Finite Group Databasebeta

← #155771

Subgroups — #155771

Order 512.

3166 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2C2×#1434C211yesmaximal, characteristic, complemented
3C2×#1434C211yesmaximal, characteristic, complemented
4#7557C212yesmaximal, complemented
5#7556C212yesmaximal, complemented
6#7556C212yesmaximal, complemented
7#7555C212yesmaximal
8#7555C212yesmaximal
9#7555C212yesmaximal
10C23×C4⋊C4C2211yescharacteristic
11C23×C22⋊C4C2211yescharacteristic
12C23×C22⋊C4C2214yes—
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#388—24no—
17C2×#388—24no—
18C2×#388C418yescomplemented
19C22×#95C2212yes—
20C22×#95C2212yes—
21C22×#95C2211yescharacteristic
22C22×#95C2211yescharacteristic
23C22×#95C2214yes—
24C22×#95C2214yes—
25C22×#95C2214yes—
26C22×#95C2212yes—
27C22×#95C2212yes—
28C22×#95C2212yes—
29C22×#95C2212yes—
30#1434—24no—
31#1434—24no—
32C26C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C24×C4C2311yescharacteristic
35C24×C4C2314yes—
36C24×C4C2314yes—
37C24×C4C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—44no—
44C22×C4⋊C4C2×C414yes—
45C22×C22⋊C4—24no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—28no—
54C22×C22⋊C4—28no—
55C22×C22⋊C4—28no—
56C22×C22⋊C4—44no—
57C22×C22⋊C4—44no—
58C22×C22⋊C4—44no—
59C22×C22⋊C4—44no—
60C22×C22⋊C4—44no—
61C22×C22⋊C4—44no—
62C22×C22⋊C4C2×C414yes—
63C22×C22⋊C4C2×C414yes—
64C22×C22⋊C4C2×C414yes—
65C22×C42—44no—
66C22×C42C2×C414yes—
67C22×C42D414yes—
68C22×C42D414yes—
69#388—232no—
70C2×#95—24no—
71C2×#95—24no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—28no—
77C2×#95—28no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95—24no—
82C2×#95—28no—
83C2×#95—28no—
84C2×#95—28no—
85C2×#95—44no—
86C2×#95—44no—
87C2×#95—44no—
88C2×#95—44no—
89C2×#95—44no—
90C2×#95—44no—
91C2×#95—44no—
92C2×#95—44no—
93C2×#95—44no—
94C2×#95—44no—
95C2×#95C2×C414yes—
96C2×#95C2×C414yes—
97C25—21no—
98C25—21no—
99C25—22no—
100C25—24no—
101C25—24no—
102C25—24no—
103C25—24no—
104C25—22no—
105C25—22no—
106C25—22no—
107C25—22no—
108C25C22×C412yes—
109C25#4114yes—
110C25C2411yescharacteristic
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4#4118yes—
181C23×C4#4118yes—
182C23×C4#4114yes—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—48no—
187C2×C22⋊C4—216no—
188C2×C22⋊C4—216no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—48no—
195C2×C22⋊C4—48no—
196C2×C22⋊C4—416no—
197C2×C22⋊C4—416no—
198C2×C22⋊C4—416no—
199C2×C22⋊C4—48no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4—416no—
204C2×C22⋊C4—416no—
205C2×C22⋊C4—416no—
206C2×C22⋊C4—416no—
207C2×C42—48no—
208C2×C42—48no—
209C2×C42—48no—
210C2×C42C22⋊C4116yescomplemented
211#95—216no—
212#95—216no—
213#95—28no—
214#95—28no—
215#95—48no—
216#95—48no—
217#95—48no—
218#95—48no—
219#95—416no—
220#95—416no—
221#95—416no—
222C24—22no—
223C24—22no—
224C24—24no—
225C24—24no—
226C24—24no—
227C24—24no—
228C24—24no—
229C24—24no—
230C24—24no—
231C24—24no—
232C24—28no—
233C24—28no—
234C24—28no—
235C24—28no—
236C24—28no—
237C24—28no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—42no—
251C24—42no—
252C24—44no—
253C24—44no—
254C24—44no—
255C24—44no—
256C24—44no—
257C24—44no—
258C24—44no—
259C24—44no—
260C24—44no—
261C24—44no—
262C24—44no—
263C24—44no—
264C24—44no—
265C24—44no—
266C24—44no—
267C24—44no—
268C24—44no—
269C24—44no—
270C24—44no—
271C24—44no—
272C24—44no—
273C24—44no—
274C24#11714yes—
275C24C23×C411yescharacteristic
276C24C22×D411yescharacteristic
277C24C22×D411yescharacteristic
278C24C2×#4112yes—
279C24C2×#4112yes—
280C24C2×#4114yes—
281C24C2×#4114yes—
282C24H(5,𝔽2)14yes—
283C24H(5,𝔽2)14yes—
284C24H(5,𝔽2)12yes—
285C24H(5,𝔽2)12yes—
286C242−1+412yes—
