Finite Group Databasebeta

← #373840

Subgroups — #373840

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2906C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic, complemented
3#21145C211yesmaximal, characteristic, complemented
4C2×#1866C212yesmaximal
5#20645C211yesmaximal, characteristic
6C2×C4×#104C211yesmaximal, characteristic
7#11329C212yesmaximal, complemented
8#11291C212yesmaximal, complemented
9#10839C211yesmaximal, characteristic, complemented
10#10832C211yesmaximal, characteristic, complemented
11#10791C212yesmaximal
12C22×#131C2212yes—
13C2×C4×#41C2211yescharacteristic
14C22×#108C2212yes—
15#2906—24no—
16#2906—24no—
17C2×#424—22no—
18C2×#424C2212yes—
19C2×#424C2212yes—
20C2×#423—21no—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8—21no—
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C22×#104C2211yescharacteristic
27C2×#416C2212yes—
28C2×C4×M16C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1950—24no—
32#1940—22no—
33#1939—21no—
34#1939—21no—
35#1866—28no—
36#1805C412yescomplemented
37#1804C412yescomplemented
38#1803C411yescharacteristic, complemented
39#1803C411yescharacteristic, complemented
40#1802C411yescharacteristic, complemented
41#1802C411yescharacteristic, complemented
42C4×#104—24no—
43C2×#341C2212yes—
44#1735C2212yes—
45C2×#339C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2211yescharacteristic
48C2×#339C2212yes—
49#1729C2212yes—
50C2×#336C2212yes—
51C2×#336C2212yes—
52C22×#41C2311yescharacteristic
53C2×#131—24no—
54C2×#131—24no—
55C2×#131—24no—
56C2×#131—24no—
57C2×#131—42no—
58C2×#131—42no—
59C22×M16—22no—
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C22×M16C2311yescharacteristic
63C23×C8—22no—
64C23×C8C2312yes—
65C23×C8C2312yes—
66C4×#41—22no—
67C4×#41—22no—
68C4×#41D414yes—
69C4×#41Q814yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C2×#117C2312yes—
78C2×#117C2311yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#108—24no—
86C2×#108—24no—
87C2×#108—42no—
88#424—216no—
89#424—24no—
90#424—24no—
91#424—24no—
92#424—42no—
93#423—28no—
94#423—22no—
95#423D414yes—
96C2×C4⋊C8—28no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8C2×C411yescharacteristic
108C2×C4⋊C8C2×C411yescharacteristic
109C2×C4⋊C8C2×C412yes—
110C2×C4⋊C8D411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D412yes—
113C2×#104—22no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104D414yes—
118C2×#104Q814yes—
119#416—24no—
120#416—24no—
121C2×#103C2×C412yes—
122C2×#103C2×C412yes—
123C2×#102C2×C412yes—
124C2×#102C2×C412yes—
125C4×M16—22no—
126C4×M16—22no—
127C2×#97—21no—
128C2×C4×C8—21no—
129#384C2×C411yescharacteristic
130#384C2×C411yescharacteristic
131#382C2×C412yes—
132#380—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C43—22no—
137C43—22no—
138#341—24no—
139#340C2×C412yes—
140#340C2×C412yes—
141#340C2×C412yes—
142#340C2×C412yes—
143#339—22no—
144#339—22no—
145#339—24no—
146#339—42no—
147#336—22no—
148#336—22no—
149#336—24no—
150#336—42no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41C2×D412yes—
158C2×#41C2×D412yes—
159C2×#41C2×Q812yes—
160C2×#41C2×Q812yes—
161C2×#41#4112yes—
162C2×#41#4112yes—
163C22×Q8C22×C411yescharacteristic
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4#4111yescharacteristic
172C23×C4#4111yescharacteristic
173C23×C4C2411yescharacteristic
174#131—48no—
175#131—48no—
176#131—48no—
177#131—48no—
178C2×M16—28no—
179C2×M16—22no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—22no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—22no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16C2×D414yes—
197C2×M16C2×D412yes—
198C2×M16C2×Q812yes—
199C22×C8—28no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—44no—
216C22×C8—44no—
217C22×C8C22×C412yes—
218C22×C8C22×C412yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8C4⋊C414yes—
224C4×Q8C4⋊C414yes—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4C4⋊C414yes—
232C4×D4C4⋊C414yes—
233#117—21no—
234#117—21no—
235#117—24no—
236#117—24no—
237#117—22no—
238#117—22no—
239#117—22no—
240#117—22no—
241#117C2×D412yes—
242#117C2×Q812yes—
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⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×Q812yes—
287C2×C42C2×Q812yes—
288C2×C42#4112yes—
289#108—48no—
290C4⋊C8—28no—
291C4⋊C8—28no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—24no—
295C4⋊C8—22no—
296C4⋊C8—24no—
297C4⋊C8—22no—
298C4⋊C8—24no—
299C4⋊C8—24no—
300C4⋊C8—44no—
301C4⋊C8—44no—
302#104—28no—
303#104—28no—
304#103—24no—
305#103—24no—
306#102—24no—
307#102—24no—
308#97—42no—
309C4×C8—42no—
310#95—22no—
311#95—22no—
312C24—21no—
313C24C2×#4111yescharacteristic
314#41—22no—
315#41—22no—
316#41—22no—
317#41—22no—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—22no—
