Finite Group Databasebeta

← #637128

Subgroups — #637128

Order 512.

2734 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Q8C211yesmaximal, characteristic
4C2×#2283C211yesmaximal, characteristic
5#16663C214yesmaximal, complemented
6#16262C214yesmaximal
7#15236C212yesmaximal
8#15221C214yesmaximal, complemented
9#15150C212yesmaximal
10#15132C214yesmaximal
11#14277C212yesmaximal, complemented
12#14261C212yesmaximal, complemented
13#14161C211yesmaximal, characteristic, complemented
14#13926C211yesmaximal, characteristic, complemented
15#13911C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2857—28no—
22#2567—28no—
23C2×#397C2218yes—
24C2×#395C2214yes—
25C4×C22⋊Q8—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2214yes—
33C2×#389C2211yescharacteristic
34C2×#387C2212yes—
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38#2283—28no—
39C2×#383C2212yes—
40C2×#382C2212yes—
41C2×#382C2212yes—
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×#376C2212yes—
54#1446C2214yes—
55#1445C2218yes—
56#1444C2218yes—
57#1443C2218yes—
58#1442C2218yes—
59#1442C2214yes—
60#1442C2218yes—
61#1442C2218yes—
62#1441C2214yes—
63#1440C2218yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C23×Q8C2311yescharacteristic
72C23×Q8C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—48no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2318yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C42—24no—
107C22×C42—24no—
108C22×C42D418yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#397—416no—
118#395—216no—
119#391—28no—
120#391—28no—
121#391—28no—
122#391—24no—
123#389—216no—
124#389—28no—
125#389—28no—
126#389—24no—
127#387—28no—
128#385—28no—
129#384—28no—
130#384—24no—
131#383—28no—
132#382—28no—
133#382—28no—
134#381—28no—
135#380—28no—
136#380—28no—
137#380—24no—
138#380—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4D418yes—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—28no—
145#376—28no—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2312yes—
165C25C2411yescharacteristic
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8#4114yes—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—42no—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4⋊Q8—416no—
210C22⋊Q8—232no—
211C22⋊Q8—432no—
212C4×Q8—416no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4#4114yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—232no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42C2×D418yes—
282C2×C42C2×D414yes—
283C2×C42C2×D414yes—
284C2×C42C2×D414yes—
285C2×C42C2×D414yes—
286C2×C42C2×D414yes—
287C2×C42#4114yes—
288#95—28no—
289#95—28no—
290#95—28no—
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—216no—
300#95—216no—
301#95—216no—
302#95—48no—
303#95—48no—
304#95—48no—
305#95—48no—
306#95C2×D418yes—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24C22×D412yes—
317C24H(5,𝔽2)14yes—
318C24C2511yescharacteristic
319C2×Q8—28no—
320C2×Q8—28no—
321C2×Q8—28no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—44no—
325C2×Q8—48no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—88no—
330C2×Q8—88no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D418yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C42−1+414yes—
425C22×C42−1+414yes—
426C22×C42−1+412yes—
427C22×C42−1+412yes—
428C4⋊C4—216no—
429C4⋊C4—28no—
430C4⋊C4—216no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—416no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—816no—
443C22⋊C4—216no—
444C22⋊C4—416no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C42—28no—
448C42—28no—
449C42—28no—
450C42—216no—
451C42—48no—
452C42—48no—
453C42—48no—
454C42#121116yes—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23#53514yes—
478C23C23×D411yescharacteristic
479C23C23×D411yescharacteristic
480C23C22×#4111yescharacteristic
481C23C22×#4111yescharacteristic
482C23C22×#4111yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+412yes—
485C23C2×2−1+412yes—
486C23C2×2−1+411yescharacteristic
487C23C2×2−1+411yescharacteristic
488C23#58512yes—
489Q8—416no—
490Q8—816no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—416no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4C2×#12114yes—
562C2×C4C2×#12114yes—
563C2×C4C2×#12114yes—
564C2×C4#52514yes—
565C2×C4#53614yes—
566C2×C4#53618yes—
567C2×C4#53614yes—
568C2×C4#53614yes—
569C2×C4#54118yes—
570C2×C4#54714yes—
571C2×C4#54814yes—
572C2×C4#55214yes—
573C2×C4#55614yes—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22C22×#12111yescharacteristic
586C22C2×#52511yescharacteristic
587C22C2×#53511yescharacteristic
588C22C2×#53611yescharacteristic
589C22C2×#53611yescharacteristic
590C22C2×#53611yescharacteristic
591C22C2×#53612yes—
592C22#344911yescharacteristic
593C22C2×#54112yes—
594C22C2×#54711yescharacteristic
595C22C2×#54811yescharacteristic
596C22C2×#55211yescharacteristic
597C22C2×#55611yescharacteristic
598C22#348812yes—
599C22#349412yes—
600C22#351512yes—
601C22#351812yes—
602C22#352212yes—
603C22#352212yes—
604C22#354412yes—
605C22#354412yes—
606C22#355512yes—
607C22#355512yes—
608C4—24no—
609C4—48no—
610C4—42no—
611C4—44no—
612C4—42no—
613C4—44no—
614C4—42no—
615C4—82no—
616C4—84no—
617C4—82no—
618C4—84no—
619C4—84no—
620C4#241518yes—
621C4#262118yes—
622C2—44no—
623C2C2×#262111yescharacteristic
624C2C2×#241511yescharacteristic
625C2#3520011yescharacteristic
626C2#3524911yescharacteristic
627C2#3525211yescharacteristic
628C2#3763211yescharacteristic
629C2#3763511yescharacteristic
630C2#3892112yes—
631C2#3893112yes—
632C2#3912512yes—
633C2#3915712yes—
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
2. C2×#2567C2semidirect
5. #16663C2semidirect
8. #15221C2semidirect
11. #14277C2semidirect
12. #14261C2semidirect
13. #14161C2semidirect
14. #13926C2semidirect
15. #13911C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#25672
C2×C4×C22⋊Q82
C2×#22832
#166632
#162622
#152362
#152212
#151502
#151322
#142772
#142612
#141612
#139262
#139112