Finite Group Databasebeta

← #6482135

Subgroups — #6482135

Order 512.

1853 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3238C211yesmaximal, characteristic
2C2×#3229C211yesmaximal, characteristic, complemented
3#35241C211yesmaximal, characteristic
4C2×#2369C211yesmaximal, characteristic, complemented
5#33111C212yesmaximal
6#33108C212yesmaximal, complemented
7#29760C212yesmaximal
8#29756C212yesmaximal, complemented
9#25464C214yesmaximal, complemented
10#25454C212yesmaximal, complemented
11#25451C212yesmaximal, complemented
12#25287C214yesmaximal, complemented
13#25278C212yesmaximal, complemented
14#25273C212yesmaximal, complemented
15#21953C211yesmaximal, characteristic, complemented
16#21394C211yesmaximal, characteristic, complemented
17#21388C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#529C2211yescharacteristic
21#3238—28no—
22#3229—28no—
23C2×#501C2212yes—
24C2×#498C2212yes—
25C2×#489C2212yes—
26C2×#487C2212yes—
27C2×#483C2212yes—
28C2×#483C2212yes—
29C2×#483C2212yes—
30C2×#480C2212yes—
31C2×#477C2212yes—
32C2×#475C2214yes—
33C2×#476C2212yes—
34C2×#432C2211yescharacteristic
35C2×#431C2211yescharacteristic
36#2857C2212yes—
37#2850C2212yes—
38#2664—22no—
39#2568C2211yescharacteristic
40#2565C2212yes—
41#2547C2211yescharacteristic
42#2546C2211yescharacteristic
43#2545C2212yes—
44#2542C2211yescharacteristic
45#2498—22no—
46#2417—21no—
47#2417—21no—
48#2369—21no—
49#2369—21no—
50#2369—22no—
51#2369C2212yes—
52#2369C2212yes—
53#2369C2214yes—
54#2323—21no—
55#2323—21no—
56C2×#391—22no—
57C2×#389—21no—
58C2×#389C2212yes—
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2212yes—
61C2×C4×C4⋊C4C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63#2085C2214yes—
64#2084C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#2043C2214yes—
70#2042C2214yes—
71#1989C2212yes—
72#1988C2212yes—
73#1983C2214yes—
74#1983C2212yes—
75#1981C2214yes—
76#1981C2212yes—
77#1949C2214yes—
78#1948C2214yes—
79#1940C2214yes—
80#1939C2212yes—
81#1939C2212yes—
82#1928C2214yes—
83#1927C2214yes—
84#1927C2214yes—
85#1926C2214yes—
86#1874C2211yescharacteristic
87#1874C2211yescharacteristic
88#1872C2211yescharacteristic
89#1872C2212yes—
90#1872C2211yescharacteristic
91#1872C2212yes—
92#1797C2212yes—
93#1797C2212yes—
94#1763C2214yes—
95C4×#103C2212yes—
96C4×#102C2212yes—
97C23×Q8—21no—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8D418yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109#529—24no—
110#529—24no—
111C2×#126—21no—
112C2×#125—21no—
113C2×#125C2311yescharacteristic
114C2×#124C2311yescharacteristic
115C2×#124C2312yes—
116C2×#123—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2311yescharacteristic
122C2×#121C2311yescharacteristic
123C2×C4×Q8—21no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C42—21no—
138C22×C42—22no—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141#501—28no—
142#498—28no—
143#489—28no—
144#487—28no—
145#483—28no—
146#483—28no—
147#483—28no—
148#480—28no—
149#477—28no—
150#476—28no—
151#475—216no—
152#432—24no—
153#431—24no—
154C2×#107C2314yes—
155C2×#106C2314yes—
156C2×#106C2314yes—
157C2×#106C2314yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#102C2314yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×C22⋊C8C2312yes—
173C2×C22⋊C8C2311yescharacteristic
174C2×C22⋊C8C2311yescharacteristic
175C2×#97C2312yes—
176C2×C4×C8C2312yes—
177#400—22no—
178#399—22no—
179#398—22no—
180#398—22no—
181#398—22no—
182#397—22no—
183#396—22no—
184#396—22no—
185#391—21no—
186#391—21no—
187#391—24no—
188#391—24no—
189#391—24no—
190#391—24no—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#390C2312yes—
195#389—21no—
196#389—21no—
197#389—22no—
198#389—22no—
199#389—24no—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389C2314yes—
205#389C2314yes—
206#388—22no—
207#388—22no—
208#388—22no—
209#388—22no—
210#387—22no—
211#387—22no—
212#386—22no—
213#386—22no—
214#384C2312yes—
215#382C2312yes—
216#380—22no—
217#380—22no—
218#380—21no—
219#380—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—24no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4D412yes—
230C4×C4⋊C4C2314yes—
231C4×C4⋊C4C2314yes—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C4⋊C4C2311yescharacteristic
236C4×C4⋊C4C2311yescharacteristic
237C2×#95—22no—
238C2×#95—22no—
239C2×#95—21no—
