Finite Group Databasebeta

← #6446089

Subgroups — #6446089

Order 512.

1908 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C2×#3080C211yesmaximal, characteristic
3#35279C211yesmaximal, characteristic
4#34217C211yesmaximal, characteristic, complemented
5#25468C214yesmaximal, complemented
6#25467C214yesmaximal, complemented
7#25466C214yesmaximal, complemented
8#25465C214yesmaximal, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23611C211yesmaximal, characteristic
12#23610C211yesmaximal, characteristic
13#21313C211yesmaximal, characteristic, complemented
14#21174C211yesmaximal, characteristic, complemented
15#21173C211yesmaximal, characteristic, complemented
16#21144C211yesmaximal, characteristic, complemented
17#21068C211yesmaximal, characteristic
18#21067C211yesmaximal, characteristic
19#20411C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#501—28no—
24C2×#501C2214yes—
25C2×#501C2214yes—
26#3080—28no—
27C2×#479C2212yes—
28C2×#479C2212yes—
29C2×#478C2212yes—
30C2×#478C2212yes—
31C2×#428C2211yescharacteristic
32C2×#428C2211yescharacteristic
33C22×#107C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C2×#424C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#423C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#417C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C22×#97C2211yescharacteristic
45#2857—22no—
46#2628—22no—
47#2568C2212yes—
48#2567C2212yes—
49#2563C2212yes—
50#2542C2212yes—
51#2487—22no—
52#2409—22no—
53#2370C2214yes—
54#2370C2214yes—
55#2345—22no—
56#2345—22no—
57#2330—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2316C2211yescharacteristic
61#2316C2211yescharacteristic
62#2311C2211yescharacteristic
63#2311C2211yescharacteristic
64C4×#125—22no—
65#2285C2211yescharacteristic
66#2274C2211yescharacteristic
67#2274C2211yescharacteristic
68C4×#117C2211yescharacteristic
69#2085C2214yes—
70#2085C2214yes—
71#2084C2214yes—
72#2083C2214yes—
73#2068C2214yes—
74#2067C2214yes—
75#2066C2214yes—
76#2065C2214yes—
77#1988C2214yes—
78#1987C2214yes—
79#1949C2214yes—
80#1949C2214yes—
81#1948C2214yes—
82#1947C2214yes—
83#1940C2214yes—
84#1940C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1926C2214yes—
88#1925C2214yes—
89#1763C2212yes—
90#1763C2212yes—
91#1762C2212yes—
92#1762C2212yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8—21no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8—24no—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—24no—
116C2×#125D418yes—
117C2×#125C2312yes—
118C2×#125C2312yes—
119C2×C22⋊Q8—24no—
120C4×#41—24no—
121C4×#41—24no—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#501—216no—
144#501—216no—
145#479—28no—
146#479—28no—
147#478—28no—
148#478—28no—
149#428—24no—
150#428—24no—
151C2×#107—24no—
152C2×#107—24no—
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#107C2314yes—
156C2×#107C2314yes—
157C2×#106—24no—
158C2×#106—24no—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2314yes—
162C2×#106C2314yes—
163#424—24no—
164#424—24no—
165#423—24no—
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2314yes—
169C2×C4⋊C8C2314yes—
170#417—24no—
171#417—24no—
172#416—24no—
173#416Q818yescomplemented
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#102C2312yes—
179C2×#102C2312yes—
180C2×#102C2312yes—
181C2×#102C2312yes—
182C2×#97—24no—
183C2×#97C2312yes—
184C2×#97C2312yes—
185#400—24no—
186#399—24no—
187#395—24no—
188#393—24no—
189#391—24no—
190#391C2312yes—
191#391C2312yes—
192#390C2312yes—
193#390C2312yes—
194#389—22no—
195#389—22no—
196#389—24no—
197#389—22no—
198#389—22no—
199#389—24no—
200#389—24no—
201#389C2314yes—
202#389C2314yes—
203#387—24no—
204#387—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384D412yes—
212#384C2312yes—
213#384C2312yes—
214#382D412yes—
215#382C2312yes—
216#382C2312yes—
217#380—24no—
218#380—22no—
219#380—22no—
220#380—22no—
221#380—22no—
222#380D412yes—
223#380D412yes—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—24no—
227C4×C4⋊C4—24no—
228C4×C4⋊C4C2314yes—
229C4×C4⋊C4C2314yes—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C22⋊C4—21no—
235C4×C22⋊C4—21no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238#376—21no—
239C43—21no—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#340C2312yes—
244#340C2314yes—
245#340C2314yes—
246#340C2312yes—
247#340C2312yes—
248#336C2314yes—
