Finite Group Databasebeta

← #1424044

Subgroups — #1424044

Order 512.

4014 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#555C211yesmaximal, characteristic
2C2×#2731C212yesmaximal, complemented
3C2×#2546C211yesmaximal, characteristic, complemented
4#16094C212yesmaximal, complemented
5#15761C212yesmaximal, complemented
6#15319C214yesmaximal, complemented
7#15316C214yesmaximal, complemented
8#14891C214yesmaximal, complemented
9#14865C212yesmaximal, complemented
10#14450C211yesmaximal, characteristic, complemented
11#14176C212yesmaximal, complemented
12#14072C214yesmaximal, complemented
13#14026C212yesmaximal, complemented
14C2×#555—28no—
15C2×#555—216no—
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×#124C2212yes—
21C22×#121C2214yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2212yes—
24C22×#117C2214yes—
25#2731—216no—
26#2546—28no—
27C2×#398C2214yes—
28C2×#396C2214yes—
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#394C2212yes—
33C2×#393C2214yes—
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#389C2214yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#387C2214yes—
43C2×#385C2214yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45#1446C2212yes—
46#1444C2212yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2212yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2214yes—
57#1438C2212yes—
58#1438C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65C4⋊#95C2212yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2214yes—
70C4×#95C2212yes—
71C23×Q8C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C23×Q8C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2312yes—
76C23×D4C2312yes—
77C24×C4C2314yes—
78#555—2128no—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—42no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8D418yes—
86C2×#124—28no—
87C2×#124—28no—
88C2×#124—28no—
89C2×#124—216no—
90C2×#124D4116yes—
91C2×#121—216no—
92C2×#121—216no—
93C2×#121—216no—
94C2×C4×Q8—24no—
95C2×C4×Q8—42no—
96C2×C4×Q8—42no—
97C2×C4×D4—216no—
98C2×C4×D4—28no—
99C2×#117—216no—
100C2×#117—216no—
101C2×#117—216no—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#398—216no—
131#396—216no—
132#394—216no—
133#394—28no—
134#394—44no—
135#394—44no—
136#393—216no—
137#391—28no—
138#391—24no—
139#391—24no—
140#390—44no—
141#390—44no—
142#389—216no—
143#388—216no—
144#388—216no—
145#387—216no—
146#385—216no—
147C4×C4⋊C4—24no—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2414yes—
167C22×Q8—24no—
168C22×Q8—22no—
169C22×Q8—24no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—42no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8C2×D414yes—
180C22×Q8C2×D414yes—
181C22×Q8C2×D418yes—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—44no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4C2×D418yes—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C4⋊Q8—232no—
219C4⋊Q8—416no—
220C4⋊Q8—416no—
221#124—264no—
222#124—264no—
223#121—2128no—
224C4×Q8—416no—
225C4×D4—264no—
226#117—2128no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4C2×D4116yes—
255C2×C4⋊C4#4118yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D4116yes—
288C2×C22⋊C4C2×D4116yes—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42C2×D418yes—
306C2×C42C2×D414yes—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—42no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24C22×D418yes—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8#120116yes—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—216no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4—88no—
372C2×D4—88no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—232no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4C22×D418yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C42−1+414yes—
447C4⋊C4—216no—
448C4⋊C4—232no—
449C4⋊C4—216no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—416no—
456C4⋊C4—84no—
457C22⋊C4—232no—
458C22⋊C4—232no—
459C22⋊C4—232no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C42—216no—
469C42—28no—
470C42—216no—
471C42—28no—
472C42—48no—
473C42—44no—
474C42—84no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23C23×D411yescharacteristic
511C23C23×D412yes—
512C23C22×#4112yes—
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×2−1+412yes—
516C23#58514yes—
517C23#58512yes—
518Q8—416no—
519Q8—88no—
520Q8—88no—
521D4—432no—
522D4—816no—
523D4—816no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—464no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4C2×#12014yes—
589C2×C4C2×#12018yes—
590C2×C4#54818yes—
591C2×C4#55514yescomplemented
592C22—48no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—48no—
597C22—48no—
598C22—48no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22C22×#12011yescharacteristic
608C22C2×#54812yes—
609C22D4×#4114yes—
610C22C2×#55511yescharacteristic
611C22#348411yescharacteristic
612C22#348511yescharacteristic
613C22#348712yes—
614C22#348914yes—
615C22#349612yes—
616C22#349612yes—
617C22#351112yes—
618C22#352414yes—
619C22#354411yescharacteristic
620C22#354512yes—
621C22#355112yes—
622C22#356514yes—
623C4—28no—
624C4—24no—
625C4—28no—
626C4—24no—
627C4—44no—
628C4—44no—
629C4—48no—
630C4—42no—
631C4—82no—
632C4—82no—
633C4—84no—
634C4—84no—
635C2—88no—
636C2#3534711yescharacteristic
637C2#3642812yes—
638C2#4476214yes—
639C2#4527012yes—
640C2#5158612yes—
641C2#5208311yescharacteristic
642C2#5913511yescharacteristic
643C2#5984112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#555C2
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×#2731C2semidirect
3. C2×#2546C2semidirect
4. #16094C2semidirect
5. #15761C2semidirect
6. #15319C2semidirect
7. #15316C2semidirect
8. #14891C2semidirect
9. #14865C2semidirect
10. #14450C2semidirect
11. #14176C2semidirect
12. #14072C2semidirect
13. #14026C2semidirect
591. C2×C4#555semidirect

Maximal subgroups

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

subgroupindex
C22×#5552
C2×#27312
C2×#25462
#160942
#157612
#153192
#153162
#148912
#148652
#144502
#141762
#140722
#140262