Finite Group Databasebeta

← #988888

Subgroups — #988888

Order 512.

2582 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C212yesmaximal
2C2×#2439C212yesmaximal, complemented
3#17107C212yesmaximal
4#16514C212yesmaximal
5#16342C214yesmaximal, complemented
6#15965C212yesmaximal, complemented
7#15632C214yesmaximal, complemented
8#15225C212yesmaximal, complemented
9#15183C212yesmaximal, complemented
10#15128C211yesmaximal, characteristic, complemented
11#14424C212yesmaximal, complemented
12#13994C212yesmaximal, complemented
13#13910C211yesmaximal, characteristic, complemented
14#13879C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19#2485—216no—
20#2439—216no—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#394C2211yescharacteristic
25C2×#394C2214yes—
26C2×#393C2214yes—
27C2×#391C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#384C2212yes—
37C2×#384C2211yescharacteristic
38C2×#382C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46#1446C2212yes—
47#1445C2214yes—
48#1445C2214yes—
49#1444C2212yes—
50#1444C2212yes—
51#1444C2212yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2211yescharacteristic
55#1442C2214yes—
56#1441C2214yes—
57#1441C2212yes—
58#1441C2211yescharacteristic
59#1441C2214yes—
60#1440C2214yes—
61#1440C2212yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2211yescharacteristic
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C4×#95C2211yescharacteristic
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—42no—
82C2×C4×D4—28no—
83C2×C4×D4—42no—
84C2×#117—28no—
85C2×#117—28no—
86C2×#117—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116#397—216no—
117#396—216no—
118#395—44no—
119#394—24no—
120#394—216no—
121#393—216no—
122#391—216no—
123#389—28no—
124#389—28no—
125#389—28no—
126#389—28no—
127#388—216no—
128#388—48no—
129#387—216no—
130#387—216no—
131#384—28no—
132#384—42no—
133#382—28no—
134#380—28no—
135#380—28no—
136#380D4116yes—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—42no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—28no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C25C2412yes—
169C22×Q8—28no—
170C22×Q8—24no—
171C22×Q8—42no—
172C22×Q8—44no—
173C22×Q8—42no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—42no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4C2×D418yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C4⋊Q8—416no—
201C4×D4—416no—
202#117—432no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4#4114yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C2×D4116yes—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42#4114yes—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—28no—
300#95—44no—
301#95—42no—
302#95—44no—
303#95—41no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—41no—
308#95—41no—
309#95—44no—
310#95—42no—
311#95C2×D4116yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24C22×D414yes—
327C24C2511yescharacteristic
328C2×Q8—48no—
329C2×Q8—416no—
330C2×Q8—88no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×D4—48no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—44no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4C22×D414yes—
417C22×C4C22×D418yes—
418C22×C4C22×D418yes—
419C22×C4C22×D414yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+412yes—
429C22×C42−1+412yes—
430C22×C42−1+414yes—
431C4⋊C4—216no—
432C4⋊C4—216no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—416no—
436C4⋊C4—416no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—84no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C22⋊C4—416no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—88no—
454C42—28no—
455C42—416no—
456C42—44no—
457C42—44no—
458C42—48no—
459C42—48no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23C23×D412yes—
496C23C22×#4111yescharacteristic
497C23C22×#4111yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+412yes—
502C23C2×2−1+411yescharacteristic
503C23#58512yes—
504C23#58511yescharacteristic
505C23#58512yes—
506Q8—816no—
507D4—416no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—432no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4#53618yes—
590C2×C4#53914yes—
591C2×C4D4214yes—
592C2×C4#54718yes—
593C2×C4#55514yes—
594C2×C4#56314yes—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C2×#53612yes—
611C22C2×#53911yescharacteristic
612C22C2×D4211yescharacteristic
613C22C2×#54712yes—
614C22D4×#4112yes—
615C22C2×#55511yescharacteristic
616C22#348712yes—
617C22#348912yes—
618C22#349312yes—
619C22#349612yes—
620C22#350511yescharacteristic
621C22#350711yescharacteristic
622C22#350911yescharacteristic
623C22#351812yes—
624C22C2×#56311yescharacteristic
625C22#353611yescharacteristic
626C22#354011yescharacteristic
627C22#354712yes—
628C22#355111yescharacteristic
629C22#355512yes—
630C22#355911yescharacteristic
631C22#356512yes—
632C22#356512yes—
633C4—24no—
634C4—48no—
635C4—42no—
636C4—48no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—42no—
641C4—81no—
642C4—82no—
643C4—84no—
644C4—84no—
645C4—82no—
646C4#268018yes—
647C2—84no—
648C2#3950411yescharacteristic
649C2#3986712yes—
650C2#4243712yes—
651C2#4268611yescharacteristic
652C2#4510311yescharacteristic
653C2#4526112yes—
654C2#4619611yescharacteristic
655C2C2×#268011yescharacteristic
656C2#5308111yescharacteristic
657C2#5900111yescharacteristic
658C2#5922912yes—
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
2. C2×#2439C2semidirect
5. #16342C2semidirect
6. #15965C2semidirect
7. #15632C2semidirect
8. #15225C2semidirect
9. #15183C2semidirect
10. #15128C2semidirect
11. #14424C2semidirect
12. #13994C2semidirect
13. #13910C2semidirect
14. #13879C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
C2×#24392
#171072
#165142
#163422
#159652
#156322
#152252
#151832
#151282
#144242
#139942
#139102
#138792
C4×#3942