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41#12114yes—
327#41C22⋊Q814yes—
328#41#12514yes—
329#41C4⋊Q814yes—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8C2×C4⋊C412yes—
334C2×Q8C4×D412yes—
335C2×Q8C4×Q812yes—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4C2×C4⋊C412yes—
346C2×D4C4×D412yes—
347C2×D4C4×Q812yes—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C22⋊Q812yes—
388C22×C4C22⋊Q812yes—
389C22×C4#12312yes—
390C22×C4#12312yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×Q811yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C42−1+411yescharacteristic
404C22×C42−1+411yescharacteristic
405M16—22no—
406M16—22no—
407M16—22no—
408M16—24no—
409M16—24no—
410M16—22no—
411M16—416no—
412M16—42no—
413M16—41no—
414M16—42no—
415M16—41no—
416M16—41no—
417M16—41no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—416no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—42no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4C2×C4⋊C412yes—
461C4⋊C4C2×C4⋊C412yes—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C42—22no—
469C42—22no—
470C42—24no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—48no—
477C42—48no—
478C42—41no—
479C42—41no—
480C42—42no—
481C42—42no—
482C42—41no—
483C42—41no—
484C42C2×C4⋊C412yes—
485C42C2×C4⋊C412yes—
486C42C4×D412yes—
487C42C4×D412yes—
488C42#12012yes—
489C42#12012yes—
490C42C22⋊Q812yes—
491C42C22⋊Q812yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C2×C4×D411yescharacteristic
504C23C2×C22⋊Q811yescharacteristic
505C23C2×#12311yescharacteristic
506C23#54711yescharacteristic
507C23#55211yescharacteristic
508C23#55311yescharacteristic
509C23#55411yescharacteristic
510Q8—22no—
511Q8—22no—
512Q8—22no—
513Q8#38914yes—
514D4—24no—
515D4—24no—
516D4—24no—
517D4—24no—
518D4—22no—
519D4—22no—
520D4—22no—
521D4#38914yes—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—42no—
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—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—84no—
567C2×C4#38512yes—
568C2×C4#38512yes—
569C2×C4#38512yes—
570C2×C4#38512yes—
571C2×C4#44312yes—
572C2×C4#44412yes—
573C2×C4C22×C4⋊C411yescharacteristic
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C2×C4×D411yescharacteristic
576C2×C4C2×C4×Q811yescharacteristic
577C2×C4#52011yescharacteristic
578C2×C4#52011yescharacteristic
579C2×C4C2×#12011yescharacteristic
580C2×C4C2×#12111yescharacteristic
581C2×C4C2×C22⋊Q811yescharacteristic
582C2×C4C2×C22⋊Q811yescharacteristic
583C2×C4C2×#12511yescharacteristic
584C2×C4C2×C4⋊Q811yescharacteristic
585C2×C4#53511yescharacteristic
586C2×C4#53611yescharacteristic
587C2×C4#54411yescharacteristic
588C2×C4#54411yescharacteristic
589C2×C4D4211yescharacteristic
590C2×C4D4211yescharacteristic
591C2×C4#54611yescharacteristic
592C2×C4#54611yescharacteristic
593C2×C4#54711yescharacteristic
594C2×C4#54811yescharacteristic
595C2×C4#54811yescharacteristic
596C2×C4D4×Q811yescharacteristic
597C2×C4D4×Q811yescharacteristic
598C2×C4#55211yescharacteristic
599C2×C4#55311yescharacteristic
600C2×C4#55411yescharacteristic
601C8—42no—
602C8—42no—
603C8—42no—
604C8—42no—
605C8—44no—
606C8—42no—
607C8—42no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C2×#38511yescharacteristic
628C22#236011yescharacteristic
629C22#236111yescharacteristic
630C22#244011yescharacteristic
631C22#244211yescharacteristic
632C22#248611yescharacteristic
633C22#248711yescharacteristic
634C22C2×#44311yescharacteristic
635C22C2×#44411yescharacteristic
636C22#295611yescharacteristic
637C22#295611yescharacteristic
638C4—24no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—82no—
647C4—82no—
648C4#186614yescomplemented
649C4C2×#38911yescharacteristic
650C4#231711yescharacteristic
651C4#236011yescharacteristic
652C4#236111yescharacteristic
653C4#240811yescharacteristic
654C4#240911yescharacteristic
655C4#244011yescharacteristic
656C4#244211yescharacteristic
657C2—21no—
658C2—21no—
659C2—21no—
660C2—21no—
661C2—44no—
662C2#1413011yescharacteristic
663C2C2×#186611yescharacteristic
664C2#2088511yescharacteristic
665C2#2298211yescharacteristic
666C2#2298311yescharacteristic
667C2#2301511yescharacteristic
668C2#2301611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1950C2semidirect
3. #21145C2semidirect
7. #11329C2semidirect
8. #11291C2semidirect
9. #10839C2semidirect
10. #10832C2semidirect
36. #1805C4semidirect
37. #1804C4semidirect
38. #1803C4semidirect
39. #1803C4semidirect
40. #1802C4semidirect
41. #1802C4semidirect
648. C4#1866semidirect

Maximal subgroups

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

subgroupindex
C2×#29062
C2×#19502
#211452
C2×#18662
#206452
C2×C4×#1042
#113292
#112912
#108392
#108322
#107912