Finite Group Databasebeta

← #725995

Subgroups — #725995

Order 512.

4490 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic, complemented
3C2×#2536C211yesmaximal, characteristic, complemented
4C2×#2397C211yesmaximal, characteristic
5#16682C214yesmaximal, complemented
6#15466C214yesmaximal, complemented
7#15264C214yesmaximal, complemented
8#15177C214yesmaximal, complemented
9#14795C214yesmaximal, complemented
10#14282C211yesmaximal, characteristic, complemented
11#13905C211yesmaximal, characteristic, complemented
12#13872C214yesmaximal, complemented
13#13786C211yesmaximal, characteristic, complemented
14C2×#554—28no—
15C2×#554—28no—
16C2×#554—28no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2212yes—
19C22×#124C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2212yes—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2212yes—
26C22×C4×D4C2214yes—
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4C2212yes—
29#2547—28no—
30#2536—28no—
31#2397—28no—
32C2×#399C2212yes—
33C2×#399C2212yes—
34C2×#398C2212yes—
35C2×#397C2214yes—
36C2×#394C2212yes—
37C2×#392C2212yes—
38C2×#392C2212yes—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#388C2214yes—
42C2×#386C2218yes—
43C2×#382C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49#1446C2214yes—
50#1445C2212yes—
51#1445C2214yes—
52#1444C2218yes—
53#1441C2218yes—
54#1439C2218yes—
55#1439C2218yes—
56#1439C2218yes—
57#1438C2218yes—
58#1438C2212yes—
59#1437C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2214yes—
63#1434C2214yes—
64C22⋊#95C2218yes—
65C23×Q8C2311yescharacteristic
66C23×Q8C2312yes—
67C23×D4C2311yescharacteristic
68C23×D4C2311yescharacteristic
69C23×D4C2314yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2314yes—
73#554—2128no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8D418yescomplemented
77C2×#125—28no—
78C2×#125—28no—
79C2×#125—28no—
80C2×#124—28no—
81C2×#124—48no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—48no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8D418yescomplemented
93C2×C4×D4—28no—
94C2×C4×D4—216no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—216no—
103C2×C4×D4—48no—
104C2×C4×D4D418yes—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2318yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#399—28no—
134#399—28no—
135#398—28no—
136#397—216no—
137#394—28no—
138#392—28no—
139#392—28no—
140#391—28no—
141#391—28no—
142#388—216no—
143#386—416no—
144#382—24no—
145#380—28no—
146#380—216no—
147#380—48no—
148C4×C4⋊C4—24no—
149#376—24no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2312yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C25C2412yes—
163C25C2412yes—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8C2×D414yes—
173C22×Q8C2×D418yes—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—28no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4C2×D414yes—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—48no—
203C23×C4—48no—
204C23×C4C2×D418yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—232no—
217#125—264no—
218#124—432no—
219C22⋊Q8—264no—
220C22⋊Q8—2128no—
221C22⋊Q8—432no—
222C4×Q8—232no—
223C4×D4—264no—
224C4×D4—232no—
225C4×D4—464no—
226C4×D4—416no—
227C2×C4⋊C4—264no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D418yes—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4C2×D418yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—44no—
294C2×C42C2×D418yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42#4118yes—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—216no—
304#95—24no—
305#95—24no—
306#95—216no—
307#95—48no—
308#95—48no—
309#95—48no—
310#95—48no—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—22no—
327C24—22no—
328C24C22×D414yes—
329C24H(5,𝔽2)14yes—
330C242−1+418yes—
331C24C2511yescharacteristic
332C2×Q8—28no—
333C2×Q8—216no—
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—88no—
341C2×Q8—88no—
342C2×D4—216no—
343C2×D4—216no—
344C2×D4—216no—
345C2×D4—28no—
346C2×D4—232no—
347C2×D4—28no—
348C2×D4—416no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—416no—
352C2×D4—44no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—44no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—232no—
384C22×C4—216no—
385C22×C4—232no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—416no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—816no—
412C22×C4—816no—
413C22×C4C22×D414yes—
414C22×C4C22×D418yes—
415C22×C4C22×D414yes—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C2×#4118yes—
427C22×C4C2×#4114yes—
428C22×C4H(5,𝔽2)14yes—
429C4⋊C4—232no—
430C4⋊C4—232no—
431C4⋊C4—28no—
432C4⋊C4—216no—
433C4⋊C4—28no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—416no—
437C4⋊C4—416no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C22⋊C4—232no—
442C22⋊C4—216no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—432no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—816no—
450C42—216no—
451C42—216no—
452C42—48no—
453C42#120116yes—
454C23—24no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23#53618yes—
498C23C23×D411yescharacteristic
499C23C23×D411yescharacteristic
500C23C23×D411yescharacteristic
501C23C22×#4112yes—
502C23C2×H(5,𝔽2)12yes—
503C23C2×H(5,𝔽2)12yes—
504C23C2×2−1+414yes—
505C23#58512yes—
506Q8—416no—
507Q8—816no—
508D4—232no—
509D4—416no—
510D4—464no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
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—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—432no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—832no—
562C2×C4—816no—
563C2×C4C2×#12014yes—
564C2×C4C2×#12014yes—
565C2×C4C2×#12014yes—
566C2×C4#53214yes—
567C2×C4#53514yes—
568C2×C4#54618yes—
569C2×C4#55018yes—
570C22—28no—
571C22—216no—
572C22—48no—
573C22—48no—
574C22—48no—
575C22—48no—
576C22—44no—
577C22—44no—
578C22—48no—
579C22—48no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—48no—
584C22—82no—
585C22—84no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—84no—
590C22C22×#12011yescharacteristic
591C22C2×#53211yescharacteristic
592C22C2×#53511yescharacteristic
593C22C2×#53612yes—
594C22C2×#54612yes—
595C22C2×#55012yes—
596C22#348411yescharacteristic
597C22#348511yescharacteristic
598C22#348612yes—
599C22#348612yes—
600C22#348812yes—
601C22#348812yes—
602C22#349214yes—
603C22#349514yes—
604C22#353614yes—
605C22#354114yes—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—48no—
612C4—48no—
613C4—84no—
614C4—84no—
615C4—82no—
616C4#243118yes—
617C2—48no—
618C2—84no—
619C2C2×#243111yescharacteristic
620C2#3524211yescharacteristic
621C2#3551211yescharacteristic
622C2#3592612yes—
623C2#3783814yes—
624C2#4281012yes—
625C2#4513812yes—
626C2#4535612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2397C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#554C2semidirect
2. C2×#2547C2semidirect
3. C2×#2536C2semidirect
5. #16682C2semidirect
6. #15466C2semidirect
7. #15264C2semidirect
8. #15177C2semidirect
9. #14795C2semidirect
10. #14282C2semidirect
11. #13905C2semidirect
12. #13872C2semidirect
13. #13786C2semidirect
76. C2×C4⋊Q8D4semidirect
92. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25472
C2×#25362
C2×#23972
#166822
#154662
#152642
#151772
#147952
#142822
#139052
#138722
#137862