8C2×D414yes—
312C4×D4—28no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—22no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—44no—
319C4×D4—48no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4C2×D414yes—
323#117—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—48no—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—48no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—48no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—44no—
381C2×C42C2×D412yes—
382C2×C42C2×D412yes—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D414yes—
386C2×C42#4114yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#107—24no—
392#107—28no—
393#106—24no—
394#106—28no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—44no—
398C4⋊C8—44no—
399#103—24no—
400#103—28no—
401#102—24no—
402#102—28no—
403#102—28no—
404#102—24no—
405#102—24no—
406#102—44no—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—24no—
410#97—24no—
411#97—24no—
412#95—21no—
413#95—22no—
414#95—21no—
415#95—22no—
416#95—22no—
417#95—21no—
418#95—21no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D412yes—
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D412yes—
430C24—24no—
431C24—21no—
432C24—22no—
433C24—42no—
434C24—44no—
435C24—42no—
436C24—44no—
437C24C22×D411yescharacteristic
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—48no—
442C2×Q8—48no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—816no—
446C2×Q8C22×D412yes—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—48no—
458C2×D4—48no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4C22×D412yes—
463C2×D42−1+414yes—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—816no—
495C22×C4#12114yes—
496C22×C4#12114yes—
497C22×C4AGL(1,ℤ/8)18yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C2×#4112yes—
509C22×C4C2511yescharacteristic
510SD16—48no—
511D8—48no—
512C2×C8—22no—
513C2×C8—24no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—24no—
519C2×C8—22no—
520C2×C8—22no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—44no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4C22×D412yes—
549C4⋊C4C22×D412yes—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—24no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—44no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C42—24no—
566C42—24no—
567C42—24no—
568C42—42no—
569C42—42no—
570C42—44no—
571C42—82no—
572C42—82no—
573C42—82no—
574C42#12114yes—
575C42#12114yes—
576C42C22×D412yes—
577C42C22×D412yes—
578C42C2×#4112yes—
579C42C2×#4112yes—
580C23—22no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—24no—
585C23—48no—
586C23—416no—
587C23—42no—
588C23—44no—
589C23—42no—
590C23—41no—
591C23—42no—
592C23—41no—
593C23—44no—
594C23—42no—
595C23—88no—
596C23—84no—
597C23—84no—
598C23C2×#12112yes—
599C23C2×AGL(1,ℤ/8)14yes—
600C23C23×D411yescharacteristic
601Q8—24no—
602Q8—42no—
603Q8—88no—
604Q8—88no—
605D4—24no—
606D4—24no—
607D4—24no—
608D4—48no—
609D4—48no—
610D4—42no—
611D4—42no—
612D4—42no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—28no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—48no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—84no—
654C2×C4—88no—
655C2×C4—88no—
656C2×C4—84no—
657C2×C4—88no—
658C2×C4—164no—
659C2×C4C2×#12112yes—
660C2×C4C2×#12112yes—
661C2×C4C2×#12112yes—
662C2×C4C2×#12112yes—
663C2×C4C2×#12112yes—
664C2×C4C2×AGL(1,ℤ/8)14yes—
665C2×C4C2×AGL(1,ℤ/8)14yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×2−1+412yes—
669C2×C4#58512yes—
670C8—44no—
671C8—42no—
672C8—42no—
673C22—24no—
674C22—22no—
675C22—22no—
676C22—24no—
677C22—42no—
678C22—44no—
679C22—42no—
680C22—48no—
681C22—42no—
682C22—816no—
683C22—82no—
684C22—84no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22#303814yes—
689C22C22×#12111yescharacteristic
690C22#348511yescharacteristic
691C22#348611yescharacteristic
692C22C22×AGL(1,ℤ/8)12yes—
693C22#358612yes—
694C4—24no—
695C4—24no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—84no—
710C4#348912yes—
711C4#349512yes—
712C2—24no—
713C2—82no—
714C2—84no—
715C2#3563311yescharacteristic
716C2C2×#303811yescharacteristic
717C2#6042711yescharacteristic
718C2#6170612yes—
719C2#6182712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3238C2semidirect
3. C2×#2542C2semidirect
4. #34224C2semidirect
5. #28120C2semidirect
7. #27647C2semidirect
8. #27644C2semidirect
9. #25465C2semidirect
10. #25334C2semidirect
11. #24829C2semidirect
12. #24827C2semidirect
13. #24694C2semidirect
14. #24692C2semidirect
15. #21955C2semidirect
16. #21840C2semidirect
17. #20739C2semidirect
23. C2×#501C22semidirect
24. C2×#489C22semidirect
30. C2×#477C22semidirect
32. C2×#476C22semidirect
49. #2370C22semidirect
50. #2367C22semidirect
51. #2357C22semidirect
52. #2356C22semidirect
53. #2351C22semidirect
54. #2349C22semidirect
66. #2084C22semidirect
67. #2083C22semidirect
68. #2072C22semidirect
69. #2071C22semidirect
74. #1993C22semidirect
75. #1993C22semidirect
76. #1987C22semidirect
77. #1948C22semidirect
78. #1947C22semidirect
79. #1940C22semidirect
80. #1940C22semidirect
81. #1940C22semidirect
91. #1803C22semidirect
92. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32382
C2×#31142
C2×#25422
#342242
#281202
#281142
#276472
#276442
#254652
#253342
#248292
#248272
#246942
#246922
#219552
#218402
#207392