Finite Group Databasebeta

← #384469

Subgroups — #384469

Order 512.

769 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21311C211yesmaximal, characteristic
2#21304C211yesmaximal, characteristic
3#20774C211yesmaximal, characteristic
4#11386C211yesmaximal, characteristic
5#11317C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic
7#11291C211yesmaximal, characteristic
8#11258C211yesmaximal, characteristic
9#11233C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic
11#10883C211yesmaximal, characteristic
12#10877C211yesmaximal, characteristic
13#10819C211yesmaximal, characteristic
14#10809C211yesmaximal, characteristic
15#10808C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1989C411yescharacteristic, complemented
24#1989C411yescharacteristic, complemented
25#1989C411yescharacteristic, complemented
26#1989C411yescharacteristic, complemented
27#1988C411yescharacteristic, complemented
28#1988C411yescharacteristic, complemented
29#1987C411yescharacteristic, complemented
30#1987C411yescharacteristic, complemented
31#1985—21no—
32#1985—21no—
33#1984—21no—
34#1984—21no—
35#1840—21no—
36#1840—21no—
37#1839—21no—
38#1839—21no—
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45#1735C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1728C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75#533—22no—
76#533—22no—
77#533—41no—
78#533—41no—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#125C2×C411yescharacteristic
82C2×#125C2×C411yescharacteristic
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#108—22no—
95C2×#108—41no—
96C2×#108—41no—
97#424—22no—
98#424—41no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8—41no—
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8C2×C411yescharacteristic
105C2×C4⋊C8C2×C411yescharacteristic
106C2×C4⋊C8C2×C411yescharacteristic
107#405—22no—
108#405—41no—
109C2×#97—21no—
110C2×#97—22no—
111C2×#97—41no—
112C2×#97—41no—
113C2×C4×C8—21no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376—21no—
121C2×#95—21no—
122C2×#95—21no—
123C43—21no—
124#341—22no—
125#341—41no—
126#341—41no—
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#339—22no—
144#339—22no—
145#339—22no—
146#339—22no—
147#339—41no—
148#339—41no—
149#339—41no—
150#339—41no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—21no—
167#336—22no—
168#336—41no—
169#336—41no—
170#336D411yescharacteristic
171#336D411yescharacteristic
172#336D411yescharacteristic
173#336D411yescharacteristic
174C22×Q8—21no—
175C22×Q8—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4#4111yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4C2411yescharacteristic
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16#4112yes—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
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×C8C22×C411yescharacteristic
234C22×C8C22×C411yescharacteristic
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—41no—
244C4⋊Q8—41no—
245#125—22no—
246#125—22no—
247#125—41no—
248#125—41no—
249C22⋊Q8—42no—
250C22⋊Q8—42no—
251C22⋊Q8—42no—
252C22⋊Q8—42no—
253C4×Q8—42no—
254C4×Q8—42no—
255C4×Q8—42no—
256C4×Q8—42no—
257#117—21no—
258#117—21no—
259#117—21no—
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#4112yes—
272#117#4112yes—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42#4112yes—
319#108—44no—
320C4⋊C8—41no—
321C4⋊C8—41no—
322C4⋊C8—41no—
323C4⋊C8—41no—
324C4⋊C8—42no—
325C4⋊C8—42no—
326#97—42no—
327#97—44no—
328C4×C8—42no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—42no—
334#95—42no—
335C24C2×#4111yescharacteristic
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—81no—
341C2×Q8—81no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C42−1+411yescharacteristic
386C22×C42−1+411yescharacteristic
387C22×C42−1+411yescharacteristic
388C22×C42−1+411yescharacteristic
389C22×C42−1+411yescharacteristic
390M16—22no—
391M16—22no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4#11712yes—
479C4⋊C4#11712yes—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—22no—
499C42—22no—
500C42—42no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—81no—
515C42—81no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23C2×C4×D411yescharacteristic
521C23#54111yescharacteristic
522C23#54211yescharacteristic
523C23#55111yescharacteristic
524C23#55111yescharacteristic
525C23#55211yescharacteristic
526C23#55611yescharacteristic
527Q8—84no—
528Q8—84no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4#44412yes—
583C2×C4C2×#11711yescharacteristic
584C2×C4C4×#4111yescharacteristic
585C2×C4#51811yescharacteristic
586C2×C4#51911yescharacteristic
587C2×C4#51911yescharacteristic
588C2×C4#52011yescharacteristic
589C2×C4#53611yescharacteristic
590C2×C4#53711yescharacteristic
591C2×C4#53811yescharacteristic
592C2×C4#53811yescharacteristic
593C2×C4#54111yescharacteristic
594C2×C4#54111yescharacteristic
595C2×C4#54211yescharacteristic
596C2×C4#54211yescharacteristic
597C2×C4#54211yescharacteristic
598C2×C4#54211yescharacteristic
599C2×C4#54611yescharacteristic
600C2×C4#54611yescharacteristic
601C2×C4#54711yescharacteristic
602C2×C4#54811yescharacteristic
603C2×C4#55211yescharacteristic
604C2×C4#55211yescharacteristic
605C2×C4#55611yescharacteristic
606C2×C4#55611yescharacteristic
607C2×C4#56311yescharacteristic
608C2×C4#56311yescharacteristic
609C2×C4#56311yescharacteristic
610C2×C4#56311yescharacteristic
611C8—41no—
612C8—41no—
613C8—41no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—42no—
618C8—41no—
619C8—41no—
620C8—81no—
621C8—81no—
622C8—81no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22#232311yescharacteristic
636C22#234211yescharacteristic
637C22#236211yescharacteristic
638C22#275811yescharacteristic
639C22#275911yescharacteristic
640C22#276211yescharacteristic
641C22#276411yescharacteristic
642C22C2×#44411yescharacteristic
643C22#295511yescharacteristic
644C22#295611yescharacteristic
645C22#295711yescharacteristic
646C4—22no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4#232011yescharacteristic
661C4#233711yescharacteristic
662C4#235611yescharacteristic
663C4#238111yescharacteristic
664C4#269811yescharacteristic
665C4#270211yescharacteristic
666C4#271511yescharacteristic
667C4#271611yescharacteristic
668C2—21no—
669C2—21no—
670C2—21no—
671C2—21no—
672C2#1438011yescharacteristic
673C2#2213111yescharacteristic
674C2#2213211yescharacteristic
675C2#2258611yescharacteristic
676C2#2258811yescharacteristic
677C2#2308111yescharacteristic
678C2#2308211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
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
23. #1989C4semidirect
24. #1989C4semidirect
25. #1989C4semidirect
26. #1989C4semidirect
27. #1988C4semidirect
28. #1988C4semidirect
29. #1987C4semidirect
30. #1987C4semidirect

Maximal subgroups

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

subgroupindex
#213112
#213042
#207742
#113862
#113172
#112912
#112582
#112332
#109102
#108832
#108772
#108192
#108092
#108082