Finite Group Databasebeta

← #192909

Subgroups — #192909

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60128C211yesmaximal, characteristic
2#20748C211yesmaximal, characteristic
3#17816C211yesmaximal, characteristic, complemented
4#17793C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic
6#8905C212yesmaximal, complemented
7#8904C212yesmaximal, complemented
8#8903C212yesmaximal, complemented
9#8902C212yesmaximal, complemented
10#8096C211yesmaximal, characteristic, complemented
11#7898C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13#2925—22no—
14#2925C2212yes—
15#2924—22no—
16#2924—21no—
17#2923—22no—
18#2923—21no—
19#2920—21no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#432—21no—
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2906C2212yes—
26#2880—21no—
27#2875—21no—
28C22×C4×C8C2211yescharacteristic
29#2283C2211yescharacteristic
30#1826—21no—
31#1806C2212yes—
32#1806C2212yes—
33#1789—21no—
34#1759—21no—
35#1759—21no—
36#1572—22no—
37#1571C2212yescomplemented
38#1570C2212yescomplemented
39#1568—22no—
40#1568—21no—
41#1568—21no—
42#1567C2212yescomplemented
43#1566C2212yescomplemented
44#1564—22no—
45#1476C411yescharacteristic, complemented
46#1476C411yescharacteristic, complemented
47#1476C411yescharacteristic, complemented
48#1476C411yescharacteristic, complemented
49#1476C2211yescharacteristic
50#1476C2211yescharacteristic
51#1476C2211yescharacteristic
52#1476C2211yescharacteristic
53C2×Q8⋊C8C411yescharacteristic, complemented
54C2×Q8⋊C8C411yescharacteristic, complemented
55C2×D4⋊C8C411yescharacteristic, complemented
56C2×D4⋊C8C411yescharacteristic, complemented
57#1303C2212yescomplemented
58#1303C2212yescomplemented
59#1279C2212yescomplemented
60#1279C2212yescomplemented
61C22×#41D411yescharacteristic
62C2×#131—22no—
63C23×C8—21no—
64C4×#41—21no—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41—21no—
68C4×#41C2×C411yescharacteristic
69C4×#41C2×C411yescharacteristic
70C4×#41D411yescharacteristic
71C4×#41D411yescharacteristic
72C4×#41Q811yescharacteristic
73C4×#41Q811yescharacteristic
74C4×#41C2311yescharacteristic
75C4×#41C2311yescharacteristic
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8Q811yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4Q811yescharacteristic
82C2×#117—21no—
83C2×#117D411yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#433—24no—
87#433—24no—
88#433—24no—
89#432—22no—
90#432—22no—
91#432—22no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—22no—
96#432—21no—
97#432—21no—
98#432D411yescharacteristic
99#432D411yescharacteristic
100#432D412yes—
101#432D412yes—
102#431—22no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—22no—
107#431—21no—
108#431—21no—
109#431C2×C411yescharacteristic
110#431C2×C411yescharacteristic
111#431D412yes—
112#431C2311yescharacteristic
113#431C2311yescharacteristic
114#424—22no—
115#424—24no—
116#423—22no—
117#423C2312yes—
118#423C2312yes—
119C2×C4⋊C8—22no—
120C2×C4⋊C8C2×C411yescharacteristic
121C2×C4⋊C8C2×C411yescharacteristic
122C2×C4⋊C8C2312yes—
123C2×#104—22no—
124C2×#104—22no—
125#408—24no—
126#408—22no—
127#407—22no—
128#407—21no—
129#407—21no—
130C2×C22⋊C8—22no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133#405—24no—
134#405—22no—
135C4×M16—22no—
136C4×M16—21no—
137C4×M16—21no—
138C2×#97—22no—
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2×C411yescharacteristic
145C2×C4×C8C2×C411yescharacteristic
146C2×C4×C8C2×C411yescharacteristic
147C2×C4×C8C2×C411yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#383—21no—
151#382—21no—
152#381—21no—
153#380—21no—
154C4×C4⋊C4C2312yes—
155C4×C22⋊C4—21no—
156#376—21no—
157#339—22no—
158#339—22no—
159#335—24no—
160#335—22no—
161#335—22no—
