Finite Group Databasebeta

← #6445651

Subgroups — #6445651

Order 512.

2176 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic, complemented
2C2×#2906C211yesmaximal, characteristic
3#35280C211yesmaximal, characteristic
4C4×#536C211yesmaximal, characteristic
5#24854C214yesmaximal, complemented
6#24850C214yesmaximal, complemented
7#24846C212yesmaximal, complemented
8#24845C212yesmaximal, complemented
9#24838C212yesmaximal, complemented
10#24837C212yesmaximal, complemented
11#23671C212yesmaximal, complemented
12#23615C212yesmaximal, complemented
13#21311C211yesmaximal, characteristic, complemented
14#21171C211yesmaximal, characteristic, complemented
15#20974C212yesmaximal
16#20664C212yesmaximal
17C2×#1803C211yesmaximal, characteristic, complemented
18C22×#131C2212yes—
19C2×#536C2211yescharacteristic
20C2×#533C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#3078—28no—
24C2×#479C2214yes—
25C2×#477C2212yes—
26C2×#475C2212yes—
27C2×#428C2212yes—
28C2×#428C2212yes—
29#2906—28no—
30C2×#424C2214yes—
31C2×#424C2212yes—
32C2×#423C2212yes—
33C2×#423C2211yescharacteristic
34C22×C4⋊C8C2212yes—
35C22×C4⋊C8C2211yescharacteristic
36C2×#419C2212yes—
37C22×#103C2212yes—
38#2857C2212yes—
39#2857C2212yes—
40#2629C2214yes—
41#2582—21no—
42#2582—21no—
43#2538—21no—
44#2538—21no—
45#2417—22no—
46#2417—22no—
47#2327—21no—
48#2326—21no—
49#2323C2212yes—
50#2323C2212yes—
51#2322—22no—
52#2317C2211yescharacteristic
53#2316C2212yes—
54#2311C2211yescharacteristic
55#2311C2212yes—
56#2310C2211yescharacteristic
57C4×C4⋊Q8—21no—
58C4×#124—21no—
59C4×C22⋊Q8C2212yes—
60C4×C22⋊Q8C2212yes—
61C4×#123—22no—
62C2×#384C2211yescharacteristic
63C4×#117C2211yescharacteristic
64#2088C2214yes—
65#2088C2214yes—
66#2087C2214yes—
67#2086C2214yes—
68#2068C2214yes—
69#2068C2214yes—
70#2066C2214yes—
71#2066C2214yes—
72#1989C2212yes—
73#1989C2212yes—
74#1988C2211yescharacteristic
75#1988C2211yescharacteristic
76#1987C2211yescharacteristic
77#1987C2211yescharacteristic
78#1948C2212yes—
79#1948C2212yes—
80#1947C2212yes—
81#1947C2212yes—
82#1894C2214yes—
83#1894C2214yes—
84#1892C2214yes—
85#1892C2214yes—
86#1809C2214yes—
87#1809C2214yes—
88#1808C2214yes—
89#1807C2214yes—
90#1803C2212yes—
91#1803C2212yes—
92#1803C2214yes—
93#1736C2212yes—
94#1735C2212yes—
95#1735C2212yes—
96C2×#340C2212yes—
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#131—28no—
100C2×#131—28no—
101C2×#131—44no—
102C22×M16C2314yes—
103C22×M16C2312yes—
104C23×C8C2312yes—
105C23×C8C2314yes—
106#536—24no—
107#536Q818yescomplemented
108#533—24no—
109#533—24no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#125C2312yes—
114C2×#124—21no—
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C4×#41—24no—
123C4×#41—24no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—24no—
129C2×C4×Q8—24no—
130C2×C4×Q8D411yescharacteristic
131C2×C4×Q8D411yescharacteristic
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×D4D412yes—
135C2×C4×D4D411yescharacteristic
136C2×C4×D4D412yes—
137C2×C4×D4D411yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2312yes—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2312yes—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2311yescharacteristic
148C22×C42—21no—
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151#479—216no—
152#477—28no—
153#475—28no—
154#428—28no—
155#428—44no—
156C2×#107C2314yes—
157C2×#107C2314yes—
158C2×#106C2314yes—
159C2×#106C2314yes—
160#424—28no—
161#424—216no—
162#423—28no—
163#423D418yes—
164C2×C4⋊C8—24no—
165C2×C4⋊C8—28no—
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2314yes—
170C2×C4⋊C8C2314yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173#419—28no—
174C2×#103—44no—
175C2×#103C2314yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179#398—24no—
