Finite Group Databasebeta

← #106963

Subgroups — #106963

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊C4C211yesmaximal, characteristic
2C2×#1787C211yesmaximal, characteristic
3C2×#1787C211yesmaximal, characteristic
4#10854C211yesmaximal, characteristic
5#8023C211yesmaximal, characteristic
6C2×#1289C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C24×Q8C2211yescharacteristic
9C22×C4×Q8—21no—
10C22×#103—22no—
11C22×#103C414yescomplemented
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14Q8×C22⋊C4—22no—
15Q8×C22⋊C4—22no—
16Q8×C22⋊C4—22no—
17Q8×C22⋊C4C414yescomplemented
18C2×#391—22no—
19C2×#391—21no—
20C2×#380—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1787—21no—
27#1787—22no—
28#1787—22no—
29#1787—21no—
30#1787C414yescomplemented
31C2×#342—21no—
32C2×#342—21no—
33#1294—22no—
34#1289—22no—
35#1282—22no—
36C23×Q8—21no—
37C23×Q8—21no—
38C23×Q8—21no—
39C23×Q8—21no—
40C23×Q8—21no—
41C23×Q8—21no—
42C23×Q8—21no—
43C23×Q8—48no—
44C23×Q8C2×C412yes—
45C23×Q8C2×C412yes—
46C23×Q8C2×C412yes—
47C23×Q8D411yescharacteristic
48C23×Q8D412yes—
49C23×Q8D412yes—
50C23×Q8D411yescharacteristic
51C23×Q8Q812yes—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C23×C8—21no—
55C23×C8C2×C412yes—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—48no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C412yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4D411yescharacteristic
68C22×C42—22no—
69C22×C42—21no—
70C2×#103—28no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—48no—
74C2×#103—48no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8C2×C412yes—
87#391—24no—
88#391—28no—
89#391—24no—
90#391—28no—
91#380—24no—
92#380—21no—
93#380—21no—
94#380—24no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102#380C2×C412yes—
103#380C2×C412yes—
104#380D412yes—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4D412yes—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95D412yes—
117#342—42no—
118#342—42no—
119C25C2×D411yescharacteristic
120C22×Q8—28no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8—28no—
146C22×Q8—28no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×Q8—44no—
150C22×Q8—48no—
151C22×Q8—48no—
152C22×Q8—48no—
153C22×Q8—48no—
154C22×Q8—48no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—48no—
159C22×Q8—48no—
160C22×Q8—48no—
161C22×Q8C4214yes—
162C22×Q8C22⋊C412yes—
163C22×Q8C22⋊C412yes—
164C22×Q8C22⋊C414yes—
165C22×Q8C22⋊C412yes—
166C22×Q8C22⋊C412yes—
167C22×Q8C4⋊C412yes—
168C22×Q8C4⋊C414yes—
169C22×Q8C4⋊C414yes—
170C22×Q8C4⋊C412yes—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—44no—
184C23×C4—48no—
185C23×C4—41no—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×Q811yescharacteristic
196C23×C4#4111yescharacteristic
197C23×C4#4111yescharacteristic
198C23×C4#4111yescharacteristic
199C23×C4#4111yescharacteristic
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—44no—
205C22×C8—41no—
206C22×C8—41no—
207C22×C8—44no—
208C4×Q8—416no—
209C4×Q8—416no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—41no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4C22⋊C412yes—
248C2×C22⋊C4C22⋊C412yes—
249C2×C22⋊C4SD1612yes—
250C2×C22⋊C4Q1612yes—
251C2×C42—28no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—48no—
255C2×C42—44no—
256#103—416no—
257#103—48no—
258#103—48no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267#95—28no—
268#95—28no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95C22⋊C414yes—
277C24—42no—
278C24—42no—
279C24C2×C22⋊C411yescharacteristic
280C24C4×D411yescharacteristic
281C24C4×D411yescharacteristic
282C24#12011yescharacteristic
283C24#12011yescharacteristic
284C24#12111yescharacteristic
285C24C22⋊Q811yescharacteristic
286C24C2×SD1611yescharacteristic
287C24C2×Q1611yescharacteristic
288C24#13511yescharacteristic
289C24#13511yescharacteristic
290C24AGL(1,ℤ/8)11yescharacteristic
291C24#13711yescharacteristic
292C24#13711yescharacteristic
293C24#13711yescharacteristic
294C2×Q8—28no—
295C2×Q8—28no—
296C2×Q8—28no—
297C2×Q8—28no—
298C2×Q8—28no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—28no—
311C2×Q8—28no—
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8#9514yes—
344C2×Q8#9514yes—
