Finite Group Databasebeta

← #115489

Subgroups — #115489

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#461C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C22×D4⋊C8C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic, complemented
7C2×#1279C211yesmaximal, characteristic, complemented
8C23×SD16—21no—
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#461—22no—
12C2×#461—22no—
13C2×#461—22no—
14C2×#461—22no—
15C2×#461—22no—
16C2×#461C414yescomplemented
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103—21no—
20C22×C4×C8C2211yescharacteristic
21C2×#390—21no—
22C2×#384C2211yescharacteristic
23C2×#380—22no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1990—22no—
26#1990—22no—
27#1988—22no—
28#1988—22no—
29C2×#340—22no—
30C2×#340—22no—
31C2×D4⋊C8—22no—
32C2×D4⋊C8—22no—
33C2×D4⋊C8—22no—
34#1439—22no—
35C4⋊#95—21no—
36#1296—22no—
37#1279—22no—
38C23×Q8—21no—
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C22×SD16—24no—
42C22×SD16—24no—
43C22×SD16—24no—
44C22×SD16—42no—
45C22×SD16—42no—
46C22×SD16—42no—
47C22×SD16—42no—
48C23×C8—21no—
49C23×C8—21no—
50C23×C8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8D412yes—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C412yes—
74C2×C4×D4D412yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C4⋊C4D412yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—24no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#461—28no—
87#461—28no—
88#461—28no—
89#461—28no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2×C412yes—
99C2×C4⋊C8D412yes—
100C2×C4⋊C8D412yes—
101C2×C4⋊C8D412yes—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8D412yes—
111#390—42no—
112#390—42no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#380—44no—
118#380—44no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C2×#95—24no—
124C2×#95—21no—
125#340—28no—
126#340—28no—
127D4⋊C8—28no—
128C25—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—41no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×Q8—41no—
142C22×Q8—41no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4C22⋊C412yes—
155C22×D4D812yes—
156C22×D4SD1612yes—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C2×SD16—216no—
175C2×SD16—48no—
176C2×SD16—48no—
177C2×SD16—48no—
178C2×SD16—48no—
179C2×SD16—48no—
180C2×SD16—48no—
181C22×C8—24no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—42no—
188C22×C8—41no—
189C22×C8—41no—
190C22×C8—41no—
191C22×C8—41no—
192C22×C8—41no—
193C22×C8—41no—
194C22×C8—41no—
195C22×C8—41no—
196C22×C8—42no—
197C22×C8—42no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—41no—
208C4⋊Q8—41no—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—41no—
212C4⋊Q8—41no—
213C4⋊Q8C22⋊C414yescomplemented
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4C22⋊C412yes—
266C2×C4⋊C4SD1614yescomplemented
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—84no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42#4111yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—41no—
300C4⋊C8—41no—
301C4⋊C8C22⋊C414yescomplemented
302#103—216no—
303C4×C8—22no—
304C4×C8—22no—
305C4×C8—41no—
306C4×C8—41no—
307#95—48no—
308#95—41no—
309#95—41no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24#12011yescharacteristic
322C2×Q8—24no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—81no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×Q8—81no—
342C2×Q8—81no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8—81no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4#10214yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
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—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C4×D411yescharacteristic
424C22×C4C4×D411yescharacteristic
425C22×C4#12011yescharacteristic
426C22×C4#12011yescharacteristic
427C22×C4#12011yescharacteristic
428C22×C4#12011yescharacteristic
429C22×C4#12111yescharacteristic
430C22×C4#12111yescharacteristic
431C22×C4#12111yescharacteristic
432C22×C4#12111yescharacteristic
433C22×C4#12111yescharacteristic
434C22×C4#12111yescharacteristic
435C22×C4#12111yescharacteristic
436C22×C4#12111yescharacteristic
437C22×C4#12311yescharacteristic
438C22×C4#12711yescharacteristic
439C22×C4#12711yescharacteristic
440C22×C4C2×D811yescharacteristic
441C22×C4C2×SD1611yescharacteristic
442C22×C4C2×SD1612yes—
443C22×C4#13511yescharacteristic
444C22×C4#13511yescharacteristic
445C22×C4#13512yes—
446C22×C4AGL(1,ℤ/8)11yescharacteristic
447C22×C4AGL(1,ℤ/8)11yescharacteristic
448C22×C4AGL(1,ℤ/8)11yescharacteristic
449C22×C4AGL(1,ℤ/8)12yes—
450C22×C4#13711yescharacteristic
451C22×C4#13712yes—