162#334—24no—
163#334—22no—
164#334—22no—
165Q8⋊C8C2×C411yescharacteristic
166Q8⋊C8C2×C411yescharacteristic
167Q8⋊C8C2×C411yescharacteristic
168Q8⋊C8C2×C411yescharacteristic
169Q8⋊C8C2×C411yescharacteristic
170Q8⋊C8C2×C411yescharacteristic
171Q8⋊C8C2×C411yescharacteristic
172Q8⋊C8C2×C411yescharacteristic
173D4⋊C8C2×C411yescharacteristic
174D4⋊C8C2×C411yescharacteristic
175D4⋊C8C2×C411yescharacteristic
176D4⋊C8C2×C411yescharacteristic
177D4⋊C8C2×C411yescharacteristic
178D4⋊C8C2×C411yescharacteristic
179D4⋊C8C2×C411yescharacteristic
180D4⋊C8C2×C411yescharacteristic
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—24no—
184C2×#41—42no—
185C2×#41C4⋊C411yescharacteristic
186C2×#41C4⋊C411yescharacteristic
187C2×#41C2×D411yescharacteristic
188C2×#41C2×D411yescharacteristic
189C22×Q8C4⋊C411yescharacteristic
190C22×D4—21no—
191C22×D4C4⋊C411yescharacteristic
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4C2×D411yescharacteristic
196C23×C4#4111yescharacteristic
197C23×C4#4111yescharacteristic
198#131—44no—
199#131—44no—
200C2×M16—24no—
201C2×M16—22no—
202C2×M16—22no—
203C22×C8—21no—
204C22×C8—24no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—44no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8C4⋊C411yescharacteristic
219C4×Q8C4⋊C411yescharacteristic
220C4×Q8C4⋊C411yescharacteristic
221C4×Q8C4⋊C411yescharacteristic
222C4×Q8C22×C411yescharacteristic
223C4×Q8C22×C411yescharacteristic
224C4×Q8C2×Q811yescharacteristic
225C4×Q8C2×Q811yescharacteristic
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C4⋊C411yescharacteristic
247C4×D4C4⋊C411yescharacteristic
248C4×D4C4⋊C411yescharacteristic
249C4×D4C4⋊C411yescharacteristic
250C4×D4C22×C411yescharacteristic
251C4×D4C22×C411yescharacteristic
252C4×D4C2×Q811yescharacteristic
253C4×D4C2×Q811yescharacteristic
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117C4⋊C411yescharacteristic
265#117C4⋊C411yescharacteristic
266#117C2×D411yescharacteristic
267#117C2×D411yescharacteristic
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C4⋊C411yescharacteristic
272C2×C4⋊C4C4⋊C411yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×Q811yescharacteristic
297C2×C42C2×Q811yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42C2411yescharacteristic
301C4⋊C8—24no—
302C4⋊C8—24no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8C22×C411yescharacteristic
310C4⋊C8C22×C411yescharacteristic
311C4⋊C8C22×C411yescharacteristic
312C4⋊C8C22×C411yescharacteristic
313C4⋊C8C2×D412yes—
314C4⋊C8C2×D412yes—
315C4⋊C8C2×D414yes—
316#104—44no—
317#104—44no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320C22⋊C8—21no—
321C22⋊C8—21no—
322C22⋊C8—21no—
323C22⋊C8—21no—
324C22⋊C8—22no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—41no—
330C22⋊C8—41no—
331#97—24no—
332#97—21no—
333#97—22no—
334#97—21no—
335#97—22no—
336#97—41no—
337C4×C8—24no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—22no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—41no—
350C4×C8C22×C411yescharacteristic
351C4×C8C22×C411yescharacteristic
352C4×C8C22×C411yescharacteristic
353C4×C8C22×C411yescharacteristic
354#95—22no—
355#95—41no—
356#95—41no—
357#95—41no—
358C24—21no—
359C24#12311yescharacteristic
360#41—22no—
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—44no—
366#41—41no—
367#41—41no—
368#41—44no—
369#41#13514yes—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—41no—
373C2×Q8C2×C4⋊C411yescharacteristic
374C2×Q8C2×C4⋊C411yescharacteristic
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—41no—
382C2×D4—44no—
383C2×D4C2×C4⋊C411yescharacteristic
384C2×D4C2×C4⋊C411yescharacteristic
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—82no—
