Finite Group Databasebeta

← #4457224

Subgroups — #4457224

Order 512.

2206 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2319C211yesmaximal, characteristic
3#17059C212yesmaximal, complemented
4#17058C212yesmaximal, complemented
5#16699C212yesmaximal, complemented
6#16699C212yesmaximal, complemented
7#16280C214yesmaximal, complemented
8#16278C214yesmaximal, complemented
9#15232C212yesmaximal, complemented
10#15221C212yesmaximal, complemented
11#15147C212yesmaximal
12#15138C212yesmaximal
13#14282C211yesmaximal, characteristic, complemented
14#14155C211yesmaximal, characteristic, complemented
15#14150C211yesmaximal, characteristic, complemented
16#13941C211yesmaximal, characteristic
17#13848C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2563—28no—
21C2×#397C2214yes—
22C2×#397C2214yes—
23C2×#393C2214yes—
24#2319—28no—
25C2×#391C2214yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#382C2211yescharacteristic
36C2×#382C2212yes—
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×#376C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2212yes—
54#1444C2211yescharacteristic
55#1444C2211yescharacteristic
56#1444C2212yes—
57#1444C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2211yescharacteristic
61#1442C2211yescharacteristic
62#1442C2212yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#397—48no—
124#397—48no—
125#393—216no—
126#391—48no—
127#390—28no—
128#390—24no—
129#390—24no—
130#390—44no—
131#389—24no—
132#389—24no—
133#389—44no—
134#388—28no—
135#388—28no—
136#382—28no—
137#382—24no—
138#381—28no—
139#380—28no—
140#380—28no—
141#380—28no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—44no—
146#376—24no—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C25C2412yes—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4C2×D414yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C2×D418yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4⋊Q8—416no—
215C4×D4—232no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4C2×D418yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42#4114yes—
293C2×C42#4114yes—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—44no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24C22×D414yes—
338C24C2511yescharacteristic
339C2×Q8—416no—
340C2×Q8—48no—
341C2×Q8—88no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—216no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+412yes—
448C22×C42−1+412yes—
449C22×C42−1+412yes—
450C4⋊C4—28no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—88no—
464C22⋊C4—216no—
465C22⋊C4—88no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C42—44no—
474C42—44no—
475C42—48no—
476C42—48no—
477C42—84no—
478C42#123116yes—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23C23×D411yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4111yescharacteristic
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)12yes—
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23#58512yes—
511C23#58512yes—
512C23#58512yes—
513Q8—816no—
514D4—416no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—816no—
601C2×C4C2×#12314yes—
602C2×C4C2×#12314yes—
603C2×C4C2×#12314yes—
604C2×C4#53614yes—
605C2×C4#53714yes—
606C2×C4#54114yes—
607C2×C4#54114yes—
608C2×C4#54214yes—
609C2×C4#54214yes—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22C22×#12311yescharacteristic
622C22C2×#53611yescharacteristic
623C22C2×#53711yescharacteristic
624C22C2×#54111yescharacteristic
625C22C2×#54111yescharacteristic
626C22C2×#54211yescharacteristic
627C22C2×#54211yescharacteristic
628C22#348511yescharacteristic
629C22#348511yescharacteristic
630C22#349412yes—
631C22#349811yescharacteristic
632C22#350011yescharacteristic
633C22#350612yes—
634C22#351012yes—
635C22#351012yes—
636C22#354012yes—
637C22#354212yes—
638C22#354712yes—
639C22#355412yes—
640C22#355712yes—
641C22#355912yes—
642C22#356512yes—
643C22#356512yes—
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—84no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4—81no—
653C4—81no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4#262518yes—
660C2—84no—
661C2C2×#262511yescharacteristic
662C2#3553111yescharacteristic
663C2#3562611yescharacteristic
664C2#3735812yes—
665C2#3736312yes—
666C2#4252312yes—
667C2#4253012yes—
668C2#4264611yescharacteristic
669C2#4264711yescharacteristic
670C2#4265011yescharacteristic
671C2#4265111yescharacteristic
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. #17059C2semidirect
4. #17058C2semidirect
5. #16699C2semidirect
6. #16699C2semidirect
7. #16280C2semidirect
8. #16278C2semidirect
9. #15232C2semidirect
10. #15221C2semidirect
13. #14282C2semidirect
14. #14155C2semidirect
15. #14150C2semidirect
17. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23192
#170592
#170582
#166992
#162802
#162782
#152322
#152212
#151472
#151382
#142822
#141552
#141502
#139412
#138482