Finite Group Databasebeta

← #4308338

Subgroups — #4308338

Order 512.

2506 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic, complemented
2C2×C4×#117C211yesmaximal, characteristic
3#15094C214yesmaximal, complemented
4#15075C214yesmaximal, complemented
5#14978C214yesmaximal
6#14923C214yesmaximal
7#14699C212yesmaximal, complemented
8#14697C212yesmaximal
9#14687C211yesmaximal, characteristic, complemented
10#14340C211yesmaximal, characteristic
11#14340C211yesmaximal, characteristic
12#14276C211yesmaximal, characteristic, complemented
13#14276C211yesmaximal, characteristic, complemented
14#14266C212yesmaximal, complemented
15#14264C212yesmaximal
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2864—28no—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#387C2214yes—
25C2×#387C2214yes—
26C2×#387C2214yes—
27C2×#385C2214yes—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2214yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C4×#117—28no—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×#376C2211yescharacteristic
41C2×#376C2211yescharacteristic
42C2×C43C2211yescharacteristic
43C2×C43C2211yescharacteristic
44#1445C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2212yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117Q818yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42—24no—
106C22×C42—24no—
107C22×C42—24no—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#389—28no—
120#389—28no—
121#389—28no—
122#389—28no—
123#387—216no—
124#387—216no—
125#387—48no—
126#385—48no—
127#384—24no—
128#384—24no—
129#382—24no—
130#382—24no—
131#382—48no—
132#380—28no—
133#380—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—28no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—28no—
138#376—24no—
139#376—24no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C43—24no—
167C43D418yes—
168C25C2411yescharacteristic
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2×Q814yes—
181C23×C4#4114yes—
182C23×C4#4114yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197#117—232no—
198#117—232no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
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—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×Q814yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C2×Q814yes—
254C2×C22⋊C4C2×Q814yes—
255C2×C42—232no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C2×D414yes—
280C2×C42C2×D414yes—
281C2×C42C2×D418yes—
282C2×C42C2×D418yes—
283C2×C42C2×D414yes—
284C2×C42C2×Q814yes—
285C2×C42C2×Q814yes—
286C2×C42#4114yes—
287C2×C42#4114yes—
288C2×C42#4114yes—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24C22×Q812yes—
324C24C2×#4112yes—
325C24C2×#4112yes—
326C24C2511yescharacteristic
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—216no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
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—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
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—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4#125116yes—
407C22×C4C22×D414yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C22×D412yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q812yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C42−1+414yes—
431C4⋊C4—216no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—88no—
446C22⋊C4—216no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—416no—
450C22⋊C4—88no—
451C42—216no—
452C42—28no—
453C42—28no—
454C42—28no—
455C42—28no—
456C42—28no—
457C42—216no—
458C42—48no—
459C42—48no—
460C42—48no—
461C42C22⋊Q8116yes—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C2×#12514yes—
485C23C23×D411yescharacteristic
486C23C23×Q811yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C22×#4111yescharacteristic
491C23C22×#4111yescharacteristic
492C23C2×H(5,𝔽2)12yes—
493C23C2×2−1+412yes—
494C23#58512yes—
495C23#58512yes—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
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—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—232no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
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—24no—
530C2×C4—432no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
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×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4C2×C22⋊Q814yes—
578C2×C4C2×C22⋊Q814yes—
579C2×C4C2×C22⋊Q814yes—
580C2×C4#52514yes—
581C2×C4#52514yes—
582C2×C4C2×#12514yes—
583C2×C4C2×#12514yes—
584C2×C4#52914yes—
585C2×C4#52914yes—
586C2×C4#53314yes—
587C2×C4#53718yes—
588C22—24no—
589C22—28no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—44no—
597C22—42no—
598C22—42no—
599C22C22×C22⋊Q811yescharacteristic
600C22C2×#52511yescharacteristic
601C22C2×#52511yescharacteristic
602C22C22×#12511yescharacteristic
603C22C2×#52911yescharacteristic
604C22C2×#52911yescharacteristic
605C22C2×#53311yescharacteristic
606C22C2×#53712yes—
607C22#344912yes—
608C22#346012yes—
609C22#346012yes—
610C22#346112yes—
611C22#346112yes—
612C22#349412yes—
613C22#349412yes—
614C22#349512yes—
615C22#349512yes—
616C22#350312yes—
617C22#350412yes—
618C22#350811yescharacteristic
619C22#350911yescharacteristic
620C22#351011yescharacteristic
621C22#351011yescharacteristic
622C4—24no—
623C4—24no—
624C4—24no—
625C4—28no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—42no—
630C4—44no—
631C4—84no—
632C4—84no—
633C4—84no—
634C4—84no—
635C4#285118yes—
636C2—44no—
637C2#3461412yes—
638C2#3526512yes—
639C2#3526811yescharacteristic
640C2#3528011yescharacteristic
641C2#3528212yes—
642C2#3529611yescharacteristic
643C2#3530212yes—
644C2#3531411yescharacteristic
645C2#3532411yescharacteristic
646C2#3533411yescharacteristic
647C2C2×#285111yescharacteristic
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×#2864C2semidirect
3. #15094C2semidirect
4. #15075C2semidirect
7. #14699C2semidirect
9. #14687C2semidirect
12. #14276C2semidirect
13. #14276C2semidirect
14. #14266C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28642
C2×C4×#1172
#150942
#150752
#149782
#149232
#146992
#146972
#146872
#143402
#142762
#142662
#142642