Finite Group Databasebeta

← #362533

Subgroups — #362533

Order 512.

879 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×#1833C211yesmaximal, characteristic, complemented
2C2×#1827C211yesmaximal, characteristic
3#20378C211yesmaximal, characteristic
4#11300C211yesmaximal, characteristic, complemented
5#11299C211yesmaximal, characteristic, complemented
6#11099C211yesmaximal, characteristic
7#11098C211yesmaximal, characteristic
8#10961C211yesmaximal, characteristic
9#10961C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic, complemented
11#10909C211yesmaximal, characteristic, complemented
12#10804C211yesmaximal, characteristic, complemented
13#10800C211yesmaximal, characteristic, complemented
14#10758C211yesmaximal, characteristic
15#10699C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#413C2211yescharacteristic
20C2×#413C2211yescharacteristic
21C2×#410C2211yescharacteristic
22C2×#410C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29#1833—24no—
30#1827—24no—
31#1750C412yescomplemented
32#1750C412yescomplemented
33#1749C412yescomplemented
34C4×C22⋊C8C412yescomplemented
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#344C2211yescharacteristic
38C2×#344C2211yescharacteristic
39C2×#341C2211yescharacteristic
40#1735C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#338C2211yescharacteristic
46C2×#338C2211yescharacteristic
47C2×H(3,ℤ/4)C2211yescharacteristic
48C2×H(3,ℤ/4)C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1728C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C22×#41C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C4×#41—22no—
67C4×#41—22no—
68C4×#41—41no—
69C4×#41—41no—
70C2×C4×Q8—21no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×#117C2×C412yes—
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2×C412yes—
80C22×C42C2311yescharacteristic
81C2×#108—22no—
82C2×#108—21no—
83C2×#108—21no—
84C2×#108—41no—
85#427—22no—
86#427—41no—
87C2×#107—21no—
88C2×#106—21no—
89#413—22no—
90#413—22no—
91#413—41no—
92#413—41no—
93C2×#101—21no—
94C2×#101—21no—
95C2×#100—21no—
96C2×#100—21no—
97#410—22no—
98#410—22no—
99#410—41no—
100#410—41no—
101C2×#99—22no—
102C2×#99—22no—
103#408—21no—
104#408—21no—
105#408—21no—
106#408—21no—
107#408—21no—
108#408—21no—
109#408—21no—
110#408—21no—
111C2×C22⋊C8C2×C412yes—
112C2×C22⋊C8C2×C412yes—
113C2×C22⋊C8C2×C412yes—
114C2×C22⋊C8C2×C412yes—
115#405—22no—
116#405—22no—
117#405—41no—
118#405—41no—
119#405—41no—
120#405—41no—
121#405—41no—
122#405—41no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97C2×C412yes—
126C2×#97C2×C412yes—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2×C412yes—
130C2×C4×C8C2×C412yes—
131#344—22no—
132#344—22no—
133#344—22no—
134#344—22no—
135#341—21no—
136#341—21no—
137#340—21no—
138#340—21no—
139#339—22no—
140#339—22no—
141#339—41no—
142#339—41no—
143#338—22no—
144#338—41no—
145H(3,ℤ/4)—22no—
146H(3,ℤ/4)—22no—
147H(3,ℤ/4)—22no—
148H(3,ℤ/4)—22no—
149H(3,ℤ/4)—41no—
150H(3,ℤ/4)D414yes—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336C2×C411yescharacteristic
156#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#336C2×C412yes—
160#336C2×C412yes—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C22×Q8#4111yescharacteristic
174C22×D4#4112yes—
175C22×D4#4111yescharacteristic
176C23×C4—22no—
177C23×C4—21no—
178C23×C4C22×C412yes—
179C23×C4C2411yescharacteristic
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16C2×D412yes—
215C2×M16#4112yes—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8C22×C412yes—
229C22×C8C22×C412yes—
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8#4111yescharacteristic
235C22×C8#4111yescharacteristic
236C22×C8#4111yescharacteristic
237C22×C8#4111yescharacteristic
238C22×C8#4111yescharacteristic
239C22×C8#4111yescharacteristic
240C22×C8#4111yescharacteristic
241C22×C8#4111yescharacteristic
242C4×Q8—42no—
243C4×Q8—42no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—42no—
247C4×D4—42no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—24no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—42no—
266#117C2×D412yes—
