Finite Group Databasebeta

← #204591

Subgroups — #204591

Order 512.

1501 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35078C211yesmaximal, characteristic
2#20525C211yesmaximal, characteristic, complemented
3C2×#1504C211yesmaximal, characteristic, complemented
4C2×#1507C211yesmaximal, characteristic, complemented
5C2×#1501C211yesmaximal, characteristic, complemented
6C22×#327C211yesmaximal, characteristic, complemented
7#13782C211yesmaximal, characteristic, complemented
8#9684C212yesmaximal, complemented
9#9682C212yesmaximal, complemented
10#9681C212yesmaximal, complemented
11#9680C212yesmaximal, complemented
12C2×#525C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C2×#407C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2856C2212yes—
17#2850C2212yes—
18#2661—22no—
19#2543—22no—
20#2504C411yescharacteristic, complemented
21#2502C411yescharacteristic, complemented
22#2500C412yescomplemented
23#2435C411yescharacteristic, complemented
24#2433C411yescharacteristic, complemented
25#2431C412yescomplemented
26#2410—21no—
27#2410—21no—
28#2321—21no—
29#2321—21no—
30C2×#385—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C22×#95C2211yescharacteristic
34#1753—22no—
35#1752—21no—
36C8×C22⋊C4—21no—
37#1749C2212yescomplemented
38C4×C22⋊C8C2212yescomplemented
39#1507—24no—
40#1504—24no—
41#1501—24no—
42C2×#328C2211yescharacteristic, complemented
43C2×#328C2211yescharacteristic, complemented
44C2×#328C2212yescomplemented
45C2×#327—24no—
46C2×#327C2212yes—
47C2×#327C2212yes—
48C2×#327C2212yes—
49C2×#327C2212yes—
50C2×#327C2211yescharacteristic, complemented
51C2×#327C2211yescharacteristic, complemented
52C2×#327C2212yescomplemented
53#1436—22no—
54C4⋊#95C2212yescomplemented
55#1434C2212yescomplemented
56C22⋊#95—21no—
57C22×#41C2×C411yescharacteristic
58C23×D4—21no—
59C23×D4C2×C411yescharacteristic
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C22×M16—21no—
63C23×C8—21no—
64C2×#127C2×C412yescomplemented
65C2×#124C2×C412yescomplemented
66#525—22no—
67#525—22no—
68#525—22no—
69#525—22no—
70C2×#123—21no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C22⋊Q8C2311yescharacteristic
73C2×#121—21no—
74C2×#121—21no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121C2×C412yes—
82C2×#121C2×C411yescharacteristic
83C2×#121C2312yes—
84C2×#121C2311yescharacteristic
85C2×#120—21no—
86C2×#120—21no—
87C2×#120—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×#117C2×C411yescharacteristic
91C22×C4⋊C4—22no—
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4D411yescharacteristic
96C22×C22⋊C4D411yescharacteristic
97C22×C42—21no—
98C22×C42C2312yes—
99#407—22no—
100#407D414yescomplemented
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8D414yescomplemented
105C2×C22⋊C8C2311yescharacteristic
106C2×C22⋊C8C2311yescharacteristic
107C2×C22⋊C8C2312yes—
108C2×#97—22no—
109C2×C4×C8—22no—
110#398—22no—
111#397—22no—
112#394—22no—
113#394—22no—
114#394—22no—
115#392—22no—
116#392—22no—
117#392—22no—
118#391C2×C412yescomplemented
119#390—22no—
120#390—22no—
121#390C2×C412yescomplemented
122#390C2×C412yescomplemented
123#390C2×C412yescomplemented
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#386—22no—
129#386—22no—
130#386—21no—
131#386—21no—
132#386—22no—
133#386—22no—
134#385—28no—
135#385—21no—
136#385—21no—
137#384C2×C412yescomplemented
138#382C2×C412yescomplemented
139#379—21no—
140#379—21no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4C2×C412yes—
144C4×C22⋊C4C2×C411yescharacteristic
145C4×C22⋊C4C2×C411yescharacteristic
146C4×C22⋊C4D411yescharacteristic
147C4×C22⋊C4D411yescharacteristic
148C4×C22⋊C4D412yes—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—21no—
153C2×#95—22no—
154C2×#95—22no—
155C2×#95D412yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159#336—22no—
160#336—22no—
161#328—22no—
162#328—22no—
163#328—24no—
164#327—24no—
165#327—24no—
166#327—24no—
167#327—24no—
168#327—22no—
169#327—22no—
170#327—24no—
171C25—21no—
172C25C2×D411yescharacteristic
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—82no—
176C22×Q8C22×C411yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—82no—
190C22×D4—82no—
191C22×D4—82no—
192C22×D4C22×C412yes—
193C22×D4C22×C411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—82no—
203C23×C4—82no—
204C23×C4—82no—
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4#4112yes—
215C23×C4#4112yes—
216C23×C4C2411yescharacteristic
217C2×M16—24no—
218C2×M16—82no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—22no—
222C22×C8—24no—
223C22×C8—82no—
224#127—82no—
225#124—82no—
226#123—42no—
227#123—42no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234#121—24no—
235#121—24no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—22no—
240#121—21no—
241#121—21no—
242#121—22no—
243#121—22no—
244#121—21no—
245#121—21no—
246#121—42no—
247#121—42no—
248#121—42no—
249#121—42no—
250#121—82no—
251#121—82no—
252#121—82no—
253#121—82no—
254#120—44no—
255#120—42no—
256#120—42no—
257#120—44no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262#117—22no—
263#117—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—82no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
