Finite Group Databasebeta

← #359281

Subgroups — #359281

Order 512.

945 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23652C211yesmaximal, characteristic
2#20444C211yesmaximal, characteristic, complemented
3#20420C211yesmaximal, characteristic
4#20420C211yesmaximal, characteristic
5C2×#1757C211yesmaximal, characteristic
6#11363C211yesmaximal, characteristic, complemented
7#11307C211yesmaximal, characteristic, complemented
8#11302C211yesmaximal, characteristic, complemented
9#11284C211yesmaximal, characteristic, complemented
10#11180C211yesmaximal, characteristic
11#11159C211yesmaximal, characteristic
12#11132C211yesmaximal, characteristic
13#10711C211yesmaximal, characteristic, complemented
14#10711C211yesmaximal, characteristic, complemented
15#10699C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#430C2211yescharacteristic
18#2911C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21#2896C2211yescharacteristic
22#2896C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#410C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2082C414yescomplemented
32#2081C414yescomplemented
33#1773—22no—
34#1772—22no—
35#1765—22no—
36#1765—22no—
37#1764—22no—
38#1764—22no—
39#1757—22no—
40#1757—22no—
41C2×#344C2211yescharacteristic
42C2×#344C2211yescharacteristic
43C2×#344C2211yescharacteristic
44C2×#344C2211yescharacteristic
45C2×#344C2211yescharacteristic
46C2×#344C2211yescharacteristic
47#1742C2211yescharacteristic
48#1742C2211yescharacteristic
49#1740C2211yescharacteristic
50#1740C2211yescharacteristic
51#1738C2211yescharacteristic
52#1738C2211yescharacteristic
53#1733C2211yescharacteristic
54#1733C2211yescharacteristic
55#1733C2211yescharacteristic
56#1733C2211yescharacteristic
57C2×#338C2211yescharacteristic
58C2×H(3,ℤ/4)C2211yescharacteristic
59C2×H(3,ℤ/4)C2211yescharacteristic
60C2×H(3,ℤ/4)C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C2×#131—21no—
63C2×#131—21no—
64C2×#131—21no—
65C2×#131—21no—
66C2×#131—21no—
67C2×#131—21no—
68C2×#131—21no—
69C2×#131—21no—
70C2×#131—21no—
71C2×#131—21no—
72C2×#131D412yes—
73C2×#131D412yes—
74C22×M16—21no—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C22×M16C2311yescharacteristic
82C23×C8—21no—
83C23×C8C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90#430—22no—
91#430C2×C414yes—
92C2×#108—22no—
93#428—22no—
94#428—22no—
95#427—22no—
96#421—22no—
97#421—22no—
98#421—22no—
99#421—22no—
100C2×#104—22no—
101C2×#104—22no—
102#419C2×C412yes—
103#419C2×C412yes—
104#418C2×C412yes—
105#418C2×C412yes—
106#417—22no—
107#417—22no—
108#416—22no—
109#416—22no—
110#413—22no—
111#413—22no—
112C2×#101C2×C412yes—
113C2×#100C2×C412yes—
114#410—22no—
115#410—22no—
116C2×#99—22no—
117#405—21no—
118#405—21no—
119#405—21no—
120#405—21no—
121#405—21no—
122#405—21no—
123#405—21no—
124#405—21no—
125#405—21no—
126#405—21no—
127#405—21no—
128#405—21no—
129#405—21no—
130#405—21no—
131#405—21no—
132#405—21no—
133#405—21no—
134#405—21no—
135#405—21no—
136#405—21no—
137#405—21no—
138#405—21no—
139#405—21no—
140#405—21no—
141C2×#97—21no—
142C2×#97—21no—
143C2×#97—21no—
144C2×#97—21no—
145C2×#97—21no—
146C2×#97D412yes—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8D412yes—
153#344—22no—
154#344—22no—
155#344—22no—
156#344—22no—
157#344C2×C414yes—
158#344C2×C414yes—
159#343—22no—
160#343—22no—
161#342—22no—
162#342—22no—
163#341—22no—
164#341—22no—
165#339—22no—
166#339—22no—
167#339—22no—
168#339—22no—
169#338—22no—
170H(3,ℤ/4)—22no—
171H(3,ℤ/4)—22no—
172H(3,ℤ/4)C2×C414yes—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41C2×D412yes—
184C2×#41#4112yes—
185C22×Q8C22×C411yescharacteristic
186C22×D4—21no—
187C22×D4C22×C411yescharacteristic
188C23×C4—21no—
189C23×C4C2411yescharacteristic
190#131—22no—
191#131—22no—
192#131—22no—
193#131—22no—
194#131—22no—
195#131—22no—
196#131—22no—
197#131—22no—
198#131—41no—
199#131—41no—
200#131—41no—
201#131—41no—
202#131—41no—
203#131—41no—
204#131—41no—
205#131—41no—
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—22no—
217C2×M16—22no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—41no—
243C2×M16—41no—
244C2×M16—41no—
245C2×M16—41no—
246C2×M16C22×C412yes—
247C2×M16C22×C412yes—
248C2×M16C22×C412yes—
249C2×M16C22×C412yes—
250C2×M16C2×D412yes—
251C2×M16#4112yes—
252C2×M16#4112yes—
253C2×M16#4112yes—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—41no—
263C22×C8—41no—
264C22×C8—41no—
265C22×C8—41no—
266C22×C8C2×D412yes—
267C22×C8C2×D411yescharacteristic
268C22×C8C2×D411yescharacteristic
269C22×C8C2×D411yescharacteristic
270C22×C8C2×D411yescharacteristic
271C22×C8#4111yescharacteristic
272C22×C8#4111yescharacteristic
273C22×C8#4111yescharacteristic
274C22×C8#4111yescharacteristic
275C22×C8#4111yescharacteristic
276C22×C8#4111yescharacteristic
277C22×C8#4111yescharacteristic
278C22×C8#4111yescharacteristic
279C22×C8#4111yescharacteristic
280C22×C8#4111yescharacteristic
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—21no—
