Finite Group Databasebeta

← #192776

Subgroups — #192776

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34194C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic, complemented
3#21984C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1478C211yesmaximal, characteristic
6#8897C211yesmaximal, characteristic, complemented
7#8894C211yesmaximal, characteristic, complemented
8#8893C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic, complemented
10#8889C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic, complemented
12#8885C211yesmaximal, characteristic, complemented
13#8882C211yesmaximal, characteristic, complemented
14#8038C211yesmaximal, characteristic, complemented
15#7894C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2354C2211yescharacteristic
21#2354C2211yescharacteristic
22#2352—21no—
23#2352—21no—
24D4×C4⋊C4C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26#2345—21no—
27#2342—22no—
28D4×C22⋊C4—21no—
29#2319—21no—
30#2318—22no—
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#2283C2211yescharacteristic
34#2280—21no—
35C2×#380—21no—
36#2275—21no—
37#1949—22no—
38#1948C2211yescharacteristic, complemented
39#1948C2211yescharacteristic, complemented
40#1947C2211yescharacteristic, complemented
41#1947C2211yescharacteristic, complemented
42#1854—21no—
43#1854—21no—
44#1854—22no—
45#1846—21no—
46#1845C2211yescharacteristic, complemented
47#1844C2211yescharacteristic, complemented
48#1835—21no—
49#1834C2211yescharacteristic, complemented
50#1834C2211yescharacteristic, complemented
51#1823—21no—
52#1802C2211yescharacteristic
53#1802C2211yescharacteristic
54#1802C2211yescharacteristic
55#1802C2211yescharacteristic
56#1786—21no—
57C4×#102—21no—
58C4×#102—21no—
59#1478C418yescomplemented
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1296C2211yescharacteristic, complemented
65#1296C2211yescharacteristic, complemented
66#1296C2211yescharacteristic, complemented
67#1296C2211yescharacteristic, complemented
68#1279C2211yescharacteristic, complemented
69#1279C2211yescharacteristic, complemented
70#1279C2211yescharacteristic, complemented
71#1279C2211yescharacteristic, complemented
72C23×D4D411yescharacteristic
73C24×C4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4C2×C414yes—
81C2×C4×D4D411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4Q814yes—
84C2×C4×D4C2311yescharacteristic
85C2×C4×D4C2311yescharacteristic
86C2×#117—21no—
87C2×#117—21no—
88C2×#117—21no—
89C2×#117D412yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4D411yescharacteristic
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C42C2311yescharacteristic
96#431—22no—
97#431—22no—
98#431—22no—
99#431—22no—
100#431C2×C414yescomplemented
101#431C2×C414yescomplemented
102C2×#107—22no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#106—22no—
106C2×#106—21no—
107C2×#106—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#388—22no—
129#388—22no—
130#388—22no—
131#388—22no—
132#387—22no—
133#387—22no—
134#386—22no—
135#386—22no—
136#385—22no—
137#385—22no—
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#384C2311yescharacteristic
142#383—21no—
143#382—21no—
144#382D412yes—
145#382D411yescharacteristic
146#382D411yescharacteristic
147#381—22no—
148#381—21no—
149#380—22no—
150#380—22no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157#380—21no—
158#380—21no—
159#380D411yescharacteristic
160#380D411yescharacteristic
161#379—22no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2311yescharacteristic
169C4×C4⋊C4C2311yescharacteristic
170C4×C22⋊C4—22no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—22no—
173C4×C22⋊C4—22no—
174C4×C22⋊C4—21no—
175#376—21no—
176C2×#95—22no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—22no—
182#340—22no—
183#340—21no—
184#340—21no—
185#340—21no—
186#340—21no—
187D4⋊C8C2×C414yes—
188D4⋊C8C2×C414yes—
189D4⋊C8C2×C414yescomplemented
190D4⋊C8C2×C414yescomplemented
191C25—21no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—41no—
197C22×D4—41no—
198C22×D4C4⋊C414yes—
199C22×D4C2×D411yescharacteristic
200C22×D4C2×D411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—24no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—81no—
211C23×C4—81no—
212C23×C4—81no—
213C23×C4—81no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—42no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4C4⋊C414yes—
238C4×D4C4⋊C414yes—
239C4×D4C22×C412yes—
240C4×D4C22×C412yes—
241C4×D4C2×Q812yes—
242C4×D4C2×Q812yes—
243#117—81no—
244#117—81no—
245#117—81no—
246#117—81no—
247#117—81no—
248#117—81no—
249#117—82no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—81no—
265C2×C4⋊C4—81no—
266C2×C4⋊C4—81no—
267C2×C4⋊C4—81no—
268C2×C4⋊C4C4⋊C414yes—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—84no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—41no—
