Finite Group Databasebeta

← #360222

Subgroups — #360222

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#444C211yesmaximal, characteristic
2C2×#2912C211yesmaximal, characteristic, complemented
3#20937C212yesmaximal
4C2×#1816C211yesmaximal, characteristic, complemented
5C2×#1815C211yesmaximal, characteristic, complemented
6#20663C211yesmaximal, characteristic, complemented
7#11249C212yesmaximal, complemented
8#11113C212yesmaximal, complemented
9#10863C211yesmaximal, characteristic, complemented
10#10860C211yesmaximal, characteristic, complemented
11#10781C212yesmaximal, complemented
12C22×#135C2211yescharacteristic
13C22×#131C2211yescharacteristic
14C22×#131C2211yescharacteristic
15C2×#520C2212yes—
16C2×#444—24no—
17C2×#444—24no—
18C2×#444C418yescomplemented
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21#2912—24no—
22#2912—24no—
23C2×#428—22no—
24C2×#428C2212yes—
25C2×#427—21no—
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106—21no—
29C22×#106C2211yescharacteristic
30C22×#104C2211yescharacteristic
31C22×#104C2211yescharacteristic
32C2×#417C2212yes—
33C2×#417C2212yes—
34C22×#97C2211yescharacteristic
35#2311C2212yes—
36#1880—22no—
37#1879—22no—
38#1878—22no—
39#1877—22no—
40#1816—24no—
41#1815—24no—
42#1809—21no—
43#1809—21no—
44#1808—22no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47#1736C2212yes—
48C2×#340C2212yes—
49C2×#339C2212yes—
50#1729C2212yes—
51C4⋊#95C2212yes—
52C22×#41C2311yescharacteristic
53C22×#41C2311yescharacteristic
54C2×#135—22no—
55C2×#135—22no—
56C2×#135—22no—
57C2×#135—22no—
58C2×#135C2×C414yes—
59C2×#135Q814yes—
60C22×Q16D411yescharacteristic
61C22×SD16D411yescharacteristic
62C22×SD16D411yescharacteristic
63C22×D8D411yescharacteristic
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—22no—
67C2×#131—22no—
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—22no—
71C2×#131—22no—
72C2×#131C2×C414yes—
73C2×#131C2×C414yescomplemented
74C2×#131D414yes—
75C2×#131D414yes—
76C22×M16—21no—
77C22×M16—21no—
78C22×M16C2312yes—
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C23×C8—21no—
82C23×C8—21no—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85#520—24no—
86#520—24no—
87#520—24no—
88#520—42no—
89C2×C4×Q8—22no—
90C2×C4×D4—22no—
91C2×C4×D4—42no—
92C2×#117—22no—
93C2×#117C2312yes—
94C22×C4⋊C4—22no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C42C2311yescharacteristic
98#444—232no—
99C2×#108—22no—
100C2×#108—22no—
101C2×#108—22no—
102C2×#108—22no—
103C2×#108—22no—
104C2×#108C2×C414yescomplemented
105#428—216no—
106#428—24no—
107#428—24no—
108#427—28no—
109#427—22no—
110#427—22no—
111C2×#107—22no—
112C2×#107—21no—
113C2×#107—21no—
114C2×#107—22no—
115C2×#107—22no—
116C2×#107D412yes—
117C2×#106—28no—
118C2×#106—21no—
119C2×#106—21no—
120C2×#106—22no—
121C2×#106—22no—
122C2×#106D411yescharacteristic
123C2×#106D411yescharacteristic
124C2×#106D414yes—
125C2×#104—22no—
126C2×#104—22no—
127C2×#104—22no—
128C2×#104—22no—
129C2×#104C2×C414yes—
130C2×#104C2×C414yescomplemented
131#417—24no—
132#417—24no—
133#417—24no—
134#417—42no—
135C2×#103—22no—
136C2×#103—22no—
137C2×#102—22no—
138C2×#102—22no—
139C2×#97—22no—
140C2×#97—22no—
141C2×#97C2×C414yescomplemented
142#391—22no—
143#390—22no—
144#386—42no—
145#380—22no—
146#380—22no—
147C2×#95—42no—
148#343—24no—
149#343—24no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#339—42no—
155#336—22no—
156#336—22no—
157C2×#41—21no—
158C2×#41—21no—
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×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41C22×C412yes—
