Finite Group Databasebeta

← #5026209

Subgroups — #5026209

Order 512.

3518 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#563C211yesmaximal, characteristic
2C2×#2545C211yesmaximal, characteristic
3C2×#2584C211yesmaximal, characteristic, complemented
4C2×#2634C211yesmaximal, characteristic, complemented
5#16102C214yesmaximal, complemented
6#15940C214yesmaximal, complemented
7#15783C212yesmaximal, complemented
8#15762C212yesmaximal, complemented
9#15614C214yesmaximal, complemented
10#15464C212yesmaximal
11#15354C214yesmaximal, complemented
12#15262C212yesmaximal
13#14289C211yesmaximal, characteristic, complemented
14#14288C211yesmaximal, characteristic
15#14160C211yesmaximal, characteristic, complemented
16C2×#563—28no—
17C2×#563—28no—
18C2×#563—28no—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2212yes—
22C22×#126C2212yes—
23C22×#125C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×#124C2211yescharacteristic
26C22×#124C2211yescharacteristic
27C22×#123C2212yes—
28C22×C22⋊Q8C2212yes—
29C22×C22⋊Q8C2212yes—
30C22×C4×Q8C2211yescharacteristic
31#2634—28no—
32#2584—28no—
33#2545—28no—
34C2×#400C2212yes—
35C2×#400C2212yes—
36C2×#398C2212yes—
37C2×#398C2212yes—
38C2×#396C2214yes—
39C2×#395C2212yes—
40C2×#395C2212yes—
41C2×#395C2214yes—
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2214yes—
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#388C2214yes—
48C2×#387C2214yes—
49C2×#385C2214yes—
50C2×#383C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#1446C2212yes—
54#1446C2214yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2212yes—
75#1436C2212yes—
76#1436C2212yes—
77#1436C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C22⋊#95C2214yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4C2312yes—
86#563—2128no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—44no—
92C2×#126—28no—
93C2×#126—28no—
94C2×#126—28no—
95C2×#126—28no—
96C2×#126—28no—
97C2×#126—28no—
98C2×#125—24no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125D418yes—
103C2×#125D418yes—
104C2×#124—24no—
105C2×#124—24no—
106C2×#124—24no—
107C2×#124—44no—
108C2×#124D418yes—
109C2×#123—28no—
110C2×#123—28no—
111C2×#123—28no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8—28no—
116C2×C22⋊Q8—28no—
117C2×C22⋊Q8—28no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2314yes—
148C22×C42C2312yes—
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151C22×C42C2311yescharacteristic
152C22×C42C2312yes—
153C22×C42C2311yescharacteristic
154#400—28no—
155#400—28no—
156#398—28no—
157#398—28no—
158#396—216no—
159#395—28no—
160#395—28no—
161#395—48no—
162#391—24no—
163#390—24no—
164#389—28no—
165#389—216no—
166#389—28no—
167#388—216no—
168#387—216no—
169#385—48no—
170#383—24no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—24no—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C25C2412yes—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8C2×D414yes—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4C2×D414yes—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—264no—
235C4⋊Q8—416no—
236#126—264no—
237#126—264no—
238#125—232no—
239#125—232no—
240#124—232no—
241#124—416no—
242#123—264no—
243C22⋊Q8—264no—
244C22⋊Q8—264no—
245C4×Q8—416no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
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—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D418yes—
295C2×C4⋊C4C2×D418yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D414yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×D418yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
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—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24C22×D414yes—
372C24C2511yescharacteristic
373C2×Q8—28no—
374C2×Q8—216no—
375C2×Q8—28no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×Q8—84no—
386C2×Q8—88no—
387C2×Q8—88no—
388C2×D4—28no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—84no—
392C2×D4—84no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
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—24no—
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—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—216no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C42−1+418yes—
472C22×C42−1+414yes—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—48no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—88no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—88no—
500C42—216no—
501C42—44no—
502C42—44no—
503C42—48no—
504C42—84no—
505C42#120116yes—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23C23×D411yescharacteristic
528C23C23×D411yescharacteristic
529C23C23×D411yescharacteristic
530C23C2×H(5,𝔽2)12yes—
531C23C2×2−1+414yes—
532C23C2×2−1+412yes—
533C23#58512yes—
534C23#58512yes—
535Q8—416no—
536Q8—88no—
537Q8—816no—
538D4—88no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—816no—
612C2×C4C2×#12014yes—
613C2×C4C2×#12014yes—
614C2×C4C2×#12014yes—
615C22—48no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—48no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22C22×#12011yescharacteristic
625C22#348411yescharacteristic
626C22#348511yescharacteristic
627C22#348612yes—
628C22#348812yes—
629C22#348812yes—
630C22#348912yes—
631C22#349712yes—
632C22#349911yescharacteristic
633C22#349912yes—
634C22#349912yes—
635C22#350011yescharacteristic
636C22#354114yes—
637C22#355214yes—
638C22#355514yes—
639C22#356414yes—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—48no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4—82no—
653C4—84no—
654C2—84no—
655C2#3551211yescharacteristic
656C2#3566011yescharacteristic
657C2#3566611yescharacteristic
658C2#5069212yes—
659C2#5089212yes—
660C2#5107112yes—
661C2#5120912yes—
662C2#5133414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2584C2semidirect
4. C2×#2634C2semidirect
5. #16102C2semidirect
6. #15940C2semidirect
7. #15783C2semidirect
8. #15762C2semidirect
9. #15614C2semidirect
11. #15354C2semidirect
13. #14289C2semidirect
15. #14160C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5632
C2×#25452
C2×#25842
C2×#26342
#161022
#159402
#157832
#157622
#156142
#154642
#153542
#152622
#142892
#142882
#141602