Finite Group Databasebeta

← #722888

Subgroups — #722888

Order 512.

3402 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×#2434C211yesmaximal, characteristic
2C2×#2627C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic
4C2×Q8×C22⋊C4C211yesmaximal, characteristic
5#16629C214yesmaximal, complemented
6#15503C214yesmaximal, complemented
7#15318C214yesmaximal, complemented
8#14935C212yesmaximal
9#14880C214yesmaximal, complemented
10#14728C212yesmaximal
11#14159C211yesmaximal, characteristic
12#14149C211yesmaximal, characteristic
13#14043C212yesmaximal
14#13972C212yesmaximal
15#13776C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2627—28no—
23#2434—28no—
24C2×#400C2212yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#395—28no—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#395C2214yes—
32C2×#395C2212yes—
33C2×#395C2212yes—
34Q8×C22⋊C4—28no—
35C2×#391C2214yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#389C2214yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#381C2211yescharacteristic
44C2×#381C2212yes—
45C2×#381C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95C2211yescharacteristic
52C22×#95C2212yes—
53#1446C2212yes—
54#1445C2212yes—
55#1444C2214yes—
56#1444C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2211yescharacteristic
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—48no—
82C23×Q8D418yes—
83C23×Q8C2311yescharacteristic
84C23×Q8C2312yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—48no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C4×Q8—28no—
98C2×C4×Q8—48no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—48no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#400—28no—
128#398—216no—
129#398—48no—
130#395—28no—
131#395—216no—
132#395—28no—
133#395—28no—
134#395—28no—
135#395—28no—
136#391—28no—
137#391—28no—
138#391—216no—
139#389—48no—
140#385—28no—
141#385—28no—
142#385—28no—
143#385—28no—
144#381—28no—
145#381—24no—
146#381Q818yescomplemented
147#380—28no—
148#380D418yes—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—28no—
152C2×#95—28no—
153C2×#95—24no—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C25C2411yescharacteristic
182C22×Q8—232no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—28no—
193C22×Q8—28no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8—432no—
198C22×Q8—432no—
199C22×Q8—432no—
200C22×Q8C2×D414yes—
201C22×Q8C2×D418yes—
202C22×Q8C2×D414yes—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D418yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×Q814yes—
227C23×C4#4114yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C4⋊Q8—432no—
243C22⋊Q8—416no—
244C4×Q8—432no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—432no—
280C2×C4⋊C4—416no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4C2×D418yes—
299C2×C22⋊C4C2×Q814yes—
300C2×C22⋊C4C2×Q814yes—
301C2×C22⋊C4#4114yes—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95C2×D418yes—
335#95C2×Q8116yescomplemented
336C24—24no—
337C24—24no—
338C24C22×D412yes—
339C24C22×D412yes—
340C24C22×Q812yes—
341C24C2×#4112yes—
342C24C2×#4112yes—
343C242−1+412yes—
344C242−1+412yes—
345C24C2511yescharacteristic
346C2×Q8—216no—
347C2×Q8—216no—
348C2×Q8—28no—
349C2×Q8—216no—
350C2×Q8—28no—
351C2×Q8—216no—
352C2×Q8—216no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—416no—
356C2×Q8—416no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—4128no—
361C2×Q8—44no—
362C2×Q8—84no—
363C2×Q8—84no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—22no—
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—22no—
385C22×C4—22no—
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—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4#121116yes—
448C22×C4C22×D418yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q814yes—
457C22×C4C22×Q814yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)18yes—
462C22×C42−1+418yes—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—88no—
477C22⋊C4—216no—
478C22⋊C4—28no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C42—416no—
484C42—48no—
485C42—48no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23C2×#12114yes—
497C23#53314yes—
498C23#54114yes—
499C23#54714yes—
500C23#54814yes—
501C23D4×Q814yes—
502C23D4×Q814yes—
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)14yes—
509C23C2×2−1+411yescharacteristic
510C23C2×2−1+411yescharacteristic
511C23C2×2−1+414yes—
512Q8—232no—
513Q8—432no—
514Q8—416no—
515Q8—88no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—416no—
545C2×C4—432no—
546C2×C4—432no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
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—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4C2×#12114yes—
593C2×C4C2×#12114yes—
594C2×C4#53418yes—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22#242918yes—
604C22C22×#12111yescharacteristic
605C22C2×#53311yescharacteristic
606C22C2×#53412yes—
607C22C2×#54111yescharacteristic
608C22C2×#54711yescharacteristic
609C22C2×#54811yescharacteristic
610C22C2×D4×Q811yescharacteristic
611C22C2×D4×Q811yescharacteristic
612C22#348412yes—
613C22#348612yes—
614C22#349012yes—
615C22#349912yes—
616C22#350314yes—
617C22#350912yes—
618C22#352014yes—
619C22#353814yes—
620C22#355214yes—
621C4—24no—
622C4—24no—
623C4—28no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—48no—
630C4—48no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C2—28no—
636C2C2×#242911yescharacteristic
637C2#3498111yescharacteristic
638C2#3566711yescharacteristic
639C2#3574312yes—
640C2#3912112yes—
641C2#3959312yes—
642C2#4054712yes—
643C2#4300412yes—
644C2#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. #16629C2semidirect
6. #15503C2semidirect
7. #15318C2semidirect
9. #14880C2semidirect
146. #381Q8semidirect
335. #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×Q8×C22⋊C42
#166292
#155032
#153182
#149352
#148802
#147282
#141592
#141492
#140432
#139722
#137762