Finite Group Databasebeta

← #361623

Subgroups — #361623

Order 512.

3984 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#1964C212yesmaximal, complemented
3C2×#1799C211yesmaximal, characteristic, complemented
4C2×#1782C214yesmaximal, complemented
5C2×C4×#99C211yesmaximal, characteristic, complemented
6#13904C212yesmaximal, complemented
7#10966C214yesmaximal, complemented
8C23×#41C2212yes—
9C2×#547—28no—
10C2×#547—24no—
11C2×#547—24no—
12C2×#547—24no—
13C2×#547—24no—
14C22×C4⋊Q8C411yescharacteristic, complemented
15C22×#123C412yescomplemented
16C22×#123C2212yes—
17C22×C22⋊Q8C412yescomplemented
18C22×#121C411yescharacteristic, complemented
19C22×#121C2211yescharacteristic
20C22×C4×D4C411yescharacteristic, complemented
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C411yescharacteristic, complemented
23C23×C4⋊C4C2211yescharacteristic
24C22×#99C2212yes—
25C22×#99C2218yes—
26C22×#99C2212yes—
27C2×#381C2218yes—
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2212yes—
30#1964—28no—
31#1799—24no—
32#1782—216no—
33C4×#99—24no—
34C2×#342C2212yes—
35C2×#342C2212yes—
36#1445—24no—
37C4⋊#95—24no—
38C22×#41—24no—
39C22×#41—24no—
40C22×#41—28no—
41C22×#41—28no—
42C22×#41—28no—
43C22×#41—28no—
44C22×#41—48no—
45C23×Q8C2×C412yes—
46C23×D4C2×C412yes—
47C23×D4C2×C411yescharacteristic
48C23×D4C2312yes—
49C23×D4C2311yescharacteristic
50C24×C4C2×C412yes—
51C24×C4C2×C412yes—
52C24×C4C2×C411yescharacteristic
53C24×C4C2312yes—
54C24×C4C2311yescharacteristic
55#547—232no—
56#547—232no—
57#547—232no—
58#547—232no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—24no—
63C2×#123—28no—
64C2×#123—28no—
65C2×#123—28no—
66C2×#123—28no—
67C2×#123—28no—
68C2×#123—24no—
69C2×#123—24no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—28no—
73C2×#121—24no—
74C2×#121—24no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121D418yes—
82C2×C4×D4—24no—
83C2×C4×D4—22no—
84C2×C4×D4—24no—
85C2×C4×D4—22no—
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4—24no—
89C2×C4×D4D414yes—
90C2×C4×D4D414yes—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4C2×C412yes—
100C22×C4⋊C4C2×C412yes—
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C4⋊C4C2×C412yes—
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4D414yes—
107C22×C4⋊C4D414yes—
108C22×C22⋊C4C2×C412yes—
109C22×C22⋊C4C2×C412yes—
110C22×C22⋊C4C2×C411yescharacteristic
111C22×C22⋊C4C2×C412yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C42—22no—
116C22×C42C2×C411yescharacteristic
117C2×#99—24no—
118C2×#99—24no—
119C2×#99—24no—
120C2×#99—216no—
121C2×#99—28no—
122C2×#99—216no—
123C2×#99—48no—
124C2×#99D418yescomplemented
125#381—216no—
126#381—48no—
127#380—24no—
128#380—24no—
129C4×C22⋊C4—24no—
130C4×C22⋊C4—24no—
131C2×#95—216no—
132C2×#95—22no—
133C2×#95—22no—
134#342—24no—
135#342—24no—
136C25C22×C412yes—
137C25C22×C411yescharacteristic
138C25C2411yescharacteristic
139C2×#41—232no—
140C2×#41—232no—
141C2×#41—432no—
142C2×#41—432no—
143C2×#41—432no—
144C2×#41—432no—
145C2×#41—416no—
146C2×#41—416no—
147C2×#41—416no—
148C2×#41—48no—
149C2×#41—48no—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—28no—
154C22×Q8—44no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—24no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4C22⋊C414yes—
174C22×D4C2×D418yes—
175C22×D4C2×D412yes—
176C22×D4C2×D412yes—
177C22×D4C2×D414yes—
178C22×D4#4114yes—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C22⋊C414yes—
203C23×C4C22×C412yes—
204C23×C4C22×C412yes—
205C23×C4C22×C412yes—
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D418yes—
212C23×C4C2×D412yes—
213C23×C4C2×D414yes—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C4⋊Q8—216no—
218C4⋊Q8—216no—
219#123—232no—
220#123—232no—
221#123—264no—
222C22⋊Q8—264no—
223#121—216no—
224#121—216no—
225#121—28no—
226#121—28no—
227#121—28no—
228#121—28no—
229C4×D4—216no—
230C4×D4—216no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—28no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C22⋊C412yes—
261C2×C4⋊C4C22⋊C412yes—
262C2×C4⋊C4C22⋊C414yes—
263C2×C4⋊C4C22⋊C412yes—
264C2×C4⋊C4C22⋊C412yes—
265C2×C4⋊C4C22⋊C418yes—
266C2×C4⋊C4C22⋊C414yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4C22⋊C414yes—
289C2×C22⋊C4C22⋊C418yes—