174C2×#41C22×C412yes—
175C2×#41C2×D412yes—
176C2×#41C2×D412yes—
177C2×#41C2×Q812yes—
178C2×#41C2×Q812yes—
179C2×#41#4112yes—
180C2×#41#4112yes—
181C22×Q8C2×D411yescharacteristic
182C22×Q8C2×D411yescharacteristic
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C2×D411yescharacteristic
186C22×D4C2×D411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—42no—
193C23×C4C2411yescharacteristic
194#135—28no—
195#135—28no—
196#135—28no—
197#135—28no—
198C2×Q16—22no—
199C2×Q16—22no—
200C2×Q16C4⋊C414yes—
201C2×SD16—22no—
202C2×SD16—22no—
203C2×SD16—22no—
204C2×SD16—22no—
205C2×SD16C4⋊C414yes—
206C2×SD16C4⋊C414yes—
207C2×D8—22no—
208C2×D8—22no—
209C2×D8C4⋊C414yes—
210#131—28no—
211#131—28no—
212#131—28no—
213#131—28no—
214#131—28no—
215#131—44no—
216#131—44no—
217#131C22⋊C4116yescomplemented
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—24no—
223C2×M16—24no—
224C2×M16—24no—
225C2×M16—24no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—22no—
233C2×M16—22no—
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—42no—
237C2×M16—42no—
238C2×M16C22×C412yes—
239C2×M16C22×C412yes—
240C2×M16C2×D412yes—
241C2×M16C2×D412yes—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—24no—
250C22×C8—21no—
251C22×C8—24no—
252C22×C8—22no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8C22×C412yes—
256C22×C8C22×C412yes—
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D412yes—
260C22×C8C2×D412yes—
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D412yes—
264C22×C8C2×Q812yes—
265C22×C8#4112yes—
266C22×C8#4112yes—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×D4—44no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—44no—
273#117—28no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—44no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D412yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—21no—
306C2×C42—42no—
307C2×C42C22×C412yes—
308#108—28no—
309#108—44no—
310#107—28no—
311#107—24no—
312#107—42no—
313#107—42no—
314#106—28no—
315#106—22no—
316#106—24no—
317#106—24no—
318#106—22no—
319#104—28no—
320#104—28no—
321#103—44no—
322#103—44no—
323#102—44no—
324#102—44no—
325#97—28no—
326#95—42no—
327#95—42no—
328#95—84no—
329C24—21no—
330C24—21no—
331C24H(5,𝔽2)11yescharacteristic
332#41—22no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41—41no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41C2×C22⋊C414yes—
361#41#11714yes—
362#41C22⋊Q814yes—
363#41C22⋊Q814yes—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8C2×C4⋊C412yes—
373C2×Q8C2×C4⋊C412yes—
374C2×Q8#12112yes—
375C2×Q8#12112yes—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4C2×C4⋊C412yes—
397C2×D4C2×C4⋊C412yes—
398C2×D4#12112yes—
399C2×D4#12112yes—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—162no—
433C22×C4—162no—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×Q811yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C42−1+411yescharacteristic
447C22×C42−1+411yescharacteristic
448Q16—28no—
449SD16—28no—
450SD16—28no—
451D8—28no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—216no—
456M16—22no—
457M16—22no—
458M16—41no—
459M16—41no—
460M16—48no—
461M16—42no—
462M16—42no—
463M16—42no—
464M16—42no—
465M16C2×C22⋊C414yes—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—216no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—48no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8C2×C22⋊C414yes—
486C2×C8C2×C4⋊C412yes—
487C2×C8C2×C4⋊C412yes—
488C2×C8C4×D412yes—
489C2×C8C4×D412yes—
