Finite Group Databasebeta

← #373795

Subgroups — #373795

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2925C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic, complemented
3#21145C211yesmaximal, characteristic, complemented
4#20645C211yesmaximal, characteristic
5C2×C4×#104C211yesmaximal, characteristic
6#11125C212yesmaximal
7#11068C212yesmaximal
8#10910C212yesmaximal, complemented
9#10887C212yesmaximal, complemented
10#10839C211yesmaximal, characteristic, complemented
11#10832C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13#2925—24no—
14#2925—24no—
15C2×#433—22no—
16C2×#432—21no—
17C2×#431—21no—
18C22×#108C2212yes—
19C2×#424C2212yes—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C2×#416C2212yes—
26C2×#408C2212yes—
27C2×#405C2212yes—
28C2×C4×M16C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1950—24no—
32#1940—22no—
33#1939—21no—
34#1939—21no—
35#1805C412yescomplemented
36#1804C412yescomplemented
37#1803C411yescharacteristic, complemented
38#1803C411yescharacteristic, complemented
39#1802C411yescharacteristic, complemented
40#1802C411yescharacteristic, complemented
41C4×#104—24no—
42C2×#341C2212yes—
43#1735C2212yes—
44C2×#339C2212yes—
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2212yes—
48#1729C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C22×#41C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2312yes—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41D414yes—
60C4×#41Q814yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×#117—21no—
68C2×#117C2312yes—
69C2×#117C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#433—216no—
77#432—28no—
78#431—28no—
79C2×#108—24no—
80C2×#108—24no—
81C2×#108—42no—
82#424—24no—
83#424—42no—
84#423—22no—
85#423D414yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2×C412yes—
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D412yes—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—22no—
106#416—42no—
107#416—42no—
108C2×#103C2×C412yes—
109C2×#103C2×C412yes—
110C2×#102C2×C412yes—
111C2×#102C2×C412yes—
112#408—24no—
113#408—24no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116#405—24no—
117#405—24no—
118C4×M16—22no—
119C4×M16—22no—
120C2×#97—22no—
121C2×#97—21no—
122C2×C4×C8—22no—
123C2×C4×C8—21no—
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#382C2×C412yes—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C43—22no—
132C43—22no—
133#341—24no—
134#340C2×C412yes—
135#340C2×C412yes—
136#340C2×C412yes—
137#340C2×C412yes—
138#339—22no—
139#339—22no—
140#339—24no—
141#339—42no—
142#336—22no—
143#336—22no—
144#336—24no—
145#336—42no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41C2×D412yes—
157C2×#41C2×Q812yes—
158C22×Q8C22×C411yescharacteristic
159C22×D4—21no—
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4C2411yescharacteristic
169C2×M16—22no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—22no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16C2×D414yes—
185C2×M16#4112yes—
186C2×M16#4112yes—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8C22×C412yes—
203C22×C8C22×C412yes—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8C4⋊C414yes—
209C4×Q8C4⋊C414yes—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4C4⋊C414yes—
217C4×D4C4⋊C414yes—
218#117—21no—
219#117—21no—
220#117—24no—
221#117—24no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—22no—
226#117C2×D412yes—
227#117C2×Q812yes—
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⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×Q812yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274#108—48no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—24no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—24no—
281C4⋊C8—24no—
282C4⋊C8—24no—
283C4⋊C8—44no—
284C4⋊C8—44no—
285#104—28no—
286#104—28no—
287#103—42no—
288#103—42no—
289#102—42no—
290#102—42no—
291C22⋊C8—28no—
292C22⋊C8—28no—
293#97—28no—
294#97—42no—
295C4×C8—28no—
296C4×C8—42no—
297#95—22no—
298#95—22no—
299C24—21no—
300C24C2×#4111yescharacteristic
301#41—22no—
