Finite Group Databasebeta

← #379975

Subgroups — #379975

Order 512.

1742 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×#433C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic
3C2×#1959C211yesmaximal, characteristic, complemented
4C2×#1958C211yesmaximal, characteristic, complemented
5#14340C211yesmaximal, characteristic, complemented
6#11077C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10969C212yesmaximal
10#10843C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#433—24no—
17C2×#433—24no—
18C2×#433—24no—
19C22×C4⋊C8C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2212yes—
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2620—24no—
28#2620C418yescomplemented
29C2×#396—22no—
30C2×#394—22no—
31C2×#391C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37#1959—24no—
38#1958—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1441—22no—
44#1439—22no—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124C2×C414yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8C2×C414yes—
62C2×#121—22no—
63C2×#121—42no—
64C2×#121—42no—
65C2×#121C2×C414yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117C2×C414yes—
71C2×#117D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D412yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#433—232no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—42no—
101C2×#102—42no—
102C2×#102—42no—
103C2×#102—42no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97D414yescomplemented
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8D414yescomplemented
113#396—216no—
114#394—216no—
115#391—22no—
116#391C2×C414yes—
117#390—22no—
118#390C2×C414yes—
119#384—22no—
120#384—22no—
121#384—22no—
122#384C2×C414yes—
123#382—22no—
124#382—22no—
125#380—42no—
126#380—42no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132#340—42no—
133#340—42no—
134#340—42no—
135#340—42no—
136C25—21no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C22×C412yes—
148C22×D4—24no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—82no—
161C22×D4—82no—
162C22×D4C22×C412yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4C22×C412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8C2×D414yes—
195C22×C8C2×D414yes—
196#124—28no—
197#124—28no—
198#124—44no—
199#124—44no—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202#121—48no—
203#121—48no—
204#117—28no—
205#117—28no—
206#117—28no—
207#117—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4C22⋊C418yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—82no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4—82no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4C22⋊C418yes—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42C22×C412yes—
263C2×C42C22×C412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270C4⋊C8—216no—
271#103—48no—
272#103—48no—
273#102—48no—
274#102—48no—
275C22⋊C8—216no—
276#97—28no—
277C4×C8—28no—
278#95—28no—
279#95—44no—
280#95—44no—
281#95—82no—
282#95—82no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24C2×C22⋊C412yes—
305C24C22×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—48no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—88no—
340C2×D4—84no—
341C2×D4—84no—
342C2×D4—88no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4C2×C22⋊C414yes—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C414yes—
409C22×C4#12114yes—
410C22×C4#12114yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C42−1+412yes—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C4⋊C4—216no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—48no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C22⋊C4—216no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—416no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42C4×D414yes—
462C42C4×D414yes—
463C42#12014yes—
464C42#12314yes—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23C22×C22⋊C411yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×#12111yescharacteristic
514C23#53511yescharacteristic
515C23#53511yescharacteristic
516C23#53511yescharacteristic
517C23#53611yescharacteristic
518C23#57612yes—
519C23#57712yes—
520C23#57712yes—
521C23#57812yes—
522Q8—44no—
523Q8—44no—
524Q8—82no—
525Q8—82no—
526D4—44no—
527D4—44no—
528D4—816no—
529D4—82no—
530D4—82no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4#38618yes—
587C2×C4C2×C4×D411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4#51812yes—
590C2×C4#52012yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4C2×#12311yescharacteristic
593C2×C4#53512yes—
594C2×C4#53512yes—
595C2×C4#53512yes—
596C2×C4#53712yes—
597C2×C4#53712yes—
598C2×C4#53712yes—
599C2×C4#54012yes—
600C2×C4#54012yes—
601C2×C4#54212yes—
602C2×C4#54212yes—
603C8—44no—
604C8—44no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—84no—
619C22—82no—
620C22—84no—
621C22—162no—
622C22—161no—
623C22—161no—
624C22—161no—
625C22—161no—
626C22—162no—
627C22C2×#38611yescharacteristic
628C22#230911yescharacteristic
629C22#231111yescharacteristic
630C22#261911yescharacteristic
631C22#261911yescharacteristic
632C22#262011yescharacteristic
633C22#262011yescharacteristic
634C22#289712yes—
635C22#289812yes—
636C22#306012yes—
637C22#306112yes—
638C22#306212yes—
639C22#306312yes—
640C22#322011yescharacteristic
641C22#322111yescharacteristic
642C22#322211yescharacteristic
643C22#322311yescharacteristic
644C22#322412yes—
645C22#322412yes—
646C22#322512yes—
647C22#322512yes—
648C22#324312yes—
649C22#324412yes—
650C4—28no—
651C4—42no—
652C4—44no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—162no—
657C4—162no—
658C4#232412yes—
659C4#232412yes—
660C4#261512yes—
661C4#262412yes—
662C2—82no—
663C2—162no—
664C2#1425711yescharacteristic
665C2#2097612yes—
666C2#2181311yescharacteristic
667C2#2181411yescharacteristic
668C2#2187112yes—
669C2#2981312yes—
670C2#2981611yescharacteristic
671C2#2981711yescharacteristic
672C2#2984911yescharacteristic
673C2#2985011yescharacteristic
674C2#2985712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#433C2semidirect
3. C2×#1959C2semidirect
4. C2×#1958C2semidirect
5. #14340C2semidirect
6. #11077C2semidirect
7. #11076C2semidirect
8. #10970C2semidirect
10. #10843C2semidirect
11. #10841C2semidirect
28. #2620C4semidirect
109. C2×#97D4semidirect
112. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#26202
C2×#19592
C2×#19582
#143402
#110772
#110762
#109702
#109692
#108432
#108412