Finite Group Databasebeta

← #641654

Subgroups — #641654

Order 512.

2382 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C212yesmaximal
2#16825C212yesmaximal
3#16824C212yesmaximal
4#16521C214yesmaximal
5#15256C212yesmaximal, complemented
6#15251C212yesmaximal, complemented
7#15228C212yesmaximal, complemented
8#15225C212yesmaximal, complemented
9#15218C212yesmaximal, complemented
10#15133C212yesmaximal, complemented
11#14682C211yesmaximal, characteristic
12#14328C214yesmaximal, complemented
13#13951C212yesmaximal
14C4×#380C211yesmaximal, characteristic
15#13756C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2441—216no—
21C2×#399C2214yes—
22C2×#398C2214yes—
23C2×#396C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387C2214yes—
30C2×#384C2212yes—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38#1445C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2214yes—
65#1434C2211yescharacteristic
66#1434C2211yescharacteristic
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×#125—28no—
80C2×#125—44no—
81C2×#125—44no—
82C2×#117—42no—
83C2×#117—42no—
84C2×#117—42no—
85C2×#117—42no—
86C2×#117D418yes—
87C2×#117Q818yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42—42no—
108C22×C42—42no—
109C22×C42Q818yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118#399—216no—
119#398—216no—
120#396—216no—
121#389—28no—
122#389—28no—
123#389—216no—
124#389—44no—
125#389—44no—
126#387—216no—
127#384—28no—
128#382—42no—
129#382—42no—
130#380—28no—
131#380—28no—
132C4×C4⋊C4—42no—
133C4×C4⋊C4—42no—
134C4×C22⋊C4—28no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C25C2411yescharacteristic
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4C2×D414yes—
178C23×C4C2×Q814yes—
179C23×C4C2×Q814yes—
180C23×C4C2×Q814yes—
181C23×C4C2×Q814yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196#125—432no—
197#117—416no—
198#117—416no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
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×D418yes—
239C2×C4⋊C4C2×Q818yes—
240C2×C4⋊C4#4118yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4C2×D418yes—
255C2×C22⋊C4C2×Q818yes—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—416no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42C2×D418yes—
276C2×C42C2×Q814yes—
277C2×C42C2×Q814yes—
278C2×C42C2×Q818yes—
279C2×C42C2×Q814yes—
280C2×C42C2×Q814yes—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—41no—
301#95—41no—
302#95—41no—
303#95—41no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C22×D412yes—
319C24C22×Q812yes—
320C24C22×Q812yes—
321C24C2511yescharacteristic
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4C4⋊Q8116yes—
410C22×C4C22×D418yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q818yes—
417C22×C4C22×Q812yes—
418C22×C4C22×Q812yes—
419C22×C4C22×Q812yes—
420C22×C4C22×Q812yes—
421C22×C4C22×Q812yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4114yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+412yes—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—416no—
434C4⋊C4—48no—
435C4⋊C4—416no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—88no—
448C42—28no—
449C42—28no—
450C42—44no—
451C42—44no—
452C42—48no—
453C42—48no—
454C42—48no—
455C42—48no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23C2×C4⋊Q814yes—
479C23C23×D411yescharacteristic
480C23C23×Q811yescharacteristic
481C23C23×Q811yescharacteristic
482C23C22×#4112yes—
483C23C2×H(5,𝔽2)11yescharacteristic
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)11yescharacteristic
486C23C2×2−1+411yescharacteristic
487C23#58512yes—
488C23#58512yes—
489C23#58512yes—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—232no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4C2×C4⋊Q814yes—
575C2×C4C2×C4⋊Q814yes—
576C2×C4#53314yes—
577C2×C4#53414yes—
578C2×C4#53714yes—
579C2×C4#54314yes—
580C2×C4#54314yes—
581C2×C4#54414yes—
582C2×C4#54414yes—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22C22×C4⋊Q811yescharacteristic
595C22C2×#53311yescharacteristic
596C22C2×#53411yescharacteristic
597C22C2×#53711yescharacteristic
598C22C2×#54311yescharacteristic
599C22C2×#54311yescharacteristic
600C22C2×#54411yescharacteristic
601C22C2×#54411yescharacteristic
602C22#345811yescharacteristic
603C22#345811yescharacteristic
604C22#345811yescharacteristic
605C22D4×#4112yes—
606C22Q8×#4112yes—
607C22#350012yes—
608C22#350412yes—
609C22#350512yes—
610C22#350612yes—
611C22#351512yes—
612C22#351712yes—
613C22#353912yes—
614C22#353912yes—
615C22#356212yes—
616C22#356512yes—
617C4—24no—
618C4—24no—
619C4—48no—
620C4—44no—
621C4—44no—
622C4—48no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—82no—
628C4—84no—
629C4—84no—
630C4#265918yes—
631C2—44no—
632C2C2×#265911yescharacteristic
633C2#3493811yescharacteristic
634C2#3503311yescharacteristic
635C2#3520211yescharacteristic
636C2#3533011yescharacteristic
637C2#3748111yescharacteristic
638C2#3756711yescharacteristic
639C2#3996112yes—
640C2#3996212yes—
641C2#3998512yes—
642C2#3998612yes—
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
5. #15256C2semidirect
6. #15251C2semidirect
7. #15228C2semidirect
8. #15225C2semidirect
9. #15218C2semidirect
10. #15133C2semidirect
12. #14328C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
#168252
#168242
#165212
#152562
#152512
#152282
#152252
#152182
#151332
#146822
#143282
#139512
C4×#3802
#137562