345C2×Q8#9918yes—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4C2×C4211yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C4⋊C411yescharacteristic
409C22×C4C2×C4⋊C411yescharacteristic
410C22×C4C2×C4⋊C411yescharacteristic
411C22×C4#11711yescharacteristic
412C22×C4#11711yescharacteristic
413C22×C4C4×D411yescharacteristic
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4#12011yescharacteristic
420C22×C4#12011yescharacteristic
421C22×C4#12111yescharacteristic
422C22×C4#12111yescharacteristic
423C22×C4#12111yescharacteristic
424C22×C4C22⋊Q811yescharacteristic
425C22×C4C22⋊Q811yescharacteristic
426C22×C4C22⋊Q811yescharacteristic
427C22×C4#12311yescharacteristic
428C22×C4#12311yescharacteristic
429C22×C4#12311yescharacteristic
430C22×C4#12311yescharacteristic
431C2×C8—44no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—44no—
435C2×C8—82no—
436C2×C8—82no—
437C2×C8—82no—
438C2×C8—82no—
439C4⋊C4—416no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—41no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4#10314yescomplemented
466C42—416no—
467C42—44no—
468C42—44no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23C4×C22⋊C411yescharacteristic
492C23#37911yescharacteristic
493C23#38011yescharacteristic
494C23#38511yescharacteristic
495C23#38511yescharacteristic
496C23#38611yescharacteristic
497C23#38611yescharacteristic
498C23C2×#10311yescharacteristic
499C23#41611yescharacteristic
500C23#41711yescharacteristic
501C23#41911yescharacteristic
502C23C4×SD1611yescharacteristic
503C23C4×SD1611yescharacteristic
504C23C4×Q1611yescharacteristic
505C23C4×Q1611yescharacteristic
506C23#44011yescharacteristic
507C23#44011yescharacteristic
508C23#44111yescharacteristic
509C23#44111yescharacteristic
510C23#44811yescharacteristic
511C23#44811yescharacteristic
512C23#44911yescharacteristic
513C23#44911yescharacteristic
514C23#45111yescharacteristic
515C23#45111yescharacteristic
516C23#45211yescharacteristic
517C23#45211yescharacteristic
518C23#46011yescharacteristic
519C23#46211yescharacteristic
520C23#46411yescharacteristic
521C23#46411yescharacteristic
522C23#47511yescharacteristic
523C23#47711yescharacteristic
524C23#47911yescharacteristic
525C23#47911yescharacteristic
526Q8—28no—
527Q8—28no—
528Q8—28no—
529Q8—24no—
530Q8—24no—
531Q8—24no—
532Q8—48no—
533Q8—48no—
534Q8—48no—
535Q8—48no—
536Q8—44no—
537Q8—44no—
538Q8—44no—
539Q8#34218yescomplemented
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
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—41no—
572C2×C4—41no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—82no—
591C2×C4—84no—
592C2×C4—82no—
593C2×C4—84no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—162no—
601C2×C4—162no—
602C2×C4—162no—
603C2×C4—162no—
604C2×C4C2×#9511yescharacteristic
605C2×C4C4×C22⋊C411yescharacteristic
606C2×C4#38011yescharacteristic
607C2×C4#38111yescharacteristic
608C2×C4#38511yescharacteristic
609C2×C4#38511yescharacteristic
610C2×C4#38611yescharacteristic
611C2×C4#38611yescharacteristic
612C2×C4C2×#9914yes—
613C8—82no—
614C8—82no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22C22⋊#9511yescharacteristic
656C22C4×#10311yescharacteristic
657C22#176311yescharacteristic
658C22#178711yescharacteristic
659C22#178811yescharacteristic
660C22#180311yescharacteristic
661C22#180511yescharacteristic
662C22#186311yescharacteristic
663C22#186311yescharacteristic
664C22#186511yescharacteristic
665C22#186511yescharacteristic
666C22#187811yescharacteristic
667C22#187811yescharacteristic
668C22#188011yescharacteristic
669C22#188011yescharacteristic
670C22#189812yes—
671C22#190512yes—
672C4—21no—
673C4—28no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—44no—
680C4—44no—
681C4—82no—
682C4—84no—
683C4—162no—
684C4—162no—
685C4C2×#34212yes—
686C4#189812yes—
687C2—21no—
688C2—21no—
689C2—21no—
690C2—21no—
691C2—21no—
692C2—21no—
693C2—21no—
694C2—21no—
695C2—21no—
696C2—21no—
697C2—21no—
698C2—21no—
699C2#925812yes—
700C2#926212yes—
701C2#1073511yescharacteristic
702C2#1078611yescharacteristic
703C2#1078911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. C22×#103C4semidirect
17. Q8×C22⋊C4C4semidirect
30. #1787C4semidirect
465. C22⋊C4#103semidirect
539. Q8#342semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C42
C2×#17872
#108542
#80232
C2×#12892
C2×#12822