Finite Group Databasebeta

← #153897

Subgroups — #153897

Order 512.

3486 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#387C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3C2×#1441C212yesmaximal
4C2×#1436C212yesmaximal
5C2×C4×#95C211yesmaximal, characteristic
6#7557C212yesmaximal
7#7554C212yesmaximal
8#7554C212yesmaximal
9#7554C212yesmaximal
10C23×C4⋊C4C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2212yes—
14C23×C42C2212yes—
15C23×C42C2211yescharacteristic
16C2×#387—24no—
17C2×#387—24no—
18C2×#387C418yescomplemented
19C2×C4×C4⋊C4—24no—
20C2×C4×C4⋊C4—28no—
21C22×#95C2214yes—
22C22×#95C2214yes—
23C22×#95C2214yes—
24C22×#95C2212yes—
25C22×#95C2211yescharacteristic
26C22×#95C2211yescharacteristic
27C22×#95C2211yescharacteristic
28C22×#95C2211yescharacteristic
29C22×#95C2212yes—
30C22×#95C2212yes—
31C22×#95C2214yes—
32#1441C4116yescomplemented
33#1436—28no—
34C4×#95—24no—
35C24×C4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C24×C4C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2314yes—
40C24×C4C2314yes—
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C22×C4⋊C4—28no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4C2×C418yes—
56C22×C4⋊C4C2×C414yes—
57C22×C4⋊C4C2×C418yes—
58C22×C4⋊C4C2×C414yes—
59C22×C42—24no—
60C22×C42—24no—
61C22×C42—24no—
62C22×C42—28no—
63C22×C42—24no—
64C22×C42—28no—
65C22×C42—24no—
66C22×C42—24no—
67C22×C42—24no—
68C22×C42—24no—
69C22×C42—44no—
70C22×C42—44no—
71C22×C42C2×C414yes—
72C22×C42D414yes—
73C22×C42D414yes—
74#387—232no—
75C4×C4⋊C4—232no—
76C2×#95—24no—
77C2×#95—24no—
78C2×#95—28no—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—28no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—28no—
93C2×#95—28no—
94C2×#95—28no—
95C2×#95—28no—
96C2×#95—44no—
97C2×#95—44no—
98C2×#95—44no—
99C2×#95—44no—
100C2×#95C2×C4116yes—
101C2×#95C2×C418yes—
102C2×#95C2×C414yes—
103C2×#95C2×C414yes—
104C2×#95C2×C414yes—
105C2×#95C2×C414yes—
106C2×#95C2×C4116yes—
107C2×#95D418yes—
108C2×#95Q818yes—
109C25C2411yescharacteristic
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
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—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C22×C414yes—
175C23×C4C22×C414yes—
176C23×C4C22×C418yes—
177C23×C4C22×C418yes—
178C23×C4C22×C412yes—
179C23×C4C22×C414yes—
180C23×C4C22×C412yes—
181C23×C4C22×C412yes—
182C23×C4C2×D418yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D414yes—
190C23×C4C2×Q818yes—
191C23×C4C2×Q814yes—
192C23×C4#4118yes—
193C23×C4#4118yes—
194C23×C4#4114yes—
195C23×C4#4114yes—
196C23×C4#4114yes—
197C2×C4⋊C4—216no—
198C2×C4⋊C4—232no—
199C2×C4⋊C4—216no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—232no—
203C2×C4⋊C4—216no—
204C2×C4⋊C4—216no—
205C2×C4⋊C4—216no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—216no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C42116yescomplemented
212C2×C42—216no—
213C2×C42—28no—
214C2×C42—216no—
215C2×C42—28no—
216C2×C42—216no—
217C2×C42—216no—
218C2×C42—416no—
219C2×C42—416no—
220C2×C42—416no—
221C2×C42—416no—
222C2×C42—48no—
223C2×C42—48no—
224C2×C42—48no—
225C2×C42—48no—
226C2×C42C22⋊C4116yes—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—232no—
232#95—216no—
233#95—216no—
234#95—216no—
235#95—416no—
236#95—416no—
237#95—416no—
238C24C23×C411yescharacteristic
239C24C23×C412yes—
240C24C22×D411yescharacteristic
241C24C22×D411yescharacteristic
242C24C22×D412yes—
243C24C22×Q812yes—
244C24C2×#4112yes—
245C24C2×#4112yes—
246C24C2×#4114yes—
247C24C2×#4112yes—
248C24C2×#4114yes—
249C24H(5,𝔽2)12yes—
250C24H(5,𝔽2)12yes—
251C242−1+412yes—
252C242−1+412yes—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—28no—
292C22×C4—28no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—41no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
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—41no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
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×C4C2×C4214yes—
422C22×C4C2×C4214yes—
423C22×C4C2×C4214yes—
424C22×C4C2×C22⋊C414yes—
425C22×C4C2×C22⋊C414yes—
426C22×C4C2×C22⋊C414yes—
427C22×C4#11714yes—
