Finite Group Databasebeta

← #6498345

Subgroups — #6498345

Order 512.

1865 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic, complemented
2C2×#2970C211yesmaximal, characteristic
3#35279C211yesmaximal, characteristic
4#35138C211yesmaximal, characteristic
5#33059C212yesmaximal, complemented
6#33053C212yesmaximal, complemented
7#29852C212yesmaximal, complemented
8#29852C212yesmaximal, complemented
9#28652C214yesmaximal, complemented
10#28611C214yesmaximal, complemented
11#28108C212yesmaximal, complemented
12#28105C212yesmaximal, complemented
13#28098C212yesmaximal, complemented
14#28095C212yesmaximal, complemented
15#21838C211yesmaximal, characteristic, complemented
16#21811C211yesmaximal, characteristic, complemented
17#20823C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#491—28no—
22C2×#491C2212yes—
23C2×#491C2212yes—
24C2×#491C2214yes—
25#2970—28no—
26C2×#446C2212yes—
27C2×#435C2212yes—
28C2×#435C2212yes—
29C2×C8×D4C2212yes—
30C2×#432C2211yescharacteristic
31C2×#432C2211yescharacteristic
32C2×#432C2211yescharacteristic
33C22×#103C2212yes—
34C22×#103C2212yes—
35C2×C4×M16C2211yescharacteristic
36C22×#97C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2857C2212yes—
39#2857C2212yes—
40#2851C2212yes—
41#2851C2212yes—
42#2634—21no—
43#2633—21no—
44#2628—21no—
45#2628—21no—
46#2611—21no—
47#2611—21no—
48#2584C2212yes—
49#2568C2212yes—
50#2567—21no—
51#2563—21no—
52#2562C2212yes—
53#2542C2212yes—
54#2487—21no—
55#2487—21no—
56#2487C2211yescharacteristic
57#2487C2211yescharacteristic
58#2486C2211yescharacteristic
59#2486C2211yescharacteristic
60#2484—21no—
61#2409C2212yes—
62#2409C2212yes—
63#2406—21no—
64#2322—21no—
65#2322—21no—
66#2317—21no—
67#2311—21no—
68#2285C2211yescharacteristic
69#2283C2211yescharacteristic
70#2068C2214yes—
71#2068C2214yes—
72#2066C2214yes—
73#2066C2214yes—
74#2036C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2035C2214yes—
78#1992C2212yes—
79#1992C2212yes—
80#1989C2214yes—
81#1981C2214yes—
82#1960C2212yes—
83#1960C2212yes—
84#1935C2214yes—
85#1935C2214yes—
86#1933C2214yes—
87#1933C2214yes—
88#1894C2214yes—
89#1894C2214yes—
90#1892C2214yes—
91#1892C2214yes—
92#1847C2212yes—
93#1836C2212yes—
94#1824C2212yes—
95#1823C2212yes—
96#1808C2212yes—
97#1807C2212yes—
98#1787C2212yes—
99#1787C2212yes—
100#1776C2214yes—
101C22×#41D411yescharacteristic
102C23×Q8C2312yes—
103C22×M16C2312yes—
104C23×C8C2312yes—
105C23×C8C2312yes—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—24no—
115C2×#125—24no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×#123—21no—
119C2×#123—21no—
120C2×C22⋊Q8C2314yes—
121C2×C22⋊Q8C2314yes—
122C4×#41—24no—
123C4×#41—24no—
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117—21no—
133C2×#117D411yescharacteristic
134C2×#117D411yescharacteristic
135C2×#117D411yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#491—28no—
146#491—28no—
147#491—216no—
148#446—28no—
149#435—28no—
150#435—28no—
151C8×D4—28no—
152#432—24no—
153#432—24no—
154#432—24no—
155C2×#107C2314yes—
156C2×#106C2314yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×#103—28no—
161C2×#103—28no—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2312yes—
172C2×C22⋊C8C2312yes—
173C4×M16—24no—
174C2×#97—24no—
175C2×#97D418yes—
176C2×#97C2311yescharacteristic
177C2×#97C2312yes—
178C2×#97C2311yescharacteristic
179C2×#97C2311yescharacteristic
180C2×#97C2311yescharacteristic
181C2×#97C2312yes—
182C2×C4×C8C2312yes—
183#400—22no—
184#400—22no—
185#400—22no—
186#400—21no—
187#400—21no—
188#399—21no—
189#399—21no—
190#399—22no—
191#399—22no—
192#398—22no—
193#398—21no—
194#398—21no—
195#397—21no—
196#397—21no—
197#395—22no—
198#395—22no—
199#395—22no—
200#394—22no—
201#393—22no—
202#393—22no—
203#391—22no—
204#391C2314yes—
205#391C2314yes—
206#390—22no—
207#389—22no—
208#389—22no—
209#389—22no—
210#389—22no—
211#389—22no—
212#389—22no—
213#389C2312yes—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#388—22no—
218#388—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#386—22no—
224#386—22no—
225#386—21no—
226#386—21no—
227#385—22no—
228#385—22no—
229#385—22no—
230#385—22no—
231#385—22no—
232#385—22no—
233#385—21no—
234#385—21no—
235#384—22no—
236#384C2312yes—
237#383—21no—
238#382—22no—
239#382—21no—
240#382C2312yes—
241#381—21no—
242#381—21no—
243#381—21no—
244#380—22no—
245#380—22no—
246#380—22no—
247#380—22no—
248#380—21no—
249#380C2312yes—
250#380C2312yes—
251#380C2312yes—
252#380C2312yes—
253#380C2311yescharacteristic
254#380C2312yes—
255#380C2311yescharacteristic
256C4×C4⋊C4—21no—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2312yes—
259C4×C22⋊C4—21no—
260#376—21no—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267C2×#41—24no—
268C2×#41—24no—
269C2×#41—42no—
270C22×Q8—22no—
271C22×Q8—24no—
272C22×Q8—24no—
273C22×Q8—44no—
274C22×Q8—44no—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2414yes—
277C22×D4C2×D411yescharacteristic
278C22×D4C2×D411yescharacteristic