452SD16—432no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—44no—
456C2×C8—42no—
457C2×C8—44no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—44no—
467C2×C8—82no—
468C2×C8—81no—
469C2×C8—81no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—816no—
500C42—81no—
501C42—81no—
502C42—81no—
503C42—81no—
504C42C2×C22⋊C411yescharacteristic
505C42C2×C22⋊C411yescharacteristic
506C42C4×D411yescharacteristic
507C42C4×D411yescharacteristic
508C42#12011yescharacteristic
509C42#12011yescharacteristic
510C42#12011yescharacteristic
511C42#12111yescharacteristic
512C42#12111yescharacteristic
513C42#12111yescharacteristic
514C42#12111yescharacteristic
515C42#12311yescharacteristic
516C42#12311yescharacteristic
517C42#12311yescharacteristic
518C42#12411yescharacteristic
519C42#12411yescharacteristic
520C23—24no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—81no—
539C23—81no—
540C23—82no—
541C23—81no—
542C23—81no—
543C23#38611yescharacteristic
544C23#39211yescharacteristic
545C23#39211yescharacteristic
546C23#44711yescharacteristic
547C23#44811yescharacteristic
548C23#44911yescharacteristic
549C23#44911yescharacteristic
550C23#45012yes—
551C23#45212yes—
552C23#45512yes—
553C23#45612yes—
554Q8—44no—
555Q8—44no—
556Q8—82no—
557Q8—82no—
558Q8—82no—
559Q8—82no—
560Q8—82no—
561Q8—82no—
562D4—24no—
563D4—24no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4#37911yescharacteristic
609C2×C4#38611yescharacteristic
610C2×C4#38611yescharacteristic
611C2×C4#38611yescharacteristic
612C2×C4#39011yescharacteristic
613C2×C4#39011yescharacteristic
614C2×C4#39211yescharacteristic
615C2×C4#39211yescharacteristic
616C2×C4#39411yescharacteristic
617C2×C4#39411yescharacteristic
618C2×C4#39411yescharacteristic
619C2×C4#39411yescharacteristic
620C2×C4C2×#10211yescharacteristic
621C2×C4#41611yescharacteristic
622C2×C4#41711yescharacteristic
623C2×C4#41811yescharacteristic
624C2×C4C4×SD1612yes—
625C2×C4#44012yes—
626C2×C4#44711yescharacteristic
627C2×C4#44711yescharacteristic
628C2×C4#44812yes—
629C2×C4#44912yes—
630C2×C4#44911yescharacteristic
631C2×C4#44911yescharacteristic
632C2×C4#45011yescharacteristic
633C2×C4#45011yescharacteristic
634C2×C4#45211yescharacteristic
635C2×C4#45211yescharacteristic
636C2×C4#46012yes—
637C2×C4#46112yes—
638C2×C4#46312yes—
639C2×C4#46412yes—
640C2×C4#46512yes—
641C2×C4#46511yescharacteristic
642C2×C4#46511yescharacteristic
643C2×C4#46611yescharacteristic
644C2×C4#46611yescharacteristic
645C2×C4#46812yes—
646C2×C4#46811yescharacteristic
647C2×C4#46811yescharacteristic
648C2×C4#46911yescharacteristic
649C2×C4#46911yescharacteristic
650C2×C4#48011yescharacteristic
651C2×C4#48111yescharacteristic
652C2×C4#48211yescharacteristic
653C2×C4#48211yescharacteristic
654C2×C4#49211yescharacteristic
655C2×C4#49511yescharacteristic
656C2×C4#49611yescharacteristic
657C2×C4#49711yescharacteristic
658C8—44no—
659C8—44no—
660C8—82no—
661C22—48no—
662C22—44no—
663C22—82no—
664C22—82no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22—84no—
672C22#143811yescharacteristic
673C22#163912yes—
674C22#164212yes—
675C22#164312yes—
676C22#164612yes—
677C22#164712yes—
678C22#165012yes—
679C22#165112yes—
680C22#165412yes—
681C22#187411yescharacteristic
682C22#187511yescharacteristic
683C22#187712yes—
684C22#188212yes—
685C22#199911yescharacteristic
686C22#199911yescharacteristic
687C22#200012yes—
688C22#200011yescharacteristic
689C22#200011yescharacteristic
690C22#200411yescharacteristic
691C22#200511yescharacteristic
692C22#200712yes—
693C22#201411yescharacteristic
694C22#201511yescharacteristic
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—82no—
700C4—84no—
701C4—84no—
702C4—162no—
703C4#178611yescharacteristic
704C4#178811yescharacteristic
705C4#188012yes—
706C4#189311yescharacteristic
707C4#189511yescharacteristic
708C4#196811yescharacteristic
709C4#197311yescharacteristic
710C4#201211yescharacteristic
711C4#201311yescharacteristic
712C4#203311yescharacteristic
713C4#203311yescharacteristic
714C4#203411yescharacteristic
715C4#203411yescharacteristic
716C4#205512yes—
717C2—84no—
718C2#898512yes—
719C2#898812yes—
720C2#1006211yescharacteristic
721C2#1006411yescharacteristic
722C2#1006611yescharacteristic
723C2#1006811yescharacteristic
724C2#1023112yes—
725C2#1023212yes—
726C2#1096911yescharacteristic
727C2#1101711yescharacteristic
728C2#1102311yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1988C2semidirect
6. C2×#1296C2semidirect
7. C2×#1279C2semidirect
16. C2×#461C4semidirect
213. C4⋊Q8C22⋊C4semidirect
266. C2×C4⋊C4SD16semidirect
301. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4612
C2×#19902
C2×#19882
C22×D4⋊C82
#139062
C2×#12962
C2×#12792