Finite Group Databasebeta

← #516723

Subgroups — #516723

Order 512.

2894 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic
3C2×C4×#117C211yesmaximal, characteristic
4#14712C211yesmaximal, characteristic, complemented
5#14707C211yesmaximal, characteristic, complemented
6#14697C212yesmaximal
7#14277C211yesmaximal, characteristic, complemented
8#14276C211yesmaximal, characteristic, complemented
9#14264C212yesmaximal
10#14118C214yesmaximal
11#14116C214yesmaximal
12#14113C214yesmaximal, complemented
13#14111C214yesmaximal, complemented
14#13783C212yesmaximal, complemented
15C4×#389C212yesmaximal, complemented
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2314—28no—
20#2314C4116yescomplemented
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yes—
29C2×#387C2214yes—
30C2×#385C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C4×#117—28no—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2214yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×#376C2211yescharacteristic
46C2×#376C2211yescharacteristic
47C2×C43C2211yescharacteristic
48C2×C43C2211yescharacteristic
49#1444C2214yes—
50#1444C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C4×#95C2214yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C2×#117—24no—
77C2×#117—24no—
78C2×#117—24no—
79C2×#117C2×C418yes—
80C2×#117D418yes—
81C2×#117Q818yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C42—24no—
101C22×C42—24no—
102C22×C42—24no—
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#389—28no—
117#389—28no—
118#389—28no—
119#389—28no—
120#389—48no—
121#388—216no—
122#388C2×C4132yes—
123#387—216no—
124#387C2×C4132yescomplemented
125#385—48no—
126#384—24no—
127#384C2×C418yescomplemented
128#382—24no—
129#382C2×C418yescomplemented
130#380—28no—
131#380C2×C4116yes—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—48no—
136C4×C4⋊C4C2×C4116yescomplemented
137C4×C22⋊C4—28no—
138C4×C22⋊C4—28no—
139#376—24no—
140#376—24no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C43—24no—
164C43—24no—
165C25C2411yescharacteristic
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C22×C414yes—
178C23×C4C2×D414yes—
179C23×C4C2×Q814yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194#117—232no—
195#117C4⋊C4164yescomplemented
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C22×C418yes—
230C2×C4⋊C4C22×C418yes—
231C2×C4⋊C4C22×C4116yes—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×Q818yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4C22×C4116yes—
247C2×C22⋊C4C22×C418yes—
248C2×C22⋊C4C2×D418yes—
249C2×C22⋊C4C2×Q818yes—
250C2×C42—232no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C22×C4116yes—
280C2×C42C22×C414yes—
281C2×C42C22×C414yes—
282C2×C42C2×D414yes—
283C2×C42C2×D414yes—
284C2×C42C2×Q814yes—
285C2×C42C2×Q814yes—
286C2×C42#4114yes—
287C2×C42#4114yes—
288C2×C42#4114yes—
289C2×C42#4114yes—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95C22×C4116yes—
310#95C22×C4116yes—
311#95C22×C4116yes—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24C23×C412yes—
321C24C22×D412yes—
322C24C22×Q812yes—
323C24C2511yescharacteristic
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—216no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4C2×C4⋊C4116yes—
402C22×C4C23×C418yes—
403C22×C4C23×C418yes—
404C22×C4C23×C412yes—
405C22×C4C23×C414yes—
406C22×C4C23×C414yes—
407C22×C4C23×C412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×Q814yes—
413C22×C4C22×Q814yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C42−1+414yes—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—416no—
433C4⋊C4—416no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—416no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4C2×C4⋊C4132yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—416no—
444C22⋊C4—88no—
445C22⋊C4C2×C4⋊C4132yes—
446C42—216no—
447C42—28no—
448C42—28no—
449C42—28no—
450C42—28no—
451C42—416no—
452C42—48no—
453C42—48no—
454C42—48no—
455C42—48no—
456C42—88no—
457C42C2×C4⋊C4116yes—
458C42C2×C4⋊C4116yes—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23C22×C4⋊C414yes—
482C23C24×C411yescharacteristic
483C23C23×D411yescharacteristic
484C23C23×Q811yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4111yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C2×H(5,𝔽2)12yes—
490C23C2×2−1+412yes—
491C23#58512yes—
492C23#58512yes—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—232no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—432no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4C22×C4⋊C418yes—
575C2×C4C22×C4⋊C418yes—
576C2×C4C22×C4⋊C414yes—
577C2×C4C22×C4⋊C414yes—
578C2×C4C4×#4114yes—
579C2×C4C4×#4114yes—
580C2×C4#52514yes—
581C2×C4#52514yes—
582C2×C4#53314yes—
583C2×C4#53314yes—
584C2×C4#54218yes—
585C22—28no—
586C22—24no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22C23×C4⋊C411yescharacteristic
597C22C2×C4×#4111yescharacteristic
598C22C2×C4×#4111yescharacteristic
599C22C4×H(5,𝔽2)12yes—
600C22C4×2−1+412yes—
601C22C2×#52511yescharacteristic
602C22C2×#52511yescharacteristic
603C22C2×#53311yescharacteristic
604C22C2×#53311yescharacteristic
605C22C2×#54212yes—
606C22#345512yes—
607C22#346012yes—
608C22#346112yes—
609C22#348812yes—
610C22#348912yes—
611C22#349412yes—
612C22#349512yes—
613C22#350312yes—
614C22#350412yes—
615C22#350811yescharacteristic
616C22#350911yescharacteristic
617C22#351211yescharacteristic
618C22#351211yescharacteristic
619C4—28no—
620C4—28no—
621C4—28no—
622C4—24no—
623C4—48no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—48no—
628C4—42no—
629C4—42no—
630C4—84no—
631C4—84no—
632C4#229618yes—
633C2—44no—
634C2C2×#229611yescharacteristic
635C2C4×#54212yes—
636C2#3425012yes—
637C2#3428011yescharacteristic
638C2#3428111yescharacteristic
639C2#3525911yescharacteristic
640C2#3526012yes—
641C2#3528212yes—
642C2#3528311yescharacteristic
643C2#3530911yescharacteristic
644C2#3533611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14712C2semidirect
5. #14707C2semidirect
7. #14277C2semidirect
8. #14276C2semidirect
12. #14113C2semidirect
13. #14111C2semidirect
14. #13783C2semidirect
15. C4×#389C2semidirect
20. #2314C4semidirect
124. #387C2×C4semidirect
127. #384C2×C4semidirect
129. #382C2×C4semidirect
136. C4×C4⋊C4C2×C4semidirect
195. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×C4×#1172
#147122
#147072
#146972
#142772
#142762
#142642
#141182
#141162
#141132
#141112
#137832
C4×#3892