Finite Group Databasebeta

← #359276

Subgroups — #359276

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2080C211yesmaximal, characteristic
2#20427C211yesmaximal, characteristic, complemented
3#20411C211yesmaximal, characteristic
4C4×#421C211yesmaximal, characteristic
5C4×#413C211yesmaximal, characteristic
6#11350C211yesmaximal, characteristic, complemented
7#11320C211yesmaximal, characteristic, complemented
8#11305C211yesmaximal, characteristic, complemented
9#11281C211yesmaximal, characteristic, complemented
10#11177C211yesmaximal, characteristic
11#11150C211yesmaximal, characteristic
12#11086C211yesmaximal, characteristic
13C4×#344C211yesmaximal, characteristic, complemented
14#10705C211yesmaximal, characteristic, complemented
15C4×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#413C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C4×#117C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#2080C418yescomplemented
32#1768—22no—
33#1767—22no—
34#1763—22no—
35#1762—22no—
36#1759—22no—
37C4×#104—22no—
38C4×#101—22no—
39C4×#100—22no—
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42C2×#344C2211yescharacteristic
43C2×#344C2211yescharacteristic
44#1742C2211yescharacteristic
45#1742C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51#1735C2211yescharacteristic
52#1733C2211yescharacteristic
53#1733C2211yescharacteristic
54C2×H(3,ℤ/4)C2211yescharacteristic
55C2×H(3,ℤ/4)C2211yescharacteristic
56C2×H(3,ℤ/4)C2211yescharacteristic
57C2×H(3,ℤ/4)C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60C22×#41C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C4×#41—21no—
70C4×#41—21no—
71C4×#41—21no—
72C4×#41—21no—
73C4×#41—22no—
74C4×#41—22no—
75C2×C4×Q8D412yes—
76C2×C4×D4—22no—
77C2×C4×D4D412yes—
78C2×#117—21no—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#424—22no—
87#424—22no—
88#424—22no—
89#424C2×C414yescomplemented
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92#421—22no—
93#421—22no—
94#421—22no—
95#421C2×C414yes—
96C2×#104—22no—
97C2×#104—22no—
98#416—22no—
99#416—22no—
100#416—22no—
101#416C2×C414yes—
102C2×#103—21no—
103C2×#103—21no—
104C2×#102—21no—
105C2×#102—21no—
106#413—22no—
107#413—22no—
108#413—22no—
109#413C2×C414yes—
110C2×#101—21no—
111C2×#101—21no—
112C2×#100—21no—
113C2×#100—21no—
114C4×M16—22no—
115C4×M16—22no—
116C4×M16—21no—
117C4×M16—21no—
118C4×M16—22no—
119C4×M16—22no—
120C4×M16—21no—
121C4×M16—21no—
122C2×#97—22no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97—22no—
127C2×#97—21no—
128C2×#97—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136#376—21no—
137#376D412yes—
138C43—21no—
139C43D412yes—
140#344—22no—
141#344—22no—
142#344—22no—
143#344C2×C414yescomplemented
144#343—22no—
145#343—22no—
146#341—22no—
147#341—22no—
148#341—22no—
149#341C2×C414yescomplemented
150#340—22no—
151#340—22no—
152#339—22no—
153#339—22no—
154H(3,ℤ/4)—22no—
155H(3,ℤ/4)—22no—
156H(3,ℤ/4)—22no—
157H(3,ℤ/4)C2×C414yescomplemented
158#336—22no—
159#336—22no—
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×#41—21no—
170C2×#41C22×C412yes—
171C2×#41#4112yes—
172C22×Q8C2×D411yescharacteristic
173C22×D4—21no—
174C22×D4C2×D411yescharacteristic
175C23×C4—21no—
176C23×C4—22no—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C2×M16—21no—
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×M16C22×C412yes—
210C2×M16C22×C412yes—
211C2×M16C22×C412yes—
212C2×M16#4112yes—
213C2×M16#4112yes—
214C2×M16#4112yes—
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×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8C22×C412yes—
234C22×C8#4112yes—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
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—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—41no—
264#117—41no—
265#117—42no—
266#117C22×C412yes—
267#117C22×C412yes—
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⋊C4C2×D412yes—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—24no—
287C2×C42—41no—
288C2×C42—41no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42#4112yes—
297C2×C42#4111yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42#4112yes—
300C2×C42#4111yescharacteristic
301C2×C42#4111yescharacteristic
302C4⋊C8—28no—
303C4⋊C8—28no—
304#104—28no—
