Finite Group Databasebeta

← #1007280

Subgroups — #1007280

Order 512.

2826 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C212yesmaximal, complemented
2C2×#2440C212yesmaximal
3#17085C212yesmaximal, complemented
4#16522C214yesmaximal, complemented
5#16254C212yesmaximal
6#15511C212yesmaximal, complemented
7#15430C214yesmaximal
8#15143C212yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#15135C211yesmaximal, characteristic
11#14575C212yesmaximal
12#14132C212yesmaximal, complemented
13#14130C211yesmaximal, characteristic
14#13973C212yesmaximal
15C4×#393C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C22⋊Q8C2214yes—
18C22×C4×Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2567—216no—
21#2440—216no—
22C2×#399C2212yes—
23C2×#398C2214yes—
24C2×#398C2214yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#393C2211yescharacteristic
31C2×#391C2214yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#385C2214yes—
38C2×#385C2212yes—
39C2×#384C2212yes—
40C2×#384C2214yes—
41C2×#384C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47#1446C2211yescharacteristic
48#1445C2212yes—
49#1445C2214yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2211yescharacteristic
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2211yescharacteristic
65#1439C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2314yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—44no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8—48no—
87C2×C4×Q8—28no—
88C2×C4×Q8—42no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119#399—44no—
120#398—216no—
121#398—48no—
122#395—216no—
123#395—216no—
124#395—216no—
125#395—48no—
126#395—48no—
127#393Q818yescomplemented
128#391—216no—
129#389—28no—
130#389—28no—
131#389—28no—
132#389—28no—
133#389D4116yes—
134#385—216no—
135#385—44no—
136#384—216no—
137#384—28no—
138#384—42no—
139#380—28no—
140#380Q818yescomplemented
141C4×C4⋊C4—28no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—42no—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C25C2411yescharacteristic
172C22×Q8—24no—
173C22×Q8—28no—
174C22×Q8—28no—
175C22×Q8—24no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8C2×Q814yes—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4C2×Q814yes—
194C23×C4C2411yescharacteristic
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C4⋊Q8—432no—
209C22⋊Q8—464no—
210C4×Q8—416no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×D4116yes—
259C2×C4⋊C4C2×Q818yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4C2×Q818yes—
276C2×C22⋊C4C2×Q818yes—
277C2×C22⋊C4C2×Q814yes—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42C2×D418yes—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—41no—
306#95—44no—
307#95—42no—
308#95—41no—
309#95—41no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95C2×D4116yes—
315#95C2×Q818yes—
316#95C2×Q814yes—
317#95C2×Q818yes—
318C24—22no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—22no—
323C24—21no—
324C24—22no—
325C24—21no—
326C24C22×Q812yes—
327C24C22×Q812yes—
328C24H(5,𝔽2)12yes—
329C24C2511yescharacteristic
330C2×Q8—216no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—48no—
334C2×Q8—44no—
335C2×Q8—416no—
336C2×Q8—416no—
337C2×Q8—88no—
338C2×Q8—88no—
339C2×Q8—88no—
340C2×Q8—88no—
341C2×Q8—88no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—88no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C22×D418yes—
421C22×C4C22×D418yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×Q814yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q818yes—
429C22×C4C22×Q812yes—
430C22×C4C22×Q814yes—
431C22×C4C22×Q812yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+414yes—
436C22×C42−1+412yes—
437C22×C42−1+414yes—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—216no—
441C4⋊C4—416no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—416no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C4⋊C4—88no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C22⋊C4—216no—
456C22⋊C4—216no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—416no—
461C42—216no—
462C42—416no—
463C42—48no—
464C42—44no—
465C42—44no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23Q8214yes—
489C23C23×D412yes—
490C23C23×Q811yescharacteristic
491C23C23×Q811yescharacteristic
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)12yes—
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×2−1+411yescharacteristic
497C23C2×2−1+412yes—
498C23C2×2−1+412yes—
499C23#58511yescharacteristic
500Q8—416no—
501Q8—832no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—416no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—816no—
584C2×C4#53618yes—
585C2×C4#54414yes—
586C2×C4D4214yes—
587C2×C4D4×Q818yes—
588C2×C4D4×Q818yes—
589C2×C4#56314yes—
590C22—28no—
591C22—24no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22C2×#53612yes—
602C22C2×#54411yescharacteristic
603C22C2×D4211yescharacteristic
604C22C2×D4×Q812yes—
605C22C2×D4×Q812yes—
606C22C2×Q8211yescharacteristic
607C22#348412yes—
608C22#348912yes—
609C22#349812yes—
610C22#350111yescharacteristic
611C22#350111yescharacteristic
612C22#350312yes—
613C22#350312yes—
614C22#350311yescharacteristic
615C22C2×#56311yescharacteristic
616C22#353711yescharacteristic
617C22#353811yescharacteristic
618C22#354412yes—
619C22#354412yes—
620C22#354412yes—
621C22#354911yescharacteristic
622C22#355912yes—
623C22#355911yescharacteristic
624C4—28no—
625C4—42no—
626C4—48no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—48no—
632C4—44no—
633C4—48no—
634C4—44no—
635C4—81no—
636C4—84no—
637C4#267718yes—
638C2—44no—
639C2#3961712yes—
640C2#3962311yescharacteristic
641C2#4248212yes—
642C2#4297311yescharacteristic
643C2#4487011yescharacteristic
644C2#4494112yes—
645C2#4615211yescharacteristic
646C2C2×#267711yescharacteristic
647C2#5303111yescharacteristic
648C2#5899811yescharacteristic
649C2#5915112yes—
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
1. C2×#2567C2semidirect
3. #17085C2semidirect
4. #16522C2semidirect
6. #15511C2semidirect
8. #15143C2semidirect
9. #15137C2semidirect
12. #14132C2semidirect
127. #393Q8semidirect
140. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#24402
#170852
#165222
#162542
#155112
#154302
#151432
#151372
#151352
#145752
#141322
#141302
#139732
C4×#3932