267#117C2×D412yes—
268#117#4112yes—
269#117#4112yes—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—42no—
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C412yes—
297#108—42no—
298#108—42no—
299#107—42no—
300#106—42no—
301#101—42no—
302#101—42no—
303#100—42no—
304#100—42no—
305#99—44no—
306#99—44no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315#97—24no—
316#97—42no—
317#97—42no—
318#97—42no—
319C4×C8—24no—
320C4×C8—42no—
321C4×C8—42no—
322C4×C8—42no—
323C24#11712yes—
324C24C2×#4111yescharacteristic
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—82no—
378C22×C4—81no—
379C22×C4#11712yes—
380C22×C4C23×C411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C42−1+411yescharacteristic
394M16—22no—
395M16—22no—
396M16—22no—
397M16—22no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
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—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8#11711yescharacteristic
461C2×C8#11711yescharacteristic
462C2×C8#11711yescharacteristic
463C2×C8#11711yescharacteristic
464C2×C8#11711yescharacteristic
465C2×C8#11711yescharacteristic
466C2×C8#11711yescharacteristic
467C2×C8#11711yescharacteristic
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—44no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—44no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42C4×D414yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23C2×#11711yescharacteristic
515C23#52911yescharacteristic
516C23#52911yescharacteristic
517C23#54611yescharacteristic
518C23#55111yescharacteristic
519C23#55111yescharacteristic
520C23#55211yescharacteristic
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—88no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4C2×#11711yescharacteristic
565C2×C4C2×#11711yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C4×#4111yescharacteristic
568C2×C4#51811yescharacteristic
569C2×C4#52011yescharacteristic
570C2×C4#52911yescharacteristic
571C2×C4#52911yescharacteristic
572C2×C4#52911yescharacteristic
573C2×C4#52911yescharacteristic
574C2×C4#53211yescharacteristic
575C2×C4#53211yescharacteristic
576C2×C4#53811yescharacteristic
577C2×C4#53811yescharacteristic
578C2×C4#53911yescharacteristic
579C2×C4#53911yescharacteristic
580C2×C4#54611yescharacteristic
581C2×C4#54611yescharacteristic
582C2×C4#54811yescharacteristic
583C2×C4#55111yescharacteristic
584C2×C4#55111yescharacteristic
585C2×C4#55211yescharacteristic
586C2×C4#55211yescharacteristic
587C2×C4#55311yescharacteristic
588C2×C4#55311yescharacteristic
589C2×C4#55311yescharacteristic
590C2×C4#55911yescharacteristic
591C2×C4#55911yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—21no—
600C8—42no—
601C8—42no—
602C8—42no—
603C8—42no—
604C8—81no—
605C8—81no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—82no—
620C22—82no—
621C22#228311yescharacteristic
622C22#234211yescharacteristic
623C22#234311yescharacteristic
624C22#255711yescharacteristic
625C22#255811yescharacteristic
626C22#257511yescharacteristic
627C22#257611yescharacteristic
628C4—44no—
629C4—42no—
630C4—42no—
631C4—81no—
632C4—82no—
633C4—82no—
634C4—161no—
635C4—161no—
636C4#229311yescharacteristic
637C4#231811yescharacteristic
638C4#235211yescharacteristic
639C4#235711yescharacteristic
640C4#255711yescharacteristic
641C4#255811yescharacteristic
642C4#257511yescharacteristic
643C4#257611yescharacteristic
644C2—21no—
645C2—21no—
646C2—21no—
647C2—21no—
648C2—21no—
649C2—21no—
650C2—82no—
651C2#1391711yescharacteristic
652C2#2078711yescharacteristic
653C2#2078811yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2283
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1833C2semidirect
4. #11300C2semidirect
5. #11299C2semidirect
10. #10910C2semidirect
11. #10909C2semidirect
12. #10804C2semidirect
13. #10800C2semidirect
15. #10699C2semidirect
31. #1750C4semidirect
32. #1750C4semidirect
33. #1749C4semidirect
34. C4×C22⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18332
C2×#18272
#203782
#113002
#112992
#110992
#110982
#109612
#109102
#109092
#108042
#108002
#107582
#106992