Finite Group Databasebeta

← #358421

Subgroups — #358421

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20438C211yesmaximal, characteristic, complemented
2#20420C211yesmaximal, characteristic
3C2×#1765C211yesmaximal, characteristic
4C2×#1757C211yesmaximal, characteristic
5C2×#1747C211yesmaximal, characteristic, complemented
6#11306C211yesmaximal, characteristic, complemented
7#11305C211yesmaximal, characteristic, complemented
8#11262C211yesmaximal, characteristic, complemented
9#11261C211yesmaximal, characteristic, complemented
10#11177C211yesmaximal, characteristic
11#11176C211yesmaximal, characteristic
12#11082C211yesmaximal, characteristic
13#11081C211yesmaximal, characteristic
14#10711C211yesmaximal, characteristic, complemented
15#10707C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C2×#421C2211yescharacteristic
19C22×#104C2211yescharacteristic
20#2896C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#413C2211yescharacteristic
23C2×#410C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C22×#97C2211yescharacteristic
30#2285C2211yescharacteristic
31#1771—22no—
32#1771—22no—
33#1765—22no—
34#1765C418yescomplemented
35#1764—22no—
36#1757—24no—
37#1747—24no—
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42#1742C2211yescharacteristic
43#1740C2211yescharacteristic
44#1738C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47#1735C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#338C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54C2×H(3,ℤ/4)C2211yescharacteristic
55C2×H(3,ℤ/4)C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58C22×#41C2311yescharacteristic
59C2×#131—22no—
60C2×#131—21no—
61C2×#131—21no—
62C2×#131—21no—
63C2×#131—21no—
64C2×#131—22no—
65C2×#131—22no—
66C2×#131C2×C414yes—
67C22×M16—21no—
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8—21no—
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#423—22no—
84#423—22no—
85C2×C4⋊C8—22no—
86#421—22no—
87#421—22no—
88C2×#104—22no—
89C2×#104—21no—
90C2×#104—21no—
91C2×#104—22no—
92C2×#104C2×C414yes—
93#419—21no—
94#418—21no—
95#417—22no—
96#416—22no—
97#416—22no—
98#416C2×C414yes—
99C2×#103—21no—
100C2×#102—21no—
101#413—22no—
102#413—22no—
103C2×#101—21no—
104C2×#100—21no—
105#410—22no—
106#410—22no—
107C2×#99—22no—
108#405—22no—
109#405—22no—
110#405—22no—
111#405—22no—
112#405—22no—
113#405—21no—
114#405—21no—
115#405—22no—
116#405—21no—
117#405—21no—
118#405C2×C414yes—
119#405D414yes—
120C2×#97—21no—
121C2×#97—21no—
122C2×#97—22no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97C2×C414yes—
128C2×#97D412yes—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8D412yes—
134#384—21no—
135#382—21no—
136#380—22no—
137C4×C4⋊C4—22no—
138#344—22no—
139#344—22no—
140#344—22no—
141#344—22no—
142#343—22no—
143#342—22no—
144#341—22no—
145#341—22no—
146#341C2×C414yes—
147#340—22no—
148#339—22no—
149#339—21no—
150#339—21no—
151#339—22no—
152#339C2×C414yes—
153#338—22no—
154H(3,ℤ/4)—22no—
155H(3,ℤ/4)—22no—
156H(3,ℤ/4)—22no—
157#336—22no—
158#336—22no—
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×#41C22×C412yes—
170C2×#41#4112yes—
171C22×Q8#4111yescharacteristic
172C22×D4—21no—
173C22×D4#4111yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C2411yescharacteristic
177#131—24no—
178#131—24no—
179#131—24no—
180#131—24no—
181#131—42no—
182#131—42no—
183#131—42no—
184#131—42no—
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—24no—
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×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—41no—
218C2×M16—41no—
219C2×M16—42no—
220C2×M16C22×C412yes—
221C2×M16C22×C412yes—
222C2×M16C2×D412yes—
223C2×M16#4112yes—
224C2×M16#4112yes—
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×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—24no—
237C22×C8—41no—
238C22×C8—41no—
239C22×C8—42no—
240C22×C8C22×C412yes—
241C22×C8C22×C412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8#4112yes—
247C22×C8#4111yescharacteristic
248C22×C8#4111yescharacteristic
249C22×C8#4112yes—
250C22×C8#4111yescharacteristic
251C22×C8#4111yescharacteristic
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—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117C22×C412yes—
271#117C2×D412yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4#4112yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4#4112yes—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42C22×C412yes—
291C2×C42C2×D412yes—
292C2×C42#4111yescharacteristic
