Finite Group Databasebeta

← #374163

Subgroups — #374163

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2C2×#2855C211yesmaximal, characteristic
3C2×#1901C212yesmaximal
4C2×#1728C211yesmaximal, characteristic
5#11204C212yesmaximal, complemented
6#11045C212yesmaximal, complemented
7#10963C212yesmaximal, complemented
8#10913C212yesmaximal, complemented
9#10739C212yesmaximal
10C23×M16C2212yes—
11C22×C22⋊Q8C2211yescharacteristic
12C23×C42C2211yescharacteristic
13#2920—28no—
14C2×#432—21no—
15C2×#432—21no—
16C2×#431—21no—
17C2×#431—21no—
18C2×#423C2212yes—
19C22×#101C2214yes—
20C2×#407C2212yes—
21C2×C4×M16C2212yes—
22#2855—21no—
23#2855—21no—
24#2855—21no—
25#2855—21no—
26#2855—21no—
27#2855—21no—
28#2855—21no—
29#2855—21no—
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#387C412yes—
35C2×#387C412yes—
36C2×#385C412yes—
37C2×#381C412yes—
38C2×#380C2211yescharacteristic
39#1901—28no—
40C2×#343C2214yes—
41C2×#343C2214yes—
42C2×#343C2214yes—
43#1728—21no—
44#1728—21no—
45#1728C2212yes—
46#1728C2212yes—
47C2×#336—21no—
48C2×#336—21no—
49C23×Q8C2311yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C22×M16—22no—
54C22×M16—28no—
55C22×M16—24no—
56C22×M16—24no—
57C22×M16—44no—
58C22×M16C2314yes—
59C22×M16C2314yes—
60C23×C8—22no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—24no—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C42—24no—
76C22×C42—24no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—28no—
90#432—28no—
91#431—28no—
92#431—28no—
93#423—28no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×#101—28no—
97C2×#101—28no—
98C2×#101—44no—
99#407—24no—
100#407D418yes—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C4×M16—28no—
104C2×#97—22no—
105C2×C4×C8—22no—
106#391—21no—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391—21no—
112#391—21no—
113#391—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#387—22no—
123#387—22no—
124#387—22no—
125#387—22no—
126#387—22no—
127#387—22no—
128#387—22no—
129#387—22no—
130#385—22no—
131#385—22no—
132#385—22no—
133#385—22no—
134#381—22no—
135#381—22no—
136#381—22no—
137#381—22no—
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142C2×#95C2×C412yes—
143C2×#95C2×C412yes—
144C2×#95C2×C412yes—
145C2×#95C2×C412yes—
146C2×#95C2×C412yes—
147C2×#95C2×C412yes—
148C2×#95C2×C412yes—
149#343—28no—
150#343—28no—
151#343—44no—
152#336—22no—
153#336—22no—
154#336—22no—
155#336—22no—
156C25C22×C411yescharacteristic
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—24no—
162C22×Q8—22no—
163C22×Q8—22no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C412yes—
188C23×C4C22×C412yes—
189C23×C4C22×C412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D414yes—
195C23×C4#4114yes—
196C23×C4C2411yescharacteristic
197C2×M16—28no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—24no—
206C2×M16—24no—
207C2×M16—24no—
208C2×M16—416no—
209C2×M16—416no—
210C2×M16—44no—
211C2×M16—44no—
212C2×M16C2×D418yes—
213C2×M16#4118yes—
214C22×C8—28no—
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—44no—
218C22×C8—44no—
219C22⋊Q8—42no—
220C22⋊Q8—42no—
221C22⋊Q8—44no—
222C22⋊Q8—44no—
223C22⋊Q8—42no—
224C22⋊Q8—42no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C4⋊C8—28no—
286C4⋊C8—28no—
287#101—416no—
288C22⋊C8—28no—
289C22⋊C8—28no—
290#97—28no—
291C4×C8—28no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C4×D414yes—
315C24C23×C411yescharacteristic
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—44no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4#10114yescomplemented
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C4×D414yes—
409C22×C4#12012yes—
410C22×C4#12012yes—
411C22×C4#12114yes—
412C22×C4#12114yes—
413C22×C4#12312yes—
414C22×C4#12312yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4112yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422M16—28no—
423M16—28no—
424M16—28no—
425M16—28no—
426M16—416no—
427M16—44no—
428M16—44no—
429M16—44no—
430M16—44no—
431C2×C8—28no—
432C2×C8—28no—
433C2×C8—416no—
434C2×C8—44no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—28no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—28no—
491C23—21no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23#38612yes—
498C23#38612yes—
499C23C2×#10112yes—
500C23#41312yes—
501C23C22×C22⋊C411yescharacteristic
502C23C2×C4×D412yes—
503C23C4×#4112yes—
504C23#52012yes—
505Q8—82no—
506Q8—82no—
507Q8—82no—
508Q8—82no—
509C2×C4—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4#38612yes—
571C2×C4#38612yes—
572C2×C4C2×#10112yes—
573C2×C4C2×#10112yes—
574C2×C4#41312yes—
575C2×C4#41312yes—
576C2×C4C2×#12011yescharacteristic
577C2×C4C2×#12112yes—
578C2×C4C2×#12311yescharacteristic
579C2×C4#52512yes—
580C2×C4#53212yes—
581C2×C4#53511yescharacteristic
582C2×C4#53611yescharacteristic
583C2×C4#53712yes—
584C2×C4#54614yes—
585C2×C4#54714yes—
586C2×C4#55114yes—
587C2×C4#55214yes—
588C8—48no—
589C8—48no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—416no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22#190114yes—
611C22C2×#38611yescharacteristic
612C22#229711yescharacteristic
613C22#231111yescharacteristic
614C22#235914yes—
615C22C22×#10111yescharacteristic
616C22C2×#41311yescharacteristic
617C22C2×#41311yescharacteristic
618C22C2×#41311yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#241112yes—
634C4#241312yes—
635C4#243012yes—
636C4#249812yes—
637C2—24no—
638C2—24no—
639C2—42no—
640C2#1413511yescharacteristic
641C2C2×#190111yescharacteristic
642C2#2098911yescharacteristic
643C2#2142511yescharacteristic
644C2#2142611yescharacteristic
645C2#2187911yescharacteristic
646C2#2188011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11204C2semidirect
6. #11045C2semidirect
7. #10963C2semidirect
8. #10913C2semidirect
397. C22×C4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#28552
C2×#19012
C2×#17282
#112042
#110452
#109632
#109132
#107392