410C22×C4C2×C4⋊C411yescharacteristic
411C22×C4C2×C4⋊C411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C4×D411yescharacteristic
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4#12011yescharacteristic
417C22×C4#12011yescharacteristic
418C22×C4C22⋊Q811yescharacteristic
419C22×C4C22⋊Q811yescharacteristic
420C22×C4C22⋊Q811yescharacteristic
421C22×C4C22⋊Q811yescharacteristic
422C22×C4#12311yescharacteristic
423C22×C4#12311yescharacteristic
424C22×C4#12311yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428M16—48no—
429M16—42no—
430M16—42no—
431M16—42no—
432M16—42no—
433C2×C8—24no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—48no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—81no—
455C2×C8—81no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4C2×C4⋊C411yescharacteristic
469C4⋊C4C2×C4⋊C411yescharacteristic
470C4⋊C4C2×C4⋊C411yescharacteristic
471C4⋊C4C2×C4⋊C411yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—82no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—42no—
499C42C2×C4⋊C411yescharacteristic
500C42C2×C4⋊C411yescharacteristic
501C42C4×D411yescharacteristic
502C42C4×D411yescharacteristic
503C42C4×D411yescharacteristic
504C42C4×D411yescharacteristic
505C42#12011yescharacteristic
506C42#12011yescharacteristic
507C42C22⋊Q811yescharacteristic
508C42C22⋊Q811yescharacteristic
509C42C22⋊Q811yescharacteristic
510C42C22⋊Q811yescharacteristic
511C42C23×C411yescharacteristic
512C42C22×D411yescharacteristic
513C42C22×D411yescharacteristic
514C42C22×Q811yescharacteristic
515C23—22no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—81no—
523C23#38511yescharacteristic
524C23#38511yescharacteristic
525C23C2×#12311yescharacteristic
526Q8—21no—
527Q8—21no—
528Q8—21no—
529Q8—42no—
530Q8#42712yes—
531D4—24no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4#42712yes—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4#38511yescharacteristic
589C2×C4#38511yescharacteristic
590C2×C4#38511yescharacteristic
591C2×C4#38511yescharacteristic
592C2×C4#38511yescharacteristic
593C2×C4#38511yescharacteristic
594C2×C4C22×C4⋊C411yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×#12011yescharacteristic
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4C2×C22⋊Q811yescharacteristic
600C2×C4C2×#13512yes—
601C2×C4#57512yes—
602C8—22no—
603C8—22no—
604C8—41no—
605C8—41no—
606C8—41no—
607C8—41no—
608C8—41no—
609C8—41no—
610C8—42no—
611C8—44no—
612C8—42no—
613C8—82no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—21no—
621C22—21no—
622C22—44no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—84no—
629C22#186112yes—
630C22C2×#38511yescharacteristic
631C22#309112yes—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—162no—
648C4C2×#42711yescharacteristic
649C4#291111yescharacteristic
650C4#295711yescharacteristic
651C4#295711yescharacteristic
652C4#300212yes—
653C2—22no—
654C2—22no—
655C2—41no—
656C2—42no—
657C2—81no—
658C2#2084311yescharacteristic
659C2C2×#186111yescharacteristic
660C2#2088611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17816C2semidirect
4. #17793C2semidirect
6. #8905C2semidirect
7. #8904C2semidirect
8. #8903C2semidirect
9. #8902C2semidirect
10. #8096C2semidirect
11. #7898C2semidirect
37. #1571C22semidirect
38. #1570C22semidirect
42. #1567C22semidirect
43. #1566C22semidirect
45. #1476C4semidirect
46. #1476C4semidirect
47. #1476C4semidirect
48. #1476C4semidirect
53. C2×Q8⋊C8C4semidirect
54. C2×Q8⋊C8C4semidirect
55. C2×D4⋊C8C4semidirect
56. C2×D4⋊C8C4semidirect
57. #1303C22semidirect
58. #1303C22semidirect
59. #1279C22semidirect
60. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#601282
#207482
#178162
#177932
C2×#14762
#89052
#89042
#89032
#89022
#80962
#78982