300C2×C42C22×C412yes—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×Q812yes—
305C2×C42#4112yes—
306C2×C42#4112yes—
307C2×C42C2411yescharacteristic
308#107—81no—
309#107—81no—
310#107—82no—
311#106—81no—
312#106—81no—
313#106—82no—
314C4⋊C8—24no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8C22×C412yes—
322C4⋊C8C22×C412yes—
323#102—42no—
324#102—42no—
325#102—42no—
326#102—42no—
327C22⋊C8—24no—
328C22⋊C8—24no—
329C22⋊C8—42no—
330C4×C8—24no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—42no—
334C4×C8C22×C412yes—
335C4×C8C22×C412yes—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—21no—
341#95—22no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—21no—
349#95—21no—
350#95—41no—
351#95—41no—
352C24—21no—
353C24—21no—
354C24—24no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—81no—
360C24—81no—
361C24#12011yescharacteristic
362C2×D4—28no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—24no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4C2×C4⋊C412yes—
378C2×D4C2×C4⋊C412yes—
379C2×D4#13512yes—
380C2×D4#13512yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4C2×C4⋊C412yes—
433C22×C4C4×D412yes—
434C22×C4C4×D412yes—
435C22×C4#12011yescharacteristic
436C22×C4#12011yescharacteristic
437C22×C4#12011yescharacteristic
438C22×C4C22⋊Q812yes—
439C22×C4C22⋊Q812yes—
440C22×C4#12312yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4C2×C4⋊C412yes—
496C4⋊C4C2×C4⋊C412yes—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—82no—
518C42—82no—
519C42—82no—
520C42—82no—
521C42C2×C4⋊C412yes—
522C42C4×D412yes—
523C42C4×D412yes—
524C42C22⋊Q812yes—
525C42C22⋊Q812yes—
526C42#12312yes—
527C42C23×C411yescharacteristic
528C42C22×Q811yescharacteristic
529C42C2×#4111yescharacteristic
530C42C2×#4111yescharacteristic
531C23—22no—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—48no—
542C23—48no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23#38512yes—
559C23C2×#12011yescharacteristic
560D4—22no—
561D4—22no—
562D4—22no—
563D4—22no—
564D4—22no—
565D4—48no—
566D4—41no—
567D4—41no—
568D4#42714yes—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—164no—
629C2×C4—164no—
630C2×C4#38512yes—
631C2×C4#38512yes—
632C2×C4#38512yes—
633C2×C4C22×C4⋊C411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×C22⋊Q811yescharacteristic
637C2×C4C2×C22⋊Q811yescharacteristic
638C2×C4C2×#12311yescharacteristic
639C2×C4C2×#13511yescharacteristic
640C2×C4C2×#13511yescharacteristic
641C2×C4#57511yescharacteristic
642C2×C4#57511yescharacteristic
643C2×C4#57611yescharacteristic
644C2×C4#57611yescharacteristic
645C2×C4#57711yescharacteristic
646C2×C4#57711yescharacteristic
647C8—42no—
648C8—42no—
649C8—42no—
650C8—42no—
651C8—81no—
652C8—81no—
653C22—22no—
654C22—24no—
655C22—22no—
656C22—22no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—42no—
662C22—41no—
663C22—41no—
664C22—42no—
665C22—41no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—816no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22C2×#38511yescharacteristic
676C22#300211yescharacteristic
677C22#300211yescharacteristic
678C22#300811yescharacteristic
679C22#300911yescharacteristic
680C22#301911yescharacteristic
681C22#302011yescharacteristic
682C4—22no—
683C4—21no—
684C4—21no—
685C4—22no—
686C4—22no—
687C4—42no—
688C4—42no—
689C4—82no—
690C4—81no—
691C4—82no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—161no—
696C4—161no—
697C4C2×#42711yescharacteristic
698C4#291111yescharacteristic
699C4#294611yescharacteristic
700C4#294611yescharacteristic
701C4#308111yescharacteristic
702C4#308111yescharacteristic
703C4#309111yescharacteristic
704C4#309111yescharacteristic
705C2—42no—
706C2—44no—
707C2—82no—
708C2#1886611yescharacteristic
709C2#1886711yescharacteristic
710C2#1887811yescharacteristic
711C2#1887911yescharacteristic
712C2#2084311yescharacteristic
713C2#2086411yescharacteristic
714C2#2087611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21985C2semidirect
3. #21984C2semidirect
6. #8897C2semidirect
7. #8894C2semidirect
8. #8893C2semidirect
9. #8890C2semidirect
10. #8889C2semidirect
11. #8886C2semidirect
12. #8885C2semidirect
13. #8882C2semidirect
14. #8038C2semidirect
15. #7894C2semidirect
38. #1948C22semidirect
39. #1948C22semidirect
40. #1947C22semidirect
41. #1947C22semidirect
46. #1845C22semidirect
47. #1844C22semidirect
49. #1834C22semidirect
50. #1834C22semidirect
59. #1478C4semidirect
64. #1296C22semidirect
65. #1296C22semidirect
66. #1296C22semidirect
67. #1296C22semidirect
68. #1279C22semidirect
69. #1279C22semidirect
70. #1279C22semidirect
71. #1279C22semidirect
100. #431C2×C4semidirect
101. #431C2×C4semidirect
189. D4⋊C8C2×C4semidirect
190. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341942
#219852
#219842
#207322
C2×#14782
#88972
#88942
#88932
#88902
#88892
#88862
#88852
#88822
#80382
#78942