Finite Group Databasebeta

← #621874

Subgroups — #621874

Order 512.

3369 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic
2C2×#2614C211yesmaximal, characteristic
3C2×#2310C211yesmaximal, characteristic, complemented
4C22×#391C211yesmaximal, characteristic, complemented
5#15883C214yesmaximal, complemented
6#15853C214yesmaximal, complemented
7#15852C214yesmaximal, complemented
8#15851C214yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14912C212yesmaximal
11#14684C211yesmaximal, characteristic, complemented
12#14263C211yesmaximal, characteristic, complemented
13#14256C211yesmaximal, characteristic
14#14135C212yesmaximal
15#13962C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#123C2211yescharacteristic
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2853—28no—
24#2614—28no—
25C2×#395C2212yes—
26C2×#395C2212yes—
27C2×#393C2212yes—
28C2×#393C2212yes—
29#2310—28no—
30C2×#391—28no—
31C2×#391C2214yes—
32C2×#391C2214yes—
33C2×#391C2214yes—
34C2×#391C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#381C2212yes—
45C2×#381C2211yescharacteristic
46C2×#381C2211yescharacteristic
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1445C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C23×Q8—24no—
76C23×Q8—24no—
77C23×Q8—28no—
78C23×Q8—44no—
79C23×Q8D418yes—
80C23×Q8C2312yes—
81C23×Q8C2312yes—
82C23×D4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#123—24no—
89C2×#123—24no—
90C2×#123—24no—
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—28no—
94C2×#123—44no—
95C2×#117—28no—
96C2×#117—28no—
97C2×#117—44no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C42—24no—
120C22×C42—24no—
121C22×C42—24no—
122C22×C42C2314yes—
123C22×C42C2314yes—
124#395—28no—
125#395—28no—
126#393—28no—
127#393—28no—
128#391—216no—
129#391—216no—
130#391—216no—
131#391—216no—
132#388—216no—
133#387—216no—
134#386—28no—
135#386—44no—
136#385—28no—
137#385—44no—
138#381—24no—
139#381—24no—
140#381—24no—
141#381—24no—
142#381—28no—
143#381D418yes—
144C2×#95—28no—
145C2×#95—28no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—232no—
175C22×Q8—232no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—416no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—432no—
190C22×Q8—432no—
191C22×Q8C2×D418yes—
192C22×Q8C2×D418yes—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4#4114yes—
222C23×C4#4118yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236#123—416no—
237#123—416no—
238#123—416no—
239#117—432no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—416no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×D418yes—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—416no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D4116yes—
325C24—24no—
326C24—24no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24C22×D412yes—
335C24C22×D412yes—
336C24C2×#4112yes—
337C24C2×#4114yes—
338C24H(5,𝔽2)14yes—
339C24C2511yescharacteristic
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—216no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—4128no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—48no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4#121116yes—
447C22×C4#123116yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D418yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C42−1+418yes—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—88no—
466C22⋊C4—216no—
467C22⋊C4—416no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—88no—
477C42—416no—
478C42—416no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23C2×#12114yes—
501C23C2×#12314yes—
502C23#53414yes—
503C23#54618yes—
504C23#55118yes—
505C23C23×D411yescharacteristic
506C23C23×D411yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4112yes—
509C23C2×H(5,𝔽2)12yes—
510C23C2×2−1+414yes—
511C23#58514yes—
512Q8—432no—
513Q8—432no—
514D4—88no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—232no—
518C2×C4—232no—
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—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—432no—
542C2×C4—44no—
543C2×C4—416no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
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—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—816no—
591C2×C4C2×#12118yes—
592C2×C4C2×#12318yes—
593C22—24no—
594C22—28no—
595C22—216no—
596C22—28no—
597C22—24no—
598C22—216no—
599C22—81no—
600C22—82no—
601C22—84no—
602C22—81no—
603C22—82no—
604C22—81no—
605C22—84no—
606C22#241218yes—
607C22C22×#12111yescharacteristic
608C22C22×#12311yescharacteristic
609C22C2×#53411yescharacteristic
610C22#346114yes—
611C22C2×#54612yes—
612C22C2×#55112yes—
613C22#348612yes—
614C22#348712yes—
615C22#348914yes—
616C22#349514yes—
617C22#350714yes—
618C22#354414yes—
619C22#354514yes—
620C4—28no—
621C4—28no—
622C4—48no—
623C4—42no—
624C4—42no—
625C4—48no—
626C4—81no—
627C4—84no—
628C4—82no—
629C4—81no—
630C4—82no—
631C4—81no—
632C4—84no—
633C2—28no—
634C2—161no—
635C2C2×#241211yescharacteristic
636C2#3518212yes—
637C2#3566112yes—
638C2#3718711yescharacteristic
639C2#3725811yescharacteristic
640C2#3777914yes—
641C2#3796714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2310C2semidirect
4. C22×#391C2semidirect
5. #15883C2semidirect
6. #15853C2semidirect
7. #15852C2semidirect
8. #15851C2semidirect
9. #14919C2semidirect
11. #14684C2semidirect
12. #14263C2semidirect
15. #13962C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×#26142
C2×#23102
C22×#3912
#158832
#158532
#158522
#158512
#149192
#149122
#146842
#142632
#142562
#141352
#139622