Finite Group Databasebeta

← #109593

Subgroups — #109593

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1749C211yesmaximal, characteristic
2#14697C211yesmaximal, characteristic
3C2×#1294C212yesmaximal
4C2×#1294C212yesmaximal
5#7554C211yesmaximal, characteristic
6C23×C42C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C22×#97—21no—
9C2×#389C411yescharacteristic, complemented
10C2×#389C411yescharacteristic, complemented
11C2×#385—21no—
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C22×#95—21no—
15C22×#95—21no—
16C22×#95C2212yes—
17#1749—24no—
18C2×#336—21no—
19#1436—21no—
20#1436—21no—
21C4⋊#95C411yescharacteristic, complemented
22C4⋊#95C411yescharacteristic, complemented
23C22⋊#95—21no—
24#1294—24no—
25#1294C418yescomplemented
26C24×C4—22no—
27C24×C4D411yescharacteristic
28C24×C4D411yescharacteristic
29C24×C4C2311yescharacteristic
30C23×C8—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C42—22no—
40C22×C42—22no—
41C22×C42—42no—
42C22×C42—42no—
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8C2×C414yes—
51C2×C22⋊C8D414yes—
52C2×C22⋊C8Q814yes—
53C2×#97—24no—
54C2×#97—42no—
55C2×#97—42no—
56#389—42no—
57#389—42no—
58#385—44no—
59#380—22no—
60#380—22no—
61#380—22no—
62#380C2×C414yes—
63C2×#95—21no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—42no—
79C2×#95—42no—
80C2×#95—42no—
81C2×#95—42no—
82C2×#95C2×C414yes—
83#336—24no—
84C25C2×D411yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×Q812yes—
133C23×C4#4112yes—
134C23×C4#4112yes—
135C23×C4#4112yes—
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4C4212yes—
162C2×C4⋊C4C4212yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—41no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—48no—
182C2×C42—48no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C22⋊C8—24no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C22⋊C8C4⋊C418yescomplemented
191#97—48no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—42no—
196#95—48no—
197#95—48no—
198#95—42no—
199#95—42no—
200#95—42no—
201#95—42no—
202#95—42no—
203#95—41no—
204#95—42no—
205#95—41no—
206#95—42no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24C2×C22⋊C411yescharacteristic
216C24C4×D412yes—
217C24#12112yes—
218C24#12411yescharacteristic
219C24C4⋊Q811yescharacteristic
220C24AGL(1,ℤ/8)11yescharacteristic
221C24AGL(1,ℤ/8)11yescharacteristic
222C24AGL(1,ℤ/8)12yes—
223C24#13711yescharacteristic
224C24#13711yescharacteristic
225C24#13712yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—28no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4#9912yes—
354C22×C4#9912yes—
355C22×C4#10412yes—
356C22×C4#10412yes—
357C22×C4C2×C4211yescharacteristic
358C22×C4C2×C4⋊C412yes—
359C22×C4#11711yescharacteristic
360C22×C4#11712yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4C4×Q812yes—
365C22×C4C4×Q812yes—
366C22×C4#12011yescharacteristic
367C22×C4C22⋊Q812yes—
368C22×C4C22⋊Q812yes—
369C22×C4#12311yescharacteristic
370C22×C4#12312yes—
371C22×C4#12512yes—
372C22×C4#12612yes—
373C2×C8—28no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—44no—
379C2×C8—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42—24no—
393C42—24no—
394C42—44no—
395C42—44no—
396C42—42no—
397C42—42no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—22no—
437C23—22no—
438C23C4×C22⋊C411yescharacteristic
439C23#38611yescharacteristic
440C23#38812yes—
441C23#38912yes—
442C23#39111yescharacteristic
443C23C2×#9911yescharacteristic
444C23C2×#9911yescharacteristic
445C23#41011yescharacteristic
446C23#41011yescharacteristic
447C23#41711yescharacteristic
448C23#41711yescharacteristic
449C23#41811yescharacteristic
450C23#41911yescharacteristic
451C23C2×#10411yescharacteristic
452C23C2×#10411yescharacteristic
453C23#42111yescharacteristic
454C23#42111yescharacteristic
455C23#44012yes—
456C23#44012yes—
457C23#44112yes—
458C23#44212yes—
459C23#46812yes—
460C23#46812yes—
461C23#46912yes—
462C23#47012yes—
463C23#48912yes—
464C23#49011yescharacteristic
465C23#49111yescharacteristic
466C23#50111yescharacteristic
467C23#50112yes—
468C23#50111yescharacteristic
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4C4×C4⋊C412yes—
569C2×C4#38512yes—
570C2×C4#38712yes—
571C2×C4#38712yes—
572C8—48no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22#143611yescharacteristic
606C22#150411yescharacteristic
607C22#150411yescharacteristic
608C22#150511yescharacteristic
609C22#150611yescharacteristic
610C22#150711yescharacteristic
611C22#150811yescharacteristic
612C22#150911yescharacteristic
613C22#150911yescharacteristic
614C22#151011yescharacteristic
615C22#151111yescharacteristic
616C22#151211yescharacteristic
617C22#151211yescharacteristic
618C22#151311yescharacteristic
619C22#151311yescharacteristic
620C22#151411yescharacteristic
621C22#151511yescharacteristic
622C22C4×#9911yescharacteristic
623C22C4×#9911yescharacteristic
624C22C4×#10411yescharacteristic
625C22#175911yescharacteristic
626C22#176211yescharacteristic
627C22#176311yescharacteristic
628C22#187111yescharacteristic
629C22#187111yescharacteristic
630C22#189411yescharacteristic
631C22#189511yescharacteristic
632C22#189811yescharacteristic
633C22#189911yescharacteristic
634C22#193512yes—
635C22#193612yes—
636C22#194312yes—
637C22#194412yes—
638C22#198611yescharacteristic
639C22#198611yescharacteristic
640C22#199211yescharacteristic
641C22#199311yescharacteristic
642C22#199611yescharacteristic
643C22#199611yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—88no—
653C4—82no—
654C4—82no—
655C4—161no—
656C4—161no—
657C2—22no—
658C2—22no—
659C2—22no—
660C2—22no—
661C2#821611yescharacteristic
662C2#821711yescharacteristic
663C2#821811yescharacteristic
664C2#821911yescharacteristic
665C2#911511yescharacteristic
666C2#911611yescharacteristic
667C2#911711yescharacteristic
668C2#911811yescharacteristic
669C2#988711yescharacteristic
670C2#988711yescharacteristic
671C2#988811yescharacteristic
672C2#988911yescharacteristic
673C2#1088111yescharacteristic
674C2#1088611yescharacteristic
675C2#1091211yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#389C4semidirect
10. C2×#389C4semidirect
21. C4⋊#95C4semidirect
22. C4⋊#95C4semidirect
25. #1294C4semidirect
190. C22⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17492
#146972
C2×#12942
#75542