287C242−1+412yes—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4C2×C22⋊C414yes—
445C22×C4C2×C22⋊C414yes—
446C22×C4C2×C22⋊C414yes—
447C22×C4#11714yes—
448C4⋊C4—416no—
449C22⋊C4—232no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—464no—
453C42—416no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—24no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—48no—
464C23—48no—
465C23—48no—
466C23—44no—
467C23—48no—
468C23—48no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—48no—
473C23—44no—
474C23—48no—
475C23—44no—
476C23—48no—
477C23—48no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—42no—
486C23—42no—
487C23—84no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23C22×C22⋊C411yescharacteristic
508C23C2×#11711yescharacteristic
509C23C2×#11711yescharacteristic
510C23C4×#4112yes—
511C23C4×#4112yes—
512C23#51812yes—
513C23#51812yes—
514C23#52012yes—
515C23#52012yes—
516C23#52512yes—
517C23#52512yes—
518C23#52512yes—
519C23#52512yes—
520C23#53212yes—
521C23#53212yes—
522C23#53412yes—
523C23#53412yes—
524C23#53512yes—
525C23#53512yes—
526C23#53511yescharacteristic
527C23#53511yescharacteristic
528C23#53611yescharacteristic
529C23#53611yescharacteristic
530C23#53712yes—
531C23#53712yes—
532C23#53814yes—
533C23#53814yes—
534C23#53814yes—
535C23#53814yes—
536C23#53814yes—
537C23#53814yes—
538C23#53914yes—
539C23#53914yes—
540C23#54014yes—
541C23#54014yes—
542C23#54214yes—
543C23#54214yes—
544C23#55114yes—
545C23#55114yes—
546C23#55114yes—
547C23#55214yes—
548C23#55214yes—
549C23#55214yes—
550C23#55314yes—
551C23#55314yes—
552C23#55314yes—
553C23#55514yes—
554C23#55614yes—
555C23#55914yes—
556C23#56014yes—
557C23#56214yes—
558C23#56214yes—
559C23#56314yes—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—416no—
636C22—82no—
637C22—82no—
638C22—88no—
639C22—88no—
640C22—88no—
641C22—88no—
642C22—88no—
643C22—82no—
644C22—82no—
645C22#228011yescharacteristic
646C22#229711yescharacteristic
647C22#229711yescharacteristic
648C22#230811yescharacteristic
649C22#230911yescharacteristic
650C22#231111yescharacteristic
651C22#231111yescharacteristic
652C22#231812yes—
653C22#231812yes—
654C22#233314yes—
655C22#233714yes—
656C22#234312yes—
657C22#234612yes—
658C22#235312yes—
659C22#235712yes—
660C22#237314yes—
661C22#237614yes—
662C22#260014yes—
663C22#260114yes—
664C22#260214yes—
665C22#260312yes—
666C22#260412yes—
667C22#260914yes—
668C22#261012yes—
669C22#261112yes—
670C22#263112yes—
671C22#263112yes—
672C22#263212yes—
673C22#263312yes—
674C22#263514yes—
675C22#263614yes—
676C22#266114yes—
677C22#266214yes—
678C22#273514yes—
679C22#274114yes—
680C22#274314yes—
681C22#274514yes—
682C22#275114yes—
683C22#275614yes—
684C22#276114yes—
685C22#277014yes—
686C22#277314yes—
687C22#277814yes—
688C22#277914yes—
689C22#278114yes—
690C22#278614yes—
691C22#279314yes—
692C22#280314yes—
693C22#280814yes—
694C22#281312yes—
695C22#281312yes—
696C22#281412yes—
697C22#281412yes—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—88no—
703C4—88no—
704C4—82no—
705C4—82no—
706C2—84no—
707C2#1394011yescharacteristic
708C2#1395711yescharacteristic
709C2#1416611yescharacteristic
710C2#1428811yescharacteristic
711C2#1429011yescharacteristic
712C2#1434511yescharacteristic
713C2#1448414yes—
714C2#1450014yes—
715C2#1465811yescharacteristic
716C2#1701714yes—
717C2#1702014yes—
718C2#1702814yes—
719C2#1704514yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC24C23×C4
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1434C2semidirect
3. C2×#1434C2semidirect
4. #7557C2semidirect
5. #7556C2semidirect
6. #7556C2semidirect
18. C2×#388C4semidirect
210. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
C2×#14342
#75572
#75562
#75552