279C22×D4C2×D411yescharacteristic
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C23×C4C2411yescharacteristic
291C23×C4C2411yescharacteristic
292C2×M16—24no—
293C2×M16—44no—
294C22×C8—24no—
295C22×C8—44no—
296C22×C8—44no—
297C22×C8C2×D418yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C4⋊Q8—24no—
303C4⋊Q8—24no—
304C4⋊Q8—24no—
305C4⋊Q8—24no—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308#125—24no—
309#125—24no—
310#125—24no—
311#125—24no—
312#125—44no—
313#125—44no—
314#123—44no—
315#123—44no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319C22⋊Q8—44no—
320C4×Q8—24no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—24no—
330C4×D4—22no—
331C4×D4—22no—
332#117—24no—
333#117—24no—
334#117—24no—
335#117—44no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4—44no—
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C22⋊C4C2×D411yescharacteristic
387C2×C22⋊C4C2×D411yescharacteristic
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D414yes—
401C2×C42#4114yes—
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107—44no—
407#106—44no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411#103—28no—
412#103—28no—
413#103—28no—
414#103—28no—
415#103—44no—
416#103—44no—
417#103—44no—
418#103—44no—
419C22⋊C8—24no—
420C22⋊C8—24no—
421C22⋊C8—24no—
422#97—24no—
423#97—22no—
424#97—22no—
425#97C2×D414yes—
426#97C2×D414yes—
427#97C2×D418yes—
428C4×C8—24no—
429#95—21no—
430#95—21no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—24no—
437#95—24no—
438#95—24no—
439#95—24no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448C24C22×D411yescharacteristic
449C24C22×D411yescharacteristic
450C24C22×D411yescharacteristic
451#41—416no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—24no—
460C2×Q8—41no—
461C2×Q8—48no—
462C2×Q8—48no—
463C2×Q8—44no—
464C2×Q8—84no—
465C2×Q8—88no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—41no—
474C2×D4#13714yes—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—42no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—44no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—82no—
507C22×C4—84no—
508C22×C4#12114yes—
509C22×C4#12114yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D412yes—
521C22×C4C22×D411yescharacteristic
522C22×C4C2×#4112yes—
523C22×C4C2511yescharacteristic
524M16—48no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8—48no—
532C2×C8—48no—
533C2×C8C22×D414yes—
534C2×C8C22×D414yes—
535C4⋊C4—22no—
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—22no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—84no—
566C4⋊C4—84no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—44no—
574C22⋊C4—42no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—42no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—84no—
581C22⋊C4—84no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—21no—
587C42—21no—
588C42—21no—
589C42—21no—
590C42—44no—
591C42—82no—
592C42#12114yes—
593C42#12114yes—
594C42C22×D412yes—
595C42C22×D412yes—
596C42C2×#4112yes—
597C42C2×#4112yes—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—21no—
602C23—22no—
603C23—21no—
604C23—22no—
605C23—21no—
606C23—22no—
607C23—21no—
608C23—21no—
609C23C2×#12112yes—
610C23C2×#13712yes—
611C23C2×#13712yes—
612C23#57512yes—
613C23C23×D411yescharacteristic
614Q8—44no—
615Q8—44no—
616Q8—44no—
617Q8—44no—
618Q8—44no—
619Q8—88no—
620D4—28no—
621D4—44no—
622D4—44no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—28no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—84no—
674C2×C4—88no—
675C2×C4—84no—
676C2×C4C2×#12112yes—
677C2×C4C2×#12112yes—
678C2×C4C2×#12112yes—
679C2×C4C2×#12112yes—
680C2×C4C2×#12112yes—
681C2×C4C2×#13712yes—
682C2×C4C23×D411yescharacteristic
683C2×C4C22×#4111yescharacteristic
684C2×C4#58514yes—
685C8—24no—
686C8—24no—
687C8—42no—
688C8—42no—
689C22—22no—
690C22—24no—
691C22—24no—
692C22—22no—
693C22—22no—
694C22—24no—
695C22—41no—
696C22—42no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—42no—
701C22—42no—
702C22—41no—
703C22#305614yes—
704C22C22×#12111yescharacteristic
705C22#350012yes—
706C22C22×#13711yescharacteristic
707C22C2×#57511yescharacteristic
708C22#358612yes—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—84no—
725C4—84no—
726C4#344912yes—
727C4#351012yes—
728C2—24no—
729C2—42no—
730C2—81no—
731C2#3532711yescharacteristic
732C2C2×#305611yescharacteristic
733C2#6045511yescharacteristic
734C2#6257712yes—
735C2#6257912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#491C2semidirect
5. #33059C2semidirect
6. #33053C2semidirect
7. #29852C2semidirect
8. #29852C2semidirect
9. #28652C2semidirect
10. #28611C2semidirect
11. #28108C2semidirect
12. #28105C2semidirect
13. #28098C2semidirect
14. #28095C2semidirect
15. #21838C2semidirect
16. #21811C2semidirect
17. #20823C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4912
C2×#29702
#352792
#351382
#330592
#330532
#298522
#286522
#286112
#281082
#281052
#280982
#280952
#218382
#218112
#208232