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—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
302C2×C22⋊C4C22⋊C412yes—
303C2×C22⋊C4C22⋊C412yes—
304C2×C22⋊C4C22⋊C411yescharacteristic
305C2×C42—21no—
306C2×C42—42no—
307C2×C42—82no—
308C2×C42—82no—
309C2×C42C22×C412yes—
310C2×C42C22×C412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—22no—
318C22⋊C8C2×D412yes—
319C22⋊C8C2×D412yes—
320C22⋊C8C2×D414yes—
321#97—82no—
322C4×C8—82no—
323#95—28no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—21no—
330#95—21no—
331#95—22no—
332#95—21no—
333#95—21no—
334#95—42no—
335#95—82no—
336#95—82no—
337#95C22⋊C412yes—
338#95C22⋊C412yes—
339#95C22⋊C412yes—
340#95C22⋊C412yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—82no—
352C24—82no—
353C24C2×C22⋊C411yescharacteristic
354C24C2×C22⋊C411yescharacteristic
355C24#12412yes—
356C24#12712yes—
357C24C22×D411yescharacteristic
358#41—84no—
359#41—84no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—42no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—44no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—88no—
399C2×D4—84no—
400C2×D4—84no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—88no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—88no—
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C412yes—
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C412yes—
455C22×C4C4×D412yes—
456C22×C4C4×D412yes—
457C22×C4C4×D412yes—
458C22×C4C4×D412yes—
459C22×C4#12112yes—
460C22×C4#12112yes—
461C22×C4#12112yes—
462C22×C4#12112yes—
463C22×C4#12412yes—
464C22×C4#12712yes—
465C22×C4C23×C411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471M16—84no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—84no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—88no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4#10414yes—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—84no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—21no—
528C23—21no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—24no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—164no—
560C23—164no—
561C23#39012yes—
562C23#39012yes—
563C23C2×#10412yes—
564C23#42112yes—
565C23C22×C22⋊C411yescharacteristic
566C23C2×#12411yescharacteristic
567C23C2×#12711yescharacteristic
568C23#57612yes—
569C23#57712yes—
570Q8—81no—
571Q8—81no—
572D4—82no—
573D4—82no—
574D4—81no—
575D4—81no—
576D4—84no—
577D4—88no—
578D4—84no—
579D4—82no—
580D4—82no—
581D4—82no—
582D4—81no—
583D4—82no—
584D4—81no—
585D4—82no—
586D4—84no—
587D4—164no—
588D4—164no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—84no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—84no—
648C2×C4—816no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—82no—
652C2×C4#39012yes—
653C2×C4#39012yes—
654C2×C4C2×#10414yes—
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C2×C4×D411yescharacteristic
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×#12111yescharacteristic
659C8—42no—
660C8—42no—
661C8—44no—
662C22—21no—
663C22—21no—
664C22—22no—
665C22—22no—
666C22—24no—
667C22—22no—
668C22—22no—
669C22—24no—
670C22—22no—
671C22—21no—
672C22—22no—
673C22—21no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—44no—
678C22—44no—
679C22—48no—
680C22—42no—
681C22—42no—
682C22—81no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—81no—
692C22—81no—
693C22—82no—
694C22—168no—
695C22#198014yes—
696C22C2×#39011yescharacteristic
697C22#288312yes—
698C22#289712yes—
699C22C22×#10411yescharacteristic
700C22C2×#42111yescharacteristic
701C22#313612yes—
702C22#315411yescharacteristic
703C22#315511yescharacteristic
704C4—24no—
705C4—21no—
706C4—21no—
707C4—22no—
708C4—21no—
709C4—21no—
710C4—22no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—82no—
715C4—82no—
716C4—84no—
717C4—81no—
718C4—82no—
719C4—81no—
720C4—81no—
721C2—21no—
722C2—21no—
723C2—24no—
724C2—22no—
725C2—42no—
726C2—161no—
727C2—162no—
728C2#1754712yes—
729C2#1759712yes—
730C2#2120611yescharacteristic
731C2#2121311yescharacteristic
732C2C2×#198011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35078C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20525C2semidirect
3. C2×#1504C2semidirect
4. C2×#1507C2semidirect
5. C2×#1501C2semidirect
6. C22×#327C2semidirect
7. #13782C2semidirect
8. #9684C2semidirect
9. #9682C2semidirect
10. #9681C2semidirect
11. #9680C2semidirect
20. #2504C4semidirect
21. #2502C4semidirect
22. #2500C4semidirect
23. #2435C4semidirect
24. #2433C4semidirect
25. #2431C4semidirect
37. #1749C22semidirect
38. C4×C22⋊C8C22semidirect
42. C2×#328C22semidirect
43. C2×#328C22semidirect
44. C2×#328C22semidirect
50. C2×#327C22semidirect
51. C2×#327C22semidirect
52. C2×#327C22semidirect
54. C4⋊#95C22semidirect
55. #1434C22semidirect
64. C2×#127C2×C4semidirect
65. C2×#124C2×C4semidirect
100. #407D4semidirect
104. C2×C22⋊C8D4semidirect
118. #391C2×C4semidirect
121. #390C2×C4semidirect
122. #390C2×C4semidirect
123. #390C2×C4semidirect
137. #384C2×C4semidirect
138. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350782
#205252
C2×#15042
C2×#15072
C2×#15012
C22×#3272
#137822
#96842
#96822
#96812
#96802