Finite Group Databasebeta

← #97537

Subgroups — #97537

Order 512.

1081 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2732C211yesmaximal, characteristic
2#11207C211yesmaximal, characteristic, complemented
3#11050C211yesmaximal, characteristic, complemented
4#11049C211yesmaximal, characteristic
5C2×#1272C211yesmaximal, characteristic, complemented
6C2×#1271C211yesmaximal, characteristic, complemented
7C2×#1270C211yesmaximal, characteristic
8C22×C22⋊Q8—21no—
9C22×#103—21no—
10C22×#102—21no—
11#2732—22no—
12#2732—22no—
13#2732—22no—
14#2732C414yescomplemented
15C2×#399—21no—
16C2×#398—21no—
17C2×#396—21no—
18C2×#391C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#386—21no—
22C2×#385—21no—
23C2×#384C2211yescharacteristic
24C2×#340—21no—
25C2×#340—21no—
26C2×#340—21no—
27C2×#340—21no—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1272—22no—
32#1271—22no—
33#1270—22no—
34C23×Q8D411yescharacteristic
35C23×D4D411yescharacteristic
36C24×C4—21no—
37C23×C8—21no—
38C23×C8—21no—
39C23×C8—21no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—42no—
43C2×C22⋊Q8—42no—
44C2×C22⋊Q8—42no—
45C2×C22⋊Q8—42no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C42C2311yescharacteristic
63C2×#103—42no—
64C2×#103—42no—
65C2×#103—42no—
66C2×#102—42no—
67C2×#102—42no—
68C2×#102—42no—
69#399—24no—
70#399—24no—
71#398—24no—
72#398—24no—
73#396—24no—
74#396—24no—
75#391—21no—
76#391—21no—
77#391C2×C412yes—
78#391D412yes—
79#390—21no—
80#390—21no—
81#390C2×C412yes—
82#390D412yes—
83#389—21no—
84#389—21no—
85#389C2×C412yes—
86#389D412yes—
87#386—24no—
88#386—24no—
89#385—24no—
90#385—24no—
91#384—42no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—21no—
101#336—21no—
102#336—21no—
103#336—21no—
104#336—21no—
105#336—21no—
106C25—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8C22⋊C412yes—
115C22×Q8D812yes—
116C22×Q8SD1612yes—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—41no—
121C22×D4—41no—
122C22×D4—41no—
123C22×D4—41no—
124C22×D4C22⋊C412yes—
125C22×D4SD1612yes—
126C22×D4Q1612yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—24no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—41no—
136C23×C4—81no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—41no—
147C22×C8—41no—
148C22×C8—41no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—41no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—41no—
157C22×C8—41no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8—41no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8—41no—
164C22⋊Q8—48no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167C22⋊Q8—48no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—81no—
237C2×C22⋊C4—81no—
238C2×C22⋊C4—81no—
239C2×C22⋊C4—81no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246#103—48no—
247#102—48no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24#12011yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—81no—
281C2×Q8—81no—
282C2×Q8—81no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×Q8—81no—
286C2×Q8#10214yescomplemented
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—81no—
297C2×D4—81no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4#10314yescomplemented
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—48no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4C2×D811yescharacteristic
371C22×C4C2×SD1611yescharacteristic
372C22×C4C2×SD1611yescharacteristic
373C22×C4C2×Q1611yescharacteristic
374C22×C4#13511yescharacteristic
375C22×C4#13511yescharacteristic
376C22×C4AGL(1,ℤ/8)11yescharacteristic
377C22×C4AGL(1,ℤ/8)11yescharacteristic
378C22×C4AGL(1,ℤ/8)11yescharacteristic
379C22×C4#13711yescharacteristic
380C22×C4#13711yescharacteristic
381C22×C4#13711yescharacteristic
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—81no—
392C2×C8—81no—
393C2×C8—81no—
394C2×C8—81no—
395C2×C8—81no—
396C2×C8—81no—
397C2×C8—81no—
398C2×C8—81no—
399C2×C8—81no—
400C2×C8—81no—
401C2×C8—81no—
402C2×C8—81no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—84no—
432C22⋊C4—48no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C23—24no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23#37911yescharacteristic
483C23#44711yescharacteristic
484C23#44811yescharacteristic
485C23#44911yescharacteristic
486C23#45011yescharacteristic
487C23#45111yescharacteristic
488C23#45211yescharacteristic
489C23#45311yescharacteristic
490C23#45311yescharacteristic
491C23#45411yescharacteristic
492C23#45511yescharacteristic
493C23#45511yescharacteristic
494C23#45611yescharacteristic
495C23B(4,𝔽2)11yescharacteristic
496C23#45811yescharacteristic
497Q8—44no—
498Q8—82no—
499Q8—82no—
500Q8—82no—
501D4—44no—
502D4—82no—
503D4—82no—
504D4—82no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—816no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—161no—
581C2×C4—161no—
582C2×C4—161no—
583C2×C4—161no—
584C2×C4C2×#10211yescharacteristic
585C2×C4C2×#10311yescharacteristic
586C2×C4#41611yescharacteristic
587C2×C4#41711yescharacteristic
588C2×C4#41811yescharacteristic
589C2×C4#41911yescharacteristic
590C2×C4#44711yescharacteristic
591C2×C4#44811yescharacteristic
592C2×C4#44911yescharacteristic
593C2×C4#45011yescharacteristic
594C2×C4#45111yescharacteristic
595C2×C4#45211yescharacteristic
596C8—82no—
597C8—82no—
598C8—82no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22#159511yescharacteristic
615C22#159611yescharacteristic
616C22#159711yescharacteristic
617C22#160111yescharacteristic
618C22#160211yescharacteristic
619C22#160311yescharacteristic
620C22#160711yescharacteristic
621C22#160811yescharacteristic
622C22#160911yescharacteristic
623C22#161311yescharacteristic
624C22#161411yescharacteristic
625C22#161511yescharacteristic
626C22#162111yescharacteristic
627C22#162211yescharacteristic
628C22#162711yescharacteristic
629C22#163111yescharacteristic
630C22#163211yescharacteristic
631C22#163711yescharacteristic
632C22#164111yescharacteristic
633C22#164211yescharacteristic
634C22#164311yescharacteristic
635C22#164411yescharacteristic
636C22#164911yescharacteristic
637C22#165111yescharacteristic
638C22#165611yescharacteristic
639C22#166111yescharacteristic
640C22#166111yescharacteristic
641C22#166211yescharacteristic
642C22#178611yescharacteristic
643C22#178711yescharacteristic
644C22#178811yescharacteristic
645C22#179011yescharacteristic
646C22#179211yescharacteristic
647C22#179211yescharacteristic
648C22#179311yescharacteristic
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—21no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—162no—
667C2—84no—
668C2#773511yescharacteristic
669C2#775311yescharacteristic
670C2#775611yescharacteristic
671C2#777411yescharacteristic
672C2#778711yescharacteristic
673C2#778911yescharacteristic
674C2#780111yescharacteristic
675C2#782311yescharacteristic
676C2#840311yescharacteristic
677C2#840511yescharacteristic
678C2#840711yescharacteristic
679C2#844411yescharacteristic
680C2#844711yescharacteristic
681C2#845011yescharacteristic
682C2#850811yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11207C2semidirect
3. #11050C2semidirect
5. C2×#1272C2semidirect
6. C2×#1271C2semidirect
14. #2732C4semidirect
286. C2×Q8#102semidirect
302. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#27322
#112072
#110502
#110492
C2×#12722
C2×#12712
C2×#12702