Finite Group Databasebeta

← #195246

Subgroups — #195246

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic
3#21759C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#14281C211yesmaximal, characteristic
6#9046C214yesmaximal, complemented
7#9040C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic
9C2×#1280C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12#3189—24no—
13#3189C418yescomplemented
14C2×#484—21no—
15C2×#484C2212yes—
16C2×#467—22no—
17C2×#451—21no—
18C2×#451C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2329—22no—
22#2329—22no—
23#2329—22no—
24#2329C2214yes—
25C2×#391—21no—
26C2×#389—21no—
27C2×#387—21no—
28C2×#387—21no—
29C2×#385—21no—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33C22×#95C2211yescharacteristic
34#1892—22no—
35#1872—21no—
36#1872—21no—
37#1824C2212yes—
38#1787C2212yes—
39#1520—24no—
40C2×#328C2214yes—
41#1446C2212yes—
42#1445C2212yes—
43#1441—22no—
44C22⋊#95—21no—
45C22⋊#95—21no—
46#1294C2214yes—
47#1280C2214yes—
48C23×Q8D411yescharacteristic
49C23×Q8C2311yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C22×Q16—22no—
54C23×C8—21no—
55C23×C8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8C2×C414yes—
68C2×C22⋊Q8C2×C414yes—
69C2×C22⋊Q8D414yes—
70C2×C22⋊Q8D412yes—
71C2×C22⋊Q8C2312yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C42—21no—
86C22×C42—21no—
87#484—24no—
88#484—28no—
89#484C2×C418yescomplemented
90#467—216no—
91#451—28no—
92#451—24no—
93#451C2×C418yescomplemented
94C2×#107—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8C2×C414yes—
105C2×C22⋊C8C2312yes—
106C2×C22⋊C8C2312yes—
107#391—24no—
108#391—24no—
109#389—24no—
110#389—24no—
111#387—24no—
112#387—24no—
113#387—24no—
114#387—24no—
115#385—24no—
116#385—24no—
117#380—21no—
118#380—21no—
119#380—22no—
120#380—22no—
121#380D412yes—
122#380C2312yes—
123C4×C22⋊C4—24no—
124C4×C22⋊C4—24no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D412yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141#340—22no—
142#340—22no—
143#328—28no—
144C25C2×D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—81no—
153C22×Q8—81no—
154C22×Q8—81no—
155C22×Q8—81no—
156C22×Q8C22⋊C414yes—
157C22×Q8C22×C412yes—
158C22×Q8C2×D412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—81no—
174C23×C4—81no—
175C23×C4—81no—
176C23×C4—81no—
177C23×C4—81no—
178C23×C4—81no—
179C23×C4Q1614yescomplemented
180C23×C4C22×C412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C2×Q16—28no—
193C2×Q16—82no—
194C2×Q16—82no—
195C22×C8—22no—
196C22×C8—24no—
197C22×C8—22no—
198C22×C8—42no—
199C22×C8—81no—
200C22×C8—81no—
201C22×C8—81no—
202C22×C8—81no—
203C22⋊Q8—24no—
204C22⋊Q8—24no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C22⋊Q8—42no—
218C22⋊Q8C22⋊C418yes—
219C22⋊Q8C22×C414yes—
220C22⋊Q8C2×D414yes—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—81no—
242C2×C4⋊C4—81no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4—81no—
246C2×C4⋊C4—81no—
247C2×C4⋊C4C22⋊C414yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—82no—
270C2×C42—82no—
271#107—28no—
272#103—28no—
273#103—28no—
274#103—82no—
275#103—82no—
276C22⋊C8—24no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—42no—
281C22⋊C8C22×C414yes—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—82no—
295#95—82no—
296#95—82no—
297#95—82no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24C4×D412yes—
303C24#12011yescharacteristic
304C24#12011yescharacteristic
305C24#12112yes—
306C24C2×Q1612yes—
307C24#13512yes—
308C24#13712yes—
309C24#13712yes—
310C24C22×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—81no—
324C2×Q8—81no—
325C2×Q8—82no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8C2×C22⋊C414yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—84no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C412yes—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4#12011yescharacteristic
412C22×C4#12011yescharacteristic
413C22×C4#12112yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12312yes—
417C22×C4#12312yes—
418C22×C4C2×Q1614yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425Q16—88no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—84no—
433C2×C8—84no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4C2×C22⋊C414yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—28no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—84no—
462C42—84no—
463C42—84no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23#38612yes—
489C23#38612yes—
490C23C4×Q1614yes—
491C23#44114yes—
492C23#45112yes—
493C23#45112yes—
494C23#45212yes—
495C23#45212yes—
496C23#46214yes—
497C23#46414yes—
498C23C2×C4×D411yescharacteristic
499C23C2×#12011yescharacteristic
500C23C2×#12111yescharacteristic
501C23C22×Q1611yescharacteristic
502C23C2×#13511yescharacteristic
503C23C2×#13711yescharacteristic
504C23C2×#13711yescharacteristic
505Q8—44no—
506Q8—44no—
507Q8—44no—
508Q8—82no—
509Q8—84no—
510Q8—82no—
511Q8—82no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—82no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—82no—
566C2×C4—84no—
567C2×C4—82no—
568C2×C4—84no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4#38612yes—
582C2×C4#38612yes—
583C2×C4C22×C22⋊C411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C8—44no—
588C8—82no—
589C22—21no—
590C22—22no—
591C22—21no—
592C22—22no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—21no—
608C22—22no—
609C22—21no—
610C22—22no—
611C22—21no—
612C22—22no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22#187814yes—
618C22#187814yes—
619C22C2×#38611yescharacteristic
620C22C2×C4×Q1611yescharacteristic
621C22C2×#44111yescharacteristic
622C22C2×#45111yescharacteristic
623C22C2×#45211yescharacteristic
624C22#301911yescharacteristic
625C22#302011yescharacteristic
626C22#302611yescharacteristic
627C22#302611yescharacteristic
628C22C2×#46211yescharacteristic
629C22C2×#46411yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—44no—
637C4—44no—
638C4—44no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—162no—
645C2—21no—
646C2—24no—
647C2—24no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2#1828711yescharacteristic
652C2#1828911yescharacteristic
653C2#1829511yescharacteristic
654C2#1829711yescharacteristic
655C2C2×#187811yescharacteristic
656C2#2094311yescharacteristic
657C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9046C2semidirect
7. #9040C2semidirect
13. #3189C4semidirect
89. #484C2×C4semidirect
93. #451C2×C4semidirect
179. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C2×#23292
#217592
C2×#15202
#142812
#90462
#90402
#80232
C2×#12802