Finite Group Databasebeta

← #409887

Subgroups — #409887

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2092C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1936C211yesmaximal, characteristic
5#15239C211yesmaximal, characteristic, complemented
6#15144C211yesmaximal, characteristic
7#11378C211yesmaximal, characteristic, complemented
8#11377C211yesmaximal, characteristic, complemented
9#11123C211yesmaximal, characteristic
10#11123C211yesmaximal, characteristic
11#10927C211yesmaximal, characteristic
12#10925C211yesmaximal, characteristic
13#10863C211yesmaximal, characteristic, complemented
14#10862C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#393—22no—
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#382C2211yescharacteristic
34#2092—24no—
35#2057—24no—
36#1993—24no—
37#1936—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49#1443—22no—
50#1442—22no—
51#1439—22no—
52C4⋊#95C2211yescharacteristic
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C23×Q8—21no—
60C23×D4C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—41no—
70C2×C4⋊Q8—41no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—41no—
95C2×#107—41no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—41no—
109C2×#102—41no—
110C2×#102Q814yescomplemented
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—41no—
114C2×#97—41no—
115C2×#97—41no—
116C2×#97D414yes—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—41no—
120#393—48no—
121#390—22no—
122#390—22no—
123#390—22no—
124#390—22no—
125#390—41no—
126#390—41no—
127#389—22no—
128#389—22no—
129#382—22no—
130#382—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95D412yes—
141C2×#95D412yes—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#336—22no—
151#336—22no—
152#336D414yes—
153C25—21no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—44no—
157C22×Q8—42no—
158C22×Q8—82no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C2×Q812yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2411yescharacteristic
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×Q812yes—
208C22×C8#4112yes—
209C22×C8#4112yes—
210C22×C8#4112yes—
211C22×C8#4112yes—
212C22×C8#4112yes—
213C4⋊Q8—28no—
214C4⋊Q8—44no—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4C2×Q812yes—
262C2×C4⋊C4#4112yes—
263C2×C4⋊C4#4112yes—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292#107—44no—
293#107—44no—
294#107—44no—
295#106—44no—
296#102—28no—
297#102—44no—
298#102—44no—
299#102—44no—
300#97—44no—
301#97—44no—
302C4×C8—44no—
303#95—24no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—82no—
313#95—82no—
314C24—22no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24C22×D411yescharacteristic
326C2×Q8—48no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—88no—
330C2×Q8—88no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4#12712yes—
416C22×C4#12712yes—
417C22×C4AGL(1,ℤ/8)14yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×Q811yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4H(5,𝔽2)11yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C42−1+411yescharacteristic
431C22×C42−1+411yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8C22⋊Q814yes—
458C2×C8#12414yes—
459C2×C8#12414yes—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—48no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—84no—
490C22⋊C4—88no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—88no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—82no—
512C23—82no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23C2×#12711yescharacteristic
530C23#53211yescharacteristic
531C23#53211yescharacteristic
532C23#54611yescharacteristic
533C23#54611yescharacteristic
534C23#54711yescharacteristic
535C23D4×Q811yescharacteristic
536C23C2×AGL(1,ℤ/8)12yes—
537C23#57512yes—
538C23#57612yes—
539C23#57812yes—
540Q8—816no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—81no—
546D4—81no—
547D4—81no—
548D4—81no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4C2×C22⋊Q811yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4C2×#12411yescharacteristic
611C2×C4C2×#12411yescharacteristic
612C2×C4#52911yescharacteristic
613C2×C4#52911yescharacteristic
614C2×C4#53211yescharacteristic
615C2×C4#53211yescharacteristic
616C2×C4#53311yescharacteristic
617C2×C4#53311yescharacteristic
618C2×C4#53411yescharacteristic
619C2×C4#53611yescharacteristic
620C2×C4#53911yescharacteristic
621C2×C4#53911yescharacteristic
622C2×C4#54211yescharacteristic
623C2×C4#54211yescharacteristic
624C2×C4#54611yescharacteristic
625C2×C4#54611yescharacteristic
626C2×C4#54711yescharacteristic
627C2×C4D4×Q811yescharacteristic
628C2×C4#55111yescharacteristic
629C2×C4#55111yescharacteristic
630C2×C4#55211yescharacteristic
631C2×C4#55211yescharacteristic
632C2×C4#55411yescharacteristic
633C2×C4#55411yescharacteristic
634C2×C4#55611yescharacteristic
635C2×C4#55611yescharacteristic
636C8—24no—
637C8—24no—
638C8—24no—
639C8—42no—
640C22—84no—
641C22—84no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—162no—
646C22—162no—
647C22—161no—
648C22—161no—
649C22—161no—
650C22—161no—
651C22#252111yescharacteristic
652C22#252111yescharacteristic
653C22#253911yescharacteristic
654C22#254211yescharacteristic
655C22#255611yescharacteristic
656C22#255611yescharacteristic
657C22#285811yescharacteristic
658C22#315111yescharacteristic
659C22#315311yescharacteristic
660C22#315511yescharacteristic
661C22#315611yescharacteristic
662C22#317112yes—
663C22#317512yes—
664C22#317512yes—
665C22#318012yes—
666C22#331312yes—
667C22#331812yes—
668C22#332512yes—
669C22#332712yes—
670C22#334312yes—
671C22#334712yes—
672C22#338612yes—
673C22#338912yes—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—161no—
682C4—161no—
683C4—161no—
684C4—161no—
685C4#239611yescharacteristic
686C4#248811yescharacteristic
687C4#248811yescharacteristic
688C4#249211yescharacteristic
689C4#249211yescharacteristic
690C4#256411yescharacteristic
691C4#260411yescharacteristic
692C4#266311yescharacteristic
693C2—162no—
694C2#1506511yescharacteristic
695C2#2807712yes—
696C2#2807912yes—
697C2#2835311yescharacteristic
698C2#2835911yescharacteristic
699C2#2854411yescharacteristic
700C2#2854511yescharacteristic
701C2#2895212yes—
702C2#2895912yes—
703C2#3324711yescharacteristic
704C2#3325111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2057C2semidirect
3. C2×#1993C2semidirect
5. #15239C2semidirect
7. #11378C2semidirect
8. #11377C2semidirect
13. #10863C2semidirect
14. #10862C2semidirect
15. #10696C2semidirect
110. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20922
C2×#20572
C2×#19932
C2×#19362
#152392
#151442
#113782
#113772
#111232
#109272
#109252
#108632
#108622
#106962