180#398—24no—
181#396—24no—
182#396—24no—
183#391—21no—
184#391—22no—
185#391—21no—
186#391—21no—
187#391—22no—
188#391—22no—
189#391C2312yes—
190#391C2312yes—
191#390—21no—
192#390—22no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389C2314yes—
200#389C2314yes—
201#389C2312yes—
202#389C2312yes—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#386—22no—
212#386—24no—
213#385—22no—
214#385—22no—
215#385—22no—
216#384—21no—
217#384—21no—
218#384—21no—
219#384C2312yes—
220#384C2312yes—
221#382—21no—
222#381—22no—
223#380—22no—
224#380—22no—
225#380—24no—
226#380C2312yes—
227#380C2314yes—
228#380C2312yes—
229#380C2314yes—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235C4×C4⋊C4C2311yescharacteristic
236C4×C22⋊C4—21no—
237C4×C22⋊C4—21no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240#376—21no—
241C43—21no—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41C2×Q814yes—
259C2×#41#4114yes—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8C2×D411yescharacteristic
269C22×Q8C2×Q814yes—
270C22×Q8C2412yes—
271C22×D4C2×D412yes—
272C22×D4C2×D411yescharacteristic
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—21no—
276C23×C4C2×D412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D412yes—
279C23×C4C2×D411yescharacteristic
280C23×C4C2412yes—
281C23×C4C2411yescharacteristic
282#131—432no—
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—28no—
286C2×M16—28no—
287C2×M16—44no—
288C2×M16C2×D418yes—
289C22×C8—24no—
290C22×C8—28no—
291C22×C8—24no—
292C22×C8—24no—
293C22×C8—28no—
294C22×C8—44no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2414yes—
298C4⋊Q8—28no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303#125—24no—
304#125—24no—
305#124—28no—
306#123—216no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C22⋊Q8—84no—
312C22⋊Q8—84no—
313C22⋊Q8C2×Q818yes—
314C22⋊Q8C2×Q818yes—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×D4—28no—
326C4×D4—28no—
327C4×D4—24no—
328C4×D4—24no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—24no—
333#117—24no—
334#117—24no—
335#117—24no—
336#117—22no—
337#117—22no—
338#117C2×Q814yes—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—44no—
362C2×C4⋊C4—82no—
363C2×C4⋊C4—82no—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2×D412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2411yescharacteristic
375C2×C4⋊C4C2411yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C22⋊C4C2×D412yes—
388C2×C42—21no—
389C2×C42—24no—
390C2×C42—24no—
391C2×C42—22no—
392C2×C42—21no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—44no—
400C2×C42C2×D412yes—
401C2×C42C2×D412yes—
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D414yes—
405C2×C42#4114yes—
406C2×C42C2412yes—
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409#107—28no—
410#107—44no—
411#106—28no—
412#106—44no—
413C4⋊C8—28no—
414C4⋊C8—28no—
415C4⋊C8—22no—
416C4⋊C8—22no—
417C4⋊C8—24no—
418C4⋊C8C2×D418yes—
419C4⋊C8C2×D418yes—
420#103—28no—
421#103—28no—
422#103—44no—
423#103—44no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—22no—
429#95—22no—
430#95—22no—
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—
440C24C22×D412yes—
441C24C22×D411yescharacteristic
442#41—28no—
443#41—28no—
444#41—28no—
445#41—28no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—28no—
451C2×Q8—28no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8C22⋊Q814yes—
467C2×Q8C22×Q814yes—
468C2×D4—24no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4C22⋊Q818yes—
473C2×D4C22⋊Q814yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—44no—
502C22×C4—81no—
503C22×C4—81no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4#12114yes—
507C22×C4#12114yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×Q814yes—