293C2×C42#4111yescharacteristic
294C2×C42#4112yes—
295C4⋊C8—28no—
296#104—28no—
297#104—24no—
298#104—24no—
299#103—24no—
300#103—24no—
301#102—24no—
302#102—24no—
303#101—24no—
304#100—24no—
305#99—28no—
306#97—24no—
307#97—24no—
308#97—24no—
309#97—24no—
310#97—24no—
311#97—42no—
312#97—41no—
313#97—41no—
314C4×C8—24no—
315C4×C8—24no—
316C4×C8—24no—
317C4×C8—42no—
318C4×C8—41no—
319C4×C8—41no—
320#95—24no—
321C24—21no—
322C24C2×#4111yescharacteristic
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8#11712yes—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4#11712yes—
357C22×C4—21no—
358C22×C4—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—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—82no—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388M16—22no—
389M16—22no—
390M16—22no—
391M16—22no—
392M16—22no—
393M16—22no—
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—44no—
413M16—44no—
414C2×C8—22no—
415C2×C8—22no—
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—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
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—44no—
440C2×C8—44no—
441C2×C8#11712yes—
442C2×C8#11712yes—
443C2×C8C4×D412yes—
444C2×C8C4×D412yes—
445C2×C8C4×D412yes—
446C2×C8C4×D412yes—
447C2×C8#12112yes—
448C2×C8#12112yes—
449C2×C8#12312yes—
450C2×C8#12312yes—
451C2×C8#12411yescharacteristic
452C2×C8#12411yescharacteristic
453C2×C8#12411yescharacteristic
454C2×C8#12411yescharacteristic
455C2×C8#12612yes—
456C2×C8#12612yes—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—84no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42#11712yes—
493C42#11712yes—
494C23—22no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—82no—
499C23C4×#4111yescharacteristic
500C23#52511yescharacteristic
501C23#52511yescharacteristic
502C23#52911yescharacteristic
503C23#52911yescharacteristic
504C23#52911yescharacteristic
505C23#52911yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510D4—22no—
511D4—22no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—41no—
517D4—41no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4C2×#11711yescharacteristic
542C2×C4C2×#11711yescharacteristic
543C2×C4C2×#11711yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C4×#4111yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12311yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52511yescharacteristic
551C2×C4#52511yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4C2×#12411yescharacteristic
554C2×C4C2×#12611yescharacteristic
555C2×C4#52911yescharacteristic
556C2×C4#52911yescharacteristic
557C2×C4#52911yescharacteristic
558C2×C4#52911yescharacteristic
559C2×C4#52911yescharacteristic
560C2×C4#52911yescharacteristic
561C2×C4#52911yescharacteristic
562C2×C4#52911yescharacteristic
563C2×C4#52911yescharacteristic
564C2×C4#52911yescharacteristic
565C2×C4#52911yescharacteristic
566C2×C4#52911yescharacteristic
567C2×C4#53211yescharacteristic
568C2×C4#53211yescharacteristic
569C8—42no—
570C8—42no—
571C8—42no—
572C8—42no—
573C8—42no—
574C8—42no—
575C8—42no—
576C8#38812yes—
577C8#38812yes—
578C8#38812yes—
579C8#38812yes—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—42no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22#229011yescharacteristic
592C22#229011yescharacteristic
593C22#229811yescharacteristic
594C22#239211yescharacteristic
595C22#239511yescharacteristic
596C22#239811yescharacteristic
597C22#240111yescharacteristic
598C4—42no—
599C4—42no—
600C4—42no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4#228311yescharacteristic
607C4C2×#38811yescharacteristic
608C4#229011yescharacteristic
609C4#229311yescharacteristic
610C4#239211yescharacteristic
611C4#239511yescharacteristic
612C4#239811yescharacteristic
613C4#240111yescharacteristic
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—82no—
621C2#1384211yescharacteristic
622C2#2383412yes—
subgroupG/H
centerC2×C4C2×#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2398
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #20438C2semidirect
5. C2×#1747C2semidirect
6. #11306C2semidirect
7. #11305C2semidirect
8. #11262C2semidirect
9. #11261C2semidirect
14. #10711C2semidirect
15. #10707C2semidirect
34. #1765C4semidirect

Maximal subgroups

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

subgroupindex
#204382
#204202
C2×#17652
C2×#17572
C2×#17472
#113062
#113052
#112622
#112612
#111772
#111762
#110822
#110812
#107112
#107072