Finite Group Databasebeta

← #367691

Subgroups — #367691

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1937C211yesmaximal, characteristic
2#21022C211yesmaximal, characteristic, complemented
3#20872C211yesmaximal, characteristic
4#20684C211yesmaximal, characteristic
5C2×C4×#104C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11125C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10911C211yesmaximal, characteristic, complemented
12#10878C211yesmaximal, characteristic, complemented
13#10859C211yesmaximal, characteristic, complemented
14#10818C211yesmaximal, characteristic, complemented
15C4×#336C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2296C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1937—24no—
32#1917—22no—
33#1916—22no—
34#1865C412yescomplemented
35#1864C412yescomplemented
36#1863C412yescomplemented
37#1862C412yescomplemented
38#1818—22no—
39#1817—22no—
40C4×#104—24no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44#1735C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C4×#41—22no—
71C4×#41—22no—
72C4×#41—41no—
73C4×#41—41no—
74C2×C4×Q8C2×C412yes—
75C2×C4×D4—22no—
76C2×C4×D4C2×C412yes—
77C2×#117—21no—
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42—22no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—41no—
89#427—41no—
90#427—41no—
91C2×#107C2×C412yes—
92C2×#106C2×C412yes—
93#424—22no—
94#424—41no—
95C2×C4⋊C8—22no—
96C2×#104—22no—
97C2×#104—22no—
98C2×#104—22no—
99C2×#104—22no—
100C2×#104—22no—
101C2×#104D414yes—
102#416—22no—
103#416—22no—
104#416—22no—
105#416—22no—
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C412yes—
108C2×#103C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C412yes—
111C2×#102C2×C411yescharacteristic
112#408—22no—
113#408—22no—
114#408—41no—
115#408—41no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—22no—
118C4×M16—22no—
119C4×M16—41no—
120C2×#97—21no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—21no—
124C2×C4×C8—22no—
125C2×C4×C8—22no—
126C2×C4×C8—22no—
127C2×C4×C8—41no—
128#389C2×C412yes—
129#389C2×C412yes—
130#385—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C43—22no—
134C43—41no—
135#341—22no—
136#341—22no—
137#341—22no—
138#340C2×C412yes—
139#340C2×C412yes—
140#339—22no—
141#339—22no—
142#339—22no—
143#339—22no—
144#339—22no—
145#339—41no—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—21no—
153#336—21no—
154#336—22no—
155#336C2×C412yes—
156#336C2×C412yes—
157#336D412yes—
158#336D412yes—
159C2×#41—21no—
160C2×#41—21no—
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×#41C2×D412yes—
170C2×#41#4112yes—
171C22×Q8C22×C411yescharacteristic
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4#4112yes—
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×M16C2×D412yes—
203C2×M16#4112yes—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
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×C8—21no—
229C22×C8—21no—
230C22×C8C22×C412yes—
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C2×D412yes—
234C22×C8C2×D412yes—
235C22×C8#4112yes—
236C22×C8#4112yes—
237C4×Q8—24no—
238C4×Q8—42no—
239C4×D4—24no—
240C4×D4—42no—
241C4×D4—44no—
242C4×D4—44no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—42no—
256#117—42no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C412yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—28no—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C22×C412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D412yes—
293C2×C42#4112yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296#108—44no—
297#107—42no—
298#106—42no—
299C4⋊C8—44no—
300#104—28no—
301#104—28no—
302#103—24no—
303#103—22no—
304#103—22no—
305#102—24no—
306#102—22no—
307#102—22no—
308C22⋊C8—44no—
309C22⋊C8—44no—
310#97—42no—
311C4×C8—44no—
312C4×C8—42no—
313C4×C8—44no—
314#95—22no—
315#95—22no—
316C24—21no—
317C24C2×#4111yescharacteristic
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—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8C4×D412yes—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4C4×D412yes—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—44no—
379C22×C4—82no—
380C22×C4C4×D412yes—
381C22×C4C4×D412yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C42−1+411yescharacteristic
395C22×C42−1+411yescharacteristic
396M16—22no—
397M16—22no—
398M16—22no—
399M16—22no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8#11712yes—
454C2×C8#11712yes—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—82no—
466C22⋊C4—44no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—44no—
485C42—416no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42#12314yes—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23C2×C4×D411yescharacteristic
503C23#52911yescharacteristic
504C23#52911yescharacteristic
505C23#54611yescharacteristic
506C23#55111yescharacteristic
507C23#55211yescharacteristic
508C23#55611yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—41no—
512Q8—41no—
513D4—22no—
514D4—22no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—41no—
520D4—41no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
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—88no—
556C2×C4#44314yes—
557C2×C4C2×#11711yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C4×#4111yescharacteristic
560C2×C4C4×#4111yescharacteristic
561C2×C4#51811yescharacteristic
562C2×C4#51911yescharacteristic
563C2×C4C2×#12311yescharacteristic
564C2×C4#52511yescharacteristic
565C2×C4#52911yescharacteristic
566C2×C4#52911yescharacteristic
567C2×C4#53211yescharacteristic
568C2×C4#53211yescharacteristic
569C2×C4#53911yescharacteristic
570C2×C4#53911yescharacteristic
571C2×C4#54211yescharacteristic
572C2×C4#54211yescharacteristic
573C2×C4D4211yescharacteristic
574C2×C4#54611yescharacteristic
575C2×C4#54711yescharacteristic
576C2×C4#54711yescharacteristic
577C2×C4#54711yescharacteristic
578C2×C4#54811yescharacteristic
579C2×C4#55111yescharacteristic
580C2×C4#55111yescharacteristic
581C2×C4#55211yescharacteristic
582C2×C4#55211yescharacteristic
583C2×C4#55311yescharacteristic
584C2×C4#55311yescharacteristic
585C8—22no—
586C8—22no—
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—81no—
593C8—81no—
594C22—22no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—22no—
601C22—22no—
602C22—44no—
603C22—42no—
604C22—82no—
605C22—81no—
606C22—81no—
607C22#229311yescharacteristic
608C22#235811yescharacteristic
609C22#236211yescharacteristic
610C22#241911yescharacteristic
611C22#246611yescharacteristic
612C22#251511yescharacteristic
613C22#252711yescharacteristic
614C22C2×#44312yes—
615C22#295612yes—
616C4—28no—
617C4—24no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—82no—
624C4—82no—
625C4C4×#12311yescharacteristic
626C4#233211yescharacteristic
627C4#234411yescharacteristic
628C4#235311yescharacteristic
629C4#242211yescharacteristic
630C4#246111yescharacteristic
631C4#255311yescharacteristic
632C4#255811yescharacteristic
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—82no—
638C2#1400811yescharacteristic
639C2#2129211yescharacteristic
640C2#2129711yescharacteristic
641C2#2288812yes—
642C2#2308012yes—
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 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21022C2semidirect
6. #11317C2semidirect
7. #11221C2semidirect
11. #10911C2semidirect
12. #10878C2semidirect
13. #10859C2semidirect
14. #10818C2semidirect
15. C4×#336C2semidirect
34. #1865C4semidirect
35. #1864C4semidirect
36. #1863C4semidirect
37. #1862C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19372
#210222
#208722
#206842
C2×C4×#1042
#113172
#112212
#111912
#111252
#109292
#109112
#108782
#108592
#108182
C4×#3362