249#336C2314yes—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41C2×Q814yes—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—44no—
259C22×Q8—84no—
260C22×Q8C2411yescharacteristic
261C22×D4—21no—
262C22×D4C2411yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2411yescharacteristic
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8C2×Q814yes—
284C22×C8#4114yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295#125—28no—
296#125—28no—
297#125—24no—
298#125—24no—
299#125C2×D418yes—
300#125C2×D418yes—
301C22⋊Q8—88no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—44no—
311C4×D4—44no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—44no—
322#117C2×Q814yes—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
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—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—24no—
379C2×C42—24no—
380C2×C42—44no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D414yes—
384C2×C42#4114yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—28no—
390#107—28no—
391#107—24no—
392#107—24no—
393#106—28no—
394#106—28no—
395#106—24no—
396#106—24no—
397C4⋊C8—28no—
398C4⋊C8—28no—
399C4⋊C8—24no—
400C4⋊C8—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103C2×Q818yescomplemented
405#102—24no—
406#102—24no—
407#102—24no—
408#102C2×Q818yescomplemented
409#97—24no—
410#97—24no—
411#95—22no—
412#95—22no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95C2×D414yes—
422#95C2×D414yes—
423C24—21no—
424C24C22×D411yescharacteristic
425#41—28no—
426#41—28no—
427#41—44no—
428#41—44no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—48no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—84no—
438C2×Q8—816no—
439C2×Q8C22×Q812yes—
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4C22×Q812yes—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4#12114yes—
476C22×C4#12114yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×Q812yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C42−1+412yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491M16—44no—
492M16—44no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8C22×Q814yes—
506C2×C8C2×#4114yes—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4C22×D414yes—
541C4⋊C4C22×D414yes—
542C4⋊C4C22×Q814yes—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—42no—
558C42—42no—
559C42—42no—
560C42—42no—
561C42—82no—
562C42—82no—
563C42—84no—
564C42—84no—
565C42#12114yes—
566C42#12114yes—
567C42C22×D412yes—
568C42C22×D412yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23C2×#12112yes—
579C23#54712yes—
580C23D4×Q812yes—
581C23C23×D411yescharacteristic
582Q8—22no—
583Q8—22no—
584Q8—41no—
585Q8—41no—
586Q8—88no—
587Q8—88no—
588D4—22no—
589D4—22no—
590D4—48no—
591D4—41no—
592D4—41no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4C2×#12112yes—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4#53314yes—
639C2×C4#54114yes—
640C2×C4#54712yes—
641C2×C4#54814yes—
642C2×C4D4×Q812yes—
643C2×C4D4×Q814yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C23×Q811yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C8—24no—
651C8—24no—
652C8—42no—
653C8—42no—
654C22—22no—
655C22—22no—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—44no—
660C22—81no—
661C22—81no—
662C22—82no—
663C22#242914yes—
664C22C22×#12111yescharacteristic
665C22C2×#54711yescharacteristic
666C22C2×D4×Q811yescharacteristic
667C22#358512yes—
668C22#358612yes—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—81no—
680C4—82no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4#242914yes—
685C4C2×#53311yescharacteristic
686C4C2×#54111yescharacteristic
687C4C2×#54811yescharacteristic
688C4C2×D4×Q811yescharacteristic
689C2—24no—
690C2—82no—
691C2C2×#242911yescharacteristic
692C2#6042611yescharacteristic
693C2#6042711yescharacteristic
694C2#6129412yes—
695C2#6138012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
4. #34217C2semidirect
5. #25468C2semidirect
6. #25467C2semidirect
7. #25466C2semidirect
8. #25465C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
13. #21313C2semidirect
14. #21174C2semidirect
15. #21173C2semidirect
16. #21144C2semidirect
173. #416Q8semidirect
404. #103C2×Q8semidirect
408. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#30802
#352792
#342172
#254682
#254672
#254662
#254652
#236612
#236602
#236112
#236102
#213132
#211742
#211732
#211442
#210682
#210672
#204112