300#117—21no—
301#117C22×C412yes—
302#117C22×C412yes—
303#117#4112yes—
304#117#4112yes—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4#4112yes—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42C2×D412yes—
323#108—28no—
324#107—24no—
325#106—24no—
326#104—28no—
327#104—28no—
328#103—24no—
329#103—24no—
330#102—24no—
331#102—24no—
332#101—24no—
333#100—24no—
334#99—28no—
335#97—22no—
336#97—22no—
337#97—41no—
338#97—41no—
339#97—41no—
340#97—41no—
341#97—41no—
342#97—41no—
343#97—41no—
344#97—41no—
345#97—41no—
346#97—41no—
347C4×C8—22no—
348C4×C8—22no—
349C4×C8—41no—
350C4×C8—41no—
351C4×C8—41no—
352C4×C8—41no—
353C4×C8—41no—
354C4×C8—41no—
355C4×C8—41no—
356C4×C8—41no—
357C4×C8—41no—
358C4×C8—41no—
359C24—21no—
360C24C2×#4111yescharacteristic
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—22no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8C4×D412yes—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4C4×D412yes—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422M16—22no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—22no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450M16—42no—
451M16—42no—
452M16—42no—
453M16—42no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8C4×D414yes—
479C2×C8#12111yescharacteristic
480C2×C8#12111yescharacteristic
481C2×C8#12111yescharacteristic
482C2×C8#12111yescharacteristic
483C2×C8#12311yescharacteristic
484C2×C8#12311yescharacteristic
485C2×C8#12311yescharacteristic
486C2×C8#12311yescharacteristic
487C2×C8#12311yescharacteristic
488C2×C8#12311yescharacteristic
489C2×C8#12311yescharacteristic
490C2×C8#12311yescharacteristic
491C2×C8#12311yescharacteristic
492C2×C8#12311yescharacteristic
493C2×C8#12311yescharacteristic
494C2×C8#12311yescharacteristic
495C2×C8#12411yescharacteristic
496C2×C8#12411yescharacteristic
497C2×C8#12411yescharacteristic
498C2×C8#12411yescharacteristic
499C2×C8#12611yescharacteristic
500C2×C8#12611yescharacteristic
501C2×C8#12611yescharacteristic
502C2×C8#12611yescharacteristic
503C2×C8#12611yescharacteristic
504C2×C8#12611yescharacteristic
505C2×C8#12611yescharacteristic
506C2×C8#12611yescharacteristic
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—81no—
547C23—82no—
548C23C4×#4111yescharacteristic
549C23#52511yescharacteristic
550C23#52911yescharacteristic
551C23#52911yescharacteristic
552C23#52911yescharacteristic
553C23#52911yescharacteristic
554C23#53211yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—41no—
566D4—41no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×C4×D411yescharacteristic
588C2×C4C4×#4111yescharacteristic
589C2×C4C4×#4111yescharacteristic
590C2×C4C4×#4111yescharacteristic
591C2×C4C4×#4111yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4C2×#12311yescharacteristic
594C2×C4C2×#12311yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52511yescharacteristic
598C2×C4#52511yescharacteristic
599C2×C4C2×#12411yescharacteristic
600C2×C4C2×#12611yescharacteristic
601C2×C4C2×#12611yescharacteristic
602C2×C4#52911yescharacteristic
603C2×C4#52911yescharacteristic
604C2×C4#52911yescharacteristic
605C2×C4#52911yescharacteristic
606C2×C4#52911yescharacteristic
607C2×C4#52911yescharacteristic
608C2×C4#52911yescharacteristic
609C2×C4#52911yescharacteristic
610C2×C4#52911yescharacteristic
611C2×C4#52911yescharacteristic
612C2×C4#52911yescharacteristic
613C2×C4#53211yescharacteristic
614C8—24no—
615C8—42no—
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C8—42no—
621C8#39711yescharacteristic
622C8#39711yescharacteristic
623C8#39711yescharacteristic
624C8#39711yescharacteristic
625C8#39711yescharacteristic
626C8#39711yescharacteristic
627C8#39711yescharacteristic
628C8#39711yescharacteristic
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22C4×#12311yescharacteristic
641C22C4×#12411yescharacteristic
642C22C4×#12611yescharacteristic
643C22#239211yescharacteristic
644C22#239511yescharacteristic
645C22#239511yescharacteristic
646C22#240111yescharacteristic
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4C4×#12111yescharacteristic
655C4C4×#12311yescharacteristic
656C4C4×#12311yescharacteristic
657C4C4×#12611yescharacteristic
658C4C2×#39711yescharacteristic
659C4#239511yescharacteristic
660C4#239811yescharacteristic
661C4#240111yescharacteristic
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—82no—
670C2C4×#39711yescharacteristic
subgroupG/H
centerC8#397
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2C4×#397
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20444C2semidirect
6. #11363C2semidirect
7. #11307C2semidirect
8. #11302C2semidirect
9. #11284C2semidirect
13. #10711C2semidirect
14. #10711C2semidirect
15. #10699C2semidirect
31. #2082C4semidirect
32. #2081C4semidirect

Maximal subgroups

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

subgroupindex
#236522
#204442
#204202
C2×#17572
#113632
#113072
#113022
#112842
#111802
#111592
#111322
#107112
#106992