302#41—22no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308#41—22no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8C2×C4⋊C412yes—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4C2×C4⋊C412yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C22⋊Q812yes—
375C22×C4C22⋊Q812yes—
376C22×C4#12312yes—
377C22×C4#12312yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C42−1+411yescharacteristic
391C22×C42−1+411yescharacteristic
392M16—22no—
393M16—22no—
394M16—22no—
395M16—24no—
396M16—24no—
397M16—22no—
398M16—42no—
399M16—41no—
400M16—42no—
401M16—41no—
402M16—41no—
403M16—41no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—41no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4C2×C4⋊C412yes—
446C4⋊C4C2×C4⋊C412yes—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C42—22no—
454C42—22no—
455C42—24no—
456C42—24no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—48no—
462C42—48no—
463C42—41no—
464C42—41no—
465C42—42no—
466C42—42no—
467C42—41no—
468C42—41no—
469C42C2×C4⋊C412yes—
470C42C2×C4⋊C412yes—
471C42C4×D412yes—
472C42C4×D412yes—
473C42#12012yes—
474C42#12012yes—
475C42C22⋊Q812yes—
476C42C22⋊Q812yes—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23C2×C4×D411yescharacteristic
489C23C2×C22⋊Q811yescharacteristic
490C23C2×#12311yescharacteristic
491C23#55211yescharacteristic
492C23#55311yescharacteristic
493C23#55511yescharacteristic
494C23#55611yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499D4—22no—
500D4—22no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—41no—
506D4—41no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—48no—
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—84no—
552C2×C4#38512yes—
553C2×C4#38512yes—
554C2×C4#38512yes—
555C2×C4#38512yes—
556C2×C4#44312yes—
557C2×C4#44412yes—
558C2×C4C22×C4⋊C411yescharacteristic
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C4×#4111yescharacteristic
561C2×C4C4×#4111yescharacteristic
562C2×C4#51811yescharacteristic
563C2×C4#51911yescharacteristic
564C2×C4C2×#12011yescharacteristic
565C2×C4C2×C22⋊Q811yescharacteristic
566C2×C4#52511yescharacteristic
567C2×C4#52511yescharacteristic
568C2×C4#53311yescharacteristic
569C2×C4#53311yescharacteristic
570C2×C4#53711yescharacteristic
571C2×C4#53711yescharacteristic
572C2×C4#54411yescharacteristic
573C2×C4#54411yescharacteristic
574C2×C4#54611yescharacteristic
575C2×C4#54611yescharacteristic
576C2×C4#54711yescharacteristic
577C2×C4#54711yescharacteristic
578C2×C4#54811yescharacteristic
579C2×C4#54811yescharacteristic
580C2×C4#55011yescharacteristic
581C2×C4#55011yescharacteristic
582C2×C4#55211yescharacteristic
583C2×C4#55311yescharacteristic
584C2×C4#55511yescharacteristic
585C2×C4#55611yescharacteristic
586C8—42no—
587C8—42no—
588C8—42no—
589C8—42no—
590C8—44no—
591C8—42no—
592C8—42no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×#38511yescharacteristic
613C22#236211yescharacteristic
614C22#236311yescharacteristic
615C22#243911yescharacteristic
616C22#244111yescharacteristic
617C22#248511yescharacteristic
618C22#248511yescharacteristic
619C22C2×#44311yescharacteristic
620C22C2×#44411yescharacteristic
621C22#295611yescharacteristic
622C22#295611yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—82no—
632C4—82no—
633C4#186614yes—
634C4#229611yescharacteristic
635C4#231611yescharacteristic
636C4#236211yescharacteristic
637C4#236311yescharacteristic
638C4#240711yescharacteristic
639C4#240711yescharacteristic
640C4#243911yescharacteristic
641C4#244111yescharacteristic
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2—82no—
647C2#1412911yescharacteristic
648C2C2×#186611yescharacteristic
649C2#2088511yescharacteristic
650C2#2308011yescharacteristic
651C2#2308111yescharacteristic
652C2#2311311yescharacteristic
653C2#2311411yescharacteristic
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
8. #10910C2semidirect
9. #10887C2semidirect
10. #10839C2semidirect
11. #10832C2semidirect
35. #1805C4semidirect
36. #1804C4semidirect
37. #1803C4semidirect
38. #1803C4semidirect
39. #1802C4semidirect
40. #1802C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29252
C2×#19502
#211452
#206452
C2×C4×#1042
#111252
#110682
#109102
#108872
#108392
#108322