Finite Group Databasebeta

← #4380483

Subgroups — #4380483

Order 512.

3514 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2817C211yesmaximal, characteristic
3C2×#2583C211yesmaximal, characteristic
4C2×#2621C211yesmaximal, characteristic
5#16262C214yesmaximal
6#15881C212yesmaximal, complemented
7#15849C212yesmaximal, complemented
8#15847C214yesmaximal, complemented
9#15783C212yesmaximal, complemented
10#15762C212yesmaximal, complemented
11#15318C214yesmaximal, complemented
12#14892C214yesmaximal
13#14292C211yesmaximal, characteristic
14#14138C211yesmaximal, characteristic
15#13870C211yesmaximal, characteristic
16C2×#544—28no—
17C2×#544—28no—
18C2×#544—28no—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125C2212yes—
23C22×#125C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×#117C2212yes—
29C22×#117C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31#2817—28no—
32#2621—28no—
33#2583—28no—
34C2×#400C2212yes—
35C2×#400C2212yes—
36C2×#398C2212yes—
37C2×#398C2214yes—
38C2×#397C2214yes—
39C2×#397C2214yes—
40C2×#395C2212yes—
41C2×#395C2212yes—
42C2×#395C2212yes—
43C2×#395C2214yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2212yes—
47C2×#389C2214yes—
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#382C2211yescharacteristic
51C2×#381C2211yescharacteristic
52C2×#381C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54#1446C2214yes—
55#1446C2212yes—
56#1445C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1444C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1436C2214yes—
71#1436C2212yes—
72#1436C2212yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2212yes—
77#1436C2212yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C22⋊#95C2212yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88#544—2128no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—44no—
97C2×#125—28no—
98C2×#125—28no—
99C2×#125—28no—
100C2×#125—28no—
101C2×#125—28no—
102C2×#125—28no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—28no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8D418yes—
114C2×C22⋊Q8D418yes—
115C2×#117—28no—
116C2×#117—24no—
117C2×#117—28no—
118C2×#117—28no—
119C2×#117—44no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4D418yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2314yes—
140C22×C4⋊C4C2314yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2312yes—
147C22×C42C2312yes—
148C22×C42C2312yes—
149C22×C42C2311yescharacteristic
150C22×C42C2311yescharacteristic
151C22×C42C2312yes—
152#400—28no—
153#400—28no—
154#398—28no—
155#398—216no—
156#397—216no—
157#397—216no—
158#395—28no—
159#395—28no—
160#395—28no—
161#395—48no—
162#391—24no—
163#391—24no—
164#391—44no—
165#389—216no—
166#385—28no—
167#385—28no—
168#382—24no—
169#381—24no—
170#381—44no—
171C4×C22⋊C4—24no—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C25C2411yescharacteristic
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—42no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8C2×D414yes—
215C22×Q8C2×D414yes—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—28no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C4⋊Q8—264no—
245C4⋊Q8—264no—
246C4⋊Q8—416no—
247#125—264no—
248#125—264no—
249C22⋊Q8—264no—
250C22⋊Q8—232no—
251C22⋊Q8—232no—
252C22⋊Q8—416no—
253#117—264no—
254#117—416no—
255C2×C4⋊C4—232no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D418yes—
305C2×C4⋊C4C2×D418yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D418yes—
309C2×C4⋊C4C2×D418yes—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4C2×D418yes—
326C2×C22⋊C4C2×D418yes—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C22×D412yes—
370C24C2511yescharacteristic
371C2×Q8—216no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—82no—
381C2×Q8—82no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×Q8—84no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×Q8—84no—
389C2×Q8—88no—
390C2×Q8—88no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—216no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—84no—
463C22×C4#120116yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+418yes—
478C22×C42−1+418yes—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—48no—
491C4⋊C4—416no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C42—216no—
504C42—216no—
505C42—28no—
506C42—44no—
507C42—84no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C2×#12014yes—
523C23C23×D411yescharacteristic
524C23C23×D411yescharacteristic
525C23C23×D411yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23C2×2−1+414yes—
528C23C2×2−1+414yes—
529C23#58512yes—
530Q8—88no—
531Q8—88no—
532Q8—88no—
533Q8—816no—
534C2×C4—232no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—416no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—88no—
608C2×C4C2×#12014yes—
609C2×C4C2×#12014yes—
610C2×C4#53618yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—44no—
615C22—44no—
616C22—48no—
617C22—48no—
618C22C22×#12011yescharacteristic
619C22C2×#53612yes—
620C22#348411yescharacteristic
621C22#348511yescharacteristic
622C22#348612yes—
623C22#348612yes—
624C22#348612yes—
625C22#349612yes—
626C22#349612yes—
627C22#349912yes—
628C22#349912yes—
629C22#354414yes—
630C22#355214yes—
631C22#355514yes—
632C22#356314yes—
633C4—24no—
634C4—24no—
635C4—28no—
636C4—44no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—48no—
643C4—82no—
644C4—82no—
645C4—84no—
646C4—82no—
647C2—44no—
648C2#3524211yescharacteristic
649C2#3534711yescharacteristic
650C2#3566611yescharacteristic
651C2#4243614yes—
652C2#4642612yes—
653C2#4721112yes—
654C2#4956814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #15881C2semidirect
7. #15849C2semidirect
8. #15847C2semidirect
9. #15783C2semidirect
10. #15762C2semidirect
11. #15318C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#28172
C2×#25832
C2×#26212
#162622
#158812
#158492
#158472
#157832
#157622
#153182
#148922
#142922
#141382
#138702