428C22×C4C4×D4116yes—
429C22×C4C4×D418yes—
430C22×C4C4×D418yes—
431C22×C4C4×D418yes—
432C22×C4C4×Q8116yes—
433C22×C4C4×Q818yes—
434C22×C4#12014yes—
435C22×C4#12118yes—
436C22×C4C22⋊Q818yes—
437C22×C4#12314yes—
438C22×C4#12414yes—
439C22×C4C4⋊Q814yes—
440C4⋊C4—264no—
441C4⋊C4—232no—
442C4⋊C4—416no—
443C42—232no—
444C42—432no—
445C42—432no—
446C23C22×C4211yescharacteristic
447C23C22×C22⋊C411yescharacteristic
448C23C2×#11711yescharacteristic
449C23C2×C4×D412yes—
450C23C2×C4×D412yes—
451C23C2×C4×D412yes—
452C23C2×C4×D414yes—
453C23C2×C4×Q814yes—
454C23C2×C4×Q812yes—
455C23C4×#4112yes—
456C23C4×#4114yes—
457C23C4×#4114yes—
458C23C4×#4112yes—
459C23#51812yes—
460C23#51912yes—
461C23#52012yes—
462C23#52012yes—
463C23#52012yes—
464C23#52012yes—
465C23C2×#12011yescharacteristic
466C23C2×#12112yes—
467C23C2×C22⋊Q812yes—
468C23C2×#12311yescharacteristic
469C23#52512yes—
470C23#52512yes—
471C23#52514yes—
472C23C2×#12411yescharacteristic
473C23#52914yes—
474C23#52914yes—
475C23C2×C4⋊Q811yescharacteristic
476C23#53212yes—
477C23#53212yes—
478C23#53314yes—
479C23#53511yescharacteristic
480C23#53511yescharacteristic
481C23#53611yescharacteristic
482C23#53611yescharacteristic
483C23#53712yes—
484C23#53712yes—
485C23#54012yes—
486C23#54112yes—
487C23#54212yes—
488C23#54212yes—
489C23#54614yes—
490C23#54614yes—
491C23#54714yes—
492C23D4×Q814yes—
493C23#55114yes—
494C23#55114yes—
495C23#55214yes—
496C23#55214yes—
497C23#55214yes—
498C23#55214yes—
499C23#55314yes—
500C23#55314yes—
501C23#55414yes—
502C23#55414yes—
503C23#55614yes—
504C23#55714yes—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—216no—
508C2×C4—216no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—216no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—216no—
526C2×C4—28no—
527C2×C4—216no—
528C2×C4—216no—
529C2×C4—216no—
530C2×C4—216no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—216no—
536C2×C4—216no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—84no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4C4×C22⋊C418yes—
590C2×C4#38618yes—
591C2×C4#39118yes—
592C22C2×C4×C22⋊C411yescharacteristic
593C22C42×D412yes—
594C22C42×Q812yes—
595C22#227412yes—
596C22C2×#38611yescharacteristic
597C22#229012yes—
598C22C2×#39111yescharacteristic
599C22#229512yes—
600C22#229711yescharacteristic
601C22#229711yescharacteristic
602C22C4×#12012yes—
603C22C4×#12114yes—
604C22C4×#12312yes—
605C22C4×C22⋊Q814yes—
606C22C4×#12412yes—
607C22C4×C4⋊Q812yes—
608C22#231111yescharacteristic
609C22#231111yescharacteristic
610C22#231412yes—
611C22#232312yes—
612C22#232312yes—
613C22#232412yes—
614C22#232412yes—
615C22#235914yes—
616C22#235914yes—
617C22#236014yes—
618C22#236014yes—
619C22#236414yes—
620C22#236514yes—
621C22#236814yes—
622C22#237014yes—
623C22#237014yes—
624C22#237114yes—
625C22#237314yes—
626C22#237314yes—
627C22#241112yes—
628C22#241312yes—
629C22#241714yes—
630C22#243012yes—
631C22#243411yescharacteristic
632C22#243511yescharacteristic
633C22#245714yes—
634C22#246214yes—
635C22#246514yes—
636C22#246814yes—
637C22#249812yes—
638C22#250211yescharacteristic
639C22#250311yescharacteristic
640C22#251514yes—
641C22#252014yes—
642C22#252214yes—
643C22#252414yes—
644C22#256212yes—
645C22#256512yes—
646C22#256612yes—
647C22#258412yes—
648C22#260114yes—
649C22#260312yes—
650C22#260412yes—
651C4—28no—
652C4—28no—
653C4—28no—
654C4—416no—
655C4—416no—
656C4—44no—
657C4—44no—
658C4—82no—
659C4—82no—
660C2C4×#38611yescharacteristic
661C2C4×#39111yescharacteristic
662C2#1377711yescharacteristic
663C2#1382514yes—
664C2#1413511yescharacteristic
665C2#1413911yescharacteristic
666C2#1414414yes—
667C2#1416011yescharacteristic
668C2#1416611yescharacteristic
669C2#1418612yes—
670C2#1419012yes—
671C2#1419814yes—
672C2#1420314yes—
673C2#1424412yes—
674C2#1425012yes—
subgroupG/H
center, Frattini, socle, prime radical, cosocleC25C24
commutatorC23C22×C42
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C2×#387C4semidirect
32. #1441C4semidirect
211. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C22×#3872
C22×C4×C4⋊C42
C2×#14412
C2×#14362
C2×C4×#952
#75572
#75542