305#104—28no—
306#103—24no—
307#103—24no—
308#102—24no—
309#102—24no—
310#101—24no—
311#101—24no—
312#100—24no—
313#100—24no—
314#97—42no—
315#97—42no—
316#97—42no—
317#97—42no—
318#97—41no—
319#97—41no—
320C4×C8—42no—
321C4×C8—42no—
322C4×C8—41no—
323C4×C8—41no—
324#95—24no—
325#95—42no—
326C24—21no—
327C24C2×#4111yescharacteristic
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8C4×D412yes—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4C4×D412yes—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—82no—
377C22×C4—84no—
378C22×C4#12312yes—
379C22×C4#12312yes—
380C22×C4#12312yes—
381C22×C4#12312yes—
382C22×C4#12612yes—
383C22×C4#12612yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398M16—22no—
399M16—22no—
400M16—22no—
401M16—22no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—22no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419M16—41no—
420M16—41no—
421M16—41no—
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—22no—
432C2×C8—22no—
433C2×C8—22no—
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—41no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—42no—
488C42—42no—
489C42—84no—
490C42—84no—
491C42C4×D414yes—
492C42#12112yes—
493C42#12112yes—
494C42#12312yes—
495C42#12312yes—
496C42#12411yescharacteristic
497C42#12411yescharacteristic
498C42#12411yescharacteristic
499C42#12411yescharacteristic
500C42#12612yes—
501C42#12612yes—
502C23—22no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—82no—
507C23C4×#4111yescharacteristic
508C23C2×#12311yescharacteristic
509C23C2×#12311yescharacteristic
510C23C2×#12611yescharacteristic
511C23#52911yescharacteristic
512C23#52911yescharacteristic
513C23#52911yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518D4—22no—
519D4—22no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—41no—
525D4—41no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4#39712yes—
552C2×C4#39712yes—
553C2×C4#39712yes—
554C2×C4#39712yes—
555C2×C4C2×C4×D411yescharacteristic
556C2×C4C2×C4×D411yescharacteristic
557C2×C4C4×#4111yescharacteristic
558C2×C4C4×#4111yescharacteristic
559C2×C4C4×#4111yescharacteristic
560C2×C4C4×#4111yescharacteristic
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52511yescharacteristic
565C2×C4#52511yescharacteristic
566C2×C4#52511yescharacteristic
567C2×C4C2×#12411yescharacteristic
568C2×C4C2×#12611yescharacteristic
569C2×C4#52911yescharacteristic
570C2×C4#52911yescharacteristic
571C2×C4#52911yescharacteristic
572C2×C4#52911yescharacteristic
573C2×C4#52911yescharacteristic
574C2×C4#52911yescharacteristic
575C2×C4#52911yescharacteristic
576C2×C4#52911yescharacteristic
577C2×C4#52911yescharacteristic
578C2×C4#52911yescharacteristic
579C2×C4#52911yescharacteristic
580C2×C4#52911yescharacteristic
581C2×C4#53211yescharacteristic
582C2×C4#53211yescharacteristic
583C8—42no—
584C8—42no—
585C8—42no—
586C8—42no—
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22C4×#12311yescharacteristic
603C22C4×#12311yescharacteristic
604C22C4×#12611yescharacteristic
605C22C2×#39711yescharacteristic
606C22#239511yescharacteristic
607C22#239511yescharacteristic
608C22#240111yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—44no—
614C4—42no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4C4×#12111yescharacteristic
620C4C4×#12311yescharacteristic
621C4C4×#12411yescharacteristic
622C4C4×#12611yescharacteristic
623C4#239211yescharacteristic
624C4#239511yescharacteristic
625C4#239811yescharacteristic
626C4#240111yescharacteristic
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—82no—
634C2C4×#39711yescharacteristic
635C2#2364912yes—
subgroupG/H
centerC2×C4C2×#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#397
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20427C2semidirect
6. #11350C2semidirect
7. #11320C2semidirect
8. #11305C2semidirect
9. #11281C2semidirect
13. C4×#344C2semidirect
14. #10705C2semidirect
15. C4×H(3,ℤ/4)C2semidirect
31. #2080C4semidirect
89. #424C2×C4semidirect
143. #344C2×C4semidirect
149. #341C2×C4semidirect
157. H(3,ℤ/4)C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20802
#204272
#204112
C4×#4212
C4×#4132
#113502
#113202
#113052
#112812
#111772
#111502
#110862
C4×#3442
#107052
C4×H(3,ℤ/4)2