518C22×C4C22×Q812yes—
519C22×C4C2×#4114yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C4H(5,𝔽2)12yes—
523C22×C42−1+412yes—
524C22×C4C2511yescharacteristic
525M16—216no—
526M16—416no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—416no—
536C2×C8—48no—
537C2×C8C22×D418yes—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—28no—
543C4⋊C4—28no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—24no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—24no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—44no—
559C4⋊C4—42no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4—82no—
572C4⋊C4—84no—
573C4⋊C4—84no—
574C4⋊C4C22×Q812yes—
575C4⋊C4C22×Q812yes—
576C22⋊C4—24no—
577C22⋊C4—24no—
578C22⋊C4—28no—
579C22⋊C4—28no—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—44no—
583C22⋊C4—42no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C22⋊C4—84no—
587C22⋊C4—84no—
588C42—24no—
589C42—24no—
590C42—24no—
591C42—22no—
592C42—22no—
593C42—44no—
594C42—44no—
595C42—44no—
596C42—42no—
597C42—42no—
598C42—42no—
599C42—81no—
600C42—81no—
601C42#12114yes—
602C42#12114yes—
603C42C22×D412yes—
604C42C22×D412yes—
605C42C2×#4112yes—
606C42C2×#4112yes—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—22no—
612C23—22no—
613C23—22no—
614C23—22no—
615C23—22no—
616C23—22no—
617C23C2×#12112yes—
618C23C2×C22⋊Q814yes—
619C23C2×C22⋊Q812yes—
620C23#53712yes—
621C23C23×D411yescharacteristic
622Q8—28no—
623Q8—24no—
624Q8—24no—
625Q8—48no—
626Q8—42no—
627Q8—42no—
628Q8—84no—
629Q8—84no—
630D4—216no—
631D4—28no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—28no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—48no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—48no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—44no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—164no—
684C2×C4C2×#12112yes—
685C2×C4C2×#12112yes—
686C2×C4C2×#12112yes—
687C2×C4C2×#12112yes—
688C2×C4C2×#12112yes—
689C2×C4C2×C22⋊Q814yes—
690C2×C4C2×C22⋊Q812yes—
691C2×C4#53712yes—
692C2×C4#54614yes—
693C2×C4D4×Q814yes—
694C2×C4#55214yes—
695C2×C4#55414yes—
696C2×C4#57712yes—
697C2×C4#57812yes—
698C2×C4C23×D411yescharacteristic
699C2×C4C23×Q811yescharacteristic
700C2×C4C22×#4111yescharacteristic
701C2×C4C22×#4111yescharacteristic
702C2×C4C2×H(5,𝔽2)11yescharacteristic
703C2×C4C2×2−1+411yescharacteristic
704C8—28no—
705C8—42no—
706C8—42no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22#241714yes—
721C22C22×#12111yescharacteristic
722C22C22×C22⋊Q811yescharacteristic
723C22C2×#53711yescharacteristic
724C22C2×#57711yescharacteristic
725C22C2×#57811yescharacteristic
726C22#358511yescharacteristic
727C22#358611yescharacteristic
728C4—22no—
729C4—22no—
730C4—22no—
731C4—24no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—81no—
741C4—84no—
742C4—81no—
743C4#241714yes—
744C4#306314yes—
745C4C2×#54611yescharacteristic
746C4C2×D4×Q811yescharacteristic
747C4C2×#55211yescharacteristic
748C4C2×#55411yescharacteristic
749C2—24no—
750C2—44no—
751C2C2×#241711yescharacteristic
752C2C2×#306311yescharacteristic
753C2#6045611yescharacteristic
754C2#6049711yescharacteristic
755C2#6049811yescharacteristic
756C2#6092411yescharacteristic
757C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. C2×#3078C2semidirect
5. #24854C2semidirect
6. #24850C2semidirect
7. #24846C2semidirect
8. #24845C2semidirect
9. #24838C2semidirect
10. #24837C2semidirect
11. #23671C2semidirect
12. #23615C2semidirect
13. #21311C2semidirect
14. #21171C2semidirect
17. C2×#1803C2semidirect
107. #536Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29062
#352802
C4×#5362
#248542
#248502
#248462
#248452
#248382
#248372
#236712
#236152
#213112
#211712
#209742
#206642
C2×#18032