290C2×C22⋊C4C2×D418yes—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D412yes—
293C2×C42—22no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—48no—
297C2×C42—44no—
298C2×C42C22⋊C414yes—
299#99—216no—
300#99—432no—
301#99—48no—
302#95—432no—
303#95—44no—
304#95—44no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—44no—
317C24—44no—
318C24C2×C22⋊C414yes—
319C24C2×C22⋊C412yes—
320C24C4×D414yes—
321C24C23×C411yescharacteristic
322C24C22×D412yes—
323C24C22×D411yescharacteristic
324C24C22×D411yescharacteristic
325C24C22×D412yes—
326C24C22×D412yes—
327C24C2×#4112yes—
328C24H(5,𝔽2)14yes—
329#41—4128no—
330#41—464no—
331#41—464no—
332C2×Q8—216no—
333C2×Q8—416no—
334C2×Q8—416no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—216no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—216no—
349C2×D4—216no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—416no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—44no—
367C2×D4#12118yes—
368C2×D4#12414yes—
369C2×D4#12714yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—24no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4C2×C22⋊C414yes—
435C22×C4C2×C22⋊C414yes—
436C22×C4C2×C22⋊C414yes—
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C412yes—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C4×D414yes—
448C22×C4#12014yes—
449C22×C4#12014yes—
450C22×C4#12014yes—
451C22×C4#12014yes—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—216no—
461C4⋊C4—28no—
462C4⋊C4—88no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—28no—
468C22⋊C4—216no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C42—24no—
477C42—24no—
478C42—88no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—28no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—48no—
498C23—42no—
499C23—42no—
500C23—48no—
501C23—42no—
502C23—42no—
503C23—48no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—88no—
516C23—88no—
517C23#39014yes—
518C23C22×C22⋊C411yescharacteristic
519C23C22×C22⋊C411yescharacteristic
520C23C22×C22⋊C411yescharacteristic
521C23C2×C4×D412yes—
522C23#51812yes—
523C23C2×#12012yes—
524C23C2×#12012yes—
525C23C2×#12112yes—
526C23C2×#12411yescharacteristic
527C23C2×#12711yescharacteristic
528C23#53512yes—
529C23#53512yes—
530C23D4214yes—
531C23D4214yes—
532C23#54614yes—
533C23#54614yes—
534Q8—432no—
535D4—216no—
536D4—28no—
537D4—28no—
538D4—48no—
539D4—432no—
540D4—416no—
541D4—48no—
542D4—48no—
543D4—48no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—432no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—432no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—416no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—416no—
589C2×C4—48no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—816no—
596C2×C4—84no—
597C2×C4#37912yes—
598C2×C4#37914yes—
599C2×C4#37912yes—
600C2×C4#39014yes—
601C2×C4#41014yes—
602C22—21no—
603C22—21no—
604C22—22no—
605C22—22no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—22no—
610C22—28no—
611C22—28no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—42no—
626C22—42no—
627C22—48no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—88no—
632C22—84no—
633C22—84no—
634C22—88no—
635C22C2×#37911yescharacteristic
636C22C2×#39011yescharacteristic
637C22#230911yescharacteristic
638C22D4×C22⋊C412yes—
639C22D4×C22⋊C412yes—
640C22#240814yes—
641C22#243312yes—
642C22#243512yes—
643C22C2×#41012yes—
644C22#288412yes—
645C4—24no—
646C4—22no—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—88no—
654C4—164no—
655C4#178214yescomplemented
656C2—24no—
657C2—22no—
658C2—21no—
659C2—22no—
660C2—21no—
661C2—22no—
662C2—44no—
663C2—84no—
664C2#1390211yescharacteristic
665C2C2×#178211yescharacteristic
666C2#2058711yescharacteristic
667C2#2240614yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#547C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1964C2semidirect
3. C2×#1799C2semidirect
4. C2×#1782C2semidirect
5. C2×C4×#99C2semidirect
6. #13904C2semidirect
7. #10966C2semidirect
14. C22×C4⋊Q8C4semidirect
15. C22×#123C4semidirect
17. C22×C22⋊Q8C4semidirect
18. C22×#121C4semidirect
20. C22×C4×D4C4semidirect
22. C23×C4⋊C4C4semidirect
124. C2×#99D4semidirect
655. C4#1782semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#19642
C2×#17992
C2×#17822
C2×C4×#992
#139042
#109662