490C2×C8C4×D414yes—
491C2×C8#12012yes—
492C2×C8#12012yes—
493C2×C8#12112yes—
494C2×C8#12112yes—
495C2×C8#12114yes—
496C2×C8#12314yes—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C22⋊C4—48no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—162no—
514C22⋊C4—162no—
515C42—22no—
516C42—22no—
517C42—48no—
518C42—42no—
519C42—41no—
520C42—41no—
521C42—42no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23#52011yescharacteristic
535C23#53511yescharacteristic
536C23#53711yescharacteristic
537C23D4211yescharacteristic
538C23#54611yescharacteristic
539C23#55311yescharacteristic
540C23#55411yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—22no—
546Q8—41no—
547Q8—41no—
548Q8#38014yes—
549D4—24no—
550D4—24no—
551D4—24no—
552D4—24no—
553D4—22no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—41no—
563D4—41no—
564D4#38014yes—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—162no—
599C2×C4—168no—
600C2×C4—162no—
601C2×C4C22×C22⋊C411yescharacteristic
602C2×C4C22×C4⋊C411yescharacteristic
603C2×C4C2×#11711yescharacteristic
604C2×C4C2×C4×D411yescharacteristic
605C2×C4C2×C4×D411yescharacteristic
606C2×C4#52011yescharacteristic
607C2×C4C2×#12011yescharacteristic
608C2×C4C2×#12111yescharacteristic
609C2×C4C2×#12111yescharacteristic
610C2×C4C2×#12111yescharacteristic
611C2×C4C2×#12111yescharacteristic
612C2×C4C2×C22⋊Q811yescharacteristic
613C2×C4C2×C22⋊Q811yescharacteristic
614C2×C4C2×#12311yescharacteristic
615C2×C4#53611yescharacteristic
616C2×C4#53711yescharacteristic
617C2×C4D4211yescharacteristic
618C2×C4#54611yescharacteristic
619C2×C4#54711yescharacteristic
620C2×C4#54711yescharacteristic
621C2×C4#54811yescharacteristic
622C2×C4#54811yescharacteristic
623C2×C4D4×Q811yescharacteristic
624C2×C4D4×Q811yescharacteristic
625C2×C4#55211yescharacteristic
626C2×C4#55211yescharacteristic
627C2×C4#55311yescharacteristic
628C2×C4#55411yescharacteristic
629C8—22no—
630C8—22no—
631C8—24no—
632C8—42no—
633C8—82no—
634C8#38614yes—
635C8#38614yes—
636C22—22no—
637C22—22no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—22no—
643C22—22no—
644C22—24no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—42no—
650C22—44no—
651C22—81no—
652C22—81no—
653C22—82no—
654C22#231111yescharacteristic
655C22D4×C4⋊C411yescharacteristic
656C22#235211yescharacteristic
657C22#240811yescharacteristic
658C22#241611yescharacteristic
659C22#241711yescharacteristic
660C22#248711yescharacteristic
661C22#328812yes—
662C22#328912yes—
663C4—24no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—84no—
669C4—162no—
670C4C2×#38011yescharacteristic
671C4C2×#38611yescharacteristic
672C4D4×C4⋊C411yescharacteristic
673C4#235211yescharacteristic
674C4#240911yescharacteristic
675C4#241611yescharacteristic
676C4#241711yescharacteristic
677C4#248611yescharacteristic
678C2—21no—
679C2—21no—
680C2—21no—
681C2—21no—
682C2—44no—
683C2—82no—
684C2#1387211yescharacteristic
685C2#2269312yes—
686C2#2429212yes—
687C2#2479911yescharacteristic
688C2#2480011yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2912C2semidirect
4. C2×#1816C2semidirect
5. C2×#1815C2semidirect
6. #20663C2semidirect
7. #11249C2semidirect
8. #11113C2semidirect
9. #10863C2semidirect
10. #10860C2semidirect
11. #10781C2semidirect
18. C2×#444C4semidirect
73. C2×#131C2×C4semidirect
104. C2×#108C2×C4semidirect
130. C2×#104C2×C4semidirect
141. C2×#97C2×C4semidirect
217. #131C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4442
C2×#29122
#209372
C2×#18162
C2×#18152
#206632
#112492
#111132
#108632
#108602
#107812