Finite Group Databasebeta

← #722646

Subgroups — #722646

Order 512.

3322 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C2×#2627C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic
4C2×#2368C211yesmaximal, characteristic
5#16293C214yesmaximal, complemented
6#15858C214yesmaximal, complemented
7#15318C214yesmaximal, complemented
8#14935C212yesmaximal
9#14816C214yesmaximal, complemented
10#14786C212yesmaximal
11#14130C212yesmaximal
12#13992C212yesmaximal
13#13922C211yesmaximal, characteristic
14#13908C211yesmaximal, characteristic
15#13776C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21#2627—28no—
22#2434—28no—
23C2×#400C2212yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#395—28no—
27C2×#395C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32#2368—28no—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2214yes—
36C2×#389C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#381C2211yescharacteristic
44C2×#381C2212yes—
45C2×#381C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C22×#95C2212yes—
53C22×#95C2211yescharacteristic
54#1446C2214yes—
55#1445C2214yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73#1434C2214yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2211yescharacteristic
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—48no—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—48no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8D418yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—48no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124#400—28no—
125#398—216no—
126#398—48no—
127#395—28no—
128#395—216no—
129#395—28no—
130#395—28no—
131#395—216no—
132#391—28no—
133#391—28no—
134#391—48no—
135#389—48no—
136#387—28no—
137#387—28no—
138#385—28no—
139#385—28no—
140#385—28no—
141#385—28no—
142#381—28no—
143#381—24no—
144#381Q818yescomplemented
145#380—24no—
146#380—28no—
147C4×C4⋊C4—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C2×#95—28no—
152C2×#95D418yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C25C2411yescharacteristic
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—28no—
191C22×Q8—28no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×Q8—432no—
196C22×Q8—432no—
197C22×Q8—432no—
198C22×Q8—416no—
199C22×Q8C2×D414yes—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D418yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×Q814yes—
224C23×C4#4114yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C4⋊Q8—432no—
240C22⋊Q8—232no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—432no—
272C2×C4⋊C4—416no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D418yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4C2×D418yes—
294C2×C22⋊C4C2×Q814yes—
295C2×C22⋊C4C2×Q814yes—
296C2×C22⋊C4#4114yes—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C2×D414yes—
330#95C2×D414yes—
331#95C2×D418yes—
332#95C2×Q8116yescomplemented
333C24—24no—
334C24—24no—
335C24C22×D412yes—
336C24C22×D412yes—
337C24C22×Q812yes—
338C24C2×#4112yes—
339C24C2×#4112yes—
340C242−1+412yes—
341C242−1+412yes—
342C24C2511yescharacteristic
343C2×Q8—216no—
344C2×Q8—216no—
345C2×Q8—28no—
346C2×Q8—48no—
347C2×Q8—416no—
348C2×Q8—416no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—4128no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—84no—
357C2×Q8—84no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—216no—
364C22×C4—28no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
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—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4#121116yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q814yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)18yes—
456C22×C42−1+418yes—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C22⋊C4—216no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C42—48no—
477C42—48no—
478C42—84no—
479C42—84no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23C2×#12114yes—
491C23#53314yes—
492C23#54114yes—
493C23#54714yes—
494C23#54814yes—
495C23D4×Q814yes—
496C23D4×Q814yes—
497C23C23×D411yescharacteristic
498C23C23×D411yescharacteristic
499C23C23×Q811yescharacteristic
500C23C22×#4111yescharacteristic
501C23C22×#4111yescharacteristic
502C23C2×H(5,𝔽2)14yes—
503C23C2×2−1+414yes—
504C23C2×2−1+411yescharacteristic
505C23C2×2−1+411yescharacteristic
506Q8—432no—
507Q8—432no—
508Q8—88no—
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—28no—
518C2×C4—28no—
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—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—416no—
539C2×C4—432no—
540C2×C4—432no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—416no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—88no—
585C2×C4C2×#12114yes—
586C2×C4C2×#12114yes—
587C2×C4#53618yes—
588C22—24no—
589C22—28no—
590C22—28no—
591C22—28no—
592C22—28no—
593C22—28no—
594C22—24no—
595C22—28no—
596C22#242918yes—
597C22C22×#12111yescharacteristic
598C22C2×#53311yescharacteristic
599C22C2×#53612yes—
600C22C2×#54111yescharacteristic
601C22C2×#54711yescharacteristic
602C22C2×#54811yescharacteristic
603C22C2×D4×Q811yescharacteristic
604C22C2×D4×Q811yescharacteristic
605C22#348412yes—
606C22#348412yes—
607C22#349012yes—
608C22#349912yes—
609C22#350314yes—
610C22#350912yes—
611C22#352014yes—
612C22#353814yes—
613C22#355214yes—
614C4—24no—
615C4—24no—
616C4—28no—
617C4—48no—
618C4—44no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—82no—
627C4—84no—
628C2—28no—
629C2C2×#242911yescharacteristic
630C2#3531611yescharacteristic
631C2#3534811yescharacteristic
632C2#3574312yes—
633C2#3803712yes—
634C2#3959212yes—
635C2#4054712yes—
636C2#4300712yes—
637C2#4389112yes—
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. #16293C2semidirect
6. #15858C2semidirect
7. #15318C2semidirect
9. #14816C2semidirect
144. #381Q8semidirect
332. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#26272
C22×#3952
C2×#23682
#162932
#158582
#153182
#149352
#148162
#147862
#141302
#139922
#139222
#139082
#137762