Finite Group Databasebeta

← #379227

Subgroups — #379227

Order 512.

1029 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23730C211yesmaximal, characteristic
2C2×#1937C211yesmaximal, characteristic
3C2×#1929C211yesmaximal, characteristic
4#21051C211yesmaximal, characteristic
5C2×#1922C211yesmaximal, characteristic, complemented
6#21025C211yesmaximal, characteristic, complemented
7C2×#1772C211yesmaximal, characteristic, complemented
8#11376C211yesmaximal, characteristic, complemented
9#11326C211yesmaximal, characteristic, complemented
10#11125C211yesmaximal, characteristic
11#10929C211yesmaximal, characteristic
12#10923C211yesmaximal, characteristic
13#10859C211yesmaximal, characteristic, complemented
14#10839C211yesmaximal, characteristic, complemented
15C4×#336C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2297C2211yescharacteristic
33#2285C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#2097C411yescharacteristic, complemented
36#2097C411yescharacteristic, complemented
37#2096C411yescharacteristic, complemented
38#2095C411yescharacteristic, complemented
39#2094C411yescharacteristic, complemented
40#2093C411yescharacteristic, complemented
41#2092C411yescharacteristic, complemented
42#2092C411yescharacteristic, complemented
43#1937—24no—
44#1929—24no—
45#1924—22no—
46#1923—22no—
47#1922—24no—
48#1917—22no—
49#1916—22no—
50#1772—24no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62#1728C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42—22no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#108—22no—
84C2×#108—22no—
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—41no—
89#427—22no—
90#427—22no—
91#427—22no—
92#427—22no—
93C2×#107C2×C411yescharacteristic
94C2×#107C2×C411yescharacteristic
95C2×#107C2×C411yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×#106C2×C411yescharacteristic
101#424—22no—
102#424—22no—
103#423—22no—
104#423—41no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—22no—
111C2×#104—22no—
112C2×#104—22no—
113C2×#104—41no—
114#416—22no—
115#416—22no—
116#416—22no—
117#416—22no—
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#103C2×C411yescharacteristic
121C2×#103C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125C2×#102C2×C411yescharacteristic
126#408—22no—
127#408—22no—
128#408—22no—
129#408—41no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—22no—
132C2×C22⋊C8—21no—
133#405—22no—
134#405—22no—
135#405—22no—
136#405—22no—
137C2×#97—21no—
138C2×#97—21no—
139C2×C4×C8—22no—
140C2×C4×C8—22no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8—22no—
144C2×C4×C8—22no—
145C2×C4×C8—41no—
146C2×C4×C8D414yes—
147#391C2×C411yescharacteristic
148#391C2×C411yescharacteristic
149#390C2×C411yescharacteristic
150#390C2×C411yescharacteristic
151#386—22no—
152#384C2×C411yescharacteristic
153#384C2×C411yescharacteristic
154#382C2×C411yescharacteristic
155#382C2×C411yescharacteristic
156#380—22no—
157C4×C4⋊C4—22no—
158C4×C22⋊C4—22no—
159C43—22no—
160C43—41no—
161#341—22no—
162#341—22no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—22no—
167#339—22no—
168#339—22no—
169#336—22no—
170#336—22no—
171#336—22no—
172#336—22no—
173#336—22no—
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176#336C2×C411yescharacteristic
177#336C2×C411yescharacteristic
178#336D412yes—
179#336D412yes—
180#336D414yes—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4C22×C411yescharacteristic
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4#4112yes—
200C23×C4C2411yescharacteristic
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16#4112yes—
224C2×M16#4112yes—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8C22×C411yescharacteristic
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C22×C411yescharacteristic
251C22×C8C2×D412yes—
252C22×C8C2×D412yes—
253C22×C8C2×D412yes—
254C22×C8C2×D412yes—
255C22×C8C2×D412yes—
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D411yescharacteristic
258C22×C8#4112yes—
259C22×C8#4112yes—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—28no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42#4112yes—