240C2×#95D412yes—
241#340C2314yes—
242#340C2312yes—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2314yes—
249C22×Q8—28no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—21no—
253C22×Q8—82no—
254C22×Q8—82no—
255C22×Q8C2411yescharacteristic
256C22×Q8C2412yes—
257C22×D4—21no—
258C22×D4C2411yescharacteristic
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—44no—
280C4⋊Q8C2×D418yes—
281C4⋊Q8C2×D418yes—
282#126—44no—
283#125—22no—
284#125—22no—
285#125—44no—
286#124—22no—
287#124—22no—
288#124—44no—
289#123—44no—
290C22⋊Q8—22no—
291C22⋊Q8—24no—
292C22⋊Q8—22no—
293C22⋊Q8—24no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297#121—22no—
298#121—22no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—22no—
302C4×Q8—24no—
303C4×Q8—44no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—44no—
307#117—44no—
308#117—44no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D418yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—44no—
366C2×C42—44no—
367C2×C42—84no—
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D414yes—
373C2×C42#4114yes—
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—28no—
379#106—28no—
380#106—28no—
381#106—28no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384C4⋊C8—44no—
385C4⋊C8—44no—
386#103—28no—
387#103—28no—
388#103—28no—
389#103—24no—
390#103—28no—
391#103—42no—
392#103—44no—
393#102—28no—
394#102—24no—
395#102—42no—
396C22⋊C8—22no—
397C22⋊C8—22no—
398C22⋊C8—24no—
399#97—24no—
400C4×C8—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418#95C2×D412yes—
419C24—21no—
420C24C22×D411yescharacteristic
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—816no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×Q8—82no—
438C2×Q8—82no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4#12114yes—
479C22×C4#12114yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4112yes—
491C22×C4C2511yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4#13714yes—
537C4⋊C4C22×D418yes—
538C4⋊C42−1+414yes—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C42—24no—
550C42—24no—
551C42—42no—
552C42—42no—
553C42—44no—
554C42—42no—
555C42—44no—
556C42—44no—
557C42—84no—
558C42—84no—
559C42#12114yes—
560C42#12114yes—
561C42C22×D412yes—
562C42C22×D412yes—
563C42C2×#4112yes—
564C42C2×#4112yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23C2×#12112yes—
577C23#57712yes—
578C23#57712yes—
579C23C23×D411yescharacteristic
580Q8—42no—
581Q8—44no—
582Q8—42no—
583Q8—44no—
584Q8—88no—
585Q8—88no—
586Q8—84no—
587D4—42no—
588D4—42no—
589D4—84no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—28no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—416no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4C2×#12112yes—
644C2×C4C2×#12112yes—
645C2×C4C2×#12112yes—
646C2×C4C2×#12112yes—
647C2×C4C2×#12112yes—
648C2×C4C2×#13714yes—
649C2×C4C2×#13712yes—
650C2×C4#57512yes—
651C2×C4C23×D411yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×2−1+412yes—
654C2×C4#58512yes—
655C8—42no—
656C8—42no—
657C8—44no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22#304214yes—
672C22C22×#12111yescharacteristic
673C22#348511yescharacteristic
674C22#348611yescharacteristic
675C22C22×#13711yescharacteristic
676C22C2×#57511yescharacteristic
677C22C2×#57711yescharacteristic
678C22C2×#57711yescharacteristic
679C4—22no—
680C4—22no—
681C4—24no—
682C4—22no—
683C4—22no—
684C4—48no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—44no—
689C4—81no—
690C4—82no—
691C4—81no—
692C4—82no—
693C4—81no—
694C4—82no—
695C4#303714yes—
696C4#346112yes—
697C4#349712yes—
698C2—24no—
699C2—161no—
700C2#3515311yescharacteristic
701C2C2×#303711yescharacteristic
702C2C2×#304211yescharacteristic
703C2#6173111yescharacteristic
704C2#6173511yescharacteristic
705C2#6183811yescharacteristic
706C2#6184111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#3229C2semidirect
4. C2×#2369C2semidirect
6. #33108C2semidirect
8. #29756C2semidirect
9. #25464C2semidirect
10. #25454C2semidirect
11. #25451C2semidirect
12. #25287C2semidirect
13. #25278C2semidirect
14. #25273C2semidirect
15. #21953C2semidirect
16. #21394C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32382
C2×#32292
#352412
C2×#23692
#331112
#331082
#297602
#297562
#254642
#254542
#254512
#252872
#252782
#252732
#219532
#213942
#213882