309C2×C42#4112yes—
310#108—28no—
311#108—44no—
312#107—22no—
313#107—22no—
314#107—22no—
315#107—22no—
316#106—22no—
317#106—22no—
318#106—22no—
319#106—22no—
320C4⋊C8—44no—
321C4⋊C8—42no—
322C4⋊C8—42no—
323#104—44no—
324#104—44no—
325#103—22no—
326#103—22no—
327#103—22no—
328#103—22no—
329#102—22no—
330#102—22no—
331#102—22no—
332#102—22no—
333C22⋊C8—44no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336#97—42no—
337#97—42no—
338C4×C8—28no—
339C4×C8—44no—
340C4×C8—42no—
341C4×C8—42no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346C24—21no—
347C24C2×#4111yescharacteristic
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C22×C42−1+411yescharacteristic
425M16—22no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8C4×D412yes—
485C2×C8C4×D412yes—
486C2×C8C4×D412yes—
487C2×C8C4×D412yes—
488C2×C8#12712yes—
489C2×C8#12712yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—44no—
503C4⋊C4—82no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—416no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42#12314yes—
533C23—22no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23C2×C4×D411yescharacteristic
541C23#53211yescharacteristic
542C23#53211yescharacteristic
543C23#55111yescharacteristic
544C23#55211yescharacteristic
545C23#55211yescharacteristic
546C23#55611yescharacteristic
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—81no—
556D4—81no—
557D4—81no—
558D4—81no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4#44314yes—
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C4×#4111yescharacteristic
599C2×C4C4×#4111yescharacteristic
600C2×C4#51911yescharacteristic
601C2×C4#52011yescharacteristic
602C2×C4C2×#12311yescharacteristic
603C2×C4#52511yescharacteristic
604C2×C4#52911yescharacteristic
605C2×C4#52911yescharacteristic
606C2×C4C2×#12711yescharacteristic
607C2×C4#53211yescharacteristic
608C2×C4#53611yescharacteristic
609C2×C4#53711yescharacteristic
610C2×C4#54111yescharacteristic
611C2×C4#54211yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54611yescharacteristic
614C2×C4#54711yescharacteristic
615C2×C4#54711yescharacteristic
616C2×C4#54711yescharacteristic
617C2×C4#54711yescharacteristic
618C2×C4#54811yescharacteristic
619C2×C4#54811yescharacteristic
620C2×C4#55111yescharacteristic
621C2×C4#55211yescharacteristic
622C2×C4#55211yescharacteristic
623C2×C4#55611yescharacteristic
624C8—22no—
625C8—22no—
626C8—22no—
627C8—22no—
628C8—22no—
629C8—22no—
630C8—42no—
631C8—42no—
632C8—42no—
633C8—81no—
634C8—81no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—82no—
644C22—81no—
645C22—82no—
646C22—81no—
647C22—81no—
648C22C4×#12711yescharacteristic
649C22#235911yescharacteristic
650C22#236211yescharacteristic
651C22#249811yescharacteristic
652C22#251911yescharacteristic
653C22#252111yescharacteristic
654C22#260411yescharacteristic
655C22C2×#44312yes—
656C22#295612yes—
657C4—28no—
658C4—24no—
659C4—44no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4C4×#12311yescharacteristic
669C4#232311yescharacteristic
670C4#235911yescharacteristic
671C4#236211yescharacteristic
672C4#250111yescharacteristic
673C4#251911yescharacteristic
674C4#252111yescharacteristic
675C4#254011yescharacteristic
676C2—21no—
677C2—21no—
678C2—21no—
679C2—21no—
680C2—161no—
681C2#1423111yescharacteristic
682C2#2170511yescharacteristic
683C2#2171011yescharacteristic
684C2#2294812yes—
685C2#2308012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#1922C2semidirect
6. #21025C2semidirect
7. C2×#1772C2semidirect
8. #11376C2semidirect
9. #11326C2semidirect
13. #10859C2semidirect
14. #10839C2semidirect
15. C4×#336C2semidirect
35. #2097C4semidirect
36. #2097C4semidirect
37. #2096C4semidirect
38. #2095C4semidirect
39. #2094C4semidirect
40. #2093C4semidirect
41. #2092C4semidirect
42. #2092C4semidirect

Maximal subgroups

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

subgroupindex
#237302
C2×#19372
C2×#19292
#210512
C2×#19222
#210252
C2×#17722
#113762
#113262
#111252
#109292
